Dependencies

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) ] .

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.

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 ] .

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

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 13.7.1.

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 13.6.55.

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

This formulation has the same value \(S(\rho ) = -\sum _i \lambda _i \log \lambda _i\) as in Definition 13.4.1.

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

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 13.6.56 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 13.6.57, \(\bar\rho \) 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 13.6.61. 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 1.8 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 3.18.2.

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 \(P^2 = P\).

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 \(T_\varphi =T\circ T_\phi \).

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 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 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 ] .

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.

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 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 2.1.2.

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

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) ] .

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.

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

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 \(\ker ((\mathbb {1}_A / d_A) \otimes \rho _R) \subseteq \ker \rho \).

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.

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 (64).

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

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 13.6.35.

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 [ con26p ] .

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 \(\rho _{ABC}\) 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 \(I\) 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, \(2^{I(X:Y)_P}\leq \operatorname{rank}_{\mathbb {R}}P\).

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

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 (71),

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 (77). 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 (71),

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 13.4.4 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 \(\mathbb {C}^{d_A} \otimes \mathbb {C}^{d_B}\).

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 (71), the chosen complementary preparation term factors as

after the canonical reassociation of the three tensor factors. This follows from the chosen support completion in (77), not from [ HJPW04 , equation (10) ] .

For the reference in (71), 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 (65), 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 13.4.63 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.

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

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 13.6.77.

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 13.6.4, 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 \(\rho \).

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 ] .

In the setting of Theorem 13.11.1.5, the two marginals of \(\rho _c\) satisfy

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

Under the hypotheses of Theorem 13.12.1.1, 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 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 \(\rho _{ABC}\) 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 13.6.70. 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 [ con26q ] .

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

from Theorem 13.6.72. 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 13.6.72 and 13.6.73. 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.

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 13.6.58, 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 13.6.58, 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 13.6.58, 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 13.6.58, the common decomposition of Theorem 13.6.65 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 13.6.66. 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 13.6.58, there is \(F_0\in \mathbf F\) with \(A_0=A_{F_0}\).

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 \(r\),

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 10.8.15, 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 \(\sigma \) 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 \(\sigma \),

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.

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 \(\rho \in M_{d}(\mathbb {C})\) be a density matrix and let \(P_1,\ldots ,P_k\in M_{d}(\mathbb {C})\) be orthogonal projections satisfying \(\sum _{i=1}^k P_i=\mathbb {1}\). Define

Then

This is [ Wol12 , Chapter 8, Eq. (8.56) ] .

Let \(T,T':M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be trace-preserving and Hermiticity-preserving linear maps with \(T'(X)=\operatorname{tr}[X]\, Y\) for some fixed \(Y\in M_{D'}(\mathbb {C})\). If \(T-\varepsilon T'\) is positive for some \(\varepsilon \geq 0\), then for all density operators \(\rho _1,\rho _2\in M_{D}(\mathbb {C})\),

See [ Wol12 , Chapter 8, Theorem 8.17 ] .

If \(\rho \) is Hermitian, then

If \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) is a Hilbert–Schmidt contraction, then

For positive definite matrices \(\rho ,\sigma \) and a positive definite matrix \(\tau \) of unit trace,

Let \(\rho \) and \(\sigma \) be positive semidefinite matrices such that \(\ker \sigma \subseteq \ker \rho \), and let \(\tau \) be positive semidefinite with \(\operatorname{tr}\tau =1\). Then

For positive definite matrices \(\rho ,\sigma \) on a tensor product of a system factor and an ancilla factor,

where \(\operatorname{tr}_C\) is the partial trace over the ancilla factor. This is the positive-definite base case; the source inequality on the support domain \(\ker \sigma \subseteq \ker \rho \) is Theorem 13.4.35.

For positive semidefinite \(\rho ,\sigma \) on a tensor product of a system factor and an ancilla factor of dimension \(d_C\), with the support condition \(\ker \sigma \subseteq \ker \rho \),

where \(\operatorname{tr}_C\) is the partial trace over the ancilla factor.

