Dependencies

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

Let \(T\) 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 \(\mu \).

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

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) ] .

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

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

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 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.

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.

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.

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

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.

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 \(T\).

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.

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 \(X\in S\).

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 \(M_{D}(\mathbb {C})\).

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

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

Equivalently, this is the matrix of \(T\) under the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\): the matrix-units special case of Definition 3.18.2.

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

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

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

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).

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.

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-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 \((A,B)\mapsto \operatorname{tr}(AB)\).

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 Kraus map is unital if \(\sum _{i=0}^{d-1} K_i K_i^\dagger = \mathbb {1}\). Equivalently, \(E(\mathbb {1}) = \mathbb {1}\).

A linear map \(E\) is unital if \(E(\mathbb {1})=\mathbb {1}\).

For a unitary matrix \(U \in \mathcal{U}(D)\), the unitary channel is \(T(\rho )=U\rho U^\dagger \). It is automatically a quantum channel.

For a finite-dimensional complex endomorphism \(T\), let

We use the explicit convention \(\mu (T)=0\) when the displayed spectrum is empty. If it is nonempty, the spectrum is finite, so the supremum is an attained maximum and \(\mu (T){\lt}1\).

Let \(\Phi :\mathcal A\simeq \mathcal B\) match the simple summands by an equivalence \(\sigma :I\simeq J\), with \(D_i=E_{\sigma (i)}\). Suppose that there are unitaries \(U_i\) satisfying \(\Phi (0,\ldots ,0,X,0,\ldots ,0)_{\sigma (i)} =U_i\iota _i(X)U_i^*\). Then \(\Phi \) preserves the total block trace.

A coordinatewise family of completely positive maps between paired summands determines a direct-sum Kraus map. For direct-sum Kraus maps \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal C\), one has \(\widehat{S\circ T}=\widehat S\circ \widehat T\), and \(S\circ T\) is again a direct-sum Kraus map. If an endomorphism \(T:\mathcal A\to \mathcal A\) has an extension satisfying

then \(T\) satisfies the corresponding inequality in every summand. Finally, if \(F:\mathcal A\to \mathcal A\) is a trace-preserving direct-sum Kraus map, then its trace adjoint satisfies, for every \(X\in \mathcal A\),

in every summand.

For maps \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal C\), trace adjoints satisfy

If \(T\) preserves the total trace, then \(T^*(\mathbb {1}_{\mathcal B})=\mathbb {1}_{\mathcal A}\). If \(\widehat T\) admits a Kraus representation, then so does \(\widehat{T^*}\); in particular, both \(T\) and \(T^*\) send families of positive semidefinite matrices to positive semidefinite families. These statements are only the adjoint and positivity part of the classification argument; none of its multiplicative or blockwise conclusions is asserted here.

Let \(\Phi :\mathcal A\simeq \mathcal B\) be a star-algebra isomorphism between finite products of full matrix algebras. If \(\Phi \) preserves the total block trace, then \(\Phi ^*=\Phi ^{-1}\).

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

This is the Jordan-block calculation in [ Wol12 , Chapter 8, proof of Equation (8.104) ] . Let \(D\ge 1\), let \(N\in M_{D}(\mathbb C)\) be the strict upper-shift matrix \(N_{i,j}=1\) iff \(j=i+1\) (and \(0\) otherwise), and \(J_{D}(\lambda )=\lambda I+N\) the associated Jordan block. Then \(N^{k}_{i,j}=1\) iff \(j=i+k\) (with \(N^{k}=0\) for \(k\ge D\)), and for every \(n\in \mathbb N\),

Let \(K_0,\ldots ,K_{d-1}\) be a trace-preserving Kraus family. The map \(E(X)=\sum _iK_iXK_i^\dagger \) is a quantum channel.

For every \(A\in M_D(\mathbb C)\) with \(D\ge 1\) and every \(n\in \mathbb N\),

Let \(V\) be finite-dimensional and let \(f:V\to V\) have bounded orbits. Then

Let \(f:V\to V\) have bounded orbits. If \(x\in \operatorname{range}(f-\mathbb {1}_V)\), then \(A_N(x)\longrightarrow 0\).

Let \(T\) be an endomorphism of a finite-dimensional complex vector space whose every eigenvalue has modulus at most one. Then \(T^{n}(Y)\to 0\) for every \(Y\) in the non-peripheral subspace:

If \(M\succeq 0\), \(\rho {\gt}0\), and \(\operatorname{tr}(\rho M)=0\), then \(M=0\).

Positive and trace-preserving matrix endomorphisms are closed under concatenation, natural powers, and finite-dimensional pointwise limits. Positive Schwarz maps are closed under the same operations. Moreover, the trace-pairing adjoint of the identity map is the identity map, trace-pairing adjoints commute with natural powers, and pointwise convergence passes to the trace-pairing adjoints. Every assertion is orientation-specific: applying the Schwarz closure to \(T^*\) requires a Schwarz hypothesis for \(T^*\).

For \(\rho \succeq 0\) on \(\mathbb {C}^D\), the support projection of \(\rho \) has the rank of \(\rho \).

One has \(E_\sigma (X)=\operatorname{tr}(\sigma X)\operatorname{tr}(\sigma )^{-1}\mathbb {1}\).

For every \(c\in \mathbb {C}\), one has \(E_\sigma (c\mathbb {1})=c\mathbb {1}\).

For every \(X\), the matrix \(E_\sigma (X)\) is fixed by \(T^*\).

For every \(X\), there exists \(c\in \mathbb {C}\) such that \(E_\sigma (X)=c\mathbb {1}\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\) and let \(T\in M_{D}(\mathbb {C})\). If \(T\) commutes with every linear operator on \(\mathbb {C}^D\) that commutes with all members of \(S\), then \(T\in S\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), let \(W\subseteq \mathbb {C}^D\) be irreducible under \(S\), and let \(f\) be an intertwiner from \(W\) into a subspace \(W'\). Then there exists a real number \(c\geq 0\) such that, for all \(x,y\in W\),

Under the hypotheses of Lemma 10.5.13, if \(f\) does not vanish on \(W\), then the factor \(c\) in (43) may be chosen positive.

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), let \(W,W'\subseteq \mathbb {C}^D\) be irreducible under \(S\), and let \(f:\mathbb {C}^D\to \mathbb {C}^D\) be linear. Suppose that \(f(W)\subseteq W'\) and \(f(Ax)=A(fx)\) for every \(A\in S\) and \(x\in W\). Then either \(f\) is zero on \(W\), or \(f\) is injective on \(W\) and maps \(W\) onto \(W'\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\) acting on \(\mathbb {C}^D\). A complex subspace \(W\subseteq \mathbb {C}^D\) is invariant under every member of \(S\) if and only if it is the underlying complex subspace of a submodule of \(\mathbb {C}^D\) over \(S\). Under this identification, an invariant subspace is irreducible under \(S\) precisely when it is a simple \(S\)-module.

With \(S\) and \(W\) as in Lemma 10.5.3, the representation space decomposes as \(\mathbb {C}^D=W\oplus W^\perp \), and \(W^\perp \) is invariant under every member of \(S\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be irreducible under \(S\). If \(W\) and \(W'\) are not of the same type, then \(W\perp W'\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(\mathcal D_1,\mathcal D_2\) be sets of subspaces irreducible under \(S\). If no piece of \(\mathcal D_1\) is of the same type as any piece of \(\mathcal D_2\), then

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). As a module over \(S\), the space \(\mathbb {C}^D\) is semisimple: every submodule admits a complementary submodule.

