Dependencies

The bounds of Theorem 13.14.8 hold for \(S(D)\) with \(\delta =\delta (\rho _D,\sigma _D)\) and states on \(D\). For states \(\rho ,\sigma \) on \(D\otimes E\), the change of \(I(D:E)=S(D)+S(E)-S(DE)\) is at most the sum of the two moduli, at \(\delta (\rho _D,\sigma _D)\) and \(\delta (\rho ,\sigma )\), and hence at most \(8\sqrt{\delta (\rho ,\sigma )}\, \log (e\, d_D)\), independently of the dimension of \(E\).

Let \(\Delta {\gt}0\) and \(p\ge 0\) be real, and set \(r_*=2p+10\). The constants \(A,T_0\) in Theorem 13.30.6.1 can be chosen as functions of \(\Delta ,p\) before \(b,L\) are specified. They satisfy the two coefficient bounds with threshold \(40+2p\), and (337) and (338) hold with \(r=r_*\) for every \(b\ge 0\) and \(L\ge 1\).

For \(s\in [0,1]\), \(t\in [0,1]\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(A^s B^{1-s})\) is jointly concave:

Obtained from Theorem 7.7.13 by taking \(K=\mathbb {1}\).

The peripheral projection of a positive trace-preserving map is positive and trace-preserving, and then the phase-weighted map \(T_\varphi \) is positive and trace-preserving as well. If the original map is completely positive, then \(T_\phi \) and \(T_\varphi \) are quantum channels. This is the preservation assertion in the opening clause of [ Wol12 , Proposition 6.3 ] (item (ii) itself states only the composition identity).

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map. Then for all density matrices \(\rho _1,\rho _2\in M_{D}(\mathbb {C})\), \(\lVert T(\rho _1)-T(\rho _2)\rVert _{\operatorname{tr}} \leq \lVert \rho _1-\rho _2\rVert _{\operatorname{tr}}\). See [ Wol12 , Chapter 8, Eq. (8.80) ] .

Write \([k]=\{ 0,\ldots ,k-1\} \). Let \(B\subseteq [k]\) have cardinality \(r\), and fix its standard finite enumeration \([r]\simeq B\). A five-factor configuration on \([r]\) corresponds to a physical \(QYV\) configuration indexed by \(B\) and auxiliary \(C,R\) configurations indexed by \([r]\). Grouping the physical factors and applying this enumeration gives an equivalence of these coordinate sets. This is the change of traced coordinates used in [ Ope26b , Section 8, lines 520–549 ] .

For a bipartite density operator \(\rho \in M_{d_A}(\mathbb {C})\otimes M_{d_B}(\mathbb {C})\), its operator-Schmidt rank is the least integer \(r\) for which there are matrices \(A_t\in M_{d_A}(\mathbb {C})\) and \(B_t\in M_{d_B}(\mathbb {C})\) satisfying

No Hermiticity or positivity condition is imposed on the factors. This is the definition in [ DlCDN19 , Equation (1) ] ; the same formula defines the rank of an arbitrary complex bipartite matrix.

For \(\rho \succeq 0\) on \(\mathbb {C}^T\), the canonical purification is the vector in \(\mathbb {C}^T\otimes \mathbb {C}^T\) with coefficient matrix \(\sqrt\rho \). Its reduced state is \(\rho \) and its squared norm is \(\operatorname{tr}\rho \), so it is a possibly unnormalized purification of \(\rho \), and a purification in the sense of Definition 1.2.1 when \(\rho \) is a density operator. For \(s\in \mathbb {C}^T\otimes \mathbb {C}^{B_T}\) and \(s'\in \mathbb {C}^E\otimes \mathbb {C}^{B_E}\), write \(s\otimes s'\) for their product regrouped in \((\mathbb {C}^T\otimes \mathbb {C}^E)\otimes (\mathbb {C}^{B_T}\otimes \mathbb {C}^{B_E})\). Its reduced state on \(\mathbb {C}^T\otimes \mathbb {C}^E\) is the tensor product of the reduced states of \(s\) and \(s'\).

The Choi matrix of a linear map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is

an operator on \(\mathbb {C}^{d'}\otimes \mathbb {C}^{d}\). This is the correspondence of [ Wol12 , Proposition 2.1 ] .

If \(z{\gt}0\), define \(\chi =z^{-1/2}(J^\dagger \otimes I_B)\Omega \) on \(\mathcal H_R\otimes \mathcal H_B\).

For a Hermitian matrix \(\rho \) on \(D\otimes E\), \(S(D|E)_\rho =S(\rho _{DE})-S(\rho _E)\).

Given a bipartite coordinate decomposition \(A\simeq Q\times T\), apply this decomposition in each of \(m\) copies and collect the two families:

The order of the copies is the same in both families.

The set of density matrices in \(M_{D}(\mathbb {C})\) is

For real \(a,b\) and initial real weights \(r_i\), define

When \(r_i\ge 0\) are squared input amplitudes, \(x\) is a diagonal density, and \(b{\gt}0\), \(F(x)\) is the squared norm after applying the filter \((x+bI)^{-a/2}\). The quotient is defined to be zero when \(F(x)=0\).

On the original doubled space \(\mathcal H\otimes \mathcal H\), define \(\Psi =\Omega \otimes \Omega \) and \(\mathbb H=H\otimes I+I\otimes H\). In coordinates, \(\Psi _{(i,j)}=\Omega _i\Omega _j\).

Let \(d_A{\gt}0\) and let \(\rho _{BC}\) be positive semidefinite. Define

Under the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), the right partial trace removes \(C\).

This formulation has the same value \(I(A{:}B) = S(\rho _A) + S(\rho _B) - S(\rho _{AB})\) as in Definition 13.9.1.

Let \(E\) be a finite set of indices, and let \((p_i)_{i\in E}\) be real weights. Writing \(z=\sum _{i\in E}p_i\), their normalized restriction is \(p_i'=p_i/z\). At zero mass, the normalized restriction is defined to be zero.

Let \(\rho _A\) and \(\rho _{BC}\) be positive semidefinite, and define

We use the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), so that the right partial trace removes \(C\).

This is the entropy formulation of the Hayashi Markov decomposition from Definition 13.8.55.

Let \(V\) be a finite set of sites with finite local Hilbert spaces, and let \(\psi \in \bigotimes _{v\in V}\mathcal H_v\). For a region \(X\subseteq V\), let \(\rho _X=\operatorname {tr}_{X^c}|\psi \rangle \langle \psi |\). Given a set \(E\) of its eigenvalue indices, let \(P_E\) be the selected spectral projection and let \(z=\operatorname {tr}(P_E\rho _X)\). If \(z{\gt}0\), the global selected vector is

expressed in the original product basis. This vector is fixed by \(X\) and \(E\).

If \(P_A\) is the support projection of \(\rho _A\), define

Let \(\rho \) be Hermitian, with an orthonormal eigenbasis \((u_i)_i\) and real eigenvalues \((p_i)_i\). For a finite set \(E\) of eigenvalue indices, define

With the convention \(0^{-1}=0\), the normalized restriction at zero mass is the zero matrix.

Let \(\psi \in \mathcal H_A\otimes \mathcal H_B\) and let \(\rho _A=\operatorname {tr}_B|\psi \rangle \langle \psi |\). For a set \(E\) of marginal eigenvalue indices with positive mass \(z\), let \(P_E\) be the selected spectral projection. The normalized typical truncation is

This formulation has the same value \(S(\rho ) = -\sum _i \lambda _i \log \lambda _i\) as in Definition 13.6.1.

For \(\rho ,\sigma \in M_{D}(\mathbb {C})\), put \(\widehat D(\rho \Vert \sigma ) =D(\rho \Vert \sigma )\) when \(\ker \sigma \subseteq \ker \rho \), and \(\widehat D(\rho \Vert \sigma )=+\infty \) otherwise. This is the usual infinite value when the support condition fails [ Ope26a , Section 2 ] .

For \(t{\gt}0\), set \(h(t)=-t\log t\), and set \(h(0)=0\). The entropy of a probability distribution \(a=(a_z)_{z\in Z}\) on a finite set is \(H(a)=\sum _{z\in Z}h(a_z)\). For a joint probability distribution \(P\) with row and column marginals \(p\) and \(q\), put

Let \(X\) and \(Y\) be finite sets. A matrix \(P=(P_{x,y})_{x\in X,y\in Y}\) is a joint probability distribution if \(P_{x,y}\geq 0\) for every \(x\in X\) and \(y\in Y\), and

Its row and column marginals are respectively

For regional subsystems define the entropy expression

For a vector \(\psi \in \mathbb C^{E_\Lambda }\), identify its coordinates with a coefficient matrix \(M_R\) indexed by \(E_R\times E_{R^c}\). Define \(\rho _R=M_RM_R^\dagger \) and \(S_\psi (R)=S(\rho _R)\). For arbitrary \(R,T\), define the entropy combination \(I_\psi (R:T)=S_\psi (R)+S_\psi (T)-S_\psi (R\cup T)\). If \(\| \psi \| =1\), each \(\rho _R\) has trace one. For disjoint \(R,T\), the entropy combination is the quantum mutual information of the reduced state on \(R\cup T\).

Restricting a configuration to \(T,E,U\) and enumerating the first two bases gives a bijection \(E_\Lambda \simeq (\{ 1,\ldots ,d_T\} \times \{ 1,\ldots ,d_E\} )\times E_U\). Write \(\widetilde\psi \) for the same vector in these coordinates.

Let \(Q,Y,V,C,R\) be coordinate sets and let \(m\geq 0\). A configuration \(x\) assigns one coordinate in each of these five sets to each copy \(j\). The equivalence \(e_m\) groups these coordinates as

Thus its range is

The inverse places the five coordinates back in the same copy. This includes \(m=0\).

Regroup the good-copy factors, with their fixed common enumeration, by

Write \(e\) for this coordinate equivalence. The ordering is the one used for the paired merge estimate in [ Ope26b , Section 8, lines 513–549 ] .

For matrices \(\mathbb H',\mathbb H,W\) on \(\mathcal H\), define \(A_t=e^{it\mathbb H'}We^{-it\mathbb H}\) and, for \(h\ge 0\),

Truncation retains the original Gaussian mass, without division by \(\mu _h([-T,T])\). Write \(M_h\) and \(M_h^T\) when the three matrices are fixed.

For \(h{\gt}0\), define the centered Gaussian density of variance \(h\) by

Write \(\mu _h\) for the corresponding probability measure. At \(h=0\), set \(\mu _0=\delta _0\) and \(g_0(t)=0\) for every \(t\). Thus \(g_0\) is not a density of \(\mu _0\) with respect to Lebesgue measure.

Separate the two auxiliary registers by the coordinate equivalence

Both good registers precede both bad registers. Denote this equivalence by \(e\).

Let \(B\subseteq \{ 1,\ldots ,k\} \), put \(r=|B|\) and \(m=|B^c|=k-r\), and use the chosen finite-set equivalences to enumerate \(B^c\) and \(B\). For a coordinate set \(C\), write

where \(j_g\) and \(j_b\) are the inverse finite-set equivalences, followed by inclusion into the whole copy set. This is an equivalence, including when one of the copy groups is empty. No order-preserving enumeration is required.

A fixed division \([k]\simeq [m]\sqcup [r]\) of the copies induces the coordinate decomposition of the full configuration space into the configurations on the first and second groups. All physical and auxiliary factors use this same division.

A Hayashi Markov decomposition of a tripartite state \(\rho _{ABC}\) consists of a finite direct-sum decomposition

together with a unitary change of basis on \(B\), a probability vector \((p_j)_j\), and density matrices \(\rho _{A B_j^L}\) and \(\rho _{B_j^R C}\) such that, in the adapted basis, the state becomes

The terminology follows Hayashi’s presentation of quantum Markov structure  [ Hay06 ] ; the block decomposition used by the MPDO argument is the structure theorem of [ HJPW04 ] .

Let \(A\) be Hermitian, with spectral decomposition \(A=U\operatorname{diag}(\lambda _i)U^\dagger \). Its support projection is

For a finite-dimensional system \(A\), there is a finite family of effects \((M_s)_s\), with \(0\leq M_s\leq \mathbf1_A\), whose complex linear span is the full matrix algebra on \(A\). One member is the identity. For an operator \(X\) on \(A\otimes B\), define its conditional slice by

The family may be chosen from the four rank-one effects occurring in the polarization identity, scaled so that every member is bounded by the identity.

For a linear map \(\Psi _B\) and a bipartite matrix \(\rho _{AB}\), define

where \(\Sigma \) denotes the canonical exchange of the two tensor factors.

A linear map \(\mathcal S\) between matrix algebras is completely positive in rectangular Kraus form if there are finitely many operators \(A_i:H\to K\) such that

No trace-preservation normalization is imposed.

A linear map \(\mathcal{S}\) between matrix algebras is trace-preserving completely positive if it has a Kraus form \(\mathcal{S}(X)=\sum _i A_iXA_i^\dagger \) with \(\sum _i A_i^\dagger A_i=I\). The Kraus operators may be rectangular, so the input and output dimensions need not agree.

Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) be an isometry, so that \(V^\dagger V=\mathbb {1}_k\). Define \(\iota _V:M_{k}(\mathbb {C})\to M_{D}(\mathbb {C})\) by

The map \(\iota _V\) is complex-linear, multiplicative, and \(*\)-preserving. In general it is not unital: \(\iota _V(\mathbb {1}_k)=VV^\dagger \) is the projection onto the range of \(V\).

Assume in addition that the common average \(\bar\rho \) of Definition 13.8.57 is positive definite. Hayden–Jozsa–Petz–Winter instead reduce to this case by shrinking to the joint support of \(\rho _1,\ldots ,\rho _K\); that reduction is not re-derived here. By Lemma 13.8.58, \(\bar\rho \) is a positive definite fixed point of every \(F\in \mathbf F\), so Theorem 10.2.10 makes the fixed-point set of each adjoint map,

a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). The common invariant algebra is

and \(X\in A_0\) if and only if \(F^*(X)=X\) for every \(F\in \mathbf F\).

Fix \(F_0\in \mathbf F\) with \(A_0=A_{F_0}\) as in Theorem 13.8.62. Let \(P_0\) be the finite-dimensional mean-ergodic projection of the Schrödinger map \(F_0\), and define \(P_0^*\) as its trace-pairing adjoint.

Let \(\rho _1,\ldots ,\rho _K\) be density matrices in \(M_{D}(\mathbb {C})\), \(K\ge 1\). A trace-preserving completely positive Kraus family \(F\) preserves \(\rho _1,\ldots ,\rho _K\) if \(F\rho _k=\rho _k\) for every \(k\). Write

for the set of such operations – non-empty since the identity operation belongs to it – and

for their common average.

Let \(V\) be finite-dimensional and let \(f:V\to V\) have bounded orbits. The mean-ergodic projection \(P_f:V\to V\) is the projection onto \(\ker (f-\mathbb {1}_V)\) along \(\operatorname{range}(f-\mathbb {1}_V)\) in (28).

Let \(\rho \) be a matrix on a finite-dimensional space \(B\). The state-preparation map from matrices on \(A\) to matrices on \(A\otimes B\) is

This is the elementary preparation operation used in the maps \(\mathcal T_1\) and \(\mathcal S_1\) of [ CPGSV16 , Appendix C.2, lines 1527–1533 and 1551–1555 ] . It is relocated here from the MPDO renormalization chapter; its Kraus-action and conditional-expectation consequences for the density-operator setting appear in Section 1.16 below.

Let \(E_{k\ell }\) denote the matrix unit with its only nonzero entry in row \(k\) and column \(\ell \). The transfer matrix \(\widehat T\) is the matrix indexed by pairs of bond indices with entries

Equivalently, this is the matrix of \(T\) under the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\): the matrix-units special case of Definition 3.19.2.

The quantum mutual information of a bipartite state \(\rho _{AB}\) is

A matrix \(P \in M_{D}(\mathbb {C})\) is an orthogonal projection if \(P = P^\dagger \) and \(P^2 = P\).

