Dependencies

A linear map is \(k\)-positive if and only if its blockwise ampliation is positive on \(M_{D}(\mathbb {C})\otimes M_{k}(\mathbb {C})\).

Let \(K=(K_i)_{i=0}^{d-1}\) be a unital Kraus family, that is, \(\sum _i K_i K_i^\dagger =\mathbb {1}\). Then for all \(X\in M_{D}(\mathbb {C})\), \(\mathcal{K}(X^\dagger X)\ge \mathcal{K}(X)^\dagger \mathcal{K}(X)\), where \(\mathcal{K}(Y)=\sum _i K_i Y K_i^\dagger \).

For an orthogonal projection \(P\) and linear map \(E\), the corner \(PM_{D}(\mathbb {C})P\) is preserved if \(E(PXP)=PE(PXP)P\) for every \(X\).

The corner subspace associated with \(P\) is \(\{ PXP:X\in M_{D}(\mathbb {C})\} \).

Irreducibility of the corner-restricted map.

The rank of an orthogonal projection \(P\in M_{D}(\mathbb {C})\), equivalently the dimension of its range.

For \(t\ge 0\) and \(n\in \mathbb {N}\), one has \(T_{nt}=(T_t)^n\).

Every density matrix is nonzero.

If \(L(X)=\mu X\), then \(e^{tL}(X)=e^{t\mu }X\) for all \(t\in \mathbb {R}\).

If \(\mu \) is in the spectrum of \(L\), then \(e^{t\mu }\) is in the spectrum of \(e^{tL}\).

Peripheral spectral points of the generator have vanishing real part.

The number of peripheral eigenvalues is bounded by \(\dim (M_{D}(\mathbb {C}))\).

If all powers of a peripheral eigenvalue remain peripheral, then some positive power not exceeding \(\dim (M_{D}(\mathbb {C}))\) equals \(1\).

For an irreducible channel with faithful fixed point, every power of a peripheral eigenvalue is again peripheral.

Let \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be CP maps with \(T_1\le T_2\). Then there exist ancilla dimensions \(r_1,m\), Heisenberg-form Stinespring matrices \(V_1:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^{r_1}\) and \(V_2:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^m\) realizing \(T_1,T_2\) via \(T_i(A)=V_i^\dagger (A\otimes \mathbb {1})V_i\), and a rectangular contraction \(\widetilde C:\mathbb {C}^m\to \mathbb {C}^{r_1}\) with \(\widetilde C^\dagger \widetilde C\le \mathbb {1}_m\) such that

This is the explicit square corollary obtained by constructing both dilation matrices, rather than the supplied-dilation statement of Theorem 3.12.7.

Let \(I\) be a nonempty finite set, and let \(T_i,T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive linear maps such that \(\sum _{i\in I}T_i=T\). If \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) satisfies \(T(A)=V^\dagger (A\otimes \mathbb {1}_r)V\), then there are positive semidefinite operators \(P_i\in M_{r}(\mathbb {C})\) satisfying \(\sum _{i\in I}P_i=\mathbb {1}_r\) and \(T_i(A)=V^\dagger (A\otimes P_i)V\) for every \(i\in I\) and \(A\in M_{D}(\mathbb {C})\).

For finitely many complex numbers \(\theta _j\) satisfying \(|\theta _j|=1\), there is a strictly increasing sequence of positive integers \(n_i\) such that \(\theta _j^{n_i}\longrightarrow 1\) simultaneously for every \(j\).

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 \(L\ge 1\).

Let \(H\in M_{D}(\mathbb {C})\) be Hermitian and let \(A\in M_{D}(\mathbb {C})\) be invertible with \(HA=AH^{T}\) and \(A^{T}=-A\). Then every eigenvalue of \(H\) is at least two-fold degenerate.

Let \(H,V\in M_{D}(\mathbb {C})\) with \(H\) Hermitian, \(V\) unitary, \(HV^{\dagger }=V^{\dagger }H^{T}\), and \(V^{T}=-V\). Then every eigenvalue of \(H\) is at least two-fold degenerate.

The following conditions are equivalent:

1. \(\tau _{E^m}{\gt}0\) for every sufficiently large \(m\);

2. \(\mathcal S_m=M_{D}(\mathbb {C})\) for every sufficiently large \(m\).

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 \(K=\mathbb {1}\).

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

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

Let \(p,q\geq 1\) and let \(\Phi :M_{p}(\mathbb {C})\to M_{q}(\mathbb {C})\) be a unital \(*\)-homomorphism with \(\Phi (M_{p}(\mathbb {C}))'=\mathbb {C}\mathbb {1}\). Then \(\Phi \) is onto.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive with bounded orbits. Then the mean-ergodic projection \(T_\infty \) of [ Wol12 , Equation (6.14) ] is completely positive. If \(T\) is completely positive and trace-preserving, the bounded orbits are automatic and \(T_\infty \) is completely positive and trace-preserving, hence a quantum channel. Together with Corollary 8.1.14 this completes the preservation assertion of [ Wol12 , Proposition 6.3 ] for all three maps \(T_\infty \), \(T_\phi \), and \(T_\varphi \).

For a leaf of positive terminal weight, the trace adjoint \(\Phi _j^\dagger \), defined by \(\operatorname {Tr}(\Phi _j^\dagger (\rho )Z)=\operatorname {Tr}(\rho \, \Phi _j(Z))\), maps density matrices to density matrices.

Let \(\{ \Phi _i\} \) be a quantum instrument on \(M_{d}(\mathbb {C})\) with \(d\geq 1\) whose total channel is the identity. Then for every outcome \(i\) the probability \(p_i(\rho )\) is a non-negative constant, the same for every \(\rho \) of unit trace.

For \(t \ge 0\), setting \(s=t\) in (8) gives

where \(M = \sup _{u \in [0,t]} \| e^{uL}\| \).

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).

For \(t \ge 0\), if \(\| e^{sL}\| \le 1\) and \(\| e^{sL'}\| \le 1\) for every \(s \in [0,t]\), then \(\| e^{tL'} - e^{tL}\| \le t\, \| L' - L\| \).

For \(G:\mathcal W\times I\to [0,\infty ]\) and \(T\ge 0\),

In particular, the expected length is \(\int _{\mathcal W}|w|\, d\mu _T(w)=NT\).

The expected number of omitted letters is

With the same assumptions,

With the same assumptions, all integrands below are integrable and

Let \(1\le k\) with \(k+1{\lt}D\). Then there is a map on \(M_{D}(\mathbb {C})\) that is \(k\)-positive but not \((k+1)\)-positive; the cone of \(k\)-positive maps strictly contains the cone of \((k+1)\)-positive maps. Every inclusion between consecutive amplification dimensions below the top one is therefore strict.

Under the preceding hypotheses,

These two inequalities remain true without the upper bounds on \(\lVert \Psi \rVert \) and \(\lVert \Xi \rVert \); the second is then a signed trace-deficit bound. If an arbitrary operator \(\sigma \) satisfies \(\lVert \sigma -\rho _\Psi \rVert _1\leq \kappa \), then, with the original upper norm bounds,

Positivity and Hermiticity of \(\sigma \) are unnecessary. This is the error accumulation in [ Ope26a , Section 8.1 ] .

If \(P\in M_{D}(\mathbb {C})\) is positive semidefinite, then

For \(0\le p_0\le p_1\le 1\) and every integrable function \(L\) on \([0,1]\) with \(L(p)\le -\partial _p\log N(p)^2\) for \(0{\lt}p{\lt}1\), \(\int _{p_0}^{p_1}L\le \log N(p_0)^2-\log N(p_1)^2\).

Let \(E\colon M_D(\mathbb C)\to M_D(\mathbb C)\) be a positive, trace-preserving linear map. Let \(\mathcal F_E=\{ X\mid E(X)=X\} \) be the fixed-point subspace and \(r = \dim _{\mathbb C} \mathcal F_E\). Then there exist \(r\) linearly independent stationary density matrices \(\rho _1,\dots ,\rho _r\) (positive semidefinite, trace 1, fixed by \(E\)) whose \(\mathbb C\)-linear span equals \(\mathcal F_E\).

This is Wolf Corollary 6.5 (Linearly independent stationary states).

The Dyson–Phillips series \(\sum _n \widetilde{T}^{(n)}_t\) converges in operator norm for every \(t \ge 0\).

Every nonzero fixed point of an irreducible channel has nonzero trace.

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) ] .

One has \(\sum _{n=0}^{\infty }\widetilde{T}^{(n)}_t = e^{tL'}\) in tsum form.

If the identity of Theorem 6.9.2.10 holds for every \(X\in M_{D}(\mathbb {C})\), then \(A\in \mathcal{A}_B\). This is immediate from the theorem by specializing at \(X=A\); the source states only the forward direction.

Let \(T\) be a complex endomorphism on the same \(D\)-dimensional coordinate space as \(\Lambda ,N\in M_D(\mathbb C)\), and take \(\mu =\mu (T)\) from Definition 10.3. If \(\Lambda \) is diagonal, \(N\) is strictly upper triangular, and \(\lVert \Lambda \rVert _\infty =\mu \), then the coarse and refined estimates of Theorem 10.10 hold with this shared source-shaped \(\mu \).

A linear map is two-positive if it is \(2\)-positive.

The fixed-point set of a complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is \(\operatorname{Fix}(E)=\{ X\in M_{D}(\mathbb {C}):E(X)=X\} \).

For a complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), set

These are [ Wol12 , Equations (5.11)–(5.12) ] , with the source’s convention that the subscript records the side on which \(A\) multiplies the varying matrix.

A complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) satisfies the Schwarz inequality if, for every \(A\in M_{D}(\mathbb {C})\),

This is [ Wol12 , Equation (5.2) ] .

Let \(F\) be a finite set with finite coordinate sets \(I_f\), let \(A\) be another finite coordinate set, and let \(k\geq 0\). Write \(\mathcal C_k=(\prod _{f\in F}I_f)^k\). Separating the added coordinate in each copy gives the bijection

Let \(F\) be a finite set of sites with finite coordinate sets \(I_f\), let \(A\) be an additional coordinate set, and let \(k\geq 0\). Write \(\mathcal C_k\) for the configurations of \(k\) copies of the original sites, and \(\widetilde{\mathcal C}_k\) for the configurations after adjoining a site \(*\) with coordinate set \(A\). For any \(b:A\to \mathbb {C}\), define the rectangular matrix \(J_b\) by

Thus insertion retains every original coordinate and places the same vector at the additional site in every copy. This is the tensor insertion in [ Ope26b , Section 8, lines 283–308 ] .