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

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W\subseteq \mathbb {C}^D\) be invariant under every member of \(S\). Then \(W\) is the sum of a finite family of pairwise orthogonal subspaces contained in \(W\), each irreducible under \(S\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W\subseteq \mathbb {C}^D\) be invariant under every member of \(S\). Then \(W^\perp \) is also invariant under every member of \(S\): \(AW^\perp \subseteq W^\perp \) for every \(A\in S\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), let \(W\subseteq \mathbb {C}^D\) be invariant under \(S\), and let \(P\) be the orthogonal projection onto \(W\). Then \(P(Ax)=A(Px)\) for every \(A\in S\) and \(x\in \mathbb {C}^D\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be irreducible under \(S\). Then \(W\) and \(W'\) are of the same type if and only if there is an intertwiner from \(W\) into \(W'\) that is injective on \(W\) and maps \(W\) onto \(W'\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be irreducible under \(S\). If \(W\) is of the same type as \(W'\), then there is an intertwiner \(u\) from \(W\) into \(W'\) that maps \(W\) onto \(W'\) and satisfies \(\langle u(x),u(y)\rangle =\langle x,y\rangle \) for all \(x,y\in W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), let \(\mathcal D\) be a family of subspaces irreducible under \(S\), and let \(W\) be irreducible under \(S\), with \(W\leq \bigvee _{W'\in \mathcal D}W'\). Then \(W\) is of the same type as some piece of \(\mathcal D\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W\subseteq \mathbb {C}^D\) be irreducible under \(S\). Then \(W\) is of the same type as itself.

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be irreducible under \(S\). If \(W\) is of the same type as \(W'\), then \(W'\) is of the same type as \(W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W',W''\subseteq \mathbb {C}^D\) be irreducible under \(S\). If \(W\) is of the same type as \(W'\) and \(W'\) is of the same type as \(W''\), then \(W\) is of the same type as \(W''\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), let \(W\subseteq \mathbb {C}^D\) be irreducible under \(S\), and let \(f:\mathbb {C}^D\to \mathbb {C}^D\) be linear. Suppose that \(f(W)\subseteq W\) and \(f(Ax)=A(fx)\) for every \(A\in S\) and \(x\in W\). Then there is \(c\in \mathbb {C}\) such that \(fx=cx\) for every \(x\in W\).

Let \((f_{k,i,j})_{k{\lt}K,\, i{\lt}m_k,\, j{\lt}d_k}\) be an orthonormal basis of \(\mathbb {C}^D\). There are an identification of the index set of triples \((k,i,j)\) with \(\{ 0,\ldots ,D-1\} \), realizing \(\sum _kd_km_k=D\), and a unitary \(U\in M_{D}(\mathbb {C})\) whose columns are the basis vectors, such that every matrix \(A\in M_{D}(\mathbb {C})\) acting on the basis by

for matrices \(B_k\in M_{d_k}(\mathbb {C})\) satisfies

Let \(\rho ,P\succeq 0\) and \(c\in \mathbb {C}\) with \(P\preceq c\rho \), and let \(Q\) be the support projection of \(\rho \). Then \(QP=P\) and \(PQ=P\).

Let \(\rho \succeq 0\) on \(\mathbb {C}^D\), with support projection \(P=\operatorname{supp}(\rho )\). For every \(v\in \mathbb {C}^D\), if \(\rho v=0\), then \(Pv=0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, let \(\rho \succeq 0\) satisfy \(T(\rho )=\rho \), and let \(V:\mathcal H\to \mathbb {C}^D\) be an isometry onto the support of \(\rho \). Define \(\widetilde T(Y)=V^*T(VYV^*)V\). If \(T^*(A)=A\), then

This is Equation (6.53) of  [ Wol12 ] .

For \(\rho \succeq 0\) on \(\mathbb {C}^D\), the trace of the support projection of \(\rho \) is the rank of \(\rho \).

If \(AB=BA=I\) and \(X=AJB\), then for every \(n\in \mathbb N\),

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

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

The corner algebra associated with an idempotent already carries a ring structure with unit \(P\). In the matrix case it also has \(\mathbb {C}\)-module and \(\mathbb {C}\)-algebra structures and, under the additional assumption \(P^\dagger =P\), star, star-ring, and star-module structures over \(\mathbb {C}\).

If \(E\) satisfies the Schwarz inequality, then

Consequently, \(A\in \mathcal{A}(E)\) exactly when both equalities hold. This is [ Wol12 , Equations (5.13)–(5.14) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, suppose that \(T^*\) satisfies the Schwarz inequality, and let \(\rho \succeq 0\) satisfy \(T(\rho )=\rho \). With \(Q=\operatorname{supp}(\rho )\), the set

is a \(*\)-subalgebra of the corner algebra \(QM_{D}(\mathbb {C})Q\). No invertibility of \(\rho \) on the ambient space is assumed.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) have bounded orbits, let \(P_T\) be its mean-ergodic projection, and let \(P_T^*\) be the trace-pairing adjoint of \(P_T\). Then

and \(P_T^*(Y)=Y\) if and only if \(T^*(Y)=Y\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, suppose that \(T^*\) satisfies the Schwarz inequality, and suppose that there is a positive definite matrix \(\rho \) satisfying \(T(\rho )=\rho \). Let \(P_T\) be the mean-ergodic projection of \(T\). There are positive integers \(d_k,m_k\), a unitary \(U\), and density matrices \(\sigma _k\in M_{m_k}(\mathbb {C})\) such that

for every \(A\in M_{D}(\mathbb {C})\). This is the full-support conditional-expectation step in the proof of [ Wol12 , Theorem 6.14 ] .

Let \(E\) be a trace-preserving Kraus map on \(M_{D}(\mathbb {C})\) with a positive definite fixed point \(\rho {\gt}0\), \(E(\rho )=\rho \). There are \(n\in \mathbb {N}\), positive dimensions \(d_0,\ldots ,d_{n-1}\) and multiplicities \(m_0,\ldots ,m_{n-1}\) with \(\sum _kd_km_k=D\), realized by an explicit identification of the index sets, and a unitary \(U\in M_{D}(\mathbb {C})\) such that a matrix \(X\in M_{D}(\mathbb {C})\) satisfies \(E^*(X)=X\) exactly when

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

The positive definite fixed point removes the zero block: this is the unital case of the block representation in Equation (1.39) of [ Wol12 ] , invoked by [ Wol12 , Theorem 6.14 ] . The general form of [ Wol12 , Theorem 6.14 ] , with a zero block and density weights \(\rho _k\) on the Schrödinger-picture fixed points, is not asserted here.

Under the hypotheses of Theorem 10.2.10, the adjoint fixed-point \(*\)-subalgebra equals the Kraus commutant:

This is [ Wol12 , Theorem 6.13 ] .

There exist integers \(n\), block sizes \(d_1,\ldots ,d_n\geq 1\), multiplicities \(m_1,\ldots ,m_n\geq 1\), and a \(\mathbb {C}\)-algebra isomorphism from the adjoint-fixed-point \(*\)-subalgebra to \(\prod _{k=1}^{n}M_{d_k}(\mathbb {C})\) such that \(\sum _{k=1}^{n}d_km_k\leq D\).

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

If \(T\) is trace-preserving, then \(T^*\circ E_\sigma =E_\sigma \).

Under the same hypotheses, for all \(X,Y\in M_{D}(\mathbb {C})\), \(E(E(X)Y)=E(E(X)E(Y))\).

Under the same hypotheses, for all \(X,Y\in M_{D}(\mathbb {C})\), \(E(E(X)Y)=E(XE(Y))\).

Under the same hypotheses, for all \(X,Y\in M_{D}(\mathbb {C})\), \(E(XE(Y))=E(E(X)E(Y))\).

Under the same hypotheses, for all \(X,Y\in M_{D}(\mathbb {C})\), \(E(E(X)E(Y))=E(X)E(Y)\).

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

In these formulas the two direct sums are transported to the original product coordinates through the stated identification. These are the two equalities in  [ Bei12 , Lemma 2.1 ] . The operators \(R_q\) and \(S_q\) are Hermitian; they are not asserted to be unitary.