For the \(QC\) pair set \(D_C=F_Q+F_C-F_{QC}\), and for the \(VR\) pair set \(D_R=F_V+F_R-F_{VR}\). Each operator is Hermitian. On the common space write \(\widehat D_C=D_C\otimes I_{VR}\) and \(\widehat D_R=I_{QC}\otimes D_R\).

Let \(X \in M_{d \cdot d'}(\mathbb {C})\) be a bipartite matrix indexed by \((\{ 0,\ldots ,d-1\} \times \{ 0,\ldots ,d'-1\} )^2\). The left partial trace \(\operatorname{tr}_A(X)\) and right partial trace \(\operatorname{tr}_B(X)\) are the \(d' \times d'\) and \(d \times d\) matrices defined by

For an operator \(\rho \) on \(H_L\otimes H_R\), define \((\operatorname{tr}_L\rho )_{r,s}:=\sum _l\rho _{(l,r),(l,s)}\).

The completed Petz map is

Put \(Q_\tau =\mathbf1_L-P_\tau \) and \(\omega _R=\operatorname{tr}_L\sigma \). The complementary term is

Let \(\sigma \) be positive semidefinite on \(H_L\otimes H_R\), and set \(\tau =\operatorname{tr}_R\sigma \). The Petz transpose formula on the support of \(\tau \) is

This is the support formula of [ HJPW04 , Theorem 3, equation (8) ] . It is not asserted to be trace preserving on operators outside the support of \(\tau \).

The map \(T_\phi \) is the projection onto the peripheral subspace along the non-peripheral subspace. The phase-weighted peripheral map is Wolf’s asymptotic dynamics \(T_\varphi =T\circ T_\phi \).

For two registers \(A,B\), the partial swap \(F_A\) exchanges the \(A\) factors of two replicas:

For a linear map \(V:\mathcal H_Q\to \mathcal H_A\otimes \mathcal H_B\), put \(K=V\otimes V\) and define the compression

The contraction below is read from left to right: first \(K\), then \(F_A\), then \(K^\dagger \). Each external leg is an original copy of \(Q\); each internal leg bundles one copy of \(A\otimes B\).

The regrouping \((r,c,x,y)\mapsto (r,x,c,y)\) identifies \((\mathcal H_R\otimes \mathcal H_C)\otimes (\mathcal H_X\otimes \mathcal H_Y)\) with \((\mathcal H_R\otimes \mathcal H_X)\otimes (\mathcal H_C\otimes \mathcal H_Y)\). For a vector \(\Omega \) on \(RCQ\), write its two replicas with both reference registers before both buffer registers:

Thus \(\Psi \) is the regrouped \(\Omega ^{\otimes 2}\) and \(\Phi =F_R\Psi \); both copies of \(C\) and \(Q\) retain their coordinates.

The density on physical factor \(j\) is the partial trace of \(|\Omega \rangle \langle \Omega |\) in the coordinates \(e_j\). Equivalently, its entries are

where \(y\) ranges over the configurations of the other two factors.

Let the physical and auxiliary coordinate sets be finite, and let \(m\geq 0\) be the number of good copies. Write the original physical factors as \(H_0=H_Q\), \(H_1=H_Y\) and \(H_2=H_V\). For \(j\in \{ 0,1,2\} \), let

be the coordinate identification that places the \(j\)th factor first and retains the original labels on its complement. On \(m\) copies, let \(c_j\) apply this identification to every copy and collect the copies of the two factors. This is the regional decomposition used in [ Ope26b , Section 8, lines 550–560 ] .

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is positive if \(T(X) \ge 0\) whenever \(X \ge 0\), where \(X \ge 0\) means that \(X\) is positive semidefinite. For \(D' = D\) this is the notion of Definition 2.1.1.

For a complex-linear map \(P:\mathcal M\to \mathcal M\) and matrices \(H,B\in \mathcal M\), define, for every \(t\in \mathbb {R}\),

In particular, \(G_{P,H,B}(0)=P(B)\).

A vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) purifies a density operator \(\rho \in M_{d_A}(\mathbb {C})\) if \(\operatorname{tr}_B[|\psi \rangle \! \langle \psi |]=\rho \).

For matrices \(\rho ,\sigma \in M_{D}(\mathbb {C})\), define the trace-log expression

On the physical domain where \(\rho \) is a density matrix and \(\sigma \) is positive definite, this is the Umegaki relative entropy.

For \(a\le b\), set \(F_{a:b}=E_a\circ \cdots \circ E_{b-1}\), with \(F_{a:a}=\mathbb {1}_{M_{D_a}(\mathbb {C})}\). For successive rectangular Kraus families \(A_i^{s_i}\), set \(A^{\boldsymbol s}=A_a^{s_a}\cdots A_{b-1}^{s_{b-1}}\).

For a complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\), its unnormalized rectangular Choi matrix \(J(\mathcal L)\in M_{de}(\mathbb {C})\) is the reshaping

When \(d=e\), this is \(d\) times the normalized Choi matrix convention in [ Wol12 , Proposition 2.1 ] .

Let \(\mathcal H_{\mathrm{phys}}\) be a finite tensor product of local coordinate spaces, and fix auxiliary coordinate spaces \(C,R\). An added site is distinguished from the original sites; applying this construction twice retains two distinct auxiliary coordinates. For disjoint physical regions \(Q,Y\), write \(F=(Q\cup Y)^c\). Restriction of configurations gives the equivalence

The five coordinate factors, in order, are \(Q,Y,F,C,R\).

On \(k\) copies, let \(G_k\) extract the original physical copies and the two original auxiliary registers from the full configuration. For any physical coordinate equivalence \(e\), let \(E_k(e)\) apply \(e\) independently to those physical copies and act identically on both auxiliary registers. Finally let \(F_k\) be the five-factor regrouping into these three registers. The actual regional equivalence is

It maps the original full configurations to the grouped five-factor configurations. The auxiliary coordinate sets and their order are unchanged. Empty regions and zero copies are included. These are the coordinate identifications in [ Ope26b , Section 8, lines 20–37 and 80–110 ] .

For regions \(Q,Y\subseteq V\), define

For \(B\subseteq \{ 0,\ldots ,k-1\} \), define

Let \(A,C,D\) be finite coordinate sets, let \(\Omega \in \mathbb C^A\), and let \(v\in \mathbb C^{A^k}\otimes \mathbb C^{C^k}\otimes \mathbb C^{D^k}\). Write \(R_B\) for the literal excitation operator, with factors \(I-|\Omega \rangle \langle \Omega |\) on \(B\) and \(|\Omega \rangle \langle \Omega |\) on its complement, and set \(w=(R_B\otimes I)v\). First form the whole auxiliary marginal \(\rho _{CD}=\operatorname {Tr}_{A^k}|w\rangle \langle w|\). Regroup its coordinates using

The actual good auxiliary marginal is the unnormalized matrix

Thus every physical copy, the bad auxiliary copies and every exterior copy are traced out; the good auxiliary copies are retained.

Let all five coordinate sets be finite. Put \(A=Q\times (Y\times V)\), let \(\Omega \in \mathbb C^A\), and let

be arbitrary. For \(B\subseteq \{ 0,\ldots ,k-1\} \), put \(G=B^c\), \(m=|G|\) and \(b=|B|\). Form the actual component \(w=(R_B\otimes I)u\). Use the same chosen enumeration of \(G\) in all five factors and the chosen enumeration of \(B\) in each auxiliary factor.

For a retained configuration \(x\), write \(e_m(x)=(a,(c,r))\). A discarded configuration consists of a physical assignment \(d\in A^B\) and auxiliary assignments \(c'\in C^b\), \(r'\in R^b\). Reconstruct the original physical register by placing \(a\) on \(G\) and \(d\) on \(B\). Reconstruct the two auxiliary registers by placing \((c,r)\) on \(G\) and \((c',r')\) on \(B\). Let \(f_B(x,(d,c',r'))\) be the value of \(w\) at these reconstructed coordinates. Define

This is the actual unnormalized rank-one partial trace. Only the bad physical and auxiliary copies are discarded. No condition on \(\Omega \) or \(u\) is imposed in the definition.

Regroup the same component \(w\) into retained and discarded factors:

The common density \(\Gamma _B(u)\) is \(\operatorname {Tr}_{\mathcal H_{\mathrm{tr}}}|w\rangle \langle w|\). It retains the four exterior good regions and is unnormalized. The discarded factors include the good middle copies and the middle coordinates within the bad physical copies, so every middle physical coordinate is traced.

Regroup \(w\) as a vector on \(\mathcal H_{\mathrm{ret}}\otimes \mathcal H_{\mathrm{tr}}\), where

The joint marginal \(\rho _{Q_GC_G}(w)\) is the partial trace of \(|w\rangle \langle w|\) over the second factor. Thus it retains the good physical \(Q\) coordinates and the good auxiliary \(C\) coordinates of the same component, without normalization.

For any \(\Omega \in \mathbb C^A\) and \(u\in \mathbb C^{A^k}\otimes \mathcal K\), define a vector \(w_B\in \mathbb C^{A^B}\otimes \mathcal K\) by

No normalization of \(u\) or \(\Omega \) is required for this definition.

A permutation \(\sigma \in S_k\) acts simultaneously on the physical copy register and the two auxiliary copy registers.

For a physical vector \(\Omega \), let \(N_\Omega =\sum _{i=0}^{k-1}(I-|\Omega \rangle \langle \Omega |)_i\). Fix the two original auxiliary labels \(\ell _C,\ell _R\) and \(\tau \in \mathbb R\). Write \(E_C,E_R\) for their whole-copy label projections, and set

where \(S\) is the average of the simultaneous permutation matrices. Here the identity operators act on the corresponding copy registers. This is the low-defect construction in [ Ope26b , Section 8, (comparator:defect-mass) ] .

Let \(\mathcal H=\mathbb C^A\), let \(H\) be an operator on \(\mathcal H\), and let \(\Omega \in \mathcal H\). For \(k\ge 0\), define

For \(k{\gt}0\), write \(\overline H=k^{-1}H_\Sigma \).

Let \(u_d=d^{-1/2}\sum _{i=0}^{d-1}|i,i\rangle \) be the uniform auxiliary pair for \(d{\gt}0\), and set \(u_0=0\). For \(k=0\), use the empty tensor product. Regroup its literal tensor power into the two auxiliary systems and write it as \(U_k\). For a central label projector \(\Pi _\lambda \) on \((\mathbb C^d)^{\otimes k}\), define

All products and projections here act on the actual finite copy spaces.

For a physical vector \(\Omega \in X\otimes Y\otimes Z\), put

The physical and ancillary factors are reassociated in the stated coordinate order. Write \(M_X=\operatorname {tr}_Y((I_X\otimes T)\rho _{XY}Q)\).

For matrices \(\rho ,Q\) on \(X\otimes Y\) and \(T\) on \(Y\), set

For matrices \(Q\) on \(X\otimes Y\) and \(T\) on \(Y\), define

The second sum in (392) includes every basis vector, including those with zero eigenvalue. These are the operators in the amplification argument of  [ Ope26b ] .

For every physical vector \(\Omega \), define

Write \(\phi =\mathcal R_X\Omega \). Substitution of the source’s Schmidt expansion into this definition gives its swapped-state expression. The identities below do not require that expansion as a hypothesis, nor do they require \(\Omega \) to be normalized or \(p\) to be its marginal spectrum. In the source application \(p\) is the nonnegative marginal spectrum; its zero eigenvalues contribute zero to \(\eta \).

For \(\rho ,\sigma \succeq 0\), the root fidelity is

where \(\sqrt{\, \cdot \, }\) is the positive square root. It is nonnegative. This is the convention of [ Ope26a , Section 2 ] .

For square complex matrices \(\rho \) and \(\omega \) and every \(\alpha \in \mathbb {R}\), set

Real powers are defined by continuous functional calculus, with negative powers equal to zero on the zero eigenspace. Thus \(\widetilde Q_\alpha \) is total even when \(\alpha =0\) or \(\omega \) is singular. For \(\alpha {\gt}1\), on positive semidefinite inputs satisfying \(\ker \omega \subseteq \ker \rho \), this convention gives the finite sandwiched Rényi trace term. In this regime, if the support inclusion fails, the totalized value remains finite but is not the divergence trace term.

For square matrices \(\rho \) and \(\omega \) of the same size, define

The negative power is defined by functional calculus and vanishes on the kernel of \(\omega \). On trace-one positive semidefinite matrices satisfying \(\ker \omega \subseteq \ker \rho \), its logarithm is the order-two sandwiched Rényi divergence [ MLDS\(^{+}\)13 , Definition 2 ] .

For a matrix \(A\in M_{D}(\mathbb {C})\), the singular values \(s_0(A),s_1(A),\ldots \) form a finitely supported family: only finitely many are nonzero. The Schatten one-norm of \(A\) is their sum, that is, the sum of the finitely many nonzero singular values:

This is the \(p=1\) case of the Schatten \(p\)-norm [ Wol12 , Chapter 8, Section 8.1 ] . The equivalent closed formula \(\lVert A\rVert _1=\sum _{i=0}^{D-1}s_i(A)\), summing over all \(D\) singular values including trailing zeros, is part of Theorem 13.1.4.

A vector \(\psi \in \mathbb {C}^{D}\otimes \mathbb {C}^{k}\) is identified with its coefficient matrix \(C_\psi \in M_{D,k}(\mathbb {C})(\mathbb {C})\). Its Schmidt rank is \(\operatorname{SR}(\psi )=\operatorname{rank}(C_\psi )\). We write \(\operatorname{SR}(\psi )\le r\) for the corresponding bounded-rank condition.

For \(w\in H_L\) and a distinguished basis vector \(e_r\in H_R\), define the lift \(J_r w=w\otimes e_r\in H_L\otimes H_R\).

A tripartite density matrix \(\rho _{ABC}\) satisfies SSA equality if

Let \(\tau =\sum _i\lambda _i|i\rangle \! \langle i|\) be positive semidefinite. Its inverse square root on the support is

For a positive semidefinite matrix \(\rho \ge 0\), the support projection \(P\) is the orthogonal projection onto the range of \(\rho \). Via the spectral decomposition \(\rho =U\operatorname{diag}(\lambda _1,\ldots ,\lambda _D)U^\dagger \), it is

where \(\mathbf{1}_{\lambda _j{\gt}0}\) is \(1\) if \(\lambda _j{\gt}0\) and \(0\) otherwise.

For linear maps \(\Phi _A\) and \(\Psi _B\), set

The trace distance of two square matrices is \(\delta (\rho ,\sigma )=\tfrac 12\operatorname {Re}\operatorname{tr}\lvert \rho -\sigma \rvert \), half the trace norm of \(\rho -\sigma \).

The trace norm of \(A\in M_{D}(\mathbb {C})\) is its Schatten one-norm:

For \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(T_\phi \) be its peripheral spectral projection and define

This is Wolf’s notation in [ Wol12 , Chapter 8, Proposition “Convergence towards asymptotic states”, Eq. (8.112) ] .

For any matrix \(Y\in M_{D'}(\mathbb {C})\), define \(T'_Y:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) by

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is trace-preserving if \(\operatorname{tr}(T(X)) = \operatorname{tr}(X)\) for all \(X\). For \(D' = D\) this is the notion of Definition 2.1.3.

For a tripartite matrix \(\rho _{ABC}\) on \(\mathbb {C}^{d_A} \otimes \mathbb {C}^{d_B} \otimes \mathbb {C}^{d_C}\), the partial trace over \(A\) is

The partial trace over \(A\) and \(C\) is

The partial trace over \(C\) is

Define

On \((\mathcal H_A\otimes \mathcal H_C)\otimes (\mathcal H_R\otimes \mathcal H_B)\), define \(\mathrm{pre}_{acrb}=\Omega _{ab}u_{cr}\) and \(\mathrm{post}=\beta \otimes \chi \).