Let \(A\) and \(B\) be Hermitian matrices of the same size, with spectral resolutions

Write \(w_{ij}=\langle u_i,v_j\rangle \), and use the total real logarithm: \(\log 0=0\), while \(\log x=\log |x|\) for \(x{\lt}0\). Then

This is an algebraic totalized extension to arbitrary Hermitian matrices of the homogeneous trace-log identity \((\mathrm{J1})\), which Jenčová–Ruskai state for strictly positive matrices in arXiv:0903.2895v4, lines 277–287.

On pairs of positive definite matrices in \(M_{D}(\mathbb {C})\), the map \((\rho ,\sigma )\mapsto D(\rho \Vert \sigma )\) is jointly convex.

On pairs of density matrices \((\rho ,\sigma )\) in \(M_{D}(\mathbb {C})\) satisfying the support condition \(\ker \sigma \subseteq \ker \rho \), the map \((\rho ,\sigma )\mapsto D(\rho \Vert \sigma )\) is jointly convex.

For every positive semidefinite operator \(\omega _{XY}\),

No invertibility assumption is made on either marginal. This is [ HJPW04 , Equation (4) ] .

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) be positive semidefinite matrices of trace one with \(\ker \omega \subseteq \ker \rho \). Assume that for every dimension and every pair of trace-one matrices \(A\succeq 0\) and \(B\succ 0\),

Then

Thus this theorem reduces the singular-reference case to the faithful comparison. Theorem 13.12.14 supplies the hypothesis in (206).

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) satisfy \(\rho \succeq 0\), \(\omega \succ 0\), and \(\operatorname{tr}\rho =1\). Then

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

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) be positive semidefinite matrices of trace one. If \(\ker \omega \subseteq \ker \rho \), then

Let \(A\) and \(B\) be positive definite, with spectral resolutions \(A=\sum _i\alpha _i|u_i\rangle \! \langle u_i|\) and \(B=\sum _j\beta _j|v_j\rangle \! \langle v_j|\). Put \(w_{ij}=\langle u_i,v_j\rangle \). Let \(L_A\) and \(R_B\) denote left and right multiplication, \(L_A(X)=AX\) and \(R_B(X)=XB\). Then, for \(t{\gt}0\),

Both quadratic forms are continuous on \((0,\infty )\). Moreover,

and the integrand in (138) is continuous and integrable on the positive half-line. This is the positive-definite spectral route of Jenčová–Ruskai, arXiv:0903.2895v4, §4.

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) be positive semidefinite with \(\ker \omega \subseteq \ker \rho \). Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfy

Then

The logarithm is totalized by \(\log 0=0\) on the kernel.

Let \(A\) and \(B\) be positive semidefinite matrices of the same size, let \(P_B\) be the orthogonal projection onto the support of \(B\), and, for \(t{\gt}0\), set

Write \(S_t^+\) for the generalized inverse that vanishes on \(\ker S_t\). Define

With the spectral notation of Theorem 13.6.15,

Consequently,

The function in (126) is continuous on \((0,\infty )\). If \(\ker B\subseteq \ker A\), then \(A P_B=A\), so \(Q_{A P_B}(t)\) equals the quadratic form with source \(\operatorname{vec}(A^{\top })\).

This is the support-projected form of \((\mathrm{intAB})\) in Jenčová–Ruskai, arXiv:0903.2895v4, §2.1, lines 423–427, with the support convention at lines 717–720.

Let \(A\) and \(B\) be positive semidefinite matrices of the same size and suppose that \(\ker B\subseteq \ker A\). With the spectral notation of Theorem 13.6.14, define, for \(t{\gt}0\),

Thus no ordinary quotient with \(\alpha _i=\beta _j=0\) is used. Define also

The function \(I_{A,B}\) is integrable on \((0,\infty )\), and

The trace-log identity \((\mathrm{J1})\), the scalar normalization \((\mathrm{intspec})\), and its matrix form \((\mathrm{intAB})\) occur in Jenčová–Ruskai, arXiv:0903.2895v4, at lines 277–287, 406–413, and 423–427, respectively. The support-domain extension is given at lines 717–720.