For a Lindblad operator \(L_j\), the adjoint dissipator on observables is

The adjoint fixed-point set is \(\operatorname{Fix}(E^*)=\{ X\in M_{D}(\mathbb {C}):E^*(X)=X\} \), where \(E^*(X)=\sum _iK_i^\dagger XK_i\).

The adjoint generator of a Lindblad form is

The adjoint kernel is the set of observables \(A\) satisfying \(L^*(A)=0\).

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 \(r\in \mathbb R_{\geq 0}\) and \(y,z\in Y\), define the real kernel

Thus \(K_r(y,z)\) is the number of anchor labels whose radius-\(r\) balls contain both sites.

Group the coordinates of \(X\) with \(C\), and leave \(R\) and all complementary physical coordinates separate. This gives

Applying \(e\) independently to the \(k\) copies defines \(e_k\). These are the coordinate changes in [ Ope26b , Section 8, lines 283–308 ] .

For natural numbers \(d,K,k\) and \(c\in \mathbb {R}\), put

The estimates below concern positive \(k\).

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 \(r,s\), let \(P_{\mathrm{top}}=C_{r,s}^\dagger C_{r,s}\) and \(P_{\mathrm{bot}}=\mathbb {1}-P_{\mathrm{top}}\). Both are PSD (for \(P_{\mathrm{bot}}\), by Lemma 3.12.10) and \(P_{\mathrm{top}}+P_{\mathrm{bot}}=\mathbb {1}\).

Let \(\{ R_i\} _{i\in I}\) be a finite family of simple rings. The \(i\)-th block ideal in \(\prod _{k\in I}R_k\) is

For natural numbers \(r,s\), let \(C=C_{r,s}\in \mathbb {C}^{r\times (r+s)}\) be the rectangular \(r\times (r+s)\) matrix whose rows are the first \(r\) rows of the identity on \(\mathbb {C}^{r+s}\): \(C_{ij}=1\) if \(j=i{\lt}r\) and \(0\) otherwise.

A \(2 \times 2\) block matrix \(\begin{pmatrix} P & Q \\ Q^\dagger & R \end{pmatrix}\) indexed by \(\{ 0,\ldots ,D_1{-}1\} \oplus \{ 0,\ldots ,D_2{-}1\} \).

For a linear map \(E : M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\), its \(k\)-fold ampliation acts between the corresponding block-matrix algebras by

For a matrix \(U\) on \(M_{D}(\mathbb {C})\), the Breuer–Hall map on \(M_{D}(\mathbb {C})\) is

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 Cesàro mean of a linear map \(E\) at order \(N\), applied to a starting matrix \(\rho _0\), is

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a quantum channel if it is both completely positive and trace-preserving (CPTP), following  [ Wol12 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the channel determinant \(\det T\) is the determinant of the matrix of \(T\) with respect to the standard matrix-unit basis of \(M_{D}(\mathbb {C})\). This is the quantity studied in [ Wol12 , Section 6.1 ] .

The period of a channel \(E\) is the number of peripheral eigenvalues, counted without multiplicity: \(p(E):=|\mathrm{peripheral}(E)|\).

The Choi matrix of a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is

Concretely, \(\tau _{(i_1,i_2),(j_1,j_2)} = \frac{1}{D}\, (T(E_{i_2 j_2}))_{i_1 j_1}\) where \(E_{i_2 j_2}\) is the matrix unit. This is the Choi–Jamiol\- kowski convention of [ Wol12 , Proposition 2.1 ] .

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 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) with Choi–Jamiołkowski operator \(\tau \), the purity of \(\tau \) is \(\operatorname{tr}[\tau ^\dagger \tau ]\), the sum

of the squared moduli of the entries of \(\tau \); in particular it is non-negative.

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) have Choi matrix \(\tau \) on \(\mathbb {C}^{d'}\otimes \mathbb {C}^d\). For \(X\in M_{d\times k}(\mathbb {C})\), the right-factor compression is the matrix on \(\mathbb {C}^{d'}\otimes \mathbb {C}^k\) with entries

The associated vector on \(\mathbb {C}^d\otimes \mathbb {C}^k\) has coefficients \(d^{-1/2}X_{a,p}\).

For \(X\in M_{d\times k}(\mathbb {C})\), let \(R_X\) be the matrix from \(\mathbb {C}^{d'}\otimes \mathbb {C}^d\) to \(\mathbb {C}^{d'}\otimes \mathbb {C}^k\) with entries

On the cyclic index set \(\mathbb {Z}/d\mathbb {Z}\), the Choi-type map is

where \(D\) keeps the diagonal of a matrix and \(U_{k0}\) is the cyclic shift by \(k\).

For a vector \(v\) on \(\mathbb {Z}/d\mathbb {Z}\), set

For a Hermitian matrix \(A\) and \(b\in \mathbb {R}\), let \(\Pi =\mathbf1_{[b,\infty )}(A)\) be the spectral projection onto the eigenvalues at least \(b\). An eigenvalue equal to \(b\) is included.

For a density matrix \(\sigma \) on the symmetric subspace of \(k\) copies, the coherent measure is \(d\mu _\sigma (\theta )=D_k\operatorname{Tr}(\sigma P_{\theta ,k})\, d\theta \), where \(\theta =Ue\) for Haar-distributed \(U\). It is a probability measure, and \(\bigl|\int \langle \theta ^{\otimes m},G\theta ^{\otimes m}\rangle \, d\mu _\sigma \bigr|\le \| G\| \) for every operator \(G\) on \(m\) copies.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be linear. For \(X\in M_{D}(\mathbb {C})\) set

where the suprema and infima in (28) are taken in \(\overline{\mathbb {R}}\). These are [ Wol12 , Equations (6.29)–(6.30) ] . The corrected global quantities are

The trace-one condition is Wolf’s homogeneous normalization from local source lines 634–637. A pair \((X,a)\) is lower feasible when \(X\geq 0\), \(\operatorname{tr}X=1\), and \(a\in L_T(X)\); it is upper feasible when the first two conditions hold and \(a\in U_T(X)\).

The commutant of a Lindblad form is the set of matrices commuting with \(H\), with every Lindblad operator \(L_j\), and with every adjoint \(L_j^\dagger \).

A linear map \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) has commutative range if \(T(X)\) and \(T(Y)\) commute for all \(X,Y\in M_D(\mathbb {C})\). Since a family of pairwise-commuting elements generates a commutative subalgebra, this is the statement that \(T\) maps into a commutative subalgebra of \(M_D(\mathbb {C})\).

The commutator form of the Lindblad equation writes the generator as

where \([A,B] = AB - BA\). This is [ Wol12 , Equation (7.22) ] .

For \(k\ge 0\), define the bijection

Its inverse places each physical configuration and auxiliary index in the same copy.

Let the local physical coordinate sets be finite. Given an original vector \(\Omega \), a spectral selection \(E\) on \(X\) with actual mass \(z\), and a copy number \(k\ge 0\), define for \(D\subseteq \widehat V\)

Here the formula is extended to \(z=0\) by setting \(a_k(D)=k\). Positive selected mass is required when this threshold is used in the exponential cutoff estimate.

Let \(V\) be a finite set of physical sites, with local coordinate sets \(I_v\), and let \(X\subseteq V\). For an auxiliary coordinate set \(R\), define

A physical region \(B\subseteq V\) becomes \(\widehat B=B\cap X^c\), included in \(\widehat V\) and excluding the auxiliary site. These are the exterior coordinates in [ Ope26b , 07-comparators.tex, lines 240–247 and 283–308 ] . They require no choice of a vector or spectral subspace, and no coordinate set is required to be nonempty.

For every \(B\subseteq V\), evaluation of the auxiliary and physical coordinates gives a bijection

If \(X\cap B=\varnothing \), evaluation of the physical coordinates also gives a bijection

The inverses assign these values to the same auxiliary and physical sites. No local coordinate set is required to be nonempty or finite for these bijections.

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 \(D\otimes E\), \(S(D|E)_\rho =S(\rho _{DE})-S(\rho _E)\).

Let \(V\) and \(V'\) be finite-dimensional real Hilbert spaces, let \(K\subseteq V\) be a closed pointed convex cone with nonempty interior, let \(T:V\to V'\) be linear, and fix \(c\in V\) and \(b\in V'\). As in [ Wol12 , Chapter 4, equations (4.1)–(4.2) ] , define the dual cone, the primal and dual feasible sets, and their values by

The values belong to the extended real line. The conventions are \(\inf \varnothing =+\infty \) and \(\sup \varnothing =-\infty \); an objective unbounded below has infimum \(-\infty \), and one unbounded above has supremum \(+\infty \). The nonempty-interior assumption is Wolf’s convention for a conic program; the definitions and weak duality do not require it.

Following [ Wol12 , Chapter 4, lines 72–78 ] , the primal problem is strictly feasible when there is an \(x\in \operatorname {int}K\) with \(T(x)=b\). The dual problem is strictly feasible when there is a \(y\in V'\) such that \(c-T^*(y)\in \operatorname {int}K^*\). A primal optimizer is an \(x^0\in \mathcal F_p\) satisfying \(\langle c|x^0\rangle \leq \langle c|x\rangle \) for every \(x\in \mathcal F_p\); a dual optimizer is a \(y^0\in \mathcal F_d\) satisfying \(\langle b|y\rangle \leq \langle b|y^0\rangle \) for every \(y\in \mathcal F_d\).

Let \(P\) be finite. For each position \(p\), let \(R_p,C_p\) be finite row and column sets. For every label \(x\in \prod _{p\in P}(R_p\times C_p)\), fix a matrix \(M_x\) with the same output row and column sets. Define the multilinear contraction

Each \(X_p\) is an arbitrary rectangular matrix. The coefficient matrices are deterministic and unrestricted.

For finite index sets \(N,M,Q\), let \(B:\mathbb {C}^N\to \mathbb {C}^M\) be rectangular and let \(J:\mathbb {C}^N\to \mathbb {C}^Q\) satisfy \(J^\dagger J=I_N\). Its adjoint domain extension is \(\overline B=BJ^\dagger \). It satisfies

Thus local contractions remain contractions on their common ambient domain. These equalities allow empty domains, empty codomains, and zero or subnormalized vectors.

Let \(P\) be finite, let \(B_p\) be its finite branch-label sets, and let \(A_{pb},C_{pb}\) be finite Schmidt supports. For each \(p,b\), let \(\nu _{pb}\) be the standard circular complex Gaussian law on the coordinates \(\{ 0,\ldots ,k-1\} \times A_{pb}\times C_{pb}\). The nested law

is a probability measure. For \(S\subseteq P\) and a deterministic choice \(\sigma _p\in B_p\) at each \(p\in S\), define \(\pi _{S,\sigma }(\omega )_p=\omega _{p,\sigma _p}\). Its actual pushforward is

The branch choice is fixed before sampling. This realizes the fresh samples used in [ Ope26a , Theorem 5.2 ] .