Let \(\mathcal H_R=\mathbb C^E\), so that \(\dim \mathcal H_R=|E|\). Define \(J:\mathcal H_R\to \mathcal H_A\) by \(J|i\rangle =u_i\).

For a Hermitian matrix \(\rho \in M_{D}(\mathbb {C})\) with eigenvalues \(\lambda _0,\ldots ,\lambda _{D-1}\), the von Neumann entropy is

where \(0\log 0:=0\).

For \(A\in M_{D}(\mathbb {C})\) and a complex-linear map \(S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),

If \(\psi \) and \(\phi \) are orthogonal unit vectors, then \(\lVert |\psi \rangle \! \langle \psi |-|\phi \rangle \! \langle \phi |\rVert _2=\sqrt2\). Consequently the orthogonal-pure-state supremum for \(S\) is at most \(\sqrt D\, \lVert S\rVert _{2\to 2}\sqrt2\). These are precisely the estimates in Wolf Equations (8.115)–(8.116).

For a Hermitian matrix \(A\), a real function \(f\), and a unitary \(U\), the continuous functional calculus satisfies

Let \(A \in M_{m \times n}(\mathbb {C})\) and \(B \in M_{n \times m}(\mathbb {C})\). Then the charpoly-root entropy sum is invariant under the cyclic swap \(AB \mapsto BA\):

with roots counted with algebraic multiplicity.

For arbitrary \(Y\in M_{D'}(\mathbb {C})\), the rectangular (output-factor-first) Choi matrix of \(T'_Y\) is

See [ Wol12 , Chapter 8, Eq. (8.86) ] .

For a state \(\rho \) on \(D\otimes E\), \(\lvert S(D|E)_\rho \rvert \le \log \dim D\).

For states \(\tau _j\) on \(D\otimes E\) and probability weights \(p_j\),

The eigenvalues of a density matrix sum to \(1\).

The von Neumann entropy of a Hermitian matrix is the \(-x\log x\) sum over the real parts of the roots of its characteristic polynomial \(\chi _\rho \):

If two Hermitian matrices are equal, then their von Neumann entropies are equal.

For any orthogonal projection \(P\) on \(\mathcal H_A\), put \(\psi _P=(P\otimes I_B)\psi \) and \(\psi _{I-P}=((I_A-P)\otimes I_B)\psi \). Then

The zero matrix has zero von Neumann entropy: \(S(0)=0\).

Let \(A,B:\mathbb {C}^n\to \mathbb {C}^m\) be linear maps with \(B^\dagger B=A^\dagger A\). Then there is a unitary \(U\) on \(\mathbb {C}^m\) such that \(B=UA\).

If \(\Psi _B\) is trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\operatorname{id}_A\otimes \Psi _B)(\rho )\geq 0\).

The sum of two completely positive maps in rectangular Kraus form is completely positive in rectangular Kraus form.

A map in rectangular Kraus form sends positive semidefinite matrices to positive semidefinite matrices. Every trace-preserving completely positive Kraus map is a completely positive Kraus map.

The composition of two completely positive rectangular Kraus maps is completely positive. If the two families are \((A_i)_i\) and \((B_j)_j\), the composite family is \((A_iB_j)_{i,j}\).

Every \(F\in \mathbf F\) satisfies \(F\bar\rho =\bar\rho \).

Let \(\rho \) be a positive semidefinite operator on \(A \otimes R\) with reduced state \(\rho _R = \operatorname{tr}_A \rho \). Then the singular reference \((\mathbb {1}_A / d_A) \otimes \rho _R\) satisfies the support condition \(\ker ((\mathbb {1}_A / d_A) \otimes \rho _R) \subseteq \ker \rho \).

The label projectors are mutually orthogonal orthogonal projections with sum the identity, and they commute with every operator that commutes with all permutation operators. Every label observable is self-adjoint.

For a Hermitian matrix \(A\) and a unitary \(U\),

For positive definite matrices \(\rho \) and \(\tau \),

Let \(\tau \) be positive semidefinite on \(\mathcal H_S\), let \(X\) be a matrix on \(\mathcal H_S\), and let \(\overline\tau =\tau \otimes d_C^{-1}\mathbf1_C\). Then

For every rectangular operator \(V:H\to K\), the map \(X\mapsto VXV^\dagger \) is completely positive.

If \(\rho \geq 0\), then the map \(X\mapsto X\otimes \rho \) is completely positive. No trace normalization is required.

The natural lifts from \(AB\) and \(BC\) to \(ABC\) preserve products and adjoints; the left lift preserves finite sums and complex scalar multiplication, and the right lift preserves finite sums. The right lift of \(U_B\otimes \mathbb {1}_C\) is \(\mathbb {1}_A\otimes U_B\otimes \mathbb {1}_C\). If \(X_{AB}\) is supported on a middle projection \(P\) from the right, \(Y_{BC}\) is supported on a middle projection \(Q\) from the left, and \(PQ=0\), then their overlapping lifts have zero product. Finally, positivity of either lifted operator implies positivity of the corresponding local operator whenever the complementary factor is nonempty.

If \(\sigma \) has trace one, then

Let \(e_L:H_L\to H_L'\) and \(e_R:H_R\to H_R'\) be bijections, and write \(Z^{e_L\otimes e_R}\) for the corresponding simultaneous relabelling of the rows and columns of \(Z\). Then

Let \(A:H_A\to K_A\) and \(B:H_B\to K_B\) be linear maps between finite-dimensional complex spaces, and let \(X\) be an operator on \(H_A\otimes H_B\). If \(B^\dagger B=1\), then

If \(A^\dagger A=1\), then, symmetrically,

The identities include zero-dimensional spaces.

For every matrix \(X\), the support Petz map is given by (79).

Let \(P_\tau \) be the orthogonal projector onto the support of \(\tau =\operatorname{tr}_R\sigma \). Then, for every matrix \(X\),

If \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) is a positive linear map and \(H\in M_{D}(\mathbb {C})\) is Hermitian, then \(T(H)\) is Hermitian.

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian with positive part \(A^+\). There is a matrix \(\Pi \) with \(0\leq \Pi \leq \mathbb {1}\), \(\Pi ^2=\Pi \), and \(\Pi A=A^+\), namely the orthogonal projection onto the support space of \(A^+\).

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map and let \(H\in M_{D}(\mathbb {C})\) be Hermitian, with Jordan decompositions \(H=P_+-P_-\) and \(T(H)=Q_+-Q_-\). Then \(\operatorname{tr}[Q_+]\leq \operatorname{tr}[P_+]\).

Let \(S,T\) be positive semidefinite matrices and let \(x\) be a vector. If \((t\mathbf1+S)^{-1}x=(t\mathbf1+T)^{-1}x\) for every \(t{\gt}0\), then \(\sqrt S\, x=\sqrt T\, x\). More generally, for any fixed matrix \(Q\), if \(Q(t\mathbf1+S)^{-1}x=Q(t\mathbf1+T)^{-1}x\) for every \(t{\gt}0\), then \(Q\sqrt S\, x=Q\sqrt T\, x\).

In particular, for positive definite \(A,B\) and every \(t{\gt}0\), put \(\Delta _{A,B}=A\otimes (B^{-1})^{\mathsf T}\). The source-\(B\) left–right resolvent satisfies

and

These are the positive-square-root specializations of the passage from relative modular resolvents to analytic functions of the relative modular operator in Jenčová–Ruskai, arXiv:0903.2895v4, lines 658–680.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a positive linear map and let \(H\in M_{D}(\mathbb {C})\) be Hermitian. Then \(\operatorname{tr}[(TH)^+]\leq \operatorname{tr}[T(H^+)]\).

Let \(p=(p_z)_{z\in Z}\) be a probability distribution on a finite set \(Z\). Define \(h(t)=-t\log t\) for \(t{\gt}0\) and \(h(0)=0\). Then

Let \(\omega _{XY}\) be positive semidefinite. Then

Equivalently, \(\omega _{XY}\) is supported on \(\operatorname {supp}\omega _X\otimes \operatorname {supp}\omega _Y\).

Let \(X,C\in M_{D}(\mathbb {C})\) with \(X\geq 0\). If \(C\geq 0\), then \(\operatorname{tr}[CX]\geq 0\); if \(C\leq \mathbb {1}\), then \(\operatorname{tr}[CX]\leq \operatorname{tr}[X]\).

If \(A,B\geq 0\), then \(\operatorname{tr}(AB)\geq 0\).

For every matrix \(\rho \in M_{D}(\mathbb {C})\), one has \(D(\rho \Vert \rho )=0\).

For matrices \(\rho ,\sigma \in M_{D}(\mathbb {C})\),

For every matrix \(\sigma \in M_{D}(\mathbb {C})\), one has \(D(0\Vert \sigma )=0\).

Let \(\rho \) be a density matrix on \(A \otimes R\) with reduced state \(\rho _R = \operatorname{tr}_A \rho \), and suppose the support condition \(\ker ((\mathbb {1}_A / d_A) \otimes \rho _R) \subseteq \ker \rho \) holds. Then

If \(A\) and \(B\) are positive definite and \(c{\gt}0\), then

Let \(a,b{\gt}0\). Then the function

is integrable on \((0,\infty )\), and

This is the scalar normalization \((\mathrm{intspec})\) in Jenčová–Ruskai, arXiv:0903.2895v4, §2.1, lines 406–413.

For Hermitian matrices \(\rho ,\sigma \) on a finite index set and any bijection \(e\) from that set onto another finite set,

Let \(A\) and \(B\) be positive semidefinite and suppose that \(\ker B\subseteq \ker A\). If \(c{\gt}0\), then

Let \(T\) be an orthogonal projection and \(K=I_X\otimes T\). For every matrix \(A\),

If \(A\ge 0\), then \(\operatorname {tr}_Y(KAK)\le \operatorname {tr}_Y A\).

For arbitrary matrices \(A\) on \(X\otimes Y\) and \(T\) on \(Y\),

If \(A\le B\), then \(\operatorname {tr}_Y A\le \operatorname {tr}_Y B\).

For matrices \(M\) and \(P\), column-stacking vectorization satisfies

Let \(\zeta \) be a primitive \(d\)-th root of unity and let \(i,j\) range over the residues modulo \(d\). Then

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a positive linear map with \(\operatorname{tr}[T(\rho )] = c\, \operatorname{tr}[\rho ]\) for some nonnegative real \(c\) and all \(\rho \in M_{D}(\mathbb {C})\). Then for all Hermitian \(H\in M_{D}(\mathbb {C})\), \(\operatorname{tr}[(TH)^+]\leq c\cdot \operatorname{tr}[H^+]\). Generalizes Lemma 13.1.18 from the trace-preserving case \(c=1\).

If \(P_\tau \) is the orthogonal projector onto the support of a positive semidefinite matrix \(\tau \), then

The inverse square root of a positive semidefinite matrix on its support is Hermitian.

Let \(\tau \) be positive semidefinite and let \(c{\gt}0\). Then

For every finite-dimensional auxiliary space \(\mathcal H_R\),

The corresponding identity for an auxiliary left factor is

Consequently, for \(d_R{\gt}0\),

If \(\tau \) is positive definite, then its inverse square root on the support is its ordinary inverse square root: \(\tau ^{-1/2}_{\mathrm{supp}}=(\sqrt\tau )^{-1}\).

The inverse square root of a positive semidefinite matrix on its support is positive semidefinite.

If \(P_\tau \) is the orthogonal projector onto the support of a positive semidefinite matrix \(\tau \), then

Let \(A,B\) be positive semidefinite, let \(t{\gt}0\), set \(B^+=(B^{-1/2}_{\mathrm{supp}})^2\), and let \(P_B\) be the support projection of \(B\). Then

Let \(A\) and \(B\) be positive semidefinite, and write \(B^+=(B^{-1/2}_{\mathrm{supp}})^2\). Then

Let \(A,B,C,D\) be positive semidefinite matrices. Write \(B^+=(B^{-1/2}_{\mathrm{supp}})^2\) and \(D^+=(D^{-1/2}_{\mathrm{supp}})^2\), and let \(P_B\) be the orthogonal projection onto \((\ker B)^\perp \). If, for every \(t{\gt}0\),

where the inverses act as relative-modular superoperators on matrices, then

This is the square-root specialization of the support functional-calculus passage in Jenčová–Ruskai, arXiv:0903.2895v4, lines 788–793. The projection \(P_B\) is essential: the source gives the common generalized resolvents only after restriction to \((\ker B)^\perp \). Deriving this restricted equality requires the preceding singular equality argument and its kernel hypotheses; these are supplied for finite families by Theorem 13.8.35.

Let \(I\) be a finite nonempty set. For each \(i\in I\), let \(S_i\) be a positive-semidefinite matrix, let \(P_i\) be its support projection, and let \(b_i\) lie in its support. Write

Assume also that \(b\) lies in the support of \(S\), and put \(x=Gb\). Then

In particular,

Jenčová and Ruskai give the positive-definite residual expansion in equations \((\mathrm{Mj})\) and \((\mathrm{eq:Schz1})\) of the Appendix to arXiv:0903.2895v4. The support assumptions make the same expansion valid for the generalized inverses of the \(S_i\) and of \(S\).

Let \(I\) be a finite nonempty set. For each \(i\in I\), let \(A_i\) and \(B_i\) be positive-semidefinite matrices of the same size. For \(t{\gt}0\), write

Then

No kernel inclusion between \(A_i\) and \(B_i\) is required. This lemma is a positive-semidefinite support-domain extension of the positive-definite calculation in equations \((\mathrm{Mj})\), \((\mathrm{eq:Schz1})\), and \((\mathrm{eq:Schwzt})\) at lines 1313–1343 of Jenčová–Ruskai, arXiv:0903.2895v4; their generalized-inverse notation is given at lines 254–262. The paper does not state this extension. Its later singular equality theorem at lines 761–785 assumes \(\ker B_i\subseteq \ker A_i\) and is not asserted here. The subsequent singular entropy-equality passage is recorded in the TNLean paper-gap note [ con26r ] .

If \(\Phi _A\) and \(\Psi _B\) are trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\Phi _A\otimes \Psi _B)(\rho )\geq 0\).

If \(\Phi _A\) is trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\Phi _A\otimes \operatorname{id}_B)(\rho )\geq 0\).

If \(\rho ,\sigma \) are Hermitian with equal trace, then the positive and negative parts of \(\rho -\sigma \) both have trace \(\delta (\rho ,\sigma )\). For states, \(\delta (\rho ,\sigma )\le 1\), and the partial trace over a tensor factor does not increase \(\delta \).

If \(\rho _{ABC}\) is Hermitian, then \(\operatorname{tr}_A(\rho _{ABC})\), \(\operatorname{tr}_C(\rho _{ABC})\), and \(\operatorname{tr}_{AC}(\rho _{ABC})\) are all Hermitian. The same holds for bipartite partial traces \(\operatorname{tr}_A(\rho _{AB})\) and \(\operatorname{tr}_B(\rho _{AB})\).

For every operator \(M\) on \(H_L\) and every operator \(\rho \) on \(H_L\otimes H_R\), one has \(\operatorname{tr}((M\otimes \mathbb {1}_R)\rho )=\operatorname{tr}(M\operatorname{tr}_R\rho )\).

For every operator \(M\) on \(H_R\), one has \(\operatorname{tr}((\mathbb {1}_L\otimes M)\rho )=\operatorname{tr}(M\operatorname{tr}_L\rho )\).

Let \(T_\varphi :=T\circ T_\phi =T_\phi \circ T\) be Wolf’s asymptotic dynamics. For every matrix \(\rho \), every \(n\geq 0\), and every \(n{\gt}0\) in the second equality,

These are the numerator identities used in [ Wol12 , Chapter 8, Eq. (8.114) ] . They follow from \(T_\phi T=TT_\phi =T_\varphi =T_\varphi T_\phi \); the positive-iterate condition records that the zeroth power of \(T_\varphi \) is the identity.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a complex-linear map. Then

The left supremum is over distinct density matrices in \(M_{D}(\mathbb {C})\), and the right supremum is over orthogonal unit vectors in \(\mathbb {C}^D\). No positivity or trace-preservation assumption is imposed on \(T\). This is [ Wol12 , Chapter 8, Lemma 8.3, Eq. (8.81) ] .