This theorem concerns the spectral expression \(I_{A,B}\). The next theorem identifies it with the coordinate-free left-right quadratic form underlying the finite Weyl formula.

For Hermitian matrices \(\rho ,\sigma \) and a unitary \(U\),

Let \(\rho \) and \(\sigma \) be positive semidefinite matrices on \(\mathcal{H}_S\otimes \mathbb {C}^{d_C}\) such that \(\ker \sigma \subseteq \ker \rho \), and suppose that \(D(\rho \Vert \sigma )=D(\operatorname{tr}_C\rho \Vert \operatorname{tr}_C\sigma )\). For a primitive \(d_C\)-th root of unity, put \(U_{ab}=\mathbb {1}_S\otimes W(a,b)\) and

Then, for every \(a,b\),

This is a scalar equality-propagation prerequisite for [ HJPW04 , Theorem 3 and equation (8) ] ; it neither characterizes equality in joint convexity nor asserts recovery.

Let \(\rho \) and \(\sigma \) be positive semidefinite matrices on \(\mathcal{H}_S\otimes \mathbb {C}^{d_C}\) such that \(\ker \sigma \subseteq \ker \rho \), and suppose that \(D(\rho \Vert \sigma )=D(\operatorname{tr}_C\rho \Vert \operatorname{tr}_C\sigma )\). For a primitive \(d_C\)-th root of unity, put \(U_{ce}=\mathbb {1}_S\otimes W(c,e)\) and

Then

This scalar identity is associated with the finite Jensen step in the Weyl proof of data processing. It is a prerequisite for [ HJPW04 , Theorem 3 and equation (8) ] ; it neither characterizes equality in joint convexity nor asserts recovery.

Let \(\rho \) be positive semidefinite. If \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \), then \((\mathbb {1}_L\otimes P_R)\rho =\rho \).

Under the same assumptions, \(\rho (\mathbb {1}_L\otimes P_R)=\rho \).

Let \(A\succeq 0\), let \(s\in \mathbb {R}\), and let \(U\) be unitary. Then

Let \(A\succeq 0\) and \(B\succeq 0\). For every \(s\in \mathbb {R}\),

If \(A\succ 0\), then

If \(\rho \geq 0\) and \(\omega {\gt}0\), then, for every \(\alpha \in \mathbb {R}\),

If \(\rho \geq 0\) and \(\omega {\gt}0\), then

If \(\rho \geq 0\) and \(\omega {\gt}0\), then

If \(\rho \succeq 0\) and \(\omega \succeq 0\), then \(Q_2(\rho ,\omega )\geq 0\).

Let \(\rho _{AB}\succeq 0\) have positive definite marginals \(\rho _A\) and \(\rho _B\). Then

No trace normalization is required.

Let \(\rho _{AB}\succeq 0\) be any finite-dimensional bipartite operator. Then

No trace normalization or positive-dimension hypothesis is required. The statement includes the cases in which either marginal support has dimension zero.

Let \(\rho \) be a square matrix and let \(\omega \succeq 0\) have the same finite index set. Let \(e\) be a bijection onto another finite index set. Then

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) be positive semidefinite with \(\ker \omega \subseteq \ker \rho \). Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfy (189). Then

Let \(\omega \succeq 0\), let \(\rho \) be a square matrix of the same size, and let \(U\) be unitary. Then

If \(\omega \succ 0\), then, for every square matrix \(\rho \) of the same size,

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 \(H\in M_{D}(\mathbb {C})\), \(\lVert T(H)\rVert _{\operatorname{tr}}\leq c\cdot \lVert H\rVert _{\operatorname{tr}}\). Generalizes Theorem 13.1.19 from the trace-preserving case \(c=1\).