Let \(X_{AB}\in \mathbf L(H_A\otimes H_B)\) and \(Y_{BC}\in \mathbf L(H_B\otimes H_C)\), with \(Y_{BC}\) Hermitian, and suppose \([X_{AB}\otimes \mathbb {1}_C,\mathbb {1}_A\otimes Y_{BC}]=0\). There is an orthonormal decomposition \(H_B\cong \bigoplus _{q=0}^{K-1}H_{q,r}\otimes H_{q,l}\) such that all middle-factor coefficients \(X_{aa'}\) act only on \(H_{q,l}\), while all middle-factor coefficients \(Y_{cc'}\) act only on \(H_{q,r}\). More precisely, in an orthonormal basis \((e_{q,r,s})\) adapted to this decomposition,

The source lemma assumes that both overlapping operators are Hermitian; the coefficient conclusion requires Hermiticity only for \(Y_{BC}\). This is the middle-space coefficient form of [ Bei12 , Lemma 2.1 ] .

Let \(X_{AB}\in \mathbf L(H_A\otimes H_B)\) and \(Y_{BC}\in \mathbf L(H_B\otimes H_C)\). For fixed outer indices, define their middle-factor coefficient matrices by

If

then \(X_{aa'}Y_{cc'}=Y_{cc'}X_{aa'}\) for every \(a,a',c,c'\). This is the coefficientwise commutation step in the proof of [ Bei12 , Lemma 2.1 ] .

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

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

Let \(I\) and \(J\) be finite, and let \(\{ R_i\} _{i\in I}\) and \(\{ S_j\} _{j\in J}\) be families of simple rings. Write \(I_i\) and \(J_j\) for the corresponding block ideals. Suppose that \(T : \prod _{i\in I}R_i \to \prod _{j\in J}S_j\) is a ring isomorphism and that an equivalence \(\sigma :I\simeq J\) satisfies \(T(I_i)=J_{\sigma (i)}\) for every \(i\in I\). Then, for each \(i\in I\), the map

from \(R_i\) to \(S_{\sigma (i)}\) is bijective.

Let \(\rho \succeq 0\) on \(\mathbb {C}^D\), with support projection \(P=\operatorname{supp}(\rho )\), and let \(V:\mathbb {C}^D\to \mathbb {C}^k\) be a compression isometry satisfying \(VV^\dagger =\mathbb {1}_k\) and \(V^\dagger V=P\). Then \(V\rho V^\dagger \in M_{k}(\mathbb {C})\) is positive definite.

Let \(P,Q\in M_{D}(\mathbb {C})\) be matrices, and suppose that \(V_P,V_Q:\mathbb C^n\to \mathbb C^D\) satisfy

Write \(\Phi _P(X)=V_PXV_P^*\) and \(\Phi _Q(X)=V_QXV_Q^*\) for the corresponding linear isomorphisms onto the two corners. For every unitary \(W\in M_{n}(\mathbb {C})\), there is a matrix \(U\in M_{D}(\mathbb {C})\) such that

and, for every \(X\in M_{n}(\mathbb {C})\),

Let \(P,Q\in M_{D}(\mathbb {C})\) be orthogonal projections. If

is a bijective complex-linear map preserving multiplication and the adjoint, then there is a matrix \(U\in M_{D}(\mathbb {C})\) such that

and

for every \(X\in QM_{D}(\mathbb {C})Q\).

Let \(E(X)=\sum _iK_iXK_i^\dagger \) be a trace-preserving Kraus map on \(M_{D}(\mathbb {C})\), with Heisenberg-picture adjoint \(E^*(Y)=\sum _iK_i^\dagger YK_i\), let \(\rho \succeq 0\) satisfy \(E(\rho )=\rho \), and let \(Q\) be the support projection of \(\rho \). There are a sector dimension \(r\leq D\), a number \(n\) of blocks, positive dimensions \(d_0,\ldots ,d_{n-1}\) and multiplicities \(m_0,\ldots ,m_{n-1}\) with \(\sum _kd_km_k=r\), realized by an explicit identification of the index sets, and an isometry \(W:\mathbb {C}^r\to \mathbb {C}^D\) with \(W^\dagger W=\mathbb {1}_r\) and \(WW^\dagger =Q\), such that a matrix \(Y\in M_{D}(\mathbb {C})\) satisfies \(QYQ=Y\) and \(QE^*(Y)Q=Y\) exactly when

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

The right-hand side is the support-sector block representation in Equation (1.39) of  [ Wol12 ] ; the equivalence characterizes the corner-restricted set, not the ambient fixed-point space.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and suppose that \(T^*\) satisfies the Schwarz inequality. Let \(\rho \succeq 0\) satisfy \(T(\rho )=\rho \), and let \(Q=\operatorname{supp}(\rho )\). Then

is a \(*\)-subalgebra of the corner algebra \(QM_{D}(\mathbb {C})Q\). Wolf chooses a maximum-rank fixed point; the formal theorem strengthens the support choice to an arbitrary PSD fixed point while retaining exactly the source assumptions on \(T\). This is [ Wol12 , Corollary 6.6 and Equation (6.61) ] .

This is the chosen change of basis in the proof of [ Wol12 , Chapter 8, Proposition “Jordan condition number and detailed balance” ] . Let \(S,A\in M_n(\mathbb C)\), with \(S{\gt}0\) and \(SA^\dagger =AS\). There are a unitary \(U\) and a real diagonal matrix \(D\) such that, for

one has

This is a statement about Wolf’s displayed, chosen diagonalizing basis; it does not identify that factor with the infimum \(\kappa _T\).

This is the square-root conjugation in the proof of [ Wol12 , Chapter 8, Proposition “Jordan condition number and detailed balance” ] . Let \(S,A\in M_n(\mathbb C)\), with \(S{\gt}0\). If \(SA^\dagger =AS\), then \(S^{-1/2}AS^{1/2}\) is Hermitian.

This is the fixed-point observation in [ Wol12 , Chapter 8, Proposition “Jordan condition number and detailed balance” ] . Let \(\sigma {\gt}0\) and set \(\Sigma (X)=\sqrt\sigma \, X\sqrt\sigma \). If \(\Sigma T^*=T\Sigma \) and \(T^*(\mathbb {1})=\mathbb {1}\), then \(T(\sigma )=\sigma \).

This is the matrix-representation step in the proof of [ Wol12 , Chapter 8, Equation (8.110) ] . Let \(T:M_d(\mathbb C)\to M_d(\mathbb C)\) be Hermiticity preserving and let \(\Sigma :M_d(\mathbb C)\to M_d(\mathbb C)\) be linear. If \(\Sigma T^*=T\Sigma \), then

Let \(\{ D_i\} _{i\in I}\) and \(\{ E_j\} _{j\in J}\) be positive integers. A star-algebra isomorphism

determines an equivalence \(\sigma :I\simeq J\) such that the isomorphism carries the \(i\)-th block ideal onto the \(\sigma (i)\)-th block ideal and \(D_i=E_{\sigma (i)}\) for every \(i\in I\). For an automorphism of one product, the induced permutation may exchange equal-dimensional block ideals; it preserves the dimension along each matched pair.