For \(G:\mathbb {C}^D\to \mathbb {C}^{D'}\) and \(\gamma \in \mathbb {C}^B\), define \(A_{G,\gamma }:\mathbb {C}^D\to \mathbb {C}^{D'\times B}\) by \((A_{G,\gamma })_{(y,e),x}=G_{yx}\gamma _e\). For an arbitrary \(A:\mathbb {C}^D\to \mathbb {C}^{D'\times B}\) and an old inventory \(J\), its idle lift \(\mathcal L_J(A)\) is given by

This is \(A\otimes I_J\) followed by the stated output permutation. On an arbitrary vector \(\xi \), \((\mathcal L_J(A)\xi )_{y,j,e}=\sum _x A_{(y,e),x}\xi _{x,j}\). In particular,

The actual operator bounds are

Likewise \((K\otimes I_J)(v\otimes w)=Kv\otimes w\) and \(\lVert K\otimes I_J\rVert \leq \lVert K\rVert \) for arbitrary rectangular \(K\). No dimension lower bounds are needed.

Choose finite fresh registers \(B_t\) and pure vectors \(\gamma _t\in \mathbb {C}^{B_t}\). Define the finite chronological inventory \(J_0=\{ *\} \), \(J_{t+1}=J_t\times B_t\), with

Coordinate tensor products satisfy \((v\otimes w)_{ij}=v_iw_j\) and \(\lVert v\otimes w\rVert =\lVert v\rVert \lVert w\rVert \). Their actual outer products satisfy

Thus normalized \(\gamma _t\) give \(\lVert \Gamma _t\rVert =1\). Discarding any normalized pure factor \(w\) gives

including zero or subnormalized \(v\).

A dynamical semigroup \(T\) is norm-continuous if the map \(t \mapsto T_t\) is continuous in the operator norm topology. In finite dimension this coincides with strong continuity.

For the chronological products \(D_m(w)\), define

Set \(G_i=(I-k_i)^{1/2}\). For a word \(w=(i_0,\ldots ,i_{m-1})\), set \(D_w=G_{i_{m-1}}\cdots G_{i_0}\), with \(D_{\varnothing }=I\). For \(v\in \mathcal H\), define

Let

The controlled dependent partial trace discards the off-diagonal blocks between distinct \(i\) and applies \(\operatorname{tr}_{B_i}\) to the \(i\)th diagonal block.

A convex decomposition of a density operator \(\rho \in M_{d_A}(\mathbb {C})\) is a finite family of nonnegative weights \(\lambda _i\) summing to one together with density operators \(\rho _i\in M_{d_A}(\mathbb {C})\) satisfying \(\rho =\sum _i\lambda _i\rho _i\).

For density operators \(\rho ,\rho _1\) on the same space and \(c\in \mathbb {R}\), \(\rho \) has a convex decomposition with \(\rho _1\) carrying weight \(c\) when \(\rho =\sum _i\lambda _i\rho _i\) for some finite index set, with every \(\rho _i\) a density operator, every \(\lambda _i\ge 0\), \(\sum _i\lambda _i=1\), and some index \(i_1\) with \(\lambda _{i_1}=c\), \(\rho _{i_1}=\rho _1\).

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 copy permutations of a subsystem, the symmetric subspace, and the averaging projector \(\Pi _k=(k!)^{-1}\sum _{\pi \in S_k}U(\pi )\) onto it.

The restriction of \(E\) to an invariant corner \(PM_{D}(\mathbb {C})P\).

For an idempotent \(P\in M_{D}(\mathbb {C})\), the subspace \(\{ X\mid PXP=X\} \) and the corner \(\{ PXP\mid X\in M_{D}(\mathbb {C})\} \) are identified as \(\mathbb {C}\)-linear spaces by the identity on underlying matrices.

For linear maps \(S,T:M_{d_{\rm in}}(\mathbb {C})\to M_{d_{\rm out}}(\mathbb {C})\), write \(T\le S\) (in the CP order) when \(S-T\) is completely positive.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (CP) if it admits a Kraus representation: there exist operators \(\{ K_i\} _{i=0}^{r-1}\) with \(K_i \in M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

The Kraus representation also gives entrywise positivity on every positive block matrix by Theorem 2.2.1, and hence the associated completely positive map between matrix \(C^*\)-algebras in Theorem 2.2.2.

A linear map \(T:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) is completely positive if, for every natural number \(j\) and every family \(X=(X_k)_{k{\lt}r}\) with \(X_k\in M_{d_k}(\mathbb {C})\otimes M_{j}(\mathbb {C})\), entrywise positivity implies

where the \((a,b)\) slice of \((T\otimes \operatorname{id}_j)(X)\) is \(T\) applied to the family of \((a,b)\) slices of the matrices \(X_k\). This is the direct-sum analogue of rectangular Kraus complete positivity (Definition 2.5.2).

Let \(S\subseteq M_{m}(\mathbb {C})\) be a subspace and \(T:S\to M_{n}(\mathbb {C})\) a linear map. For a natural number \(k\), write \(T\otimes \mathrm{id}_k\) for the map sending a \(k\times k\) block matrix with entries in \(S\) to the block matrix obtained by applying \(T\) to every entry. Then \(T\) is completely positive on \(S\) if \(T\otimes \mathrm{id}_k\) sends every positive semidefinite such block matrix to a positive semidefinite matrix, for every \(k\) (the condition is vacuous at \(k=0\)). This makes no reference to \(S\) being an operator system; it is applied below to an operator system \(S\) because that is the situation in which the extension theorem holds.

Let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be a subspace and \(T:S\to M_{p}(\mathbb {C})\) a linear map. The level-\(j\) ampliation of \(T\) is defined by bipartite slicing: for a family \(X=(X_k)_{k{\lt}r}\) with \(X_k\in M_{d_k}(\mathbb {C})\otimes M_{j}(\mathbb {C})\) whose slice families lie in \(S\),

and \(T\) is completely positive on \(S\) if entrywise positivity \((\forall k{\lt}r,\ X_k\ge 0)\) always implies \((T\otimes \operatorname{id}_j)(X)\ge 0\), for every \(j\). This is Definition 1.14.10 applied block by block, one summand at a time.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\). A block-permutation structure for \(T\) consists of a finite index set \(\iota \), a family of orthogonal projections \(P_k\in M_{D}(\mathbb {C})\) for \(k\in \iota \), and a permutation \(\sigma \in \mathrm{Sym}(\iota )\), not necessarily a single cycle, such that, for every \(k\in \iota \) and \(X\in M_{D}(\mathbb {C})\),

In general, \(\sigma \) may have multiple disjoint cycles.

A map \(\Phi \colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is completely copositive if \(\Phi \circ \theta \) is completely positive, where \(\theta \) is transposition. It is decomposable if \(\Phi =\Phi _{\mathrm{cp}}+\Phi _{\mathrm{ccp}}\) with \(\Phi _{\mathrm{cp}}\) completely positive and \(\Phi _{\mathrm{ccp}}\) completely copositive. It is indecomposable if it is positive and not decomposable.

A witness \(W\) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) is decomposable when

This is the explicit cone in Wolf’s Equation (3.15), with partial transposition on the first factor.

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 \(F(x)=0\).

In the existing matrix-family coordinates for \(\mathcal A=\bigoplus _{k\in I}M_{d_k}(\mathbb {C})\), its density states are the families \(A=(A_k)_{k\in I}\) such that every \(A_k\succeq 0\) and \(\sum _k\operatorname{tr}(A_k)=1\).

Let

and let \(\mathcal H_A=\bigoplus _{k\in I}\mathbb {C}^{n_k}\) and \(\mathcal H_B=\bigoplus _{l\in J}\mathbb {C}^{m_l}\). Write \(\iota _A,\iota _B\) for the block-diagonal embeddings and \(\pi _A,\pi _B\) for diagonal-block compression. For a linear map \(T:\mathcal A\to \mathcal B\), its canonical full-matrix extension is

For \(X\in \mathcal A\) and \(Y\in \mathcal B\), the trace adjoint \(T^*:\mathcal B\to \mathcal A\) is defined by

This is the block-diagonal realization needed for the fixed-point and classification arguments in [ CPGSV16 , Appendix C.4, lines 1980–2003 ] . It is only an auxiliary construction: it does not assert the classification conclusion of that passage.

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, \(\Psi _{(i,j)}=\Omega _i\Omega _j\).

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and the reduced density matrix \(\operatorname{tr}_{1}[\tau ]\) of its Choi matrix \(\tau =(T\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)\) is proportional to the identity. This is the normal form in [ Wol12 , Proposition 2.9 ] .

A linear map \(T : M_{d_1}(\mathbb {C}) \to M_{d_2}(\mathbb {C})\) between matrix algebras of possibly different dimensions is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and \(T^{*}(\mathbb {1})\propto \mathbb {1}\), where \(T^{*}\) is the trace-pairing adjoint. This is the normal-form condition of [ Wol12 , Proposition 2.9 ] ; by the rectangular Choi–Jamiolkowski correspondence it is equivalent to both partial traces of the Choi matrix being proportional to the identity.

A family of linear maps \(T : \mathbb {R}\to (M_{D}(\mathbb {C}) \to _{\ell } M_{D}(\mathbb {C}))\) is a dynamical semigroup if

1. (semigroup law) \(T_{t+s} = T_t \circ T_s\) for all \(t,s \ge 0\); and

2. (initial condition) \(T_0 = \operatorname{id}\).

This is [ Wol12 , Equation (7.1) ] .

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 a fixed \(B\), the set \(\mathcal{A}_B\subseteq M_{D}(\mathbb {C})\) of \(A\)’s for which equality is attained in the two-variable operator Schwarz inequality:

Given a generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\), the exponential semigroup is

For every \(S\subseteq Y\) and every pair of real parameters \(a,\alpha \), define

There are no sign restrictions on \(a\) or \(\alpha \) in this definition.

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 a region \(D\subseteq \widehat V\), let

regarded as a subset of the original site set \(V\). The auxiliary site contributes no element to this set.

For complex square matrices \(H\) and \(A\), set \(\mathcal F_H(A)=\int _{\mathbb {R}} f(t)\, e^{itH}Ae^{-itH}\, dt\), a Bochner integral for the operator norm.

In \(\mathbb {C}^2 \otimes \mathbb {C}^2\), let \(v = |01\rangle - |10\rangle \). Equivalently, \(v_{(0,1)}=1\), \(v_{(1,0)}=-1\), and all other coefficients are zero.