Let \(\rho _{ABC}\) be a tripartite density matrix and set \(\sigma _{ABC}=(\mathbf1_A/d_A)\otimes \rho _{BC}\). Then equality holds in strong subadditivity if and only if relative-entropy data processing under the partial trace over \(C\) is saturated for this pair:

By Lemma 13.6.3, the reference on the right is \((\mathbf1_A/d_A)\otimes \rho _B\).

This is an equivalent hypothesis-free criterion with a different reference state. The exact product-marginal formulation is Theorem 13.6.48.

Let \(\rho _{ABC}\) be a tripartite density matrix. Then equality holds in strong subadditivity if and only if relative-entropy data processing under the partial trace over \(C\) is saturated for the pair \(\rho _{ABC}\) and \(\rho _A\otimes \rho _{BC}\):

Moreover, \(\operatorname{tr}_C(\rho _A\otimes \rho _{BC})=\rho _A\otimes \rho _B\). This is the product-marginal formulation in [ HJPW04 , Equations (5)–(7) ] .

Let \(\rho _{ABC}\) be a tripartite density matrix attaining equality in strong subadditivity. The raw Petz support map for the reference \(\rho _A\otimes \rho _{BC}\) recovers \(\rho _{ABC}\) from \(\rho _{AB}\). Moreover,

where \(\widehat{\mathcal R}_{\rho _{BC}}\) is a trace-preserving completely positive extension of the support formula. This is equation (11) of [ HJPW04 ] , obtained from Theorem 3 and equation (8) together with the supported form of equation (10).

No marginal is required to be invertible. If \(\rho _A\) is singular, the displayed factorization is asserted on the supported input \(\rho _{AB}\) only; it is not a global factorization of an ambient product-reference completion.

Let \(\rho \) be a Hermitian operator on \(A\otimes B\otimes C\), and let \(e_A:A'\to A\), \(e_B:B'\to B\), and \(e_C:C'\to C\) be bijections. Define \(\rho '\) by

If \(\rho \) satisfies equality in strong subadditivity, then so does \(\rho '\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). There are \(K\in \mathbb {N}\), positive dimensions \(d_0,\ldots ,d_{K-1}\) and multiplicities \(m_0,\ldots ,m_{K-1}\) with \(\sum _kd_km_k=D\), realized by an explicit identification of the index sets, and a unitary \(U\in M_{D}(\mathbb {C})\) such that a matrix \(A\in M_{D}(\mathbb {C})\) belongs to \(S\) exactly when

for some matrices \(B_k\in M_{d_k}(\mathbb {C})\); that is, up to reordering the two tensor factors of each block,

This is the block representation of a finite-dimensional \(*\)-algebra in Equation (1.39) of  [ Wol12 ] , invoked by [ Wol12 , Theorem 6.14 ] , in the unital case: a \(*\)-subalgebra contains the identity matrix, so there is no zero block.

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

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

This is the positive definite case of Theorem 13.6.4.

Let \(\rho ,\omega \in M_{D}(\mathbb {C})\) be positive semidefinite and suppose that \(\ker \omega \subseteq \ker \rho \). If \(V:\mathbb {C}^k\to \mathbb {C}^D\) satisfies \(VV^\dagger =P_{\operatorname{supp}(\omega )}\), then

No condition on \(V^\dagger V\) is needed for this identity.

Let \(A\) and \(B\) be positive semidefinite. Their positive square roots, support inverse square roots, and support projections satisfy

Moreover,

The same support projection absorbs the square root:

If \(A\) is positive definite, then \(P_A=\mathbf1\).

Let \(A,B\) be positive semidefinite, let \(t{\gt}0\), set

and let \(P_B\) be the support projection of \(B\). Then

Consequently,

This is the one-pair algebraic identification used in the singular equality argument of Jenčová–Ruskai, arXiv:0903.2895v4, lines 783–790. It does not assert that equality of relative entropies gives a common resolvent for a finite family.