For every \(A\in \mathcal A\), every \(X\in \operatorname{End}(\mathcal H_A)\), and every linear endomorphism \(T:\mathcal A\to \mathcal A\),

Let \(T:\mathcal A\to \mathcal A\) be a linear endomorphism of a finite sum of matrix algebras. For every \(A\in \mathcal A\),

Let \(\mathcal A=\bigoplus _{i\in I}M_{n_i}(\mathbb {C})\) be a finite direct sum. Suppose that \(T:\mathcal A\to \mathcal A\) is positive and preserves the total trace, that its trace adjoint satisfies the Schwarz inequality, and that \(T\) fixes a family \(\rho =(\rho _i)_i\) with every \(\rho _i\) positive definite. Then there are positive integers \(d_k,m_k\), density matrices \(\sigma _k\in M_{m_k}(\mathbb {C})\), a unitary \(U\) on \(\bigoplus _i\mathbb {C}^{n_i}\), and a reindexing of this space by \(\bigoplus _k(\mathbb {C}^{m_k}\otimes \mathbb {C}^{d_k})\) such that \(T(A)=A\) if and only if there are matrices \(X_k\in M_{d_k}(\mathbb {C})\) satisfying

Here \(\iota \) is the block-diagonal embedding. This is the finite-direct-sum fixed-point step used in [ CPGSV16 , Appendix C.4, lines 1980–1995 ] ; it does not assert the channel hypotheses for the particular sector maps in that argument. This is the full-support restriction: the support reduction and complementary zero summand of the general theorem remain open.

In the endomorphic case \(\mathcal B=\mathcal A\), write \(\mathcal H=\mathcal H_A=\mathcal H_B\) and \(\iota =\iota _A=\iota _B\). If \(T\) is positive and satisfies the Schwarz inequality on \(\mathcal A\), then \(\widehat T\) is positive and satisfies the Schwarz inequality on \(\operatorname{End}(\mathcal H)\). If \(T\) preserves the total trace \(\sum _k\operatorname{tr}(A_k)\), then \(\widehat T\) preserves the ordinary trace. Moreover, \((\widehat T)^*=\widehat{T^*}\). If \(T\) is positive and its direct-sum trace adjoint satisfies the Schwarz inequality, then the trace adjoint of \(\widehat T\) satisfies the Schwarz inequality on \(\operatorname{End}(\mathcal H)\). Finally,

Equivalently, for every \(X\in \operatorname{End}(\mathcal H)\),

Let \(\{ D_i\} _{i\in I}\) and \(\{ E_j\} _{j\in J}\) be positive integers, and set

If \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal A\) are mutually inverse completely positive trace-preserving maps, let \(\Phi :\mathcal A\simeq \mathcal B\) be the star-algebra isomorphism obtained from their trace adjoints. Write \(I_i\) and \(J_j\) for the block ideals of \(\mathcal A\) and \(\mathcal B\), respectively. Then there is an equivalence \(\sigma :I\simeq J\) such that, for every \(i\in I\),

This is the simple-summand matching conclusion used in [ CPGSV16 , Appendix C.4, line 1997 ] ; the unitary action within the paired summands is a separate conclusion.

Let \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal A\) be mutually inverse completely positive trace-preserving maps between finite products of nonzero full matrix algebras. There are an equivalence \(\sigma :I\simeq J\), equalities \(D_i=E_{\sigma (i)}\), and unitaries \(U_i\in M_{E_{\sigma (i)}}(\mathbb {C})\) such that

with the nonzero entry in (67) in position \(\sigma (i)\) and the entries on the left and right of (68) in positions \(\sigma (i)\) and \(i\), respectively. These identities hold for every \(i\in I\), \(X\in M_{D_i}(\mathbb {C})\), and \(Y\in M_{E_{\sigma (i)}}(\mathbb {C})\).

Let \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal A\) be mutually inverse completely positive trace-preserving maps between finite direct sums of full matrix algebras. Their trace adjoints are mutually inverse unital completely positive maps. Moreover, for all \(X,Y\in \mathcal B\),

and likewise for \(S^*\). Thus \(T^*\) and \(S^*\) determine mutually inverse star-algebra equivalences. This theorem does not yet identify the simple summands or their dimensions.

If \(T:\mathcal A\to \mathcal A\) is an endomorphism, then its full-matrix extension as a map between two finite sums agrees with its canonical endomorphic extension.

Retain an equivalence \(\sigma :I\simeq J\) which matches the block ideals of a star-algebra isomorphism \(\Phi :\prod _{i\in I}M_{D_i}(\mathbb {C})\simeq \prod _{j\in J}M_{E_j}(\mathbb {C})\), together with equalities \(D_i=E_{\sigma (i)}\). Then there are unitaries \(U_i\in M_{E_{\sigma (i)}}(\mathbb {C})\) such that, for every \(X\in M_{D_i}(\mathbb {C})\),

where \(\iota _i\) is the reindexing induced by the retained dimension equality. In particular, this conclusion applies to the trace-adjoint star-algebra isomorphism of any mutually inverse completely positive trace-preserving pair. This is the blockwise-unitary statement in [ CPGSV16 , Appendix C.4, line 1997 ] ; it does not assert the later MPDO multiplicity or coefficient relations.

A density matrix is convex-extreme if and only if it is a rank-one orthogonal projection. In Wolf’s terminology, these and only these are the pure density states of a full matrix algebra.

A density state on \(\bigoplus _{k\in I}M_{d_k}(\mathbb {C})\) is convex-extreme if and only if exactly one block is nonzero and that block is a rank-one orthogonal projection. These are the relative pure states of the direct sum.

The adjoint-fixed-point \(*\)-subalgebra of a trace-preserving Kraus map on \(M_{D}(\mathbb {C})\) is a semisimple ring.

There exist \(n\in \mathbb {N}\) and dimensions \(d_1,\ldots ,d_n\geq 1\) such that the adjoint-fixed-point \(*\)-subalgebra is \(\mathbb {C}\)-algebra isomorphic to \(\prod _{k=1}^{n}M_{d_k}(\mathbb {C})\).

Let \(E\) be a unital Kraus map on \(M_{D}(\mathbb {C})\) whose adjoint map \(E^*\) has a positive definite fixed point \(\rho {\gt}0\). There are \(n\in \mathbb {N}\), positive dimensions \(d_k\) and multiplicities \(m_k\) with \(\sum _kd_km_k=D\), and a unitary \(U\in M_{D}(\mathbb {C})\) such that a matrix \(X\in M_{D}(\mathbb {C})\) satisfies \(E(X)=X\) exactly when

for some matrices \(B_k\in M_{d_k}(\mathbb {C})\). This is Theorem 10.6.15 with the roles of the map and its adjoint exchanged.

Let \(E\) be a trace-preserving Kraus map whose Schrödinger-picture map has a positive definite fixed point \(\rho {\gt}0\). Then every adjoint fixed point belongs to the multiplicative domain of the adjoint Kraus family.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and unital, and suppose that, for every \(A\in M_{D}(\mathbb {C})\), it satisfies the Schwarz inequality

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

In the terminology introduced immediately before [ Wol12 , Example 5.3 ] , a Schwarz map is precisely a positive unital map satisfying (35). Thus the three assumptions reproduce the source’s Schwarz-map convention. In [ Wol12 , Theorem 6.12 ] , the trace adjoint is initially assumed to have an arbitrary full-rank fixed point. The proposition on positive fixed points then produces a positive definite fixed point \(\rho \), to which the theorem above applies. The resulting explicit block form of the algebra is given in [ Wol12 , Equation (1.39) ] .

Let \(E\) be a unital Kraus map on \(M_{D}(\mathbb {C})\) whose adjoint map \(E^*\) has a positive definite fixed point \(\rho {\gt}0\). There exist \(n\in \mathbb {N}\) and dimensions \(d_1,\ldots ,d_n\geq 1\) such that the fixed-point \(*\)-subalgebra is \(\mathbb {C}\)-algebra isomorphic to \(\operatorname{Fix}(E)\cong \prod _{k=1}^{n}M_{d_k}(\mathbb {C})\).