Let \(S:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be complex-linear and let \(\rho _1\neq \rho _2\) be density matrices. Then

This is the pointwise inequality from Wolf Lemma 8.3 used in Equation (8.115).

Let \(H\in M_{D}(\mathbb {C})\) be Hermitian with Jordan decomposition \(H=H^+-H^-\) into orthogonal positive parts, \(H^\pm \geq 0\) and \(H^+H^-=0\). Then \(\lVert H\rVert _{\operatorname{tr}}=\operatorname{tr}[H^+]+\operatorname{tr}[H^-]\).

For every \(A\in M_{D}(\mathbb {C})\), the trace norm is the sum of the square roots of the eigenvalues of the positive operator \(A^\dagger A\):

For every \(A\in M_{D}(\mathbb {C})\), with \(\lambda _0,\ldots ,\lambda _{D-1}\) the eigenvalues of the matrix \(A^\dagger A\), \(\lVert A\rVert _{\operatorname{tr}} =\sum _{i=0}^{D-1}\sqrt{\lambda _i}\).

For every tripartite matrix \(\rho _{ABC}\),

Fix a dimension \(d_C\ge 1\) and a primitive \(d_C\)-th root of unity \(\zeta \). For every matrix \(M\) on \(\mathcal{H}_S\otimes \mathbb {C}^{d_C}\), the uniform average of the conjugations by the \(d_C^2\) unitaries \(\mathbb {1}_S\otimes W(a,b)\) on the second factor is the partial trace over that factor tensored with the maximally mixed state \(\mathbb {1}_C/d_C\):

Let \(E\) be the set of indices with \(p_i{\gt}0\) and \(|-\log p_i-S|\le w\), and let \(z=\sum _{i\in E}p_i\). Then \(z\ge 1-\Pr \{ |K-S|{\gt}w\} \), and \(\mathrm e^{-S-w}\le p_i\le \mathrm e^{-S+w}\) for \(i\in E\).

Let \(I\) be a finite nonempty set. For each \(i\in I\), let \(A_i\) and \(B_i\) be positive-semidefinite matrices of the same size satisfying \(\ker B_i\subseteq \ker A_i\). For \(t{\gt}0\), write

Then

The kernel inclusions for the summands imply \(\ker (\sum _i B_i)\subseteq \ker (\sum _i A_i)\). This lemma is the source-\(A\) support-domain extension of the positive-definite residual calculation in equations \((\mathrm{Mj})\), \((\mathrm{eq:Schz1})\), and \((\mathrm{eq:Schwzt})\) at lines 1313–1343 of Jenčová–Ruskai, arXiv:0903.2895v4. The paper does not state this fixed-parameter extension separately. The lemma does not assert that equality of relative entropies makes the defect vanish.

Let \(X\) be a matrix on \(\mathcal H_S\otimes \mathbb C^{d_C}\) and let \(U_g=\mathbf1\otimes W_g\) for the \(d_C^2\) Weyl indices. Then

Let \(T\) be a complex finite-dimensional endomorphism, let \(T_\phi \) be its peripheral spectral projection, and let \(T_\varphi =T T_\phi \) be the phase-weighted peripheral map of Equation (6.13). For every positive integer \(n\),

The restriction \(n{\gt}0\) is essential: at \(n=0\) the first left-hand side is zero, whereas the right-hand side is the identity. In particular, the later condition \(d^2-1\leq n\) does not exclude this failure when \(d=1\).

If the Kraus map is trace-preserving and has a positive definite fixed point \(\rho {\gt}0\) with \(E(\rho )=\rho \), then \(\operatorname{Fix}(E^*)\) is a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\).

The set of doubly stochastic matrices in \(M_d(\mathbb {R})\) is the convex hull of the \(d\times d\) permutation matrices, and its extreme points are exactly the permutation matrices. See [ Wol12 , Chapter 8, Theorem 8.6 ] . This statement concerns only doubly stochastic matrices; the corresponding assertion of Theorem 8.6 for doubly substochastic matrices is not included here.

Let \(A\in M_{k}(\mathbb {C})\) be Hermitian, let \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfy \(V^\dagger V=\mathbb {1}_k\), and let \(f:\mathbb {R}\to \mathbb {R}\) satisfy \(f(0)=0\). Then

where both sides use the continuous functional calculus on the respective finite spectra.

If \(A\in M_{n}(\mathbb {C})\) is Hermitian and \(f:\mathbb {R}\to \mathbb {R}\), then

In particular,

If \(A\) is positive semidefinite, then for every \(r\in \mathbb {R}\),

These identities include singular matrices and matrices whose index set is empty. They are project-derived rather than statements of CPSV16.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving, Hermiticity-preserving linear map and \(Y\in M_{D'}(\mathbb {C})\) Hermitian with \(\operatorname{tr}[Y]=1\). For \(\varepsilon \geq 0\), if the rectangular Choi matrix \(\tau (T)\) satisfies

then for all density operators \(\rho _1,\rho _2\in M_{D}(\mathbb {C})\),

See [ Wol12 , Chapter 8, Eq. (8.86) ] .

The map \(T\) is completely positive, in the sense that it admits a Kraus representation \(T(X)=\sum _{j}K_jXK_j^\dagger \) with \(K_j\in M_{d'\times d}(\mathbb {C})\), if and only if \(\tau \geq 0\). This is the complete-positivity clause of [ Wol12 , Proposition 2.1 ] .

Let \(X\) and \(Y\) be finite sets and let \(P=(P_{x,y})_{x\in X,y\in Y}\) be a joint probability distribution. With natural logarithms, \(I(X:Y)_P\leq \log \operatorname{rank}_{\mathbb {R}}P\). Equivalently, with logarithms to base two, \(2^{I(X:Y)_P}\leq \operatorname{rank}_{\mathbb {R}}P\).

For every \(A\ge 0\) and every real \(b\), \(b\operatorname {rank}\Pi \le \operatorname {Re}\operatorname{tr}A\).

Let \(X_{AB}\in \mathbf L(H_A\otimes H_B)\) and \(Y_{BC}\in \mathbf L(H_B\otimes H_C)\) be Hermitian, and suppose \([X_{AB}\otimes \mathbb {1}_C,\mathbb {1}_A\otimes Y_{BC}]=0\). There are positive integers \(d_q,m_q\), a unitary identification \(H_B\cong \bigoplus _{q=0}^{K-1}H_{q,l}\otimes H_{q,r}\), and Hermitian operators \(R_q\in \mathbf L(H_A\otimes H_{q,l})\) and \(S_q\in \mathbf L(H_{q,r}\otimes H_C)\) such that

This is the explicit coordinate form of the two block-action identities in  [ Bei12 , Lemma 2.1 ] .

Suppose \(z{\gt}0\) and put \(\Omega _E=z^{-1/2}(P_E\otimes I_B)\Omega \). Then \(\| \chi \| =1\) and

In particular, every reduced state on a subsystem of \(B\) is preserved by this relabeling, as required in the marginal comparison of Section 7.

Let \(\rho ,\sigma \) be states on \(D\otimes E\), put \(\delta =\delta (\rho ,\sigma )\) and \(d_D=\dim D\), and let \(h(u)=-u\log u-(1-u)\log (1-u)\). Then

Neither bound depends on the dimension of \(E\).

Each eigenvalue of a density matrix lies in \([0,1]\).

Let \(I\) be finite, let \(x,p\in \mathbb R^I\) be nonnegative with \(\sum _i x_i=\sum _i p_i=1\), and let \(b,c{\gt}0\). Suppose that \(f:\mathbb R^I\to \mathbb R\) attains its minimum over the probability simplex at \(x\) and is Fréchet differentiable there, with

Then there exists \(0{\lt}\lambda \le 1\) such that

These are the simplex necessary conditions used in the patch-filter argument of [ Ope26a , Section 4 ] .

If \(x_i+b{\gt}0\) for every \(i\), then \(F\) is Fréchet differentiable at \(x\), with \(DF(x)v=-a\sum _i r_i(x_i+b)^{-a-1}v_i\). Suppose further that \(r_i\ge 0\), \(\sum _i r_i{\gt}0\), \(b{\gt}0\), and \(x_i\ge 0\). Then \(F(x){\gt}0\), \(p_i(x)\ge 0\), and \(\sum _i p_i(x)=1\). If also \(a{\gt}0\), \(\sum _i x_i=1\), and \(x\) minimizes the actual function \(F\) on the probability simplex, then the normalized weights \(p_i(x)\) satisfy (276)– (278) for some \(0{\lt}\lambda \le 1\).

In a family of rectangular matrices whose row and column spaces depend on a parameter, equal parameters and the corresponding identified row and column coordinates give equal matrix entries.

With the notation of (289), the algebraic identity

holds for every \(H,\Omega ,E_0,\Delta \). If \(R\ge 0\), \(\| \Omega \| =1\), and \(\Delta \ge 0\), then

No eigenvector assumption is needed for this inequality.

Suppose \(\mathcal H=\mathcal H_A\otimes \mathcal H_B\), \(\| \Omega \| =1\), \(H\Omega =E_0\Omega \), \(\Delta \ge 0\), and \(H-E_0I\ge \Delta (I-|\Omega \rangle \! \langle \Omega |)\). Let \(F=F_A\) be the physical partial swap from Definition 13.28.1, and put \(\Phi =F\Psi \) and \(\mathbb H'=F\mathbb HF^\dagger \). Then

Both vectors and both Hamiltonians act on the same doubled physical space \((\mathcal H_A\otimes \mathcal H_B)^{\otimes 2}\).

Let \(U\) be unitary on \(\mathcal H\). Then \(\| U\Omega \| =\| \Omega \| \) and

Hence \(R\ge 0\) implies that the left side of (294) is positive semidefinite. If \(H\Omega =E_0\Omega \), then \((UHU^\dagger )(U\Omega )=E_0(U\Omega )\). These statements hold for every real \(E_0,\Delta \), without a normalization assumption.

The doubled vector satisfies

Consequently, \(\| \Omega \| =1\) implies \(\| \Psi \| =1\). If \(H\Omega =E_0\Omega \), then \(\mathbb H\Psi =2E_0\Psi \).

For positive semidefinite matrices \(A_1,\dots ,A_k\), \(S(\sum _jA_j)\le \sum _jS(A_j)\). With \(A_j=p_j\rho _j\) for states \(\rho _j\) and probability weights \(p_j\) this is the mixing bound \(S(\sum _jp_j\rho _j)\le H(p)+\sum _jp_jS(\rho _j)\).

For a family of Hermitian matrices \(M_j\), the block-diagonal direct sum satisfies

If \(\rho _{BC}\) is positive semidefinite with trace one, then \(\widehat{\mathcal R}_{\sigma _{ABC}}\) is completely positive and trace preserving.

For the reference in (86),

after canonical reassociation of the three tensor factors. The raw support-map summand is the maximally mixed specialization of [ HJPW04 , equation (10) ] . The complementary summand comes from the chosen support completion in (92). This theorem does not assert a Hayashi–Koashi–Imoto decomposition.

Let \(\rho \) be a Hermitian matrix and let \(\log \rho \) be its logarithm defined through the functional calculus. Then

If \(\rho \in M_{1}(\mathbb {C})\) is Hermitian and \(\operatorname{tr}(\rho )=1\), then \(S(\rho )=0\).

For density matrices \(\omega \) and \(\tau \), \(S(\omega \otimes \tau )=S(\omega )+S(\tau )\).

For a density matrix \(\rho \in M_{D}(\mathbb {C})\) with \(D\ge 1\), one has \(S(\rho )\le \log D\).

For a density matrix \(\rho \) of rank \(r\), one has \(S(\rho )\le \log r\). This refines the dimension bound: only the nonzero eigenvalues contribute to the entropy, and there are exactly \(r\) of them.

If \(\rho _{BC}\) is positive semidefinite and \(\rho _B=\operatorname{tr}_C\rho _{BC}\), then

If \(P_B\) is the support projector of \(\rho _B\), then the support projector of \(d_A^{-1}\mathbf1_A\otimes \rho _B\) is

For the reference in (86),

For matrices \(A \in M_{m \times n}(\mathbb {C})\) and \(B \in M_{n \times m}(\mathbb {C})\) such that \(AB\) and \(BA\) are Hermitian, \(S(AB)=S(BA)\).

For any density matrix \(\rho \), \(S(\rho )\ge 0\).

For any PSD Hermitian matrix \(\rho \) with \(\operatorname{tr}(\rho ) = 1\) on an arbitrary finite index set, \(S(\rho ) \ge 0\). This is the analog of Theorem 13.6.4 for arbitrary finite index sets: it does not require the index set to be \(\{ 0,\ldots ,D{-}1\} \), so it applies to bipartite density matrices on \(\mathbb {C}^{d_A} \otimes \mathbb {C}^{d_B}\).

If \(z\ne 0\), then \(\sum _{i\in E}p_i'=1\). If \(z{\gt}0\) and \(p_i{\gt}0\), then \(p_i'{\gt}0\).

Let \(A=\sum _j M_j\) be a finite sum of Hermitian matrices. Suppose there are operators \(P_j\) such that

Then \(S(A)=\sum _j S(M_j)\). The operators \(P_j\) need only resolve the support of \(A\); their sum need not be the identity on the ambient space. This is the support form of the direct-sum entropy identity used in [ CPGSV16 , Appendix C.2, lines 1760–1770 ] .

For the reference in (86), the chosen complementary preparation term factors as

after the canonical reassociation of the three tensor factors. This follows from the chosen support completion in (92), not from [ HJPW04 , equation (10) ] .

For the reference in (86), the support Petz map for \(\operatorname{tr}_C\) factors as

after the canonical identification \((H_A\otimes H_B)\otimes H_C\cong H_A\otimes (H_B\otimes H_C)\). This is the maximally mixed specialization of [ HJPW04 , equation (10) ] . It is the tensor-product identity for the raw Petz support formula, not the Hayashi–Koashi–Imoto block decomposition.

For the reference in (80), the raw Petz map for \(\operatorname{tr}_C\) satisfies

Equivalently, for every product operator \(A_0\otimes X_B\),

The formulas use the canonical identification \((H_A\otimes H_B)\otimes H_C\cong H_A\otimes (H_B\otimes H_C)\).

This is the globally valid ambient-space form of [ HJPW04 , equation (10) ] . The literal identity-tensored formula in that equation is obtained on the support of \(\rho _A\), or after choosing an extension away from that support. For singular \(\rho _A\), the raw map on the full matrix algebra contains the compression \(X\mapsto P_AXP_A\).

Let \(X\) be an operator on \(H_A\otimes H_B\). If

then

If \(\rho _A\) is positive definite, then \(P_A=\mathbf1_A\), and this identity holds for every \(X\).

When \(\rho _A\) is singular, no global identity-tensored formula is asserted for the raw map outside the displayed support. Nor is the generic trace-preserving completion in Definition 13.6.63 asserted to factor: its complementary projection is \(\mathbf1_{AB}-P_A\otimes P_B\), which need not be the identity on \(A\) tensored with a projection on \(B\).

Set \(\rho _B=\operatorname{tr}_C\rho _{BC}\), and let \(P_A\) and \(P_B\) be the support projections of \(\rho _A\) and \(\rho _B\). Then

Each identity is understood after the same canonical reassociation of the three tensor factors.

Let \(z{\gt}0\). For every region \(B\) disjoint from \(X\), the same selected global vector satisfies

If also \(\| \psi \| =1\), then

These are the disjoint-region estimates of [ Ope26b , Section 8 ] .

If \(z{\gt}0\), then \(\| \psi _E\| =1\) and

For a density matrix \(\omega \) and a scalar \(c\), \(S(c\, \omega )=c\, S(\omega )-c\log c\).