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.

For \(r\geq 1\), let \(W_0:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) be the isometry \(W_0x=x\otimes e_0\).

A Fitting decomposition of a linear endomorphism \(f:V\to V\) on a finite-dimensional vector space over an algebraically closed field consists of:

1. \(f\) is nilpotent on the generalized \(0\)-eigenspace,

2. \(f\) is invertible on each generalized \(\mu \)-eigenspace for \(\mu \neq 0\),

3. the generalized eigenspaces span \(V\),

4. the generalized eigenspaces are linearly independent.

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 finite-dimensional spaces \(S,E\) with fixed orthonormal bases and a chosen environment basis vector \(|a\rangle \), define \(J_a:S\to S\otimes E\) by \(J_a|s\rangle =|s\rangle \otimes |a\rangle \).

Given a linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) and a fixed point \(\rho \) with \(E(\rho )=\rho \) and \(\operatorname{tr}(\rho )\neq 0\), the fixed-point projection is the rank-one map

This projects onto the span of \(\rho \) along the kernel of the trace functional. When the fixed point is unique up to scaling, this span is the full fixed-point space.

The flip operator \(F\) on \(\mathbb {C}^d \otimes \mathbb {C}^d\) is

Its matrix entries are

Let

A real matrix \(C\in M_4(\mathbb R)\) is \(M\)-selfadjoint when \(C^{\mathsf T}M=MC\).

Let \(S,E,F\) be finite-dimensional spaces, and let \(\Sigma :(S\otimes E)\otimes F\to (S\otimes F)\otimes E\) send \(|s,e,f\rangle \) to \(|s,f,e\rangle \). For a matrix \(V\) on \(S\otimes F\), define \(\widetilde V=\Sigma ^\dagger (V\otimes \mathbb {1}_E)\Sigma \).

For a complex-linear map \(E:\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define its Frobenius transport by

For finite index sets \(I\) and \(J\), define

This is a complex-linear isometric equivalence when the matrix space is equipped with the Frobenius norm.

The multiplicative domain of \(E\) is \(\mathcal{A}(E):=\mathcal{A}_R(E)\cap \mathcal{A}_L(E)\).

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.

A generator decomposition consists of a completely positive map \(\phi : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) and a matrix \(\kappa \in M_{d}(\mathbb {C})\), defining the linear map

This is [ Wol12 , Equation (7.14) ] .

For a projection \(P\), the generator preserves the compression \(PM_{D}(\mathbb {C})P\) if, for every \(X\in M_{D}(\mathbb {C})\),

Fix the original physical vector \(\Omega \), the two auxiliary labels, the copy number \(k\), and the cutoff \(\tau k\). Let \(P_0\) be the actual low-defect projection in the physical-copy and two auxiliary registers. Using the fixed regrouping \(G_k\) of Definition 23.20.1, define

This projection is defined without a regional cut, metric, or mean tree.

For the original physical Hamiltonian \(H\), define in the global coordinates

Here \(H_i\) acts on the \(i\)th physical copy. At \(k=0\), set \(\overline H_0=0\).

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.

Following Ha’s displayed decomposition [ Ha98 , pp. 594–595 ] , continue to use zero-based cyclic indices, and let \(\mathcal B_d\) be the set of pairs

This is the zero-based form of Ha’s two displayed ranges for the vectors \(\beta _{1j}\) and \(\beta _{ij}\); it is empty when \(d=3\). Define

and let

Let \(\sigma (x\otimes y)=y\otimes x\), and write \(A^\sigma \) for the corresponding exchange of both tensor factors of a bipartite matrix. This exchange is an involution. If \(A=\sum _{i,j}a_{ij}\otimes e_{ij}\) and \(\Phi :M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) is linear, define the Eom–Kye bilinear pairing by

For \(J_d=\sum _{i,j}e_{ij}\otimes e_{ij}\), its associated matrix pairing is \(\langle X,J_d\rangle =\sum _{i,j}X_{(i,i),(j,j)}\).

The Choi-type map may be transported from the cyclic basis \(\mathbb {Z}/d\mathbb {Z}\) to the standard basis indexed by \(0,\ldots ,d-1\), and \(A_\gamma \) may be transported in the opposite direction on both tensor factors. These simultaneous changes of basis identify the two forms of the pairing in Theorem 4.8.1.36; they do not add a positivity hypothesis.

Fix \(d\ge 3\) and \(\gamma {\gt}0\). Following Ha’s proof of [ Ha98 , Theorem 2.1 ] , put \(m_k=\frac32(3^k-1)\) for \(0\le k{\lt}d\), choose \(\zeta =\exp (2\pi i/3^d)\), and enumerate the \(3^d\)-th roots of unity by \(\omega _i=\zeta ^i\) for \(0\le i{\lt}3^d\). Define

If \(S e_p=e_{p+1}\) with cyclic indices, set

With \(\circ \) denoting coordinatewise multiplication, let

Condition (4): there exists a nontrivial projector \(P\) and a Lindblad form \((H,\{ L_j\} )\) for \(L\) such that \((\mathbb {1}-P)L_jP=0\) and \((\mathbb {1}-P)\kappa P=0\) for all \(j\), where \(\kappa =iH+\frac{1}{2}\sum _jL_j^\dagger L_j\).

Let \(\mathbb K\) denote either \(\mathbb {R}\) or \(\mathbb {C}\), and let \(V\) be a normed vector space over \(\mathbb K\). A linear endomorphism \(f:V\to V\) has bounded orbits if, for every \(x\in V\), the set \(\{ f^n x:n\in \mathbb {N}\} \) is bounded.

A generator \(L\) has a faithful stationary state if there exists a positive definite density matrix \(\rho _0\) with \(L(\rho _0)=0\).

Condition (3): there exists a nontrivial orthogonal projector \(P\) such that \(e^{tL}(PM_{D}(\mathbb {C})P)\subseteq PM_{D}(\mathbb {C})P\) for all \(t\ge 0\).

The commutant of the Lindblad family \(\{ L_j\} '\) equals \(\mathbb {C}\cdot \mathbb {1}\), i.e. the Lindblad operators act irreducibly on \(M_{D}(\mathbb {C})\).

Condition (1): there exists a density matrix \(\rho _0\) with nontrivial kernel such that \(e^{tL}(\rho _0)=\rho _0\) for all \(t\ge 0\).

Condition (2): there exists a density matrix \(\rho _0\) with nontrivial kernel satisfying \(L(\rho _0)=0\).

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) has Schmidt number at most \(n\) when it is a finite sum of pure-state projectors of Schmidt rank at most \(n\),

A complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) and a fixed \(B\in M_{D}(\mathbb {C})\) satisfy the per-\(B\) Schwarz hypothesis when

for every \(A\in M_{D}(\mathbb {C})\). A \(2\)-positive map satisfies this for every \(B\) (Theorem 6.9.2.5).

A completely positive map \(E\) on \(M_{D}(\mathbb {C})\) has the restricted CP spectral properties used here if there exist a positive real number \(r\), a positive definite right eigenvector \(\rho \), and a positive definite left eigenvector \(\sigma \) for the adjoint map such that

every positive semidefinite right eigenvector for eigenvalue \(r\) is a scalar multiple of \(\rho \), and the spectral radius of \(E\) is equal to \(r\). The uniqueness clause concerns only positive semidefinite Perron eigenvectors. Unlike the full nondegenerate-eigenspace statement in [ Wol12 , Theorem 6.4 ] , it does not assert that every complex eigenvector at \(r\) is proportional to \(\rho \).

A Lindblad form has traceless Kraus operators if \(\operatorname{tr}(L_j) = 0\) for every \(j\).

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 ] .

For a complex square matrix \(H\) and \(t\in \mathbb {R}\), set \(U_H(t)=e^{itH}\).

The Hermitian \(n\times n\) matrices form a finite-dimensional real Hilbert space for Wolf’s trace pairing \(\langle A|B\rangle =\operatorname {Re}\operatorname{tr}(AB)\). Its positive-semidefinite matrices form a closed proper cone \(K_{\mathrm{psd}}\) satisfying \(K_{\mathrm{psd}}^*=K_{\mathrm{psd}}\), and \(\operatorname {int}K_{\mathrm{psd}}=\{ A:A{\gt}0\} \). The last identity also covers the zero-dimensional matrix space, where positive definiteness is vacuous and the cone is the whole space.

Let \(A=U\operatorname {diag}(\lambda _0,\ldots ,\lambda _{d-1})U^\dagger \) be Hermitian with \(\lambda _0\ge \cdots \ge \lambda _{d-1}\). For \(\varepsilon \in \mathbb {R}\), define

Define the corresponding Hermitian matrix by

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

A family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) is Hilbert–Schmidt orthonormal when

This is the pairing \(\operatorname{tr}(PA^\dagger B)\) of [ Wol12 , Equation (2.18) ] in the case \(P=\mathbb {1}\).

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.

For a matrix \(U\) on \(S\otimes E\) and basis indices \(a,b\) of \(E\), set \((K_b)_{s,t}=U_{(s,b),(t,a)}\).

For an operator \(G\) on \(m\) copies and an ordered injection \(\iota \) of \(m\) copies into \(k\) copies, \(G_\iota \) is \(G\) on the copies \(\iota (1),\dots ,\iota (m)\) and the identity elsewhere. The injection average \(\mathcal T_{k,m}(G)\) is the average of \(G_\iota \) over all ordered injections; \(\| \mathcal T_{k,m}(G)\| \le \| G\| \).

A finite matrix family \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

A quantum instrument with \(n\) outcomes is a family \(\{ \Phi _i\} _{i=0}^{n-1}\) of completely positive maps on \(M_{D}(\mathbb {C})\) whose sum \(\sum _i\Phi _i\) is trace-preserving.

Associated to an instrument are the total channel \(\sum _i\Phi _i\), the unnormalized update \(\rho \mapsto \Phi _i(\rho )\) for each outcome \(i\), the outcome probability \(p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))\), and the normalized posterior state \(\Phi _i(\rho )/p_i(\rho )\) whenever \(p_i(\rho )\neq 0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a bijective complex-linear map. Write \(T^{-1}\) for the inverse linear map supplied by the associated linear equivalence; it is both a left and a right inverse of \(T\).

An invertible filtering operation on \(M_{D}(\mathbb {C})\) is a completely positive map

If \(T:M_{d_1}(\mathbb {C})\to M_{d_2}(\mathbb {C})\), pre- and postfiltering are written in Wolf’s order as \(\Phi _2\circ T\circ \Phi _1\). The matrices \(X\) are retained as part of the data; in particular, matrices which differ by a phase are not identified.