Let \(A\in M_{n}(\mathbb {C})\) be positive semidefinite and let \(v\in \mathbb {C}^n\) satisfy \(\langle v,v\rangle =1\) and \(P_{\operatorname{supp}(A)}v=v\). Then

Here \(\log A\) is defined by the continuous functional calculus with \(\log 0=0\). Compare the finite-dimensional vector-state Jensen argument in [ Bha97 , Chapter V ] .

Let \(A\) and \(B\) be Hermitian matrices such that \(\ker A\subseteq \ker B\). If \(P_A\) is the support projection of \(A\), then \(P_A B P_A=B\).

Let \(I\) be a finite nonempty set. For each \(i\in I\), let \(A_i\) and \(B_i\) be positive-semidefinite matrices satisfying \(\ker B_i\subseteq \ker A_i\). Put

and suppose that

Then, for every \(i\in I\) and \(t{\gt}0\),

Thus the shifted relative-modular resolvents agree on \((\ker B_i)^\perp \). The local support projection is essential; no ambient equality is asserted. This is the support-restricted conclusion of Jenčová–Ruskai, arXiv:0903.2895v4, lines 766–793.

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 \(\ker B_i\subseteq \ker A_i\). Put

Then the function

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

This is the finite-family support-domain form of \((\mathrm{intspec})\) and \((\mathrm{intAB})\) in Jenčová–Ruskai, arXiv:0903.2895v4, lines 406–431, with the positive-semidefinite convention and kernel hypotheses at lines 717–720 and 766–785.

Under the hypotheses and notation of Theorem 13.6.20, for every \(t{\gt}0\) one has

This is the coefficient-correct combination of the two source defects in \((\mathrm{intAB})\) of Jenčová–Ruskai, arXiv:0903.2895v4, lines 423–435.

Under the hypotheses and notation of Lemma 13.6.24, suppose that

Then, for every \(i\in I\),

This is the support-domain form of the common-resolvent equation \((\mathrm{basiceq})\) in Section 3.1 of Jenčová–Ruskai, arXiv:0903.2895v4. Its residual calculation is the one in the Appendix, equations \((\mathrm{Mj})\) and \((\mathrm{eq:Schz1})\).

Under the hypotheses of Theorem 13.6.20, suppose

Then, for every \(t{\gt}0\),

This is the source-\(B\) pointwise-vanishing passage in Jenčová–Ruskai, arXiv:0903.2895v4, lines 433–435, 652–674, and 788–790. The common projected resolvent conclusion is stated downstream in Theorem 13.6.35.

Under the hypotheses and notation of Lemma 13.6.17, suppose that the source-\(B\) defect vanishes:

Put

and let \(P_{S_i}\) be the support projection of \(S_i\). Then

This is the fixed-parameter support-domain residual step behind \((\mathrm{basiceq})\) in Jenčová–Ruskai, arXiv:0903.2895v4, lines 652–660 and 788–790. No kernel inclusion is needed for this algebraic implication.

Let \(\rho \geq 0\) be a bipartite complex matrix whose first marginal is faithful. Choose an eigenbasis in which \(\operatorname{tr}_B\rho =\operatorname{diag}(p_1,\ldots ,p_{d_A})\) with every \(p_i{\gt}0\), and define the linear map \(\Phi _\rho \) on matrix units by

Write \(\sigma =\operatorname{diag}(p_1,\ldots ,p_{d_A})\) and \(\tau =\operatorname{tr}_A\rho \). Then \(\Phi _\rho \) is completely positive and trace preserving,

and application of \(\Phi _\rho \) to the first half of \(\sum _{i,j}\sqrt{p_ip_j}\, E_{ij}\otimes E_{ij}\), followed by restoring the order of the two factors, reconstructs \(\rho \).