For \(0{\lt}t{\lt}1\) the complementary mass satisfies

If \(S(\sigma )\le M\), \(M{\gt}0\) and \(c{\gt}0\), then at \(t=e^{-cM}\), \(\operatorname{rank}Q_t\le e^{cM}\) and \(\operatorname{tr}((\mathbb {1}-Q_t)\sigma )\le 1/c\).

Suppose \(S(\sigma )\le M\), \(M{\gt}0\), \(\rho \) is Hermitian of trace one and \(\| \rho -\sigma \| _1\le 1/16\). Then \(Q=Q_{e^{-16M}}\) satisfies \(\operatorname{rank}Q\le e^{16M}\) and \(\Re \operatorname{tr}((\mathbb {1}-Q)\rho )\le 1/8\). If moreover \(\rho ,x\ge 0\) commute, \(C{\gt}0\), \(u\ge 0\), \(M=C(1+u)\), \(R\ge (16C+\log 8)(1+u)\), and \(\rho \le x+e^{-R}\mathbb {1}\), then

Let \(\rho ,\sigma \) be Hermitian matrices of equal trace and \(0\le E\le \mathbb {1}\). Then \(\Re \operatorname{tr}(E\rho )-\Re \operatorname{tr}(E\sigma )\le \delta (\rho ,\sigma )\), where \(\delta (\rho ,\sigma )=\frac12\| \rho -\sigma \| _1\). In particular, for density \(\sigma \), Hermitian trace-one \(\rho \) and \(0{\lt}t{\lt}1\),

No commutation with \(\rho \) is required.

If \(S(\sigma )\le 0\), then \(\operatorname{rank}Q_1\le 1\) and \(\operatorname{tr}((\mathbb {1}-Q_1)\sigma )=0\).

For any tripartite density matrix \(\rho _{ABC}\), equality in strong subadditivity holds if and only if \(\rho _{ABC}\) admits a quantum Markov decomposition on the middle subsystem \(B\).

This formulation introduces no new axiom: it is the same equality criterion as Theorem 13.8.78.

For any tripartite density matrix \(\rho _{ABC}\) on \(A \otimes B \otimes C\),

This formulation introduces no new axiom: it is the same strong-subadditivity statement as Theorem 13.8.4, which is proved there from Lieb concavity along the relative-entropy route [ LR73 ] .

For a tripartite density matrix \(\rho _{ABC}\) with \(\dim B = 1\), one has \(S(\rho _{ABC}) \le S(\rho _{AB}) + S(\rho _{BC})\).

Let \(\rho \) be a Hermitian matrix indexed by a finite set \(J\), and let \(e : I \to J\) be a bijection from a finite set \(I\). The reindexed matrix on \(I\) with entries \(\rho _{e(i)\, e(j)}\) has the same von Neumann entropy as \(\rho \).

Suppose that \(z=\sum _{i\in E}p_i{\gt}0\), and that real numbers \(S,w\) satisfy \(e^{-S-w}\le p_i\le e^{-S+w}\) for every \(i\in E\). Then

Let \(z{\gt}0\) and put \(\psi _{\mathrm{rem}}=((I_A-P_E)\otimes I_B)\psi \). The complementary marginal satisfies

If \(\| \psi \| =1\), then \(S(\rho _{\psi _E,B})\le S(\rho _B)/z\). These are the marginal estimates for the typical Schmidt truncation in [ Ope26b , Section 8 ] .

Let \(\psi \in \mathcal H_A\otimes \mathcal H_B\otimes \mathcal H_C\) and let \(\psi _E\) be its normalized spectral truncation on \(A\), with selected mass \(z{\gt}0\). Put \(\rho _B=\operatorname {tr}_{AC}|\psi \rangle \langle \psi |\) and \(\rho _{\psi _E,B}=\operatorname {tr}_{AC}|\psi _E\rangle \langle \psi _E|\). With the unnormalized discarded vector \(\psi _{\mathrm{rem}}=((I_A-P_E)\otimes I_{BC})\psi \), one has

If \(\| \psi \| =1\), then \(S(\rho _{\psi _E,B})\le S(\rho _B)/z\). This proves the estimates in [ Ope26b , Section 8 ] for every subsystem disjoint from the projected region.

The operator \(P_E\) is an orthogonal projection and commutes with \(\rho \). The normalized spectral restriction satisfies \(\rho _E=z^{-1}P_E\rho P_E\). If \(z\ne 0\), then \(\operatorname {tr}\rho _E=1\). If \(\rho \ge 0\) and \(z{\gt}0\), then \(\rho _E\ge 0\). If \(\rho \ge 0\) and \(\operatorname {tr}\rho =1\), then \(z\le 1\).

Let \(\rho \ge 0\), let \(S=S(\rho )\), and suppose that \(z{\gt}0\) and \(e^{-S-w}\le p_i\le e^{-S+w}\) for every \(i\in E\). Then

This gives the typical-Schmidt entropy estimates for actual reduced density matrices [ Ope26b , Section 8 ] ; zero eigenvalues outside \(E\) are permitted.

Let \(\rho \ge 0\) have trace one, with eigenvalues \(p_i\), and put \(S=S(\rho )\). For \(w\in \mathbb R\), take its canonical typical set

Suppose that

Then \(1-\delta \le z\le 1\) and \(z{\gt}0\). The actual spectral restriction \(\sigma =z^{-1}P_E\rho P_E\) is positive semidefinite with trace one and

This is a finite-dimensional consequence of the typical-set estimates in [ Ope26b , Section 8 ] , conditional on a spectral tail bound.

If \(z\ne 0\) and every selected eigenvalue is positive, then \(\operatorname {rank}\rho _E=|E|\). If \(\rho \ge 0\) and \(z{\gt}0\), then \(S(\rho _E)=S(p')\), where \(p_i'=p_i/z\) for \(i\in E\).

If \(z{\gt}0\), then \(\| \psi _E\| =1\), \(\operatorname {tr}_B|\psi _E\rangle \langle \psi _E|=z^{-1}P_E\rho _A P_E\), and \(\langle \psi ,\psi _E\rangle =\sqrt z\). If also \(\| \psi \| =1\), then

Let \(\psi \in \mathcal H_A\otimes \mathcal H_B\) be a unit vector. Form the preceding canonical set \(E\) from its actual marginal \(\rho _A\), with center \(S=S(\rho _A)\), and assume the same tail bound \(\delta {\lt}1\). Let \(\psi _E=z^{-1/2}(P_E\otimes I_B)\psi \) be the selected vector. Then

Under the hypotheses of Theorem 13.22.1.3,

These are the typical-Schmidt entropy estimates [ Ope26b , Section 8 ] .

Let \(\omega _j\) be density matrices and let \(p_j\geq 0\). Then

In particular, when the \(p_j\) form a probability distribution, this is

This is the entropy identity used in [ CPGSV16 , Appendix C.2, lines 1760–1770 ] .

Let \(k,r\) be nonnegative integers and \(0\le \tau \le 1/2\). If \(r\le \tau k\), then

These are the binomial estimates used before the inverse-compression inequality in  [ Ope26b , Section 7 ] .

Let \(\Omega \in \mathcal H\), let \(B\subseteq \{ 1,\ldots ,k\} \), and put \(w=(R_B\otimes I_{\mathcal K})v\). Suppose \(\sigma B=B\) and \((U_\sigma \otimes V_\sigma )v=v\), where \(U_\sigma \) permutes the physical copies and \(V_\sigma \) is a permutation representation on \(\mathcal K\). Then \(V_\sigma \) commutes with \(\operatorname {Tr}_{\mathcal H^{\otimes k}}|w\rangle \langle w|\).

Let \(S\) be a finite set with \(k\) elements and \(0\le \tau \le 1/2\). Then

This is the subset-count estimate preceding the inverse-compression inequality in  [ Ope26b , Section 7 ] .

In the setting of Theorem 13.16.1.5, the two marginals of \(\rho _c\) satisfy

Both compressed marginals are positive definite, including when one of their spaces has dimension zero.

For normalized finite-dimensional states,

Moreover, after enlarging the purifying space if necessary, an isometry on that space realizes overlap \(F(\rho ,\sigma )\) between purifications. In particular, if \(I(A{:}C)\le \delta \) for a pure state \(\Omega \) on \(\mathbb {C}^A\otimes \mathbb {C}^C\otimes \mathbb {C}^U\), then there are an isometry \(V:\mathbb {C}^U\to \mathbb {C}^A\otimes (\mathbb {C}^C\oplus \mathbb {C}^U)\) and unit vectors \(s\in \mathbb {C}^A\otimes \mathbb {C}^A\) and \(s'\in \mathbb {C}^C\otimes (\mathbb {C}^C\oplus \mathbb {C}^U)\) purifying \(\rho _A\) and \(\rho _C\), so that \(s\otimes s'\) purifies \(\rho _A\otimes \rho _C\), with \(\lVert (\mathbb {1}_{AC}\otimes V)\Omega -s\otimes s'\rVert \le \sqrt\delta \). This is [ Ope26a , Lemma 2.2 ] .

For a finite totally ordered family and \(P_i=\bigcup _{j{\lt}i}X_j\),

For pairwise-disjoint \(X,C,Z\), purity also gives \(I_\psi (X:C\mid Z)=I_\psi (X:C\mid (X\cup C\cup Z)^c)\).

The entropy expressions satisfy \(I_\psi (X:C\mid Z)=I_\psi (X:C\cup Z)-I_\psi (X:Z)\). For a normalized state and disjoint \(X,Z\), this implies \(I_\psi (X:C\mid Z)\le I_\psi (X:C\cup Z)\).

If \(C\) is disjoint from \(G\) and \(B\), then \(C\cup (G\cup B\cup C)^c=(G\cup B)^c\).

Let \(\psi \) be a normalized pure state on a finite product, and let \(G,B,C\) be pairwise-disjoint regions. Suppose \(G=D\mathbin {\dot\cup }\bigcup _{i\in \mathcal I_0}X_i \mathbin {\dot\cup }\bigcup _{j\in \mathcal I_1}Y_j\) is a disjoint partition with a total order on each finite index set. Put \(F=(G\cup B\cup C)^c\), \(P_i=\bigcup _{k{\lt}i}X_k\) and \(Q_j=\bigcup _{k{\lt}j}Y_k\). Assume \(I_\psi (X_i:C\cup B\cup P_i)\le \varepsilon _i\) and \(I_\psi (Y_j:C\cup F\cup Q_j)\le \delta _j\) for all indices. Then

This is the entropy estimate inside the proof of [ Ope26a , Lemma 3.2 ] , before the geometric counting argument. No nonempty-region or sign condition on the errors is required.

For every vector \(\psi \) and every region \(R\), \(S_\psi (R^c)=S_\psi (R)\).

Let \(V\) be a finite set, with finite coordinate sets \(\mathcal B_v\), and let \(\mathcal H_R=\mathbb C^{\prod _{v\in R}\mathcal B_v}\) for \(R\subseteq V\). For unit vectors \(\Omega ,\Phi \in \mathcal H_V\), write \(\rho _\Omega =|\Omega \rangle \langle \Omega |\), \(\rho _\Phi =|\Phi \rangle \langle \Phi |\), and \(\delta =\tfrac 12\| \rho _\Omega -\rho _\Phi \| _1\). Then every region \(R\subseteq V\) satisfies

This is the regional form of Lemma 2.1 used in the scanner argument of [ Ope26b , Section 9 ] . The two vectors and the region are supplied to the estimate; no further choice is made.

If \(\| \psi \| =1\) and \(R\cap T=\varnothing \), then \(S_\psi (R\cup T)\le S_\psi (R)+S_\psi (T)\).

Let \(\mathcal I\) be a finite totally ordered index set, and set \(P_i=\bigcup _{j{\lt}i}X_j\). If \(I_\psi (X_i:B\cup P_i)\le \varepsilon _i\) for each \(i\), then

Let \(\psi \) be a unit pure state, and put \(b=I_\psi (T:E)\). There are finite-dimensional spaces \(\mathcal B_T,\mathcal B_E\), an isometry \(W:\mathcal H_U\to \mathcal B_T\otimes \mathcal B_E\), and unit vectors \(s\in \mathcal H_T\otimes \mathcal B_T\) and \(s'\in \mathcal H_E\otimes \mathcal B_E\) such that

One may take \(\mathcal B_T=\mathbb C^{d_T}\) and \(\mathcal B_E=\mathbb C^{d_E}\oplus \mathcal H_U\). The isometry acts on all of \(\mathcal H_U\). This is the finite-product form of [ Ope26a , Lemma 6.4 ] .

The joint regional density matrix, expressed in the enumerated bases, is \(\rho _{TE}=\operatorname {Tr}_U |\widetilde\psi \rangle \langle \widetilde\psi |\). If \(\| \psi \| =1\), then \(\| \widetilde\psi \| =1\).

Under the hypotheses of Theorem 13.19.3, if \(L{\gt}0\) and \(I_\psi (T:E)\le L^{-60}\), the same conclusion holds with vector error at most \(L^{-30}\).

For a normalized pure state and disjoint \(X,Y\), \(I_\psi (X:Y)\le 2S_\psi (X)\). If \(D\) is disjoint from both \(C\) and \(Z\), then \(I_\psi (D:C\mid Z)\le 2S_\psi (D)\).

Let \(\psi \) be a normalized pure state on the finite tensor product. Suppose \(A=D\mathbin {\dot\cup }\bigcup _{i\in \mathcal I_0}X_i \mathbin {\dot\cup }\bigcup _{i\in \mathcal I_1}X_i\) is a disjoint partition, with a total order on each finite family. Put \(B=\Lambda \setminus A\) and \(P_i=\bigcup _{j\in \mathcal I_f,\, j{\lt}i}X_j\) for \(i\in \mathcal I_f\). If \(I_\psi (X_i:B\cup P_i)\le \varepsilon _i\) for all indices, then

This is [ Ope26b , Lemma 11.1 ] . Empty families, empty regions, and varying local dimensions are permitted.

For every real function \(f\) on the irreducible labels, the observables of \(\{ Q\} ,\{ C\} ,\{ Q,C\} \), expressed in the coordinates \(e\), are respectively

Those of \(\{ V\} ,\{ R\} ,\{ V,R\} \) are respectively

All spaces in these formulas comprise the same \(m\) good copies.

Set \(\Delta _{QC}=F_Q+F_C-F_{QC}\) and \(\Delta _{VR}=F_V+F_R-F_{VR}\). Let \(D_{QC}\) and \(D_{VR}\) be the corresponding paired-copy deficits, each extended by the identity on the other pair. In the coordinates \(e\),

For every real \(a\),

For every real \(a\), the same coordinate identification sends \(e^{aF_{\{ j\} }}\) to

For every real label function \(f\), express the five good factors in the physical–auxiliary coordinates \((Q_GY_GV_G),(C_GR_G)\). Then the observable \(L_{\{ j\} }(f)\) is

where \(L_{j,m}(f)\) belongs to the original copy action on \(H_j^{\otimes m}\).

Let \(c\geq 0\), let \(M_i\geq 0\) and \(A_i\in \mathbb R\) for \(0\leq i{\lt}5\), and suppose that \(M_i\leq c e^{A_i}\). Then \(\prod _{i=0}^4M_i^{1/5}\leq c\exp (\frac15\sum _{i=0}^4A_i)\). The common mass \(c\) may vanish.

Let \(U_g\) and \(V_g\) be permutation representations on finite-dimensional spaces \(\mathcal H\) and \(\mathcal K\). If \(v\in \mathcal H\otimes \mathcal K\) satisfies \((U_g\otimes V_g)v=v\), then

Under the hypotheses of Theorem 13.17.1.1, the matrix

is positive semidefinite, and

This is the faithful-marginal-support form of the order-two estimate obtained from [ Bei13 , Theorem 6, Equation (18) ] .