Let \(I\) be a finite index set and let \(C\in \mathbb C^{I\times I}\) be invertible. Congruence by \(C\) is the complex-linear equivalence

whose inverse is

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is irreducible if whenever \(P\) is an orthogonal projection satisfying \(E(P M_{D}(\mathbb {C}) P) \subseteq P M_{D}(\mathbb {C}) P\), then \(P = 0\) or \(P = \mathbb {1}\). This is [ Wol12 , Theorem 6.2(1) ] . The definition applies to any linear map; complete positivity is not required.

Let \(G\) be a finite group. By Maschke’s theorem and the Artin–Wedderburn theorem over \(\mathbb {C}\) there is an algebra isomorphism

and we fix one. Its factors are the labels \(\lambda \) of \(G\), and \(d_\lambda \ge 1\) is the dimension of the irreducible representation with label \(\lambda \). The central idempotent \(e_\lambda \in \mathbb {C}[G]\) and the matrix units \(e^\lambda _{ij}\) are the preimages of the identity and of the matrix units of the factor \(M_{d_\lambda }(\mathbb {C})\).

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is conditionally completely positive (CCP) if it admits a generator decomposition, i.e. there exist a CP map \(\phi \) and a matrix \(\kappa \) such that \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \). This is condition 1 of [ Wol12 , Proposition 7.2 ] .

Given a \(*\)-subalgebra \(S\subseteq M_{D}(\mathbb {C})\), a linear map \(P:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is a conditional expectation onto \(S\) if it is positive, idempotent, and unital, has range contained in \(S\), and satisfies \(P(X)=X\) for every \(X\in S\).

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is a GKSL generator if \(e^{tL}\) is a quantum channel (CPTP) for all \(t \ge 0\).

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.

The span of the Lindblad operators is closed under Hermitian conjugation, i.e. it forms a \(*\)-subspace of \(M_{D}(\mathbb {C})\).

A Naimark dilation of a POVM \(\{ E_i\} \) consists of an isometry \(V\) into a larger Hilbert space together with a projective measurement \(\{ P_i\} \) there such that \(E_i=V^\dagger P_iV\) for all \(i\).

A projection is nontrivial if it is orthogonal and neither \(0\) nor \(\mathbb {1}\).

A bipartite matrix \(\rho \) has the positive-partial-transpose (PPT) property when its first-factor partial transpose is positive semidefinite, \(\rho ^{T_1}\ge 0\).

A quantum dynamical semigroup is a norm-continuous dynamical semigroup whose time slices are quantum channels for every \(t \ge 0\).

A quantum dynamical semigroup is reducible if it has a nontrivial invariant compression.

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) is separable when it is a finite sum of Kronecker products of positive semidefinite matrices,

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is trace-annihilating if \(\operatorname{tr}(L(\rho )) = 0\) for all \(\rho \in M_{d}(\mathbb {C})\). This is the infinitesimal version of trace preservation.

A generator decomposition \((\phi , \kappa )\) satisfies the trace constraint if \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), where \(\phi ^*\) is the Hilbert–Schmidt adjoint. This is the condition in [ Wol12 , Equation (7.20) ] .

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 \(V\).

A linear map \(E : M_{D_{\rm in}}(\mathbb {C}) \to M_{D_{\rm out}}(\mathbb {C})\) is \(k\)-positive if \(E \otimes \operatorname{id}_{k}\) is positive from \(M_{D_{\rm in}}(\mathbb {C}) \otimes M_{k}(\mathbb {C})\) to \(M_{D_{\rm out}}(\mathbb {C}) \otimes M_{k}(\mathbb {C})\).

For a square complex matrix \(\rho \), the kernel projection is \(\Pi _{\ker \rho }=\mathbf1_{\{ 0\} }(\rho )\), computed through the continuous functional calculus, and the kernel completion is \(\widehat\rho =\rho +\Pi _{\ker \rho }\).

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.

A Kossakowski form consists of \(H=H^\dagger \), a fixed basis \(F_1,\ldots ,F_{d^2-1}\) of the traceless matrices in \(M_{d}(\mathbb {C})\), and a positive semidefinite matrix \(C\in M_{d^2-1}(\mathbb {C})\), defining

This is [ Wol12 , Theorem 7.1, Equation (7.23) ] . Allowing an arbitrary finite family \(\{ F_k\} \) gives the corresponding algebraic form used in the factorization argument.

The infimum of the numbers of Lindblad operators across all representations of \(L\). When such a representation exists, this is the minimum; the definition is totalized to zero otherwise.

The adjoint Kraus map is

When the \(\{ K_i\} \) are the matrices of an MPS tensor \(A\), this is the transfer map of the conjugate-transposed family \(i \mapsto (A^i)^\dagger \).

The Kraus commutant is

It is a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\).

Let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices. For a word \(w=(i_1,\ldots ,i_n)\), its word evaluation is

with the empty word evaluated as the identity matrix.

For \(\psi \in \mathbb {C}^D\) and \(m\in \mathbb N\), define

The Kraus family has eventually full vector spread if, for every sufficiently large \(m\) and every nonzero \(\psi \in \mathbb {C}^D\), \(H_m(K,\psi )=\mathbb {C}^D\). This is [ Wol12 , Theorem 6.8(2) ] .

A finite Kraus family has eventually full word span when \(S_N(K)=M_{D}(\mathbb {C})\) for all sufficiently large \(N\).

For a finite Kraus family \(K_0,\ldots ,K_{d-1}\in M_{D}(\mathbb {C})\) and \(N\in \mathbb N\), define

The empty product is the identity matrix.

The fixed-point set of a Kraus map \(E(X)=\sum _iK_iXK_i^\dagger \) is \(\operatorname{Fix}(E)=\{ X\in M_{D}(\mathbb {C}):E(X)=X\} \).

Given operators \(\{ K_i\} _{i=0}^{d-1}\) with \(K_i \in M_{D}(\mathbb {C})\), the Kraus map is

A linear map is completely positive (Definition 2.1.6) if and only if it can be written in this form, as in (1).

Let

Write \(\iota _{\mathcal A},\pi _{\mathcal A}\) and \(\iota _{\mathcal B},\pi _{\mathcal B}\) for the corresponding block-diagonal embeddings and diagonal-block compressions. The canonical full-matrix extension of \(T:\mathcal A\to \mathcal B\) is

The map \(T\) is a direct-sum Kraus map when \(\widehat T\) has a Kraus representation.

Let \(\{ A_i\} _{i=0}^{d-1}\subseteq M_{D_1}(\mathbb {C})\) and \(\{ B_i\} _{i=0}^{d-1}\subseteq M_{D_2}(\mathbb {C})\). Their rectangular mixed Kraus map is the endomorphism of \(M_{D_1\times D_2}(\mathbb {C})\) given by

Its spectral radius is denoted by \(\varrho (\mathcal M_{A,B})\).

A map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) has Kraus cardinality \(r\) if it has an exact \(r\)-operator Kraus representation. Its Kraus rank (Choi rank) is the rank of its Choi matrix, \(r=\operatorname {rank}(\tau )\); following [ Wol12 , Theorem 2.1, footnote ] , this is distinguished from the rank of \(T\) as a linear map.

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian with eigenvalues \(\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _D\) listed in decreasing order. For \(0\le k\le D\), set

This is the sum of the \(k\) largest signed eigenvalues. Indices beyond \(D\) contribute zero, so the value stabilizes at \(\operatorname{Re}\operatorname{tr}(A)\) once \(k\) reaches \(D\).

The Ky-Fan \(k\)-norm \(\| A\| _{(k)}\) is by definition the sum of the \(k\) largest singular values of \(A\). It agrees with \(S_k(A)\) exactly when \(A\) is positive semidefinite, where eigenvalues and singular values coincide; for an indefinite Hermitian \(A\) the two differ. For \(A=\operatorname{diag}(1,-2)\), one has \(S_1(A)=1\) but \(\| A\| _{(1)}=2\). The results below concern \(S_k(A)\) for arbitrary Hermitian \(A\), so on the positive-semidefinite cone they are statements about the Ky-Fan norm.

Let \(G\) act on a finite set \(X\). The permutation operator \(U(g)\) on \(\mathbb {C}^X\) sends the basis vector \(e_x\) to \(e_{gx}\), and extends linearly to an algebra homomorphism \(\mathbb {C}[G]\to \mathrm{End}(\mathbb {C}^X)\). The label projector \(\pi ^\lambda \) is the image of \(e_\lambda \). For a real function \(f\) of the labels, the label observable is \(\sum _\lambda f(\lambda )\pi ^\lambda \); for \(f(\lambda )=\log d_\lambda \) it is the observable \(F=\sum _\lambda (\log d_\lambda )\pi ^\lambda \) of  [ Ope26b , Section 6.1 ] .

A Lindblad form consists of a Hermitian matrix \(H = H^\dagger \) (the Hamiltonian) and a family of matrices \(\{ L_j\} _{j=0}^{r-1}\) (the Lindblad operators), defining the linear map

where \([A,B] = AB - BA\) and \(\{ A,B\} _+ = AB + BA\). This is [ Wol12 , Equation (7.21) ] .

The \(\mathbb {C}\)-linear span of the Lindblad operators \(\{ L_j\} \).

Let \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) be complex-linear. Its coefficient matrix \(C(\mathcal L)\in M_{e^2\times d^2}(\mathbb {C})\) in the matrix-unit bases is defined by

If a Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\) acts on the first factor while a finite-dimensional space \(R\) is left unchanged, the corresponding local Stinespring matrix is \(W=V\otimes \mathbb {1}_R\).

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) for its matrix in the normalized Pauli basis \(\{ \sigma _{0}/\sqrt{2},\sigma _{1}/\sqrt{2}, \sigma _{2}/\sqrt{2},\sigma _{3}/\sqrt{2}\} \). The channel is in diagonal Lorentz normal form when it is unital and every off-diagonal entry of \(\widehat{T'}\) is zero.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in non-diagonal Lorentz normal form when, for some \(x \in [0,1]\),

Trace preservation supplies the first row of the displayed matrix.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in singular Lorentz normal form when

equivalently, every input state is mapped to \((1+\sigma _{3})/2\). Trace preservation supplies the first row of the displayed matrix.

A linear map \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) maps the positive semidefinite cone onto itself if \(T\) is positive and every positive semidefinite matrix is the image under \(T\) of a positive semidefinite matrix. This is condition (1) of [ Wol12 , Proposition 3.6 ] .

Let \(R,K\in M_{m}(\mathbb {C})\) satisfy \(RK=KR=\mathbb {1}\). Conjugation

is a \(\mathbb {C}\)-algebra automorphism of \(M_{m}(\mathbb {C})\), with inverse \(X\mapsto KXR\).