Let \(W:\mathbb {C}^n\to \mathbb {C}^D\) satisfy \(W^\dagger W=\mathbb {1}_n\). There are an identification \(\mathbb {C}^D\simeq \mathbb {C}^{D-n}\oplus \mathbb {C}^n\) and a unitary \(U\) on \(\mathbb {C}^D\) such that, for every \(A\in M_{n}(\mathbb {C})\),

This is [ Wol12 , Chapter 8, Equation (8.104) ] . Let \(D\ge 1\), let \(J_{D}(\lambda )=\lambda I+N\) be the Jordan block of Lemma 10.1, and write \(\| \cdot \| _{\infty }\) for the largest singular value. Then, for every \(n\in \mathbb N\) and every \(k_{0}\le \min \{ n,D-1\} \),

Let \(E(X) = \sum _i K_i X K_i^\dagger \) be a unital Kraus map, so that \(\sum _i K_i K_i^\dagger = \mathbb {1}\). Then, for every \(X \in M_{D}(\mathbb {C})\),

This is [ Wol12 , Equation (5.2) ] .

Under the hypotheses above, the Kraus commutant is the largest \(*\)-subalgebra contained in the adjoint fixed-point set.

Let \(K\) be a finite matrix family with Kraus map \(E\), and let \(Q\) be an orthogonal projection. If \((\mathbb {1}-Q)K_iQ=0\) for every \(i\), then

for every \(X\in M_{D}(\mathbb {C})\).

Let \(K\) be trace preserving. If \(S_N(K)=M_{D}(\mathbb {C})\) and \(m\geq N\), then \(S_m(K)=M_{D}(\mathbb {C})\). Consequently, eventual fullness is equivalent to the existence of a positive \(N\) such that \(S_N(K)=M_{D}(\mathbb {C})\).

Let \(E\) be a complex module and let \(T,S:M_{d}(\mathbb {C})\to E\) be complex-linear maps. If

for every ray \(p\) in \(\mathbb {C}^d\), then \(T=S\).

Let \(E\) be a unital Kraus map whose adjoint map has a positive definite fixed point, and suppose \(E^2=E\). Then \(E(X)\) lies in the multiplicative domain of \(E\) for every \(X\in M_{D}(\mathbb {C})\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and suppose that \(T^*\) satisfies the Schwarz inequality. Set \(\rho _0=T_\infty (\mathbb {1})\), and let \(V:\mathcal H\to \mathbb {C}^D\) be an isometry onto the support of \(\rho _0\). Then \(\widetilde T(Y)=V^*T(VYV^*)V\) has a positive definite fixed point. Moreover, \(T(B)=B\) if and only if there is a fixed point \(Y\) of \(\widetilde T\) such that \(B=VYV^*\). There are positive integers \(d_k,m_k\), a unitary \(U\) on \(\mathcal H\), and positive definite density matrices \(\sigma _k\in M_{m_k}(\mathbb {C})\) such that

This is the maximal-support application of the full-support step in the proof of  [ Wol12 , Theorem 6.14 ] . This intermediate theorem does not itself transport the displayed decomposition back to the original space and adjoin the complementary zero summand; that final step is Theorem 10.9.6.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, and let \(T_\infty \) be its mean-ergodic projection. Then \(\rho _0=T_\infty (\mathbb {1})\) is positive semidefinite and fixed by \(T\). With \(Q_0\) the support projection of \(\rho _0\), every fixed point \(X\) of \(T\) satisfies \(Q_0XQ_0=X\). This is the maximal-support property of fixed points in Section 6.4 of  [ Wol12 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and have bounded orbits, and let \(T_\infty \) be its mean-ergodic projection. Then \(\rho _0=T_\infty (\mathbb {1})\) is positive semidefinite and fixed by \(T\). If \(Q_0\) is the support projection of \(\rho _0\), then every fixed point \(X\) of \(T\) satisfies \(Q_0XQ_0=X\). This is the bounded-orbit form of the maximal-support argument underlying [ Wol12 , Proposition 6.9 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and let \(\rho \succeq 0\) be a fixed point of \(T\) whose rank bounds the rank of every fixed-point density matrix: \(\operatorname{rank}\sigma \leq \operatorname{rank}\rho \) for every \(\sigma \succeq 0\) with \(\operatorname{tr}(\sigma )=1\) and \(T(\sigma )=\sigma \). Then the support projection \(Q\) of \(\rho \) satisfies \(QXQ=X\) for every fixed point \(X\) of \(T\).

Let \(T(X)=\sum _iK_iXK_i^\dagger \) be trace-preserving. There is a positive semidefinite fixed point \(\rho _0\) of \(T\) such that every fixed point \(X\) of \(T\) arises as \(X=\sqrt{\rho _0}\, Y\sqrt{\rho _0}\) for a corner-supported \(Y\) with \(\sqrt{\rho _0}\, Y\sqrt{\rho _0}\) fixed by \(T\).

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

Moreover,

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

Under the hypotheses of Theorem 10.8.15, the mean-ergodic projection of \(T\) satisfies

Consequently,

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

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive trace-preserving linear map whose trace adjoint \(T^*\) satisfies the Schwarz inequality. If \(T\) has a positive definite fixed point \(\rho {\gt}0\), then the trace adjoint of the mean-ergodic projection of \(T\) is a conditional expectation onto the fixed-point star-subalgebra of \(T^*\). This is the conditional-expectation step used in the proof of Wolf Theorem 6.14.

If \(N\in \mathbb {N}\) is divisible by every per-cycle period \(m(c)\), then \(T^N\) preserves every corner \(P_{c,k}M_{D}(\mathbb {C})P_{c,k}\) of the decomposition.

For every cycle \(c\in \iota \) and index \(k\in \{ 0,\ldots ,m(c){-}1\} \), the iterate \(T^{m(c)}\) preserves the corner \(P_{c,k}M_{D}(\mathbb {C})P_{c,k}\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, with \(D{\gt}0\), and let

be the span of its peripheral eigenvectors. Then

1. \(T_\phi (M_{D}(\mathbb {C}))=\mathcal X_T\),

2. there are positive semidefinite matrices \(\rho _i\) with \(\mathcal X_T=\operatorname{span}\{ \rho _i\} \),

3. \(T(\mathcal X_T)=\mathcal X_T\).

This is [ Wol12 , Proposition 6.12 (Asymptotic image) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, with \(D{\gt}0\). There is a strictly increasing sequence of positive integers \(n_i\) such that \(T^{n_i}\to T_\phi \) pointwise and in operator norm. This is [ Wol12 , Proposition 6.3(i) ] .

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

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

Let \(I\) and \(J\) be nonempty finite index sets, with \(A\in \mathbb {C}^{I\times I}\) and \(B\in \mathbb {C}^{J\times J}\). If \(A\otimes B{\gt}0\) and \(\operatorname{tr}(A)=1\), then \(A{\gt}0\).

Let \(I\) and \(J\) be finite index sets, and let \(X\in \mathbb {C}^{(I\times J)\times (I\times J)}\) be positive definite. If \(I\) is nonempty, then the matrix with entries \((\operatorname{tr}_I X)_{j,j'}=\sum _{i\in I}X_{(i,j),(i,j')}\) is positive definite.

Let \(I\) and \(J\) be finite index sets, and let \(X\in \mathbb {C}^{(I\times J)\times (I\times J)}\) be positive definite. If \(J\) is nonempty, then the matrix with entries \((\operatorname{tr}_J X)_{i,i'}=\sum _{j\in J}X_{(i,j),(i',j)}\) is positive definite.

Let \(E\) be a positive trace-preserving linear map and let \(X=E(X)\) be a fixed point. Set \(H_1=X+X^*\) and \(H_2=i(X-X^*)\), and write \(H_j=P_{j,+}-P_{j,-}\) for the canonical positive and negative parts. Then all four positive semidefinite matrices \(P_{j,\pm }\) are fixed by \(E\). This is [ Wol12 , Proposition 6.8 ] .

If \(T:M_{n}(\mathbb {C})\to M_{m}(\mathbb {C})\) is positive, then \(T(A^\dagger )=T(A)^\dagger \) for all \(A\in M_{n}(\mathbb {C})\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and have bounded orbits. Then its mean-ergodic projection \(P_T\) is positive.