For every \(A\in M_{D}(\mathbb {C})\), with \(\lvert A\rvert =\sqrt{A^\dagger A}\) the positive-semidefinite square root of \(A^\dagger A\),

This is the formula \(\lVert A\rVert _1=\operatorname{tr}[\lvert A\rvert ]\) of [ Wol12 , Chapter 8, Section 8.1 ] ; the trace of the positive-semidefinite matrix \(\lvert A\rvert \) is real.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, let \(\rho \) be a density operator, and let \(n{\gt}0\). Then

Here the displayed transfer-matrix norm equals \(\lVert T^n-T_\varphi ^n\rVert _{2\to 2}\), the Hilbert–Schmidt operator norm of the corresponding superoperator. The explicit \(n{\gt}0\) records the paper’s positive-natural convention: at \(n=0\), both superoperator powers are the identity, so the displayed right-hand side vanishes. This is the upper assertion of  [ Wol12 , Chapter 8, Proposition “Convergence towards asymptotic states”, Eqs. (8.112), (8.114)–(8.116) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map. Then for all Hermitian \(H\in M_{D}(\mathbb {C})\), \(\lVert T(H)\rVert _{\operatorname{tr}}\leq \lVert H\rVert _{\operatorname{tr}}\). See [ Wol12 , Chapter 8, Theorem 8.16 ] .

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

Also, \(\lVert A\rVert _{\operatorname{tr}}=0\) if and only if \(A=0\). Equivalently, \(\lVert A\rVert _{\operatorname{tr}}{\gt}0\) if and only if \(A\neq 0\).

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

The first inequality is the \(p=1\), \(p'=2\) case of [ Wol12 , Eq. (8.1) ] ; the second is the corresponding case of [ Wol12 , Eq. (8.7) ] . Both are used in the proof of trace-norm convergence toward asymptotic states.

For every \(c\in \mathbb {C}\) and \(A\in M_{D}(\mathbb {C})\), \(\lVert cA\rVert _{\operatorname{tr}} =\lvert c\rvert \, \lVert A\rVert _{\operatorname{tr}}\). This is the homogeneity axiom for matrix norms [ Wol12 , Chapter 8, Section 8.1 ] .

The Schatten one-norm and trace norm are the sums over the finite support of the singular-value sequence. Equivalently, the trace norm is the sum over the indices below the rank of the represented linear map.

For all \(A,B\in M_{D}(\mathbb {C})\), \(\lVert A+B\rVert _{\operatorname{tr}} \leq \lVert A\rVert _{\operatorname{tr}}+\lVert B\rVert _{\operatorname{tr}}\). Together with homogeneity (Theorem 13.1.6) and definiteness (Theorem 13.1.4), this completes the norm axioms of [ Wol12 , Chapter 8, Section 8.1 ] for the trace norm.

For all unitaries \(U,V\in M_{D}(\mathbb {C})\) and every \(A\in M_{D}(\mathbb {C})\),

The trace norm is thus unitarily invariant [ Wol12 , Chapter 8, Section 8.1 ] .

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

and the maximum is attained: some unitary \(U\) satisfies \(\operatorname{tr}[A^\dagger U]=\lVert A\rVert _{\operatorname{tr}}\). See [ Wol12 , Chapter 8, Eq. (8.11) ] .

For any tripartite matrix \(\rho _{ABC}\), \(\operatorname{tr}(\rho _{ABC})=\operatorname{tr}(\operatorname{tr}_A(\rho _{ABC}))\).

For any tripartite matrix \(\rho _{ABC}\), \(\operatorname{tr}(\rho _{ABC})=\operatorname{tr}(\operatorname{tr}_{AC}(\rho _{ABC}))\).

For any tripartite matrix \(\rho _{ABC}\), \(\operatorname{tr}(\rho _{ABC})=\operatorname{tr}(\operatorname{tr}_C(\rho _{ABC}))\).

For a linear map \(S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(\lVert S\rVert _{2\to 2}\) be its operator norm for the Hilbert–Schmidt norm. Then

where the norm on the transfer matrix is its largest-singular-value norm. The transfer representation also preserves subtraction and powers.