For a map \(\alpha :M_D(\mathbb {C})\to M_D(\mathbb {C})\), a matrix \(X_0\), and an index \(a\), define \(X_{ij}=(\alpha (E_{ja})X_0)_{ia}\).

For every \(d\in \mathbb {N}\), define \(|\Omega \rangle \! \langle \Omega |\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) using

When \(d\geq 1\), this is the maximally entangled state: a rank-one projector with \((|\Omega \rangle \! \langle \Omega |)_{(i_1,i_2),(j_1,j_2)} = \frac{1}{d}\delta _{i_1 i_2}\delta _{j_1 j_2}\) and \(\operatorname{tr}(|\Omega \rangle \! \langle \Omega |)=1\).

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).

A weighted binary tree is either a leaf with a label \(j\) or an internal vertex with a parameter \(p\in [0,1]\) and two subtrees. Its root on inputs \((A_j)\) is \(A_j\) at a leaf and \(M_l\# _pM_r\) at an internal vertex with subtree roots \(M_l\), \(M_r\). The terminal weights are \(\delta _j\) at a leaf and \((1-p)w^{(l)}+p\, w^{(r)}\) at an internal vertex.

Let \(H=\bigoplus _k H_k\) and \(K=\bigoplus _k K_k\). For each \(k\), let \((A_{k,a}:H_k\to K_k)_a\) be a finite family of operators, and let \(\widetilde A_{k,a}:H\to K\) agree with \(A_{k,a}\) on \(H_k\) and vanish on every other summand. The orthogonally controlled Kraus map is

In particular, \(\mathcal C(X)_{kl}=0\) for \(k\ne l\). This is the sector control in the definitions of \(\mathcal T_1\) and \(\mathcal S_1\) in [ CPGSV16 , Appendix C.2, lines 1523–1535 and 1548–1555 ] .

Let \(e:I\simeq J\) be an equivalence of index sets. The associated matrix reindexing is the linear map \(R_e:\mathbb {C}^{I\times I}\to \mathbb {C}^{J\times J}\) determined by

Let \(I\) be a finite index set, let \(J\) be an arbitrary index set, and let \((A_a\in \mathbb {C}^{J\times I})_a\) be a finite family of matrices. Its rectangular Kraus map is

For a matrix \(X\) on \(A\otimes B\), the right partial trace is the linear map from matrices on \(A\otimes B\) to matrices on \(A\) given by

After the retained and discarded subspins have been regrouped as \(A\otimes B\), this is the partial-trace ingredient of the maps \(\mathcal T_0\) and \(\mathcal S_0\) in [ CPGSV16 , Appendix C.2, lines 1521–1522 and 1547 ] .

For a rectangular matrix \(V:H\to K\), define the linear map

Let \(R=\sqrt\rho \). For an orthonormal basis \((e_j)_j\) of \(B\), define rectangular operators \(A_j:A\to A\otimes B\) by \(A_j(e_a)=e_a\otimes Re_j\).

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.

A multi-cycle decomposition of \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) consists of a finite cycle index set \(\iota \), a per-cycle period \(m:\iota \to \mathbb {N}_{{\gt}0}\), and a family of orthogonal projections \(P_{c,k}\in M_{D}(\mathbb {C})\) for \(c\in \iota \) and \(k\in \{ 0,\ldots ,m(c){-}1\} \). For every \(c\in \iota \) and \(k\in \{ 0,\ldots ,m(c){-}1\} \), the per-cycle cyclic action is \(T(P_{c,k+1})=P_{c,k}\). The multiplicative-domain factorizations are \(T(P_{c,k}X)=T(P_{c,k})T(X)\) and \(T(XP_{c,k})=T(X)T(P_{c,k})\).

A multi-cycle decomposition gives a single-index block-permutation structure on the disjoint-union index \(\Sigma _{c\in \iota }\{ 0,\ldots ,m(c){-}1\} \): the permutation is the product of the per-cycle cyclic shifts \(k\mapsto k+1\), whose cycle decomposition has one cycle per \(c\in \iota \); the projections are \((c,k)\mapsto P_{c,k}\); and the multiplicative-domain factorizations descend componentwise. The resulting map forgets the explicit cycle indexing.

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

The Naimark isometry \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n\) of a POVM is the Stinespring-type construction

For a POVM \(\{ E_i\} \), each effect \(E_i\ge 0\) admits a square-root factorisation \(E_i=M_i^\dagger M_i\) with \(M_i\in M_{D}(\mathbb {C})\). The operators \(M_i\) are the Naimark Kraus square roots.

The Naimark projectors on \(\mathbb {C}^D\otimes \mathbb {C}^n\) are

Following Lewenstein–Kraus–Cirac–Horodecki [ LKCH00 ] , a normalized decomposable witness is an operator of the form

The transpose remains on Wolf’s first tensor factor.

Let \(I\) be finite and let \(A=\mathbb {R}^I\), with coordinatewise multiplication and the faithful positive functional \(\varphi (x)=\sum _{i\in I}x_i\). Its strictly positive invertible cone is

and the induced Hilbert norm and Nowosad functional are

For \(w\in A\), write \(P(w)=\operatorname {alg}_{\mathbb {R}}\{ w,w^{-1}\} \) for its Laurent-polynomial subalgebra.

For a Kraus map \(E\), define

We follow the convention that \(\mathcal{A}_R(E)\) controls right multiplication and \(\mathcal{A}_L(E)\) controls left multiplication. These are Kraus-map specializations of the preceding abstract domains, with the names interchanged: here the subscript records the side on which the varying factor is appended, whereas the preceding convention records the side occupied by the fixed element.

For a finite family of matrices \(\{ C_i\} _{i\in \iota }\) and a defect block \(S\), one forms an isometric dilation whose rows are the adjoints of the matrices \(C_i\) together with the adjoint of \(S\). One also forms the scalar block-diagonal matrix whose \(\iota \)-blocks carry prescribed weights and whose defect block carries a single scalar.

A subspace \(S\subseteq M_{m}(\mathbb {C})\) is an operator system if it contains the unit and is closed under the adjoint: \(\mathbb 1\in S\) and \(X\in S\implies X^\dagger \in S\).

A subspace \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) is an operator system if it contains the unit and is closed under the entrywise adjoint: \(\mathbb 1\in S\) and \(X\in S\implies X^\dagger \in S\), where \(X^\dagger \) denotes the family \((X_k^\dagger )_{k{\lt}r}\).

Let \(V:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^r\) be a supplied Stinespring matrix, and let \(|\Omega \rangle =d'^{-1/2}\sum _{a=1}^{d'}|a,a\rangle \) be normalized. Wolf’s auxiliary operator is

or, in coordinates, \(W_{j,(i,a)}=d'^{-1/2}V_{(a,j),i}\).

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

For a matrix \(V:\mathbb C^K\to \mathbb C^J\) and a complex-linear map \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define \(\Psi (X)=V^\dagger \Phi (X)V\).

Let \(\{ \psi _k\} _{k {\lt} m}\) be an ensemble and let \(n\) be a natural number. The padded ensemble \(\{ \psi _k^{\mathrm{pad}}\} _{k {\lt} n}\) is \(\psi _k\) for \(k {\lt} m\) and the zero vector for \(m \le k {\lt} n\). For \(m \le n\) the padded family agrees with \(\psi \) on every index \(k {\lt} m\) and retains the whole ensemble; for \(n {\lt} m\) only the first \(n\) vectors survive. The intended use is \(m \le n\).

For \(\sigma \in S_m\) and \((x,c)\in Q^m\times C^m\), put

Let \(U_Q,U_C,U_{QC}\) be their permutation operators. The separate actions commute, and \(u_{QC}(\sigma )=u_Q(\sigma )u_C(\sigma )\). Let \(\Pi ^Q_\lambda ,\Pi ^C_\mu ,\Pi ^{QC}_\nu \) be the corresponding central Schur projectors, with irreducible dimensions \(d_\lambda ,d_\mu ,d_\nu \).

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 \(\tau \).

For a bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), the partial transpose over the first factor is the matrix \(\rho ^{T_1}\) with entries

transposing the first index pair \(i,j\) and fixing the second pair \(k,l\).

For a bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), the partial transpose over the second factor is the matrix \(\rho ^{T_2}\) with entries

transposing the second index pair \(k,l\) and fixing the first pair \(i,j\).

Define Pauli time reversal by \(\Theta (X)=\operatorname{tr}(X)\mathbb {1}-X\). Its Pauli transfer matrix is

For a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\), define its Pauli-block diagonal truncation by

This is the truncation in [ Wol12 , Proposition 2.10, Section 2.4 ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 984–998.

Every Hermitian matrix \(M\in M_2^\dagger (\mathbb {C})\) has unique real Pauli coordinates \(x=(x_0,x_1,x_2,x_3)\) such that

This identifies \(M_2^\dagger (\mathbb {C})\) with \(\mathbb {R}^4\). Write

for the Minkowski quadratic form, its matrix, and the closed future cone. These are the coordinates used immediately before [ Wol12 , Eq. (2.41) ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 1040–1044.

Let \(\sigma _0=\mathbb {1},\sigma _1,\sigma _2,\sigma _3\) be the Pauli matrices. The Pauli transfer matrix of a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\) is \(\widehat T\in M_4(\mathbb {C})\) with entries

We write

where \(\Delta =(\widehat T_{ij})_{i,j=1}^{3}\). Thus \(r^{\mathsf T}\) and \(v\) are exactly the two off-diagonal Pauli blocks. This is the representation used in [ Wol12 , Section 2.4, Eq. (2.39) ] .

The peripheral eigenvalues of a linear map \(E\) on a finite-dimensional space are the eigenvalues \(\lambda \) on the unit circle: \(|\lambda | = 1\). For a channel, the spectral radius is \(1\) [ Wol12 , Proposition 6.1 ] , so these coincide with the eigenvalues of maximal modulus.

The condition \(|\lambda | = 1\) is appropriate for channels, since the spectral radius of a channel is \(1\); for a general linear map with spectral radius \(r \neq 1\) the condition would generalise to \(|\lambda | = r\).

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

Let \(T\) be an endomorphism of a finite-dimensional complex vector space. Its peripheral subspace is the sum of the maximal generalized eigenspaces for eigenvalues \(\mu \) with \(|\mu |=1\); its non-peripheral subspace is the corresponding sum for the remaining eigenvalues. This is the splitting induced by the full spectral decomposition of [ Wol12 , Equation (6.5) ] , with the peripheral projection itself defined in [ Wol12 , Equation (6.12) ] .