Let \(S\subseteq M_{D}(\mathbb {C})\) be a unital \(*\)-subalgebra, and let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive complex-linear map whose image lies in \(S\) and whose restriction to \(S\) is the identity. There are positive integers \(d_k,m_k\), a unitary \(U\), and density matrices \(\rho _k\in M_{m_k}(\mathbb {C})\) such that

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

Equivalently, cyclicity of the partial trace permits the factor \(\rho _k\otimes \mathbb {1}_{d_k}\) to be placed on the right of \((U^*AU)_{kk}\). This is [ Wol12 , Proposition 1.5 and Equation (1.40) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace nonincreasing. Then the forward orbit \(\{ T^n(X):n\geq 0\} \) is bounded for every \(X\in M_{D}(\mathbb {C})\). This is the trace-nonincreasing form of [ Wol12 , Proposition 6.3 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. Then \(T\) has bounded orbits, its Cesàro averages converge pointwise to the mean-ergodic projection \(P_T\), and \(P_T\) is positive and trace-preserving. Its range is precisely the fixed-point space of \(T\):

If \(T(\mathbb {1})=\mathbb {1}\), then \(P_T(\mathbb {1})=\mathbb {1}\). This is the Cesàro projection \(T_\infty \) of [ Wol12 , Proposition 6.3 and Equation (6.14) ] .

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

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

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and unital. Then the forward orbit \(\{ T^n(X):n\geq 0\} \) is bounded for every \(X\in M_{D}(\mathbb {C})\). This is the unital form of [ Wol12 , Proposition 6.2 ] , proof.

Let \(D\geq 1\) and let \(P\in M_{D}(\mathbb {C})\) be positive semidefinite. Then

If \(T(\sigma )=\sigma \), then \(E_\sigma \circ T^*=E_\sigma \).

For every \(X\), one has \(E_\sigma (E_\sigma (X))=E_\sigma (X)\).

Let \(K\) be a trace-preserving Kraus family, let \(\rho {\gt}0\) satisfy \(T(\rho )=\rho \) for the associated channel \(T(\rho )=\sum _iK_i\rho K_i^\dagger \), and assume every adjoint fixed point of \(T^*\) is a scalar multiple of \(\mathbb {1}\). Then \(E_\rho \) is a conditional expectation onto the adjoint fixed-point \(*\)-subalgebra.

For any positive semidefinite \(\sigma \), the scalar conditional expectation \(E_\sigma \) is a completely positive map.

If \(\operatorname{tr}(\sigma )\neq 0\), then \(\operatorname{tr}(\sigma E_\sigma (X))=\operatorname{tr}(\sigma X)\) for every \(X\).

One has \(E_\sigma (\mathbb {1})=\mathbb {1}\).

This is the Kronecker identity used in the specialization of [ Wol12 , Chapter 8, Proposition “Jordan condition number and detailed balance” ] . If \(\sigma {\gt}0\) and \(\Sigma (X)=\sqrt\sigma \, X\sqrt\sigma \), then

Every \(\mathbb {C}\)-algebra automorphism \(f\) of \(M_{D}(\mathbb {C})\) is inner: there is \(X \in \mathrm{GL}_D(\mathbb {C})\) with \(f(M) = X M X^{-1}\) for all \(M\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\) and let \(\mathcal{D}\) be a finite, nonempty family of pairwise orthogonal subspaces of \(\mathbb {C}^D\), irreducible under \(S\) and pairwise of the same type. Write \(m\) for the number of pieces in \(\mathcal{D}\); all pieces share one dimension \(d\). Then the component \(C=\bigvee _{W\in \mathcal{D}}W\) has an orthonormal basis \((f_{i,j})_{0\leq i{\lt}m,\, 0\leq j{\lt}d}\) such that for every \(A\in S\) there is a matrix \(B\in M_{d}(\mathbb {C})\) satisfying

for all \(i{\lt}m\) and \(j{\lt}d\). In the adapted basis, \(A\) acts on \(C\) by the matrix \(B\) on the irreducible index, identically across the multiplicity index; after reordering the indices this is the block \(M_{d}(\mathbb {C})\otimes \mathbb {1}_m\) of  [ Wol12 , Theorem 6.14 ] . The vectors \((f_{i,j})_{0\leq j{\lt}d}\) of the \(i\)-th copy span one of the pieces of \(\mathcal{D}\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). There are positive integers \(d_k,m_k\) and an orthonormal basis \((f_{k,i,j})_{k{\lt}K,\, i{\lt}m_k,\, j{\lt}d_k}\) of \(\mathbb {C}^D\) such that every \(A\in S\) acts by

while every \(T\in M_{D}(\mathbb {C})\) commuting with all members of \(S\) acts by

Thus the same orthonormal identification realizes \(S\) on the second tensor factor and its commutant on the complementary first tensor factor. This is the finite-dimensional \(C^*\)-algebra step used in the proof of  [ Bei12 , Lemma 2.1 ] .

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). There are \(K\in \mathbb {N}\), positive dimensions \(d_0,\ldots ,d_{K-1}\) and multiplicities \(m_0,\ldots ,m_{K-1}\), and an orthonormal basis \((f_{k,i,j})_{k{\lt}K,\, i{\lt}m_k,\, j{\lt}d_k}\) of \(\mathbb {C}^D\) such that for every \(A\in S\) there are matrices \(B_k\in M_{d_k}(\mathbb {C})\) satisfying

for all \(k{\lt}K\), \(i{\lt}m_k\), and \(j{\lt}d_k\). In this basis, \(A\) acts on the \(k\)-th component as \(\mathbb {1}_{m_k}\otimes B_k\). Each row \((f_{k,i,j})_{0\leq j{\lt}d_k}\) spans a subspace irreducible under \(S\), and rows drawn from distinct components span subspaces that are never of the same type. Only the containment direction is asserted: every member of \(S\) takes this form. Equality with the block algebra, including the reverse inclusion, is Theorem 10.6.9.

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). Then \(\mathbb {C}^D\) is the supremum of a finite family of pairwise orthogonal isotypic components, and each component carries an orthonormal basis \((f_{i,j})\), indexed by a multiplicity index \(i{\lt}m\) and an irreducible index \(j{\lt}d\), such that every \(A\in S\) acts by

for a matrix \(B\in M_{d}(\mathbb {C})\) depending only on \(A\) and the component. In the adapted bases, the action of \(S\) on each component is by matrix-times-identity blocks, the form \(M_{d_k}(\mathbb {C})\otimes \mathbb {1}_{m_k}\) of  [ Wol12 , Theorem 6.14 ] .

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). Then \(\mathbb {C}^D\) is the sum of a finite family of pairwise orthogonal subspaces, each irreducible under \(S\); that is, \(\mathbb {C}^D\) is an orthogonal direct sum of irreducible invariant subspaces.

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). Then \(\mathbb {C}^D\) is the supremum of a finite family \(\mathcal{C}\) of pairwise orthogonal subspaces, the isotypic components, and each component \(C\in \mathcal{C}\) is the supremum \(C=\bigvee _{W\in \mathcal{D}_C}W\) of a finite, nonempty class \(\mathcal{D}_C\) of pairwise orthogonal subspaces irreducible under \(S\) that are pairwise of the same type. Moreover, for distinct components \(C\neq C'\), no subspace irreducible under \(S\) contained in \(C\) is of the same type as any subspace irreducible under \(S\) contained in \(C'\).

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

for matrices \(B_k\in M_{d_k}(\mathbb {C})\) depending on \(A\). Up to reordering the two tensor factors of each block, this is the containment direction of the unital case of the block representation of [ Wol12 , Theorem 6.14 ] : \(S\) is carried by \(U\) into the block algebra \(\bigoplus _k\mathbb {1}_{m_k}\otimes M_{d_k}(\mathbb {C})\). Equality, including the reverse inclusion, is Theorem 10.6.9.