Let \(\rho _{AB}\geq 0\), and choose an eigenbasis of its faithful first marginal \(\sigma =\operatorname{diag}(p_1,\ldots ,p_{d_A}){\gt}0\). Suppose also that the second marginal \(\tau \) is positive definite. For the supported-marginal channel \(\Phi _\rho \), set

Then

Here \(\sigma ^T=\sigma \) because the chosen basis diagonalizes \(\sigma \).

For \(h\ge 0\) and \(\omega ,s\in \mathbb {R}\),

In particular, all these exponential moments are finite, and the centered Gaussian is sub-Gaussian with variance parameter \(h\). For \(h{\gt}0\), the first integral also equals \(\int _{\mathbb {R}}g_h(t)e^{i\omega t}\, dt\). The frequency \(\omega \) is angular frequency, so no additional factor of \(2\pi \) occurs in the exponent.

Under the hypotheses of Theorem 13.30.3.1, let \(h\ge 0\) and suppose \(x,y\in \mathcal H\) satisfy

for every \(k\). Then

Let \(h{\gt}0\) and let \(f\) take values in a real normed vector space. Then \(g_hf\) is Lebesgue integrable if and only if \(f\) is integrable with respect to \(\mu _h\). The integrals satisfy

for every measurable set \(S\). In particular, \(\int _S g_h(t)\, dt=\mu _h(S)\).

For Hermitian \(\mathbb H',\mathbb H\), every \(h\ge 0\) and \(T\in \mathbb {R}\),

Suppose \(\mathbb H'\) and \(\mathbb H\) are Hermitian and \(h\ge 0\). For every \(T\in \mathbb {R}\),

In particular, for \(T\ge 0\),

For \(h{\gt}0\) and arbitrary matrices \(\mathbb H',\mathbb H,W\),

If the two generators are Hermitian, these density integrands are integrable. At zero variance, \(M_0(\mathbb H',\mathbb H;W)=W\). For every \(h\ge 0\), \(M_h(\mathbb H',\mathbb H;0)=0\).

Let \(\mathbb H'v=Ev\) and \(\mathbb Hw=E'w\), where \(E,E'\in \mathbb {R}\). If \(\mathbb H'\) is Hermitian, then

If both generators are Hermitian, then for every \(h\ge 0\),

The vectors need not be normalized or real. In coordinates, \(\langle v,M_hw\rangle =\sum _j\overline{v_j}(M_hw)_j\).

Let \(K\) be Hermitian, \(\| \Omega \| =1\), \(K\Omega =\mathcal E\Omega \), and \(\Delta {\gt}0\). Suppose

For an orthonormal eigenbasis \(Ku_k=\lambda _ku_k\), each \(k\) satisfies one of the following alternatives:

Thus \(\Delta \) is a lower bound on the gap; equality with the least excitation energy is not required.

Assume (316), \(\Delta {\gt}0\), and \(\mathbb H'-\mathcal E I\ge \Delta (I-|\Phi \rangle \! \langle \Phi |)\). For every \(h\ge 0\),

Only the target Hamiltonian \(\mathbb H'\) needs a gap for this bound.

The function \(t\mapsto A_t\) is continuous for arbitrary matrices \(\mathbb H',\mathbb H,W\). If \(\mathbb H'\) and \(\mathbb H\) are Hermitian, then \(\| A_t\| =\| W\| \) for every \(t\). In that case \(A_t\) is integrable against \(\mu _h\) and its restriction to \([-T,T]\), for every \(h\ge 0\) and every \(T\in \mathbb {R}\).

For every \(h\ge 0\), the function \(g_h\) is nonnegative and integrable. For \(h{\gt}0\), \(\int _{\mathbb {R}}g_h(t)\, dt=1\).

Let \(\Delta {\gt}0\) and \(r\ge 0\) be real. There exist \(A{\gt}0\) and \(T_0{\gt}0\), depending only on \(\Delta \) and \(r\), such that

For every real \(b\ge 0\) and \(L\ge 1\), put \(B_*=1+b+\log L\), \(h=AB_*\), and \(T=T_0B_*\). Then \(h{\gt}0\), \(T\ge 0\), and

These bounds also hold for any \(A{\gt}0\) and \(T_0\ge 0\) satisfying (335) and (336).

Let \(\Delta {\gt}0\) and \(r\ge 0\) be real. There are constants \(a{\gt}0\) and \(T_0{\gt}0\), depending only on \(\Delta ,r\), with the following property. Let \(\mathcal H_R,\mathcal H_C,\mathcal H_Q\) be finite-dimensional complex Hilbert spaces, and put \(\mathcal H=\mathcal H_R\otimes \mathcal H_C\otimes \mathcal H_Q\). Suppose \(H\) is an operator on \(\mathcal H\), \(\Omega \) is a unit vector, and \(E_0\in \mathbb {R}\) and \(b\ge 0\) satisfy

With \(\rho _{RC}=\operatorname {Tr}_Q|\Omega \rangle \! \langle \Omega |\), \(\rho _R=\operatorname {Tr}_C\rho _{RC}\), and \(\rho _C=\operatorname {Tr}_R\rho _{RC}\), suppose also that \(D(\rho _{RC}\Vert \rho _R\otimes \rho _C)\le b\).

Let \(U_e\) regroup the two copies of \(\mathcal H\) as \((\mathcal H_R\otimes \mathcal H_C)^{\otimes 2} \otimes \mathcal H_Q^{\otimes 2}\), and let \(S\) exchange the two \(R\) registers while fixing both \(C\) and both \(Q\) registers. Set

Write \(\Psi =U_e(\Omega \otimes \Omega )\) and \(\Phi =S\Psi \). There are a Hermitian contraction \(W\) on \(\mathcal H_Q^{\otimes 2}\) and a real \(z\ge e^{-2b}\), chosen independently of \(L\), such that \(\langle \Phi | V\Psi \rangle =z\) for \(V=\mathbb {1}_{(RC)^{\otimes 2}}\otimes W\).

For every real \(L\ge 1\), put \(B_*=1+b+\log L\), \(h=aB_*\), and \(T=T_0B_*\). Then \(h{\gt}0\) and \(T\ge 0\). The unrenormalized integral

satisfies \(\| M_L\| \le 1\) and

This is the uniform Gaussian approximation step in the reset argument of  [ Ope26a ] .

Under (316) and (319), suppose \(\Delta {\gt}0\), \(\| W\| \le 1\), and \(\langle \Phi ,W\Psi \rangle =z\in \mathbb {R}\). For every \(h\ge 0\),

For all complex matrices \(A,H\) indexed by \(n\) and all \(t\in \mathbb {R}\),

Let \(h\ge 0\), \(T\in \mathbb {R}\), and let \(H',H,W\) be arbitrary complex matrices indexed by \(n\). For the centered Gaussian measure \(\mu _h\) of variance \(h\), the actual full integral satisfies

Its unrenormalized truncation satisfies

Equivalently, \(R_e(M_h(H',H;W))=M_h(R_e(H'),R_e(H);R_e(W))\), with the same identity for \(M_h^T\). The variance, cutoff, and restricted Gaussian mass are preserved.

These identities include \(h=0\), when \(\mu _0=\delta _0\) and the full integral is \(W\). They also include \(T{\lt}0\), when \([-T,T]\) is empty and both truncated integrals vanish. No Hermiticity, positive variance, nonnegative cutoff, or nonempty-index assumption is required.

Let \(\mu \) be a measure on a measurable space \(X\) and let \(F:X\to \mathbb {C}^{n\times n}\) be any function. Then

The integrals use the convention that a nonintegrable function has integral zero. No integrability assumption is needed.

For all complex matrices \(H',H,W\) indexed by \(n\) and every real \(t\),

For \(h\ge 0\) and \(T\ge 0\), \(\mu _h(I_T^c)\le \tau _h(T)\). For \(h{\gt}0\), this gives the density estimate

Under the hypotheses of Theorem 13.30.3.3, for \(h\ge 0\) and \(T\ge 0\),

Under (316) and (319), suppose \(\Delta {\gt}0\), \(h\ge 0\), and \(T\ge 0\). With \(\varepsilon =(q_h+\tau _h(T))\| W\| \),

In particular, at positive variance the common bound is exactly

Assume (316), \(\Delta {\gt}0\), and both lower-gap inequalities

For every \(h\ge 0\),

Let \(\Delta {\gt}0\) and \(r\ge 0\) be real. There exist \(A{\gt}0\) and \(T_0{\gt}0\), depending only on \(\Delta ,r\), with the following property. For every real \(b\ge 0\) and \(L\ge 1\), put \(B_*=1+b+\log L\), \(h=AB_*\), and \(T=T_0B_*\). Then \(h{\gt}0\) and \(T\ge 0\).

In any finite-dimensional complex Hilbert space, let \(H,H'\) be self-adjoint operators and let \(W\) be a contraction. Suppose that unit vectors \(\Psi ,\Phi \) and a real number \(E_0\) satisfy

Define \(z=\langle \Phi | W\Psi \rangle \) and the unrenormalized integral

Then \(\| M_T\| \le 1\), and

This is the uniform Gaussian approximation in the reset argument of  [ Ope26a ] .

Let \(B\in M_{m,k}(\mathbb {C})\) and \(A\in M_{m',k}(\mathbb {C})\) satisfy \(B^\dagger B=A^\dagger A\), with \(m'\le m\). Then \(B=TA\) for some isometry \(T\in M_{m,m'}(\mathbb {C})\). Equivalently, if \(X\in M_{a,m}(\mathbb {C})\) and \(Y\in M_{a,m'}(\mathbb {C})\) satisfy \(XX^\dagger =YY^\dagger \) with \(m'\le m\), then \(X=YT^\dagger \) for some isometry \(T\in M_{m,m'}(\mathbb {C})\).

Let a group \(G\) act by permutations on finite sets \(Z\) and \(B\), with actions \(\alpha \) and \(\beta \). Let \(A\) be any set and \(e:Z\to A\times B\) a bijection satisfying

for every \(g\in G\) and \(z\in Z\). Let \(\pi _\alpha \) and \(\pi _\beta \) be the linear extensions of the two permutation representations to \(\mathbb C[G]\). For every \(h\in \mathbb C[G]\), \(a,a'\in A\), and \(b,b'\in B\),

Equivalently, the matrix in product coordinates is \(I_A\otimes \pi _\beta (h)\). Neither \(G\) nor \(A\) is required to be finite.

Let \(S_j\) be a finite family of subsystems and let \(a_j\in \mathbb R\). Writing \(F_{S_j,G}\) for the first-subgroup label entropy on the full space, the same coordinates give

No disjointness of the subsystems is required. This is a coordinate identity, as used in [ Ope26b , Section 8, lines 454–560 ] .

For a tripartite density matrix \(\rho _{ABC}\),

holds if and only if \(\rho _{ABC}\) admits a quantum Markov decomposition on the middle subsystem \(B\).

Let \(\rho _{ABC}\) be a tripartite density matrix satisfying

Then there are a decomposition

a unitary \(V_B\), probabilities \(p_j\), and density matrices \(\rho _{A B_j^L}\) and \(\rho _{B_j^R C}\) such that, for \(V=\mathbf1_A\otimes V_B\otimes \mathbf1_C\),

A tripartite density matrix \(\rho _{ABC}\) that admits a quantum Markov decomposition on the middle subsystem \(B\) satisfies

Let \(A\) be Hermitian on a finite-dimensional complex Hilbert space, and let \(u,s\in \mathbb R\). Then \(e^{u(A-sI)}=f(A)\) for \(f(x)=e^{u(x-s)}\).

For every density matrix \(\rho \), there is one sequence \((\lambda _k)_{k\ge 0}\) such that, for all sufficiently large \(k\),

and

Let \(\sigma \) be a density matrix on \(\mathcal H_R^{\otimes k}\) which commutes with all copy permutations. Let \(a\in \mathbb R\) and \(w\ge 0\). Set \(L=-\log \sigma \), with the logarithm assigned value zero on the kernel. If

then there is a label \(\lambda \) such that

For every density matrix \(\rho \) on \(\mathcal H_R\), for all sufficiently large \(k\) there is a label \(\lambda \) satisfying

No invertibility of \(\rho \) is assumed.

Let \(\psi \in \mathcal H_S\otimes \mathcal H_R\) be a unit vector, let \(\rho =\operatorname {tr}_S|\psi \rangle \langle \psi |\), and let \(v_k\) be its literal tensor power regrouped into \(\mathcal H_S^{\otimes k}\otimes \mathcal H_R^{\otimes k}\). There is one sequence \((\lambda _k)\) for which, for all sufficiently large \(k\),

and \(\log d_{\lambda _k}=kS(\rho )+o(k)\).

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity. Then the ambient middle subsystem has a direct-sum tensor decomposition

In unitary coordinates adapted to this decomposition there are density matrices \(\sigma _j\) and positive, not necessarily normalized operators \(\omega _j\) such that

The tensor factors appear here in the order \(b_j^R\otimes b_j^L\), the reverse of HJPW’s \(b_j^L\otimes b_j^R\) order. Directions complementary to the minimum joint support form one-dimensional tensor sectors with zero \(\omega _j\).

This is [ HJPW04 , Theorem 6, equations (13)–(14) ] , lines 493–502.

If two operators \(X,Y\) on \(A\otimes B\) have

for every member of the finite effect family, then \(X=Y\).

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity, and put \(\rho _{AB}=\operatorname{tr}_C\rho _{ABC}\). If \(\widehat{\mathcal R}:B\to B\otimes C\) is the completed recovery channel, define

Then \(\varphi \) is trace-preserving and completely positive and

For each selected effect, let

The indices with \(p_s\neq 0\) form a finite nonempty set. For each such index, positivity of \(\xi _s\) makes \(p_s\) real and positive, and

is a density operator and \(\varphi (\mu _s)=\mu _s\). Consequently a Kraus representation of \(\varphi \) gives a single preserving operation for this finite nonempty density family.

This is the conditional-family construction in [ HJPW04 , Theorem 6, lines 493–505 ] . Effects with \(p_s=0\) are omitted before normalization; no artificial conditional state is introduced for them.

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity, and let \(\mu _s\) be the finite nonempty family of normalized conditional states obtained from the active separating effects. On the minimum joint supporting subspace of this family there are tensor-product direct-sum coordinates such that

where the weights form probability distributions and all factors are density operators. The support isometry reconstructs every \(\mu _s\).

The channel

restricts to the same support and intertwines with its ambient action. In these coordinates its action on each diagonal summand is

The tensor factors are written in the reverse order from HJPW.

This is the application of HJPW Theorem 6, lines 493–505, to Properties 1 and \(2'\) from Appendix A, lines 761–816 and 853–882.

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity, and let \(V\) be the support isometry and \((e,U,\sigma _j)\) the direct-sum coordinates of Theorem 13.8.71. Put \(W=\mathbf1_A\otimes V\). Then

In the same support coordinates there are positive, not necessarily normalized operators \(\omega _j\) such that

The tensor factors are written in the reverse order from HJPW. The block equation is on the minimum joint supporting subspace; it does not assert an ambient direct-sum equivalence on subsystem \(B\).

Scope restriction (HJPW Theorem 6, equation (14)): the displayed direct sum is restricted to the minimum joint supporting subspace, whereas equation (14), lines 499–502, decomposes the ambient subsystem \(B\). This restriction is documented in the TNLean paper-gap note [ con26s ] .

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity, with the ambient decomposition

from Theorem 13.8.73. There are a finite-dimensional ancilla, a fixed pure ancilla vector, and one unitary \(U_{BCE}\) whose block-coordinate form is

On every supported sector, \(U_j\) dilates a trace-preserving completely positive map from \(b_j^R\) to \(b_j^R C\) that sends \(\sigma _j\) to a state whose \(b_j^R\) marginal is \(\sigma _j\). The resulting state is positive and has trace one. The displayed block form uses the order \(U_j\otimes \mathbf1_{b_j^L}\); in HJPW’s order it is \(\mathbf1_{b_j^L}\otimes U_j\).