The peripheral spectrum of a bounded linear operator \(T\) is the set of spectral values \(\mu \) with \(|\mu |\) equal to the spectral radius of \(T\).

The physical label operator and the two merge deficits are

The region \(Y\) is independent of \(Q\) and \(V\).

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 ] .

The Moore-Penrose pseudoinverse \(B^+\) is defined via the support-generalised inverse of \(B B^\dagger \): \(B^+ := B^\dagger \cdot (B B^\dagger )^{-1}_{\mathrm{supp}}\).

For \(\xi \in \mathcal H\otimes \mathcal K\) define the nonnegative observable

Put \(\nu _t=\bigotimes _{i\in I}\operatorname {Poisson}(t)\) and define

This samples independent counts and then a uniform compatible word. Every \(C_n\) is finite and nonempty, including at \(t=0\).

For words \(u\) and \(v\), their concatenation \(uv\) places all letters of \(v\) after all letters of \(u\). Thus \(|uv|=|u|+|v|\). Concatenation is associative, and the empty word is a two-sided identity.

For a word \(w\in I^{\{ 0,\ldots ,m-1\} }\), define \(C(w)=\sum _{j=0}^{m-1}e_{w(j)}\), where \(e_i\) is the vector equal to one at \(i\) and zero elsewhere. Thus \(C(w)_i\) is the number of occurrences of \(i\), and \(\sum _{i\in I}C(w)_i=m\) when \(I\) is finite. The empty word has count vector zero. For \(n\in \mathbb {N}^I\), put \(\mathcal W_n=\{ w\in \mathcal W:C(w)=n\} \). Every word in \(\mathcal W_n\) has length \(\sum _{i\in I}n_i\).

For a word \(w\) over \(I\sqcup J\), let \(L(w)\) retain its \(I\)-letters and let \(R(w)\) retain its \(J\)-letters, preserving their original order. Both filters send the empty word to the empty word, and \(|L(w)|+|R(w)|=|w|\). For prescribed words \(u\) over \(I\) and \(v\) over \(J\), put \(E(u,v)=\{ w:L(w)=u,\ R(w)=v\} \).

For \(a\in I\), write \([a]\) for the word with the single letter \(a\). For words \(u,v\in \mathcal W\), inserting \(a\) between \(u\) and \(v\) gives the word \(u[a]v\), of length \(|u|+|v|+1\).

A marked word is a pair \((w,j)\) with \(w\in \mathcal W\) and \(0\leq j{\lt}|w|\). The marked insertion of \((u,a,v)\) is \((u[a]v,|u|)\).

The bijection between marked words and \(\mathcal W\times I\times \mathcal W\) is

No finiteness assumption on \(I\) is needed. Repeated letters at distinct positions give distinct marked words; when \(I\) is empty, both sets in the bijection are empty.

Equip \(\mathcal W\) with its discrete \(\sigma \)-algebra and put

Let \(p\) be a predicate on an alphabet \(I\), and put \(S=\{ i\in I:p(i)\} \) and \(T=\{ i\in I:\neg p(i)\} \). Classify each letter by the bijection \(c:I\to S\sqcup T\), then define \(P_p(w)=(L(c^{\mathcal W}(w)),R(c^{\mathcal W}(w)))\). Write \(P_p(w)=(w_S,w_T)\). The empty word gives two empty words, and \(|w_S|+|w_T|=|w|\).

For a bijection \(e:I\to J\) of alphabets, let \(e^{\mathcal W}\) apply \(e\) to each letter of a finite word, in its original position. Thus \(e^{\mathcal W}(i_0,\ldots ,i_{m-1})=(e(i_0),\ldots ,e(i_{m-1}))\). This is a bijection of word spaces, preserves length, and sends the empty word to the empty word.

For a word \(w\) and \(k\geq 0\), write \(w_{{\lt}k}\) for its first \(\min (k,|w|)\) letters, and \(w_{\geq k}\) for the remaining \(\max (|w|-k,0)\) letters. In particular, \(|w_{{\lt}k}|=k\) whenever \(k\leq |w|\).

A linear map \(E : M_{n}(\mathbb {C}) \to M_{m}(\mathbb {C})\) is positive if \(E(X) \ge 0\) whenever \(X \ge 0\), where \(X \ge 0\) means that \(X\) is positive semidefinite.

A positive matrix map \(E : M_{n}(\mathbb {C}) \to M_{m}(\mathbb {C})\) determines the same linear map regarded as a positive linear map: if \(A,B\in M_{n}(\mathbb {C})\) and \(A \le B\), then \(E(A) \le E(B)\) in \(M_{m}(\mathbb {C})\).

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.

A positive operator-valued measure with \(n\) outcomes on \(\mathbb {C}^D\) is a family \(\{ E_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^D\) satisfying the resolution of identity

Given an isometry \(V:\mathbb {C}^D\to \mathbb {C}^{d'}\) with \(V^\dagger V=\mathbb {1}_D\) and a family \(\{ P_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^{d'}\) summing to the identity, the pulled-back operators \(E_i:=V^\dagger P_iV\) form a POVM. In particular, any projective measurement on the dilation (a special case of a PSD resolution of identity) pulls back to a POVM.

A linear map is primitive if its only peripheral eigenvalue is \(\lambda = 1\), i.e. \(\mathrm{peripheral}(E) = \{ 1\} \). For channels this corresponds to [ Wol12 , Theorem 6.7 ] . This definition applies to any linear endomorphism, not only to channels.

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

For a ray \(p=[v]\) in \(\mathbb {C}^d\), define

For each ray \(p\), fix a nonzero representative \(p^{\mathrm{rep}}\). Define

For \(P\in M_{D}(\mathbb {C})\), define

Define \(\mathcal I_a(X)=J_aXJ_a^\dagger =X\otimes |a\rangle \! \langle a|\) on all system matrices \(X\). Relabeling the system and environment bases by bijections \(p,q\) carries \(\mathcal I_a(X)\) to \(\mathcal I_{q(a)}(p_*X)\), where \(p_*X\) denotes the matrix in the relabeled system basis.

Given a finite family \(\{ \psi _i\} _{i \in \iota }\) of (unnormalized) vectors in \(\mathbb {C}^D\), its pure-ensemble density is \(\rho = \sum _i |\psi _i\rangle \! \langle \psi _i|\). Weights \(p_i \geq 0\) with \(\sum p_i = 1\) can be absorbed by replacing \(\psi _i \mapsto \sqrt{p_i} \psi _i\).

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 \(\operatorname{tr}_B[|\psi \rangle \! \langle \psi |]=\rho \).

A semigroup \(T\) relaxes to a density matrix \(\rho _\infty \) if \(\rho _\infty {\gt}0\) and

for every density matrix \(\rho \). A generator \(L\) has a simple faithful kernel if its kernel consists precisely of the scalar multiples of a positive-definite density matrix.

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.

Let \(S\) be a finite set. For each \(s\in S\), let \(\Omega _s\) be a measurable space and let \(I_s\) and \(L_s\) be two index sets. Given complex functions \(c_s(a,b;\cdot )\) on \(\Omega _s\), define the coefficient associated with multiindices \(i\in \prod _s I_s\) and \(l\in \prod _s L_s\) by

where \(\omega \in \prod _s\Omega _s\). The ket and bra index sets may have different cardinalities. This is the coefficient product in [ Ope26a , Equation (5.11) ] .

A real-eigenvalue block is the ordinary real Jordan block \(J_n(a)\) with eigenvalue \(a\) and ones on its first superdiagonal. If \(a\pm \tau i\) is a non-real conjugate pair, put

Its real Jordan chain of length \(m\) has \(B(a,\tau )\) on the block diagonal and \(I_2\) on the first block superdiagonal. On the product indexing of this \(2m\)-dimensional real block, its unsigned metric reverses both the chain coordinate and the two-dimensional realification coordinate. No sign characteristic is attached to a non-real block.

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 ] .

A complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) is a Hilbert–Schmidt contraction if, for every \(X\in M_{d}(\mathbb {C})\),

Let \(\tau \) be a Hermitian operator on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) with normalized eigenvectors \(\phi _i\). The reduced density operator of the \(i\)-th eigenvector on the first factor is \(\rho _i=\operatorname{tr}_2|\phi _i\rangle \! \langle \phi _i|\).

Let \(A,C,W,X,Y,Z\) be finite index sets. Choose arbitrary fixed endpoint maps \(E:\mathbb {C}^A\to \mathbb {C}^W\), \(F:\mathbb {C}^A\to \mathbb {C}^X\), \(\widetilde E:\mathbb {C}^C\to \mathbb {C}^Y\), and \(\widetilde F:\mathbb {C}^C\to \mathbb {C}^Z\). For nonnegative real weights \(\lambda ,\mu \), define

The second ambient vector uses the weights \(\mu \). For the coordinate sampled factors of Definition 15.1.1, put \(U_j^{\mathrm{amb}}=EU_j\widetilde E^\dagger \) and \(V_j^{\mathrm{amb}}=FV_j\widetilde F^\dagger \). Then

In particular, the represented ambient vector is the endpoint transport of its paired coordinate Schmidt vector. These are the ambient-coordinate forms of [ Ope26a , Equation (5.6) ] .

The reduction map on \(M_{D}(\mathbb {C})\), for \(k\in \mathbb {N}\), is

The reduction witness on \(M_{D}(\mathbb {C})\otimes M_{D}(\mathbb {C})\), for \(n\in \mathbb {N}\), is

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 ] .

Assign an original site to factor \(0,1,2,3,4\) according as it belongs to \(Q,Y,F,\{ c\} ,\{ r\} \). For a set \(S\subseteq \{ 0,1,2,3,4\} \), let \(\widetilde S\) consist of all original sites assigned to \(S\).

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

Let \(\Theta \) be the unit sphere of the one-copy space \(\mathcal V\), let \(e_a\) be a coordinate unit vector and let \(H\) be Haar probability measure on \(U(\mathcal V)\). For a matrix \(\rho \) on \(\mathcal V^{\otimes k}\), set \(q_\rho (U)=D_k\operatorname {Re}\operatorname {Tr}(\rho P_{Ue_a,k})\). Define

For \(\rho \ge 0\), the density is nonnegative and (1) is the coherent measure of  [ Ope26b , Proposition 7.4 ] .

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

For the terminal state \(\sigma _{j,u}\) of the transport tree, let \(d_j(u,U)=m_{1/4}(u)q_{\sigma _{j,u}}(U)\) and define

Neither a terminal weight nor a factor of two occurs in this definition. The parameter is clamped to \([0,1]\); a zero terminal weight gives the zero normalized derivative map, using the convention \(0^{-1}=0\).