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

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

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

There exist \(n\in \mathbb {N}\) and dimensions \(d_1,\ldots ,d_n\geq 1\) such that every \(*\)-subalgebra of \(M_{D}(\mathbb {C})\) is \(\mathbb {C}\)-algebra isomorphic to \(\prod _{k=1}^nM_{d_k}(\mathbb {C})\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, set \(\rho _0=T_\infty (\mathbb {1})\), and let \(Q_0\) be the support projection of \(\rho _0\). There are a finite-dimensional Hilbert space \(\mathcal H\) and an isometry \(V:\mathcal H\to \mathbb {C}^D\) with \(V^*V=\mathbb {1}_{\mathcal H}\) and \(VV^*=Q_0\) such that

is positive and trace preserving and has a positive definite fixed point. Moreover,

These are Equations (6.52) and (6.51), respectively, of  [ Wol12 ] . Thus the complementary fixed-point summand vanishes.

For an irreducible channel \(E\), the stationary support equals the identity: \(\operatorname{supp}(\rho _\infty )=\mathbb {1}\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive, and let \(\rho \succeq 0\) satisfy \(T(\rho )=\rho \). If \(Q\) is the support projection of \(\rho \), then, for every \(X\in M_{D}(\mathbb {C})\),

Let \(E\) be a quantum channel, let \(\rho \geq 0\) satisfy \(E(\rho )=\rho \), and let \(P\) be the support projection of \(\rho \). Then \(PE(PXP)P=E(PXP)\) for every \(X\in M_{D}(\mathbb {C})\).

Let \(P\in M_{D}(\mathbb {C})\) be an orthogonal projection and let \(A_1,\ldots ,A_d\in M_{D}(\mathbb {C})\) satisfy

Then there are \(n=\operatorname{tr}P\), matrices \(C_i\in M_{n}(\mathbb {C})\), an isometry \(V:\mathbb {C}^n\to \mathbb {C}^D\), and a linear isomorphism \(\varphi :M_{n}(\mathbb {C})\xrightarrow {\sim }PM_{D}(\mathbb {C})P\) such that

If \(T_A^*(X)=\sum _iA_i^\dagger XA_i\) and \(T_C^*(X)=\sum _iC_i^\dagger XC_i\), then

For \(M \in M_{D}(\mathbb {C})\), \(\operatorname{tr}(M N) = 0\) for all \(N \in M_{D}(\mathbb {C})\) if and only if \(M = 0\).

For every linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\), every \(\rho \in M_{D'}(\mathbb {C})\), and every \(X \in M_{D}(\mathbb {C})\), one has

If \(E : M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is positive, then its trace-pairing adjoint \(E^*\) is positive.

On any finite full matrix algebra, a linear endomorphism \(S\) is trace-preserving if and only if its trace-pairing adjoint fixes the identity:

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be trace-preserving and have bounded orbits. Then its mean-ergodic projection \(P_T\) is trace-preserving.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) have bounded orbits. If \(T(\mathbb {1})=\mathbb {1}\), then \(P_T(\mathbb {1})=\mathbb {1}\).

For every \(A\in M_D(\mathbb C)\), including \(D=0\), there are a unitary matrix \(U\) and an upper-triangular matrix \(R\) such that

Moreover, a complex number \(z\) is an eigenvalue of \(A\) if and only if \(R_{ii}=z\) for some \(i\).

Every star-algebra automorphism \(\Phi \) of a nonzero full complex matrix algebra is implemented by a unitary: there is a unitary matrix \(U\) such that \(\Phi (X)=UXU^*\) for every matrix \(X\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and let \(\rho \succeq 0\) be a fixed point of \(T\) whose rank bounds the rank of every fixed-point density matrix. Then every fixed point \(X\) of \(T\) arises as \(X=\sqrt{\rho }\, Y\sqrt{\rho }\) for a corner-supported \(Y\) with \(\sqrt{\rho }\, Y\sqrt{\rho }\) fixed by \(T\).

In the setting of Theorem 10.8.20, every fixed point \(X\) of \(T\) with \(QXQ=X\) arises as \(X=\sqrt{\rho }\, Y\sqrt{\rho }\) for a corner-supported \(Y\) with \(\sqrt{\rho }\, Y\sqrt{\rho }\) fixed by \(T\).

Let \(T(X)=\sum _iK_iXK_i^\dagger \) be trace-preserving, let \(\rho \succeq 0\) satisfy \(T(\rho )=\rho \), and let \(Q\) be the support projection of \(\rho \). The corner elements \(Y\in QM_{D}(\mathbb {C})Q\) such that \(\sqrt{\rho }\, Y\sqrt{\rho }\) is a fixed point of \(T\) form a \(*\)-subalgebra of the corner algebra \(QM_{D}(\mathbb {C})Q\).

Let \(T(X)=\sum _iK_iXK_i^\dagger \) be trace-preserving, and let \(\rho {\gt}0\) satisfy \(T(\rho )=\rho \). If \(F_T:=\{ X\in M_{D}(\mathbb {C})\mid T(X)=X\} \), then \(\rho ^{-1/2}F_T\rho ^{-1/2}\) is a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and suppose that \(T^*\) satisfies the Schwarz inequality. There are positive integers \(d_k,m_k\), positive definite density matrices \(\sigma _k\in M_{m_k}(\mathbb {C})\), a non-negative integer \(d_0\), and a unitary \(U\) associated with a decomposition

This is [ Wol12 , Theorem 6.14 and Equation (6.63) ] . The two tensor factors are interchanged from the displayed convention in the reference; the interchange is a unitary change of basis on each summand.

This is Wolf, Chapter 6, Theorem 6.16, Equations (6.66)–(6.68), local source lines 1597–1664, under the separately retained trace-adjoint-Schwarz contract required by the printed proof’s invocation of Theorem 6.14. Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, with \(D{\gt}0\). Suppose separately that \(T\) and its trace-pairing adjoint \(T^*\) satisfy the Schwarz inequality. There are zero-extended density-block coordinates \(E\), positive integers \(d_k,m_k\), density matrices \(\sigma _k\in M_{m_k}(\mathbb {C})\), a permutation \(\pi \) of the blocks, and unitaries \(V_k\in M_{d_k}(\mathbb {C})\) such that

If \(q_k:\{ 0,\ldots ,d_{\pi (k)}{-}1\} \simeq \{ 0,\ldots ,d_k{-}1\} \) is the canonical reindexing, the coordinate action \(A\) and its ambient realization are

Here \(\pi \) is explicitly the output-to-input permutation: output block \(k\) reads input block \(\pi (k)\). In the formal coordinates the tensor order is \(\sigma _k\otimes X_k\); Wolf’s displayed \(X_k\otimes \rho _k\) differs by the canonical tensor-factor swap.

The compiled statement retains the equivalence \(e_0\) and the literal zero extension \(\operatorname {fromBlocks}(0,0,0,-)\) while also proving \(n=D\). This records Equations (6.66)–(6.67) without a dependent rewrite that erases Wolf’s zero-summand form. The inverse used to classify each matched block is only the inverse on the peripheral image; no inverse of the ambient map \(T\) is asserted.

Let \(d\ge 1\), and let \(T:M_d(\mathbb C)\to M_d(\mathbb C)\) be positive and trace preserving. Write \(T_\phi \) for the peripheral spectral projection and

with \(\mu =0\) when the displayed set is empty. After the canonical identification of the transfer-matrix coordinates with \(\mathbb C^{d^2}\), there are a unitary \(U\), a diagonal matrix \(\Lambda \), and a strictly upper-triangular matrix \(N\) such that

For every \(n\in \mathbb N\) satisfying \(d^2-1\le n\),

If \(2(d^2-1)\le n\), the factor \((d^2-1)n^{d^2-1}\) may be replaced by \((d^2-1)\binom {n}{d^2-1}\).