The recovery operation determined by the chosen unitary agrees with the Petz recovery operation on every operator supported by the minimum joint supporting subspace, and

No equality of the two operations is asserted on the complementary ambient sectors.

This is [ HJPW04 , Theorem 6, equation (15) ] , lines 547–560, using Appendix A, Theorem 10, Property 2, lines 791–800, the equivalence with Property \(2'\) in lines 808–823, and the operation-level proof in lines 853–882.

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity, and let \(U_B\) and the factors \(\omega _j\) be those of Theorems 13.8.73 and 13.8.74. For each supported sector, put

This is a density matrix on \(\mathcal H_{b_j^R}\otimes \mathcal H_C\), where \(\widehat{\mathcal R}_j\) is the sector recovery operation. Read each middle-system summand in the HJPW order \(\mathcal H_{b_j^L}\otimes \mathcal H_{b_j^R}\), and set \(W=\mathbf1_A\otimes (U_B\otimes \mathbf1_C)\). Then

Thus every complementary block is zero, while the supported \(j\)th block, in the factor order \((A b_j^L)\otimes (b_j^R C)\), is \(\omega _j\otimes \widehat\rho _j\). The factors \(\omega _j\) remain unnormalized; no probabilities or normalized left factors are asserted.

This is the final substitution in [ HJPW04 , Theorem 6, equations (11), (14), and (15) ] , lines 562–570.

Let \(\rho \) be positive semidefinite on a finite-dimensional complex Hilbert space, with eigenvalues \(p_i\), and put \(M_p(u)=\sum _i p_i e^{-u\log p_i}\). For \(k\geq 0\) and \(u,s\in \mathbb R\),

The logarithm is assigned the value zero on the kernel. No trace normalization is assumed. Zero copies give the identity on the one-dimensional space of empty configurations.

For all real \(u,s\),

Let \(\rho \) be a positive semidefinite matrix of trace one on a finite-dimensional complex Hilbert space of dimension \(d\). Write \(p_i\) for its eigenvalues, \(S=-\sum _i p_i\log p_i\), and \(M_p(v)=\sum _i p_i e^{-v\log p_i}\), with zero spectral weights contributing zero. Let \(K,r\in \mathbb R\) satisfy \(\log M_p(v)\leq vS+Kv^2\) for every \(|v|\leq r\). For a nonnegative integer \(k\), let \(\rho _k=\rho ^{\otimes k}\) and \(F_k=\sum _\lambda \log (\dim \lambda )\pi _\lambda \) be the central entropy observable of the copy-permutation action. If \(u\geq 0\), \(2u\leq r\), and \(2u\leq 1\), then

These bounds retain the entropy center in the signed moment estimates of [ Ope26b , Section 8 ] .

Let \(K,r\) be real and suppose \(\log M_p(v)\leq vS+Kv^2\) whenever \(|v|\leq r\). Let \(F_k=\sum _\lambda \log (\dim \lambda )\pi _\lambda \) be the central entropy observable for the permutation action on the \(k\) copies. If \(u\geq 0\), \(2u\leq r\), and \(2u\leq 1\), then

These are the explicit independent-copy bounds in [ Ope26b , Section 8 ] .

Let \((p_i)_i\) be any finite probability distribution, allowing zero weights, and let \(S,V\) be given by (355). For every \(k\ge 0\) and \(w{\gt}0\),

For every \(k\ge 0\) and \(w{\gt}0\),

In particular, for \(k{\gt}0\) and \(w=k^{3/4}\),

Thus the mass outside this entropy window tends to zero for a fixed density matrix as the number of copies tends to infinity.

For every real function \(f\) and \(k\ge 0\),

This identity also holds for any positive semidefinite \(\rho \) whose trace need not be one. A real function on a finite spectrum defines its spectral functional calculus without a global continuity assumption.

For every real number \(V\) and every integer \(m\ge 0\), there is an integer \(K\ge 1\) such that, for all integers \(k\ge K\),

In particular, the estimate applies to a nonnegative variance \(V\). It requires no spectral or representation label assumptions.

For every pair of matrices \(H^{\prime },H\in \mathcal M\), define

Then \(V\) is differentiable and \(V^{\prime }(t)=iG(t)V(t)\) for every \(t\in \mathbb {R}\). Neither matrix is required to be Hermitian.

A completely positive map in rectangular Kraus form that preserves the matrix trace is trace-preserving completely positive.

Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfy \(V^\dagger V=\mathbb {1}_k\). For every \(A\in M_{k}(\mathbb {C})\),

Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfy \(V^\dagger V=\mathbb {1}_k\). For every \(A\in M_{k}(\mathbb {C})\),

Let \(\rho ,\sigma \in M_{D}(\mathbb {C})\) be density matrices with \(\sigma \) of full rank. Then the relative entropy is non-negative, \(D(\rho \Vert \sigma )\ge 0\).

Let \(\rho ,\sigma \in M_{D}(\mathbb {C})\) be density matrices with \(\sigma \) of full rank. Then the relative entropy vanishes exactly when the states coincide, \(D(\rho \Vert \sigma )=0\iff \rho =\sigma \). Together with nonnegativity, this is the order property that makes \(D\) a divergence.

Let \(\rho ,\sigma \in M_{D}(\mathbb {C})\) be density matrices satisfying the support condition \(\ker \sigma \subseteq \ker \rho \), that is, every vector annihilated by \(\sigma \) is annihilated by \(\rho \). Then the relative entropy is non-negative, \(D(\rho \Vert \sigma )\ge 0\).

Under the hypotheses of Definition 13.8.59, there are \(L\in \mathbb {N}\), positive dimensions \(d_0,\ldots ,d_{L-1}\) and multiplicities \(m_0,\ldots ,m_{L-1}\) with \(\sum _\ell d_\ell m_\ell =D\), and a unitary \(U\in M_{D}(\mathbb {C})\) such that a matrix \(A\in M_{D}(\mathbb {C})\) belongs to \(A_0\) exactly when

for some matrices \(B_\ell \in M_{d_\ell }(\mathbb {C})\).

Under the hypotheses of Definition 13.8.59, there are \(L\in \mathbb {N}\), positive dimensions \(d_\ell ,m_\ell \), a unitary \(U\), and density matrices \(\sigma _\ell \in M_{m_\ell }(\mathbb {C})\) such that, for every member \(\rho _k\) of the invariant family, there are matrices \(X_{\ell ,k}\in M_{d_\ell }(\mathbb {C})\) satisfying

The density matrices \(\sigma _\ell \) and the decomposition are common to the whole family.

No positivity or trace normalization is asserted for the matrices \(X_{\ell ,k}\). Thus this is the full-support fixed-point block form underlying HJPW Appendix A, lines 853–856, not the normalized decomposition of Property 1.

Under the hypotheses of Definition 13.8.59, there are finitely many operations \(F_1,\ldots ,F_M\in \mathbf F\), \(M\ge 1\), with

For every \(X\in M_{D}(\mathbb {C})\),

Let \(\rho _1,\ldots ,\rho _K\) be density operators, without a faithfulness assumption on their average. On their minimum joint supporting subspace there is one direct-sum tensor decomposition in which

where the weights form probability distributions and all displayed factors are density operators. Every trace-preserving completely positive operation preserving the compressed family acts on each diagonal summand as

In particular, every operation preserving the original family restricts to such an operation, and its action transports back through the support isometry by the intertwining identity above. The support isometry reconstructs every original \(\rho _k\) from \(\widehat\rho _k\).

This is HJPW Properties 1 and \(2'\) after the joint-support reduction in Appendix A, lines 761–816 and 853–882. The tensor factors are written in the reverse order from HJPW.

Let \(\rho _1,\ldots ,\rho _K\) be density operators and let \(Q\) be the support projection of their average. There are an integer \(r\) and an isometry \(V:\mathbb C^r\to \mathcal H\) with \(VV^\dagger =Q\) such that the compressed states \(\widehat\rho _k=V^\dagger \rho _kV\) are density operators, their average is positive definite, and

Every trace-preserving completely positive operation preserving all \(\rho _k\) compresses along the same isometry to a trace-preserving completely positive operation \(\widehat F\) preserving all \(\widehat\rho _k\). Its ambient and compressed actions intertwine:

This is the reduction to the minimum joint supporting subspace in HJPW, Appendix A, lines 761–763.

The map \(P_0^*\) is positive, unital, and idempotent, and its range is exactly \(A_0\).

Suppose in addition that every \(\rho _k\) is positive semidefinite with trace one. Under the explicit positive-definite common-average hypothesis of Definition 13.8.59, the common decomposition of Theorem 13.8.66 admits numbers \(q_{\ell |k}\geq 0\) and density matrices \(\tau _{\ell |k}\in M_{d_\ell }(\mathbb {C})\) such that, for every \(k\),

The density matrix \(\sigma _\ell \) is independent of \(k\). The tensor factors are written in the reverse order from HJPW Property 1. This is the full-support specialization of that state decomposition. No action of a preserving operation on the summands is asserted in this theorem; the next theorem supplies it under the same full-support hypothesis.

Use the decomposition of Theorem 13.8.67. For every trace-preserving completely positive operation \(F\) satisfying \(F(\rho _k)=\rho _k\) for all \(k\), and every summand \(\ell \), there is a trace-preserving completely positive operation \(F_\ell \) on \(M_{m_\ell }(\mathbb {C})\) such that \(F_\ell (\sigma _\ell )=\sigma _\ell \). If \(\iota _\ell \) denotes inclusion of the \(\ell \)-th diagonal summand and

then, for all \(A\in M_{m_\ell }(\mathbb {C})\) and \(B\in M_{d_\ell }(\mathbb {C})\),

This is the full-support specialization of HJPW Property \(2'\) (Appendix A, lines 808–816 and 860–882), with the tensor factors reversed. It does not include the joint-support reduction or the Stinespring form of Property 2.

Under the hypotheses of Definition 13.8.59, there is \(F_0\in \mathbf F\) with \(A_0=A_{F_0}\).

Let \(T_0,\mu ,r\ge 0\) and \(v,K\in \mathbb {R}\) satisfy (340). For every \(b\ge 0\), \(L\ge 1\), and \(C\ge 0\), put \(B_*=1+b+\log L\) and \(\ell =\max \{ 0,\lceil KB_*\rceil \} \), the least nonnegative integer at least \(KB_*\). Then

Let \(T_0\ge 0\), \(v\ge 0\), \(\mu {\gt}0\), and \(r\ge 0\) be real. There is a constant \(K{\gt}0\), depending only on these four parameters, such that

Let \(T_0{\gt}0\), \(v\ge 0\), \(\mu {\gt}0\), and \(r\ge 0\) be real. There is one \(K{\gt}0\) satisfying (340) such that, for every \(b\ge 0\), \(L\ge 1\), and \(C\ge 0\), the choices \(B_*=1+b+\log L\) and \(\ell =\lceil KB_*\rceil \) satisfy \(\ell \ge 1\), \(KB_*\le \ell {\lt}KB_*+1\), and (341).

If \(A\succ 0\), then, for every \(s\in \mathbb {R}\),

If \(\rho \succeq 0\), then

Let \(A\) be a Hermitian matrix on a finite index set, and let \(e\) be a bijection onto another finite index set. Then

Let \(\rho \) be positive semidefinite on \(H_L\otimes H_R\), and let \(P_L\) be the orthogonal projector onto the support of \(\operatorname{tr}_R\rho \). Then \((P_L\otimes \mathbb {1}_R)\rho =\rho \).

Under the same assumptions, \(\rho (P_L\otimes \mathbb {1}_R)=\rho \). For \(H_L=H_A\otimes H_X\) and \(H_R=H_B\), the two absorption identities are precisely the \(P_{AX}\) support-projector identities in [ CPGSV16 , Appendix D.2, lines 2228–2235 ] .

Let \(X\) be a matrix on \(H_L\otimes H_R\). For basis indices \(i,s\) and \(r\),

Let \(A:\mathbb {R}\to \mathcal M\) be continuously differentiable, and let \(M\ge 0\) satisfy \(\| A(s)\| \le M\) for every \(s\in \mathbb {R}\). For every \(X_0\in \mathcal M\), there is a differentiable path \(U:\mathbb {R}\to \mathcal M\) with \(U(0)=X_0\) such that, for every \(t\in \mathbb {R}\),

For a positive semidefinite matrix \(H\), \(\| H\| \le \operatorname{tr}H\). For a rectangular matrix \(B\), \(\| B\| ^2\le \operatorname{tr}(B^\dagger B)\).

For \(d_C\ge 1\), the matrix \(\tau _C=d_C^{-1}\mathbb {1}_C\) is positive definite and has trace one.

Let \(V\) be finite-dimensional and let \(f:V\to V\) have bounded orbits. For every \(x\in V\), the Cesàro averages satisfy

Moreover,

and \(P_fx=x\) if and only if \(fx=x\). The complex matrix specialization underlying [ Wol12 , Equation (6.14) ] is given below.

Under the hypotheses of Theorem 10.9.15, the mean-ergodic projection of \(T\) satisfies

Consequently,

Here each summand is written on \(\mathbb {C}^{m_k}\otimes \mathbb {C}^{d_k}\), so \(\operatorname{tr}_{m_k}\) traces the first factor. This is the full-support part of Equation (6.63) in [ Wol12 , Theorem 6.14 ] ; the density matrices are not yet asserted to be positive definite.

Let \(A\) and \(B\) be positive semidefinite, with \(P_A\) and \(P_B\) the orthogonal projections onto their respective ranges. Then

Here the logarithm is extended by zero on the kernel. Neither matrix is required to be positive definite, and either index set may be empty. This auxiliary identity is project-derived rather than a theorem of CPSV16.

Let \(A\) and \(B\) be positive semidefinite, and let \(f\colon \mathbb {R}\to \mathbb {R}\) be multiplicative on the non-negative reals. Then

Let \(\rho _{AB}\) be a bipartite density operator and let \(\Phi _A\) and \(\Psi _B\) be trace-preserving completely positive maps whose input and output matrix algebras may have different dimensions. Then \(I(A':B')_{(\Phi _A\otimes \Psi _B)(\rho )}\leq I(A:B)_\rho \).

Let \(\rho _{AB}\) be a bipartite density operator and let \(\Phi _A\) be a trace-preserving completely positive map whose input and output matrix algebras may have different dimensions. Then \(I(A':B)_{(\Phi _A\otimes \operatorname{id}_B)(\rho )}\leq I(A:B)_\rho \).

Let \(\rho _{AB}\) be a bipartite density operator and let \(\Psi _B\) be a trace-preserving completely positive map whose input and output matrix algebras may have different dimensions. Then \(I(A:B')_{(\operatorname{id}_A\otimes \Psi _B)(\rho )}\leq I(A:B)_\rho \).

For a bipartite density matrix \(\rho _{AB}\) on \(\mathbb {C}^{d_A} \otimes \mathbb {C}^{d_B}\) with \(d_A, d_B \ge 1\) whose single-system reduced states are obtained by partial trace, \(I(A{:}B) \le \log d_A + \log d_B\).

Let \(\rho _{AB}\succeq 0\) be a finite-dimensional bipartite operator with \(\operatorname{tr}\rho _{AB}=1\). Then

No positive-dimension or faithful-marginal hypothesis is required. The operator-Schmidt rank is the ordinary operator-Schmidt rank.

For any tripartite density matrix \(\rho _{ABC}\) on \(A \otimes B \otimes C\),

where the left-hand side is the bipartite mutual information of the reduced state \(\rho _{AB} = \operatorname{tr}_C(\rho _{ABC})\) and the right-hand side is evaluated by expanding \(I(A{:}BC) = S(\rho _A) + S(\rho _{BC}) - S(\rho _{ABC})\).

For any bipartite density matrix \(\rho _{AB}\), \(I(A{:}B) \ge 0\).