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 \(\sigma \geq 0\) with \(\operatorname{tr}(\sigma )\neq 0\), define \(E_\sigma (X):=\operatorname{tr}(\sigma X)\operatorname{tr}(\sigma )^{-1}\mathbb {1}\).

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 \(k\ge 0\), define the bijection

Let \(U_k f=f\circ e_k\) be the induced unitary between coordinate spaces.

Fix \(k\geq 0\) and an irreducible label \(\ell \) of \(\mathfrak S_k\). Let \(v_{k,\ell }\) be the original prevector in the canonical coordinates \((A\times B)^k\times (R^k\times R^k)\). The regrouping

and the fixed bijection \(e_k\) of Definition 22.49.1 give

The original state and supplied label are unchanged, as in [ Ope26b , Section 8, lines 240–254 and 283–290 ] .

Group \(X\) with \(C\) in each copy, and then separate these \(k\) pairs from the exterior copies. The resulting bijection is

These are the fixed coordinates used in [ Ope26b , Section 8, lines 240–254 and 283–290 ] .

For \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) with \(d=\min \{ d_A,d_B\} \), a Schmidt decomposition of \(\psi \) is a choice of orthonormal bases \(\{ e_j\in \mathbb {C}^{d_A}\} \), \(\{ f_j\in \mathbb {C}^{d_B}\} \) and nonnegative reals \(\lambda _j\), \(j=1,\ldots ,d\), satisfying

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 a matrix \(\tau \) on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) and a bound \(n\), the set

collects the real parts of the quadratic forms of \(\tau \) in the normalized vectors of Schmidt rank at most \(n\). For Hermitian \(\tau \), the setting of every statement below, the quadratic form is real and \(E_n(\tau )\) is the set of expectations of \(\tau \) in those vectors.

The Schmidt singular values of a vector \(\psi \in \mathbb {C}^{D}\otimes \mathbb {C}^k\) are the singular values of the coefficient matrix \(C_\psi \), viewed as a linear map \(\mathbb {C}^k\to \mathbb {C}^D\).

For a finite list \(\mathcal B=(B_1,\ldots ,B_m)\) and real thresholds \(a_B\), put \(C_j=C_{B_j}(a_{B_j})\) and set \(C=\prod _{j=1}^m C_j\), in the order of the list. The empty product is the identity. Repeated regions are permitted.

For \(a\in \mathbb R\), define

The pseudoinverse Schur complement \(P - Q R^+ Q^\dagger \).

For Hermitian data \(F_i\), the traces \(\operatorname{tr}(F_iX)\) are real; define \(T(X)_i=\operatorname{tr}(F_iX)\). Then \(T^*(y)=\sum _i y_iF_i\). Consequently the conic primal constraint is precisely \(X\geq 0\) and \(\operatorname{tr}(F_iX)=b_i\), while conic dual feasibility is precisely \(F_0-\sum _i y_iF_i\geq 0\). The two conic strict-feasibility predicates become, respectively, \(X{\gt}0\) with the trace constraints and \(F_0-\sum _i y_iF_i{\gt}0\), exactly as in [ Wol12 , Chapter 4, lines 85–105 ] .

Let \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\). Write \(F=(A+b\mathbb {1})^r\) and define

Let \(d\ge 1\). A symmetric informationally complete family in dimension \(d\) consists of \(d^2\) rank-one orthogonal projections \(P_i\in M_{d}(\mathbb {C})\) satisfying

A matrix \(P\in M_{D}(\mathbb {C})\) is a rank-one orthogonal projection if \(P=P^\dagger \), \(P^2=P\), and \(\operatorname{rank}P=1\).

The sip matrix of size \(n\) is \(P_n\). A signed sip block is \(\varepsilon P_n\), where \(\varepsilon =1\) or \(\varepsilon =-1\). Each such block is symmetric, invertible, and its own inverse.

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

An SL-filtering for \(D \times D\) matrices is a completely positive map of the form \(\Phi (X)=SXS^{\dagger }\), where \(S \in M_{D}(\mathbb {C})\) satisfies \(\det S=1\). Such maps are invertible; when \(D\geq 1\), they have Kraus rank 1.

Let \(A,C\) be finite sets with nonnegative weights \(\lambda _a,\mu _c\), and let \(k\) be a positive integer. In the paired coordinate bases, put \(|\eta \rangle =\sum _a\sqrt{\lambda _a}|a,a\rangle \) and \(|\widetilde\eta \rangle =\sum _c\sqrt{\mu _c}|c,c\rangle \). For independent standard circular complex Gaussian coordinates \(g^{(j)}_{ac}\), with \(\mathbb E|g^{(j)}_{ac}|^2=1\), set

Thus \(X_{(a,b),(c,d)}=\sqrt{\lambda _a\mu _c}\delta _{ab}\delta _{cd}\), and

This is [ Ope26a , Equation (5.6) ] .

Define \(\chi (\omega )=e\exp [-(1-(\omega /\delta )^2)^{-p}]\) for \(|\omega |{\lt}\delta \) and \(\chi (\omega )=0\) for \(|\omega |\ge \delta \).

Define \(f(t)=(2\pi )^{-1}\int _{\mathbb {R}}\chi (\omega )\cos (t\omega )\, d\omega \).

For \(X\in \operatorname{SL}(2,\mathbb {C})\), let \(L(X)\) be the real linear transformation of \(\mathbb {R}^4\) determined by Wolf’s Hermitian congruence

Its Pauli-basis entries are

The trace is real. This is the four-dimensional action in [ Wol12 , Eq. (2.41) ] ; it is not the three-dimensional adjoint action \(M\mapsto UMU^{-1}\) on traceless Pauli matrices. The special orthochronous Lorentz group is described, in the row-action convention of the source, by

Regard \(\mathrm{SO}^+(1,3)\) from (103) as a matrix group. The spinor action defines the homomorphism

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

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\). A linear operator \(f:\mathbb {C}^D\to \mathbb {C}^D\) is an intertwiner from \(W\) into \(W'\) when \(f(W)\subseteq W'\) and \(f(Ax)=A(fx)\) for every \(A\in S\) and \(x\in W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). A subspace \(W\subseteq \mathbb {C}^D\) is irreducible under \(S\) when it is nonzero, invariant under every member of \(S\), and its only invariant subspaces are \(\{ 0\} \) and \(W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be invariant under \(S\). The pieces \(W\) and \(W'\) are of the same type when there is an intertwiner from \(W\) into \(W'\) that is nonzero on \(W\).

For an irreducible channel \(E\) on \(M_{D}(\mathbb {C})\) with \(D\geq 1\), the stationary state is the unique density-matrix fixed point \(\rho _\infty \) satisfying \(E(\rho _\infty )=\rho _\infty \), \(\rho _\infty {\gt}0\), and \(\operatorname{tr}(\rho _\infty )=1\). Existence, uniqueness, and positive definiteness follow from Theorem 8.14.1.

The stationary support of an irreducible channel \(E\) is the support projection of its stationary state (Definition 10.4.2).

Given possibly rectangular Kraus operators \(K_j:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\), the Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\otimes \mathbb {C}^r\) is defined by

Thus \(V=\sum _j K_j\otimes |j\rangle \). It is an isometry precisely when the Kraus family satisfies the trace-preserving normalization.

For a Kraus family \((K_j)_{j\in J}\) indexed by an arbitrary finite set \(J\), define \((V_J)_{(i,j),k}=(K_j)_{ik}\).

For real \(C,c,\alpha \) and \(y,z\in Y\), define

Every radius \(n\) is regarded as a finite nonnegative real radius. The sum is the real infinite sum, with value zero if the series does not converge. Theorem 27.28.4 establishes convergence under the counting hypotheses used below.

For every \(k\ge 0\), define the bijection

Its inverse joins the two restricted configurations in each copy.

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.

Let \(\tau \geq 0\). Its support-restricted negative quarter power \(\tau ^{-1/4}_{\operatorname {supp}}\) acts by \(x^{-1/4}\) on every positive eigenspace and vanishes on the kernel. For a complex-linear map \(\Phi \), define

For a real parameter \(\eta \), the map \(T_\eta \) on \(M_{D}(\mathbb {C})\) is

The map is defined for every real \(\eta \); at \(\eta =0\) it reduces to \(\rho \mapsto \operatorname{tr}(\rho ) \mathbb {1}\), and the positivity threshold below assumes \(\eta {\gt}0\).

For words \(u\) over \(A\) and \(v\) over \(B\), define the list \(I(u,v)\) recursively. If either word is empty, the list contains just the other word with its tag attached. If \(u=au'\) and \(v=bv'\) are nonempty, first list the words obtained by prepending the left-tagged \(a\) to each member of \(I(u',v)\), then list the words obtained by prepending the right-tagged \(b\) to each member of \(I(u,v')\).

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

Given a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the tensor extension \(T \otimes \operatorname{id}\) acts on bipartite matrices \(X \in M_{D \times D}(\mathbb {C})\) by applying \(T\) to each “slice”:

where \(X^{(i_2,j_2)}_{ab} = X_{(a,i_2),(b,j_2)}\) is the bipartite slice.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be linear and let \(\rho \in M_{D}(\mathbb {C})\). Following [ Wol12 , Equation (6.35) ] , define

A Kraus map is trace-preserving if \(\sum _{i=0}^{d-1} K_i^\dagger K_i = \mathbb {1}\). Equivalently, the adjoint Kraus map is unital. This is the standard MPS normalization condition. In the later gauge language it is the left-canonical condition, so Kadison–Schwarz arguments are often applied to the adjoint map.

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) ] .

The trace-pairing adjoint \(E^* : M_{D'}(\mathbb {C}) \to M_{D}(\mathbb {C})\) of a linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is the adjoint for the bilinear pairing \((A,B)\mapsto \operatorname{tr}(AB)\).

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) and a trace-self-dual basis \(\{ \sigma _i\} \), define coefficients \(t_{ij}:=\operatorname{tr}(\sigma _i T(\sigma _j))\).

A basis \(\{ \sigma _i\} _i\) of \(M_{D}(\mathbb {C})\) is trace-self-dual when its coordinate functionals are given by trace pairing:

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 \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is trace-preserving if \(\operatorname{tr}(E(X)) = \operatorname{tr}(X)\) for all \(X\).

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

The transfer map associated to a finite matrix family \(A\) is the finite Kraus map of that family; the notation \(\mathcal{E}_A\) abbreviates the linear map \(M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

For a linear map \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) and a family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) used on both sides, the transfer matrix of \(T\) in \((\sigma _\alpha )\) is

which is [ Wol12 , Equation (2.20) ] with \(F_\alpha =G_\alpha \).