The statement includes \(\mu =0\) without division. When \(d=1\) and \(n=0\), both estimates reduce to \(0\le 1\); this boundary is included even though the auxiliary positive-power identity does not hold there.

Let \(d{\gt}0\). The map

is continuous. Its unit-vector domain is path connected, and hence the set of rank-one pure-state projections in \(M_{d}(\mathbb {C})\) is connected.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a trace-preserving, positive, linear map for which \(T^*\) satisfies the Schwarz inequality. For every density matrix \(\rho \) which is a maximum-rank fixed point of \(T\), the set

is a \(*\)-algebra, with the inverse taken on \(\operatorname{supp}(\rho )\). This is [ Wol12 , Corollary 6.7 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, with \(D{\gt}0\), and suppose that \(T^*\) is Schwarz. Choose the zero-extended density-block coordinates of the recurrent projection from Theorem 10.9.6, in the formal tensor-factor order \(\sigma _k\otimes X_k\), and let \(E\) and \(R\) denote their embedding and compression maps. If \(S\) is the recurrent inverse from Theorem 10.11.3, set

Then \(\bar T\) and \(\bar S\) are mutually inverse positive maps on \(\bigoplus _{k{\lt}K}M_{d_k}(\mathbb {C})\) and preserve the total trace. There is an equivalence \(\tau :\{ 0,\ldots ,K{-}1\} \simeq \{ 0,\ldots ,K{-}1\} \) such that, with Wolf’s output-to-input permutation \(\pi =\tau ^{-1}\),

Every matched block map and its inverse is positive, preserves the ordinary matrix trace, and the two compositions are identities. The conclusion compares only the full-matrix dimensions \(d_k\).

This is exactly the relative-pure-state, connectedness, and block permutation step at source lines 1641–1659 of Wolf’s proof of Theorem 6.16. It does not compare the multiplicities \(m_k\), transport the ordinary Schwarz inequality to the matched block maps, assemble the exclusion of the transpose alternative, or prove Equation (6.68); those are assembled in Theorem 10.11.7 below.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, with \(D{\gt}0\). Suppose separately that \(T\) and its trace-pairing adjoint \(T^*\) satisfy the Schwarz inequality. In the density-block coordinates and notation of Theorem 10.11.4, the zero summand has dimension zero and

If \(\tau \) is the source-to-target block permutation, then

For every \(X\in M_{d_i}(\mathbb {C})\), the raw matched block map satisfies

Transporting along \(d_i=d_{\tau (i)}\) gives an endomorphism of \(M_{d_i}(\mathbb {C})\) with the ordinary Schwarz property. Together with the matched block conclusions of Theorem 10.11.4, this endomorphism is positive, trace preserving, bijective, and has a positive inverse, exactly the hypotheses needed for the subsequent exclusion of matrix transposition.

Two printed dimension defects are kept explicit. At source line 1499, \(\rho _k\in M_{m_k}(\mathbb {C})\) and the full algebra acts on the \(d_k\) factor, so the compatible identity is \(\mathbb {1}_{d_k}\otimes \rho _k\), not the printed \(\mathbb {1}_{m_k}\otimes \rho _k\); in the formal tensor-factor order this is \(\sigma _k\otimes \mathbb {1}_{d_k}\). At source lines 1614–1616 the statement asks \(\pi \) to preserve the dimension \(d_km_k\) of the whole space \(\mathcal H_k\), whereas the pure-face argument at lines 1653–1656 proves only equality of \(d_k\). The equality of \(m_k\) in (95) supplies the missing comparison.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, with \(D{\gt}0\), and write \(I=T_\phi \) for its recurrent/peripheral projection. There are a strictly increasing sequence of positive integers \((n_i)\) and a positive trace-preserving map \(S\) such that

The map \(I\) is positive and trace preserving. Moreover, \(S\) and \(I\) have the same range. Hence \(S\) and \(T\) are mutual inverses on \(\operatorname {ran}I=\mathcal{X}_T\); this is not a global inverse claim. If \(T\) is Schwarz, then \(I\) is Schwarz. If, separately, \(T^*\) is Schwarz, then \(I^*\) is Schwarz.

This is the recurrent-projection and positive-inverse step at lines 1629–1640 of the proof of  [ Wol12 , Theorem 6.16 ] , under the corrected two-orientation contract used by that printed proof.

Let \(d{\gt}0\), and let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a positive, trace-preserving bijection whose inverse is positive. If \(T\) satisfies the Schwarz inequality in (16), then there is a unitary \(U\) such that

This is the final classification step at lines 1660–1663 of Wolf’s proof of Theorem 6.16.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, with \(D{\gt}0\), and suppose that \(T\) is Schwarz. Then \(T(\mathbb {1})=\mathbb {1}\). For \(m,d{\gt}0\), positivity of \(\mathbb {1}_m\otimes B\) implies \(B\succeq 0\). If \(\operatorname{tr}(\sigma )=1\) and \(\sigma \otimes X=\mathbb {1}_{md}\), then \(\sigma =m^{-1}\mathbb {1}_m\) and \(X=m\mathbb {1}_d\).

These auxiliary facts are used below in the matched-multiplicity comparison and the transport of the ambient Schwarz defect to an individual weighted full-matrix block map at source lines 1660–1663. This is not a modified-product argument.

Let \(D{\gt}0\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive, trace preserving, and bijective. Then \(T^{-1}\) is positive if and only if there is a unitary \(U\) for which

This is the corollary “Positive invertible maps” following [ Wol12 , Theorem 6.1 ] .

Let \(\Lambda ,N\in M_D(\mathbb C)\), where \(\Lambda \) is diagonal and \(N\) is strictly upper triangular. Suppose \(\lVert \Lambda \rVert _\infty =\mu \le 1\). If \(D-1\le n\), then

If \(2(D-1)\le n\), the factor \((D-1)n^{D-1}\) may be replaced by \((D-1)\binom {n}{D-1}\).

Let \(D\ge 1\), let \(J_D(\lambda )\) be a Jordan block, and suppose that \(0{\lt}|\lambda |\le 1\) and \(D-1\le n\). Then

At \(D=1\) the inverse-power factor in the first line is interpreted as the zeroth power and hence equals one. This is the fixed-block estimate used in the proof of Wolf’s Equations (8.106)–(8.107), before selecting the largest subperipheral block and comparing the full direct sum.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, let \(\sigma \) be a density operator with \(T(\sigma )=\sigma \), and let \(Q\) be the support projection of \(\sigma \). Then

for every density operator \(\rho \in M_{D}(\mathbb {C})\). This is [ Wol12 , Proposition 6.10 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace preserving, and let \(Q\in M_{D}(\mathbb {C})\) be a Hermitian projection. Then the following are equivalent:

1. every density operator \(\rho \preceq Q\) satisfies \(T(\rho )\preceq Q\);

2. \(T^*(Q)\succeq Q\).

This is [ Wol12 , Proposition 6.11 ] .

Under the hypotheses of Theorem 10.6, suppose in addition that \(\nu (\Lambda )\le 1\). If \(D-1\le n\), then

If \(2(D-1)\le n\), the factor \((D-1)n^{D-1}\) may be replaced by \((D-1)\binom {n}{D-1}\).

Let \(D\ge 1\), let \(\Lambda ,N\in M_D(R)\) for a ring \(R\), assume that \(\Lambda \) is diagonal and that \(N\) is strictly upper triangular, and let \(\nu \) be an arbitrary submultiplicative ring seminorm. Then, for every \(n\in \mathbb N\),

In particular, this applies to every submultiplicative norm in Wolf’s upper-triangular power lemma [ Wol12 , Chapter 8, Equation (8.105) ] .