Write \(\rho _{ij}\in M_{d_B}(\mathbb {C})\) for the blocks determined by a basis of the first factor, and define \(\mathcal R_\rho (X)=\sum _{i,j}X_{ij}\rho _{ij}\). Then

Let \(V_A:\mathcal H_A\to \mathcal K_A\) and \(V_B:\mathcal H_B\to \mathcal K_B\) be isometries between finite-dimensional complex spaces. For every operator \(X\) on \(\mathcal H_A\otimes \mathcal H_B\),

This remains valid when one or more of the spaces have dimension zero.

Let \(\rho \geq 0\) be an operator on \(H_A\otimes H_B\). Let \(V_A:\widehat H_A\to H_A\) and \(V_B:\widehat H_B\to H_B\) be isometries whose range projectors are the support projectors \(P_A\) and \(P_B\) of the two marginals. Set \(W=V_A\otimes V_B\) and \(\rho _c=W^\dagger \rho W\). Then

No marginal is required to be faithful, and zero-dimensional coordinate spaces are permitted.

Every nonzero finite-dimensional bipartite operator \(X\) satisfies

This includes zero-dimensional factors, where the nonzero hypothesis is impossible.

Let \(\rho \) be any matrix on the common space, and let \(\rho _{QC}=\operatorname {Tr}_{VR}\rho \) and \(\rho _{VR}=\operatorname {Tr}_{QC}\rho \). For all real \(a,b\),

The matrix need not be normalized, invariant under copy permutations, or independent between the two pairs. The identities are complex trace equalities, and hence also hold for their real parts.

Let \(e:K\simeq C\times R\), where \(R\) is finite. For arbitrary matrices \(A\) and \(B\) on the first factor and on \(K\), respectively, the partial trace on the \(R\) coordinate satisfies

where the coordinates of \(B\) are changed by \(e\) on both sides.

Let \(\sigma \) be positive semidefinite on \(H_L\otimes H_R\), and let \(\rho \) be any matrix on the same space such that \(\ker \sigma \subseteq \ker \rho \). Then

If \(\sigma \) is positive semidefinite with trace one, then \(\widehat{\mathcal R}_\sigma \) is completely positive and trace preserving.

Let \(\rho \) and \(\sigma \) be positive semidefinite matrices on \(H_L\otimes H_R\) such that \(\ker \sigma \subseteq \ker \rho \). Then

For every positive semidefinite \(\sigma \),

If \(P_\tau XP_\tau =X\), then \(\widehat{\mathcal R}_\sigma (X)=\mathcal R_\sigma (X)\).

The complementary preparation term \(\mathcal C_\sigma \) is completely positive.

Let \(X\) be a matrix on \(H_L\otimes H_R\), let \(w\in H_L\), and let \((e_r)_r\) be the distinguished orthonormal basis of \(H_R\). Then

Let \(\rho \) and \(\sigma \) be positive semidefinite matrices satisfying \(\ker \sigma \subseteq \ker \rho \), and suppose that

Fix a primitive \(d_C\)-th root of unity, put \(U_g=\mathbf1\otimes W_g\), and define the unweighted Weyl family

Then, for every Weyl index \(g\) and every \(t{\gt}0\),

In particular, the zero Weyl index gives the projected resolvent of the original pair \((\rho ,\sigma )\). No ambient resolvent equality is asserted. The support convention is that of Jenčová–Ruskai, arXiv:0903.2895v4, lines 717–720, and their common-resolvent equality argument is at lines 766–793. This is one analytic step toward the recovery implication of Hayden–Jozsa–Petz–Winter, Theorem 3 and equation (8); it is not the direct-sum Markov decomposition invoked in CPSV16, Lemma Lsigma3.

Let \(\rho \) and \(\sigma \) be positive definite and suppose that \(D(\rho \Vert \sigma ) =D(\operatorname{tr}_C\rho \Vert \operatorname{tr}_C\sigma )\). Put

Then

Hence the raw partial-trace Petz map satisfies \(\mathcal R_\sigma (\operatorname{tr}_C\rho )=\rho \). This is the positive-definite case only; no singular-support conclusion is asserted.

In the finite Weyl coordinates \(\mathbb C^{d_S}\otimes \mathbb C^{\mathbb Z/d_C\mathbb Z}\), let \(\rho \) and \(\sigma \) be positive semidefinite matrices satisfying \(\ker \sigma \subseteq \ker \rho \), and suppose that

Put

If \(P_\sigma \) is the orthogonal projection onto \((\ker \sigma )^\perp \), then

Consequently, \(\mathcal R_\sigma (\operatorname{tr}_C\rho )=\rho \) for the raw support Petz map. The projected relative-modular argument follows Jenčová–Ruskai, arXiv:0903.2895v4, lines 766–793, and the recovery formula is [ HJPW04 , Theorem 3, equation (8) ] . This result does not assert the middle-space direct-sum decomposition in [ CPGSV16 , Lemma Lsigma3, lines 1351–1363 ] .

Let \(\rho \) and \(\sigma \) be positive definite, and suppose that \(D(\rho \Vert \sigma )=D(\operatorname{tr}_C\rho \Vert \operatorname{tr}_C\sigma )\). For the uniformly weighted Weyl conjugates \(A_g=d_C^{-2}U_g\rho U_g^\dagger \) and \(B_g=d_C^{-2}U_g\sigma U_g^\dagger \), put \(A=\sum _gA_g\) and \(B=\sum _gB_g\). Then

The support Petz transpose map \(\mathcal R_\sigma \) is completely positive.

For every positive semidefinite \(\sigma \),

For matrices \(M_0,\ldots ,M_{m-1}\) on \(A\), their tensor product in the preceding coordinates satisfies

The formula includes \(m=0\) and requires no positivity or normalization. It is the tensor-product identity used in [ Ope26b , Section 8, lines 550–560 ] .

Let \(x,\rho \in \mathbb {C}^{I\times I}\) be commuting positive semidefinite matrices, \(Q\) an orthogonal projection, and \(C,u,R\in \mathbb {R}\). Assume \(u\ge 0\), \(R\ge (C+\log 8)(1+u)\), \(\rho \le x+e^{-R}\mathbb {1}\), \(\operatorname{rank}Q\le e^{C(1+u)}\), and \(\operatorname{tr}((\mathbb {1}-Q)\rho )\le 1/8\). Then \(T=e^{-R}\rho (x+e^{-R}\mathbb {1})^{-1}\) satisfies \(\operatorname{tr}T\le 1/4\).

The range of \(T_\phi \) is the peripheral subspace, \(T_\phi \) is idempotent, and

On a peripheral \(\mu \)-eigenvector, \(T_\phi \) acts as the identity and \(T_\varphi \) acts as multiplication by \(\mu \). These are [ Wol12 , Equations (6.12) and (6.13) ] .

Let \(\rho \) and \(\sigma \) be density operators on the finite-dimensional product \(H_L\otimes H_R\) such that \(\ker \sigma \subseteq \ker \rho \). If

then the completed partial-trace Petz channel associated with \(\sigma \) recovers \(\rho \):

This is the right-partial-trace forward implication of [ HJPW04 , Theorem 3, equation (8) ] . The additional off-support term belongs to the trace-preserving completion and vanishes on \(\operatorname{tr}_R\rho \); it is not part of the formula in the cited equation.

For an arbitrary real function \(f\) and independently chosen real label functions \(g_A\), with \(A\in \{ QC,VR,C,R\} \), write \(L_A(g_A)=\sum _\lambda g_A(\lambda )\Pi _A^\lambda \). Then

In particular, \(J\) commutes with both merge deficits.

Set \(D_{QV}=F_Q+F_V-F_{QV}\) and \(J=\mathbf1_{[0,\infty )}(G_{\mathrm{phys}})\). Then

Zero copies and zero-dimensional factors are included.

The partial swap is a Hermitian unitary and has operator norm at most one. Its compression \(W\) is Hermitian for every \(V\). If \(V^\dagger V=I\), then \(\| W\| \le 1\).

Let \(\Omega \in \mathcal H_R\otimes \mathcal H_C\otimes \mathcal H_Q\) be normalized, and let \(\rho _{RC}\) be its reduced state, with marginals \(\rho _R,\rho _C\). Suppose \(b\ge 0\) and \(D(\rho _{RC}\Vert \rho _R\otimes \rho _C)\le b\). Set \(\Psi =\Omega ^{\otimes 2}\) and \(\Phi =F_R\Psi \), grouping the two \(RC\) registers before the two \(Q\) registers. There exist a Hermitian operator \(W\) on the original doubled buffer \(\mathcal H_Q^{\otimes 2}\) with \(\| W\| \le 1\) and a real number \(z\) such that

This is the physical-buffer overlap construction in [ Ope26b ] .

Under the original-system hypotheses of Theorem 13.30.4.2, suppose \(\Delta {\gt}0\), \(b\ge 0\), and \(D(\rho _{RC}\Vert \rho _R\otimes \rho _C)\le b\), where \(\rho _{RC}=\operatorname {Tr}_Q|\Omega \rangle \! \langle \Omega |\). There are a Hermitian contraction \(W\) on \(\mathcal H_Q^{\otimes 2}\) and a real number \(z\ge e^{-2b}\) such that, with \(V=I_{(RC)^{\otimes 2}}\otimes W\), \(\langle \Phi ,V\Psi \rangle =z\). The same \(W,z\) satisfy the following statements for every \(h\ge 0\) and \(T\ge 0\). For the actual integrals \(M_h=M_h(\mathbb H',\mathbb H;V)\) and \(M_h^T=M_h^T(\mathbb H',\mathbb H;V)\),

For \(h{\gt}0\), \(\tau _h(T)=2e^{-T^2/(2h)}\). At \(h=0\), the probability integral gives \(M_0=V\) and the displayed bounds use \(\tau _0(T)=2\).

Let \(T\) be a reference register, \(F\) a Hermitian operator on \(\mathcal H_T\), and \(x,y\in \mathcal H_T\otimes \mathcal H_Q^{\otimes 2}\). Write \(\widetilde x=(I_T\otimes K)x\) and \(\widetilde y=(I_T\otimes K)y\). Then

This is an equality of complex numbers. Neither normalization nor an isometry hypothesis is needed.

For any \(\chi \in \mathcal H_L\otimes \mathcal H_M\) and unit vectors \(a\in \mathcal H_L\), \(a^{\prime }\in \mathcal H_M\), put \(\sigma _L=\operatorname {Tr}_M\vert \chi \rangle \langle \chi \vert \). Then

and the exact complex swap identity is

No normalization of \(\chi \) is required.

Let \(V:\mathcal H_Q\to \mathcal H_X\otimes \mathcal H_Y\) be any linear map, let \(\chi =(I_{RC}\otimes V)\Omega \), and view \(\chi \) across \(RX\mid CY\). Put \(\sigma =\operatorname {Tr}_{CY} \vert \chi \rangle \langle \chi \vert \) and use \(W=(V\otimes V)^\dagger F_X(V\otimes V)\). Then

Regrouping preserves inner products, \(\| \Psi \| ^2=\| \Omega \| ^4\), and a tensor product of two unit purifications is again a unit vector.

Suppose \(\| \Omega \| =1\), \(H\Omega =E_0\Omega \), \(\Delta \ge 0\), and \(H-E_0I-\Delta (I-|\Omega \rangle \! \langle \Omega |)\succeq 0\). For the operators and vectors in (325), both generators are Hermitian, \(\| \Psi \| =\| \Phi \| =1\), and

For \(E_0,\Delta \in \mathbb {R}\) and any \(\Omega \), if \(H-E_0I-\Delta (I-|\Omega \rangle \! \langle \Omega |)\) is positive semidefinite, then \(H\) is Hermitian.

Each \(\rho _j\) is positive semidefinite, for every vector \(\Omega \). Empty coordinate sets are allowed.

If \(\| \Omega \| =1\), then \(\operatorname {Tr}\rho _j=1\) for each \(j\in \{ 0,1,2\} \).

If \(\| \Omega \| =1\), then the three original marginals satisfy

The marginals may be singular. This is the nonnegativity of the physical entropy difference used in [ Ope26b , Section 8 ] .

Let \(\rho _{ABC}\) have trace one and suppose that, after reassociating the tensor factors, it factors as

Here the two factors act through their natural overlapping lifts. Then \(\rho _{ABC}\) admits a quantum Markov decomposition on the middle system \(B\).

Let \(H\) be positive semidefinite on a finite-dimensional complex Hilbert space, let \(S\) be a finite set, and let \((v_i)_{i\in S}\) be any vectors. Then

Suppose \(R\ge 0\), \(\Delta {\gt}0\), and \(\| \Omega \| =1\). Every vector \(\psi \) satisfying \(H\psi =E_0\psi \) obeys \(\psi =\langle \Omega ,\psi \rangle \Omega \). The vector \(\psi \) may be zero or unnormalized; no eigenvector equation on \(\Omega \) is required for this projection identity. If also \(H\Omega =E_0\Omega \), then

If \(P\ge 0\), then for all \(x,y\in \mathcal H\),

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. The trace adjoint \(P_T^*\) of its mean-ergodic projection is positive, unital, and idempotent. It is a retraction onto the adjoint fixed-point space:

and \(P_T^*(Y)=Y\) if and only if \(T^*(Y)=Y\). This is the adjoint projection used in the proof of [ Wol12 , Theorem 6.14 ] .

Suppose \(p_j{\gt}0\) and \(L_j\leq R_j\) for every \(j\). If \(\sum _jp_jL_j=\sum _jp_jR_j\), then \(L_j=R_j\) for every \(j\). This is the positivity argument applied to strong subadditivity in [ CPGSV16 , Appendix C.2, lines 1770–1780 ] .

Let \(\rho _{AB}\succeq 0\), with marginals \(\rho _A\) and \(\rho _B\). Then

Equivalently, the support of \(\rho _{AB}\) is contained in \(\operatorname{supp}(\rho _A)\otimes \operatorname{supp}(\rho _B)\). This remains valid when either factor has dimension zero.

Let \(P:\mathcal M\to \mathcal M\) be complex linear, preserve Hermitian matrices, and satisfy \(\| P(X)\| \le \| X\| \) for every \(X\in \mathcal M\). For every pair of Hermitian matrices \(H^{\prime },H\in \mathcal M\), put

There is a differentiable path \(U:\mathbb {R}\to \mathcal M\) with \(U(0)=I\) such that, for every \(s\in \mathbb {R}\),

Every differentiable path \(W:\mathbb {R}\to \mathcal M\) with \(W(0)=I\) and \(W^{\prime }(s)=iG_P(s)W(s)\) for every \(s\in \mathbb {R}\) equals \(U\). The same path \(U\) satisfies the following estimate simultaneously for every \(t,\varepsilon \in \mathbb {R}\): if \(\| G(s)-G_P(s)\| \le \varepsilon \) for every \(s\in J_t\), then

Let \(P:\mathcal M\to \mathcal M\) be complex linear with \(\| P(X)\| \le \| X\| \) for every \(X\in \mathcal M\). For every Hermitian matrix \(H\in \mathcal M\), every matrix \(B\in \mathcal M\), and every \(t\in \mathbb {R}\),

Neither Hermiticity of \(B\) nor Hermitian preservation by \(P\) is required for this bound.

Let \(P:\mathcal M\to \mathcal M\) be complex linear and satisfy \(P(X)^\dagger =P(X)\) for every Hermitian matrix \(X\). For every pair of Hermitian matrices \(H,B\in \mathcal M\) and every \(t\in \mathbb {R}\), the matrix \(G_{P,H,B}(t)\) is Hermitian.

For every \(H\in \mathcal M\) and every differentiability order \(k\in \mathbb {N}\cup \{ \infty ,\omega \} \), the path \(t\mapsto e^{itH}\) is \(C^k\), where \(C^\omega \) denotes real analyticity. For every complex-linear map \(P:\mathcal M\to \mathcal M\), every \(H,B\in \mathcal M\), and every such \(k\), the path \(G_{P,H,B}\) is \(C^k\). Hermiticity, Hermitian preservation, and contractivity are not needed for these regularity assertions.