Let \(\rho \) and \(\sigma \) be matrices on \(\mathcal H_S\otimes \mathbb C^{d_C}\), with \(\sigma \) positive semidefinite, and set \(\overline\sigma =(\operatorname{tr}_C\sigma )\otimes d_C^{-1}\mathbf1_C\). If the identity summand obeys the support sandwich identity

then the raw support Petz map recovers \(\rho \): \(\mathcal R_\sigma (\operatorname{tr}_C\rho )=\rho \). This is the algebraic reduction in [ HJPW04 , Theorem 3, equation (8) ] . It does not derive the support sandwich identity from equality of relative entropies.

Fix a dimension \(d\ge 1\) and a primitive \(d\)-th root of unity \(\zeta \). The cyclic shift \(X\), the clock operator \(Z\), and every Weyl operator \(W(a,b)=X^aZ^b\) are unitary.

For the positive definite finite-Weyl family, let

and define \(\operatorname{Def}_B(t)\) analogously, with the real parts of the corresponding source-\(B\) pairings. Then both defects are non-negative and continuous for \(t{\gt}0\), and

The integrand is continuous, non-negative, and integrable on \((0,\infty )\). The coefficient of the source-\(B\) defect is exactly \(t\), as prescribed by the integral formula and equality analysis of Jenčová–Ruskai, arXiv:0903.2895v4, §4.

Under the positive-definite finite-Weyl hypotheses, suppose that \(\sum _gD(A_g\Vert B_g)-D(A\Vert B)=0\). Then \(\operatorname{Def}_B(t)=0\) for every \(t{\gt}0\), and consequently \(T_g^{-1}(B_g)=T^{-1}(B)\) for every \(t{\gt}0\) and every Weyl index \(g\). In particular, equality of relative entropy under the right partial trace implies this conclusion by Theorem 13.6.12. This is the positive-definite conclusion of the equality argument in Jenčová–Ruskai, arXiv:0903.2895v4, §4 and Appendix. It makes no assertion at \(t=0\) or for singular inputs.

Let \(\rho \) and \(\sigma \) be positive definite on \(\mathcal H_S\otimes \mathbb C^{d_C}\), let \(q=d_C^{-2}\), and put

where \(U_g=\mathbf1_S\otimes W_g\). For \(t{\gt}0\), let

Then

This is the positive-definite, fixed-\(t\) identity in Jenčová–Ruskai, arXiv:0903.2895v4, Appendix, lines 1313–1343. It does not infer zero defect from equality of relative entropies and makes no assertion about singular supports.

Under the hypotheses and notation of the preceding theorem, suppose that

Then \(T_g^{-1}(B_g)=T^{-1}(B)\) for every Weyl index \(g\). In particular this holds for the identity Weyl element \(g=(0,0)\). This is the common-resolvent conclusion in Jenčová–Ruskai, arXiv:0903.2895v4, §4, lines 652–674; its squared-defect input is in Appendix, lines 1313–1343. This fixed-\(t\), positive-definite conclusion does not assert that equality of relative entropies implies the scalar hypothesis in (142).

Under the notation of Theorem 13.6.26, set \(\Gamma _t=T^{-1}(A)\). Then, for every \(t{\gt}0\),

This is the second defect family in Jenčová–Ruskai, arXiv:0903.2895v4, §4 and Appendix.

Fix a dimension \(d\ge 1\) and a primitive \(d\)-th root of unity \(\zeta \), and let \(X\) be the cyclic shift \(|i\rangle \mapsto |i+1\rangle \) and \(Z=\operatorname{diag}(\zeta ^0,\ldots ,\zeta ^{d-1})\) the clock operator. For every matrix \(M\) on \(\mathbb {C}^d\), the uniform average of the conjugations by the \(d^2\) Weyl operators \(W(a,b)=X^aZ^b\) is the completely depolarizing channel: