Dependencies

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 \(j\).

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 \(T_\varphi \).

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 \(E\colon M_D(\mathbb C)\to M_D(\mathbb C)\) 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 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 ] .

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 ] .

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 \(a\in U_T(X)\).

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.

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

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 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 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 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.

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.

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 \(V\) 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).

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) ] .

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 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 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.

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 \(X\).

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

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

The fixed-length vector span at length \(n\) is

This is \(S_n(K)|\varphi \rangle \) in the notation of  [ SPGWC10 ] .

Let \(T\) be an endomorphism of a complex vector space and let \(s\) be a finite set of complex numbers. The phase-weighted Cesàro mean of \(T\) over \(s\) is

Taking for \(s\) the eigenvalues of \(T\) of modulus one gives the averages of [ Wol12 , Equation (6.15) ] .

At a threshold \(n\), the vector-spread clause says that, for every \(m\geq n\) and every nonzero \(\psi \in \mathbb {C}^D\), \(K_m|\psi \rangle =\mathbb {C}^D\). At a threshold \(q\), the word-span clause says \(K_m=M_{D}(\mathbb {C})\) for every \(m\geq q\), while the Choi clause says \(\tau _m{\gt}0\) for every \(m\geq q\). These are the direct quantified clauses in [ Wol12 , Theorem 6.8(2–4) ] .

Let \(E\) be a quantum channel and \(\rho \in \mathcal{D}_D\). Then, for every \(N\geq 0\),

Let \(u_{1},u_{2},\dots \) be a sequence in a complex normed space with \(u_{n}\to 0\). Then

Let \(E\) be a quantum channel, \(\rho \in \mathcal{D}_D\), and \(\psi :\mathbb {N}\to \mathbb {N}\) satisfy \(\psi (k)\to \infty \). If

then \(\sigma \in \mathcal{D}_D\) and \(E(\sigma )=\sigma \).

Let \(D{\gt}0\). The completely positive maps form a closed subset of the endomorphisms of \(M_{D}(\mathbb {C})\) in the operator norm. In particular, if every \(S_N:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is completely positive and \(S_N\to P\) in the operator norm, then \(P\) is completely positive.

Let \(D\geq 1\), and let \(\rho ,\sigma \in M_{D}(\mathbb {C})\) be positive definite. There is a real number \(c{\gt}0\) such that \(\sigma -c\rho \) is positive semidefinite but not positive definite.

Let \(x_1,\dots ,x_m\) be \(m\) real numbers and \(q{\gt}1\) any integer. Then there exist integers \(n,p_1,\dots ,p_m\) such that \(1\le n\le q^m\) and \(|x_k n-p_k|\le 1/q\) for all \(k\). This is [ Wol12 , Lemma 6.1 ] .

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:

For any invertible \(C \in M_{D}(\mathbb {C})\) and any linear map \(E\) on \(M_{D}(\mathbb {C})\), write \(S_C(E)(X)=C^{-1}E(CXC^\dagger )(C^\dagger )^{-1}\) for the similarity transform by \(C\). The transforms by \(C\) and \(C^{-1}\) compose to the identity:

Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\), let \(C \in M_{D}(\mathbb {C})\) be invertible, with \(\det C \neq 0\), and let \(c {\gt} 0\). Define the similarity-transformed map by

Then \(E'\) is also irreducible.

For any invertible \(C \in M_{D}(\mathbb {C})\) and any linear map \(E\) on \(M_{D}(\mathbb {C})\), the similarity transform \(X \mapsto C^{-1}E(CXC^\dagger )(C^\dagger )^{-1}\) is irreducible if and only if \(E\) is.

Let \(E\) be a finite-dimensional complex normed space and let \(S_N\) and \(P\) be continuous linear endomorphisms of \(E\). If \(S_N(x)\to P(x)\) for every \(x\in E\), then \(S_N\to P\) in the operator norm.

Let \(s\) be a finite set of complex numbers of modulus one and let \(\lambda \in s\). Then

Let \(T\) be an endomorphism of a complex normed space and let \(s\) be a finite set of complex numbers of modulus one.

1. On the span of the eigenspaces of \(T\) belonging to the elements of \(s\), the means \(C_{N}(T,s)\) converge to the identity.

2. At a vector \(X\) with \(T^{n}(X)\to 0\), the means \(C_{N}(T,s)(X)\) converge to zero.

If some Kraus operator \(K_i \neq 0\), then the Kraus map \(X \mapsto \sum _i K_i X K_i^\dagger \) is nonzero.

Let \(T\) be a continuous map from a nonempty, compact, convex set \(S\subset \mathbb {R}^n\) into itself. Then there is an \(x\in S\) such that \(T(x)=x\).

This is [ Wol12 , Theorem 6.10 ] , including the case \(n=0\).

Let \(K\) be a compact subset of a finite-dimensional real normed space \(E\). If there is a continuous retraction \(r\colon E\to K\), then every continuous map from \(K\) to itself has a fixed point.

Let \(D {\gt} 0\). Every continuous map from the compact convex set of density matrices in \(M_{D}(\mathbb {C})\) to itself has a fixed point.

Every quantum channel \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), with \(D\geq 1\), has a nonzero positive semidefinite fixed point: there exists \(\rho \geq 0\), \(\rho \neq 0\), with \(E(\rho )=\rho \).

For \(N\geq 1\), the Cesàro mean satisfies

Let \(E\) be an irreducible quantum channel on \(M_{D}(\mathbb {C})\) with \(D {\gt} 0\), and let \(\rho \) be any density matrix. Then the Cesàro means (1) converge to the unique positive definite density-matrix fixed point of \(E\). This is the Cesàro-convergence conclusion in the forward direction of [ Wol12 , Corollary 6.3 ] , specialized to quantum channels.

Let \(E\) be an irreducible quantum channel on \(M_{D}(\mathbb {C})\) with \(D {\gt} 0\). Then there exists a unique density matrix \(\sigma \) such that \(E(\sigma ) = \sigma \). The fixed point \(\sigma \) is positive definite. This is the fixed-point conclusion in the forward direction of [ Wol12 , Corollary 6.3 ] , specialized to completely positive trace-preserving maps.

If \(E\) is a quantum channel, then \(E(\mathcal{D}_D) \subseteq \mathcal{D}_D\).

For every linear map \(T\), \(r_T(0)=+\infty \) and \(\widetilde r_T(0)=-\infty \). If \(U_T(X)=\varnothing \), then \(\widetilde r_T(X)=+\infty \).

Let \(X\geq 0\) be nonzero and suppose \(T(X)=aX\) for some \(a\in \mathbb {R}\). Then

This is the pointwise equality used at local source lines 649–651.

Let \(X\geq 0\) be nonzero. Then

Let \(E\) be completely positive, let \(\rho \geq 0\) satisfy \(E(\rho )=\rho \), and let \(Q\) be the support projection of \(\rho \). Then \(Q\) is an orthogonal projection and

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

Let \(D\geq 1\). A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (in the Kraus sense) if and only if its Choi matrix \(\tau \ge 0\). This is the \(d=d'\) specialization of [ Wol12 , Proposition 2.1 ] .

Every rectangular Kraus completely positive map is positive. In particular, every square completely positive map is positive.

\(\mathcal{D}_D\) is compact in the entrywise topology on \(M_{D}(\mathbb {C})\).

\(\mathcal{D}_D\) is convex.

For \(D \ge 1\), \(\mathcal{D}_D \neq \varnothing \).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive trace-preserving linear map with \(D{\gt}0\). Then there exists a nonzero positive semidefinite matrix \(\rho \) such that \(T(\rho )=\rho \). This is the eigenvalue-\(1\) assertion of [ Wol12 , Proposition 6.1 ] .

Let \(D \ge 1\) and let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\). Suppose \(\rho , \sigma \ge 0\) are nonzero positive semidefinite matrices satisfying \(E(\rho ) = r_1\rho \) and \(E(\sigma ) = r_2\sigma \) for real scalars \(r_1, r_2 {\gt} 0\). Then \(r_1 = r_2\).

Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\), let \(A \ge 0\) be nonzero, and let \(t {\gt} 0\). Then

This is the completely positive specialization of the forward implication in  [ Wol12 , Theorem 6.2(3) ] .

Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\), let \(A \ge 0\) be nonzero, and let \(t {\gt} 0\). Then the finite exponential truncation satisfies

This is the finite-sum core of the completely positive specialization of [ Wol12 , Theorem 6.2(3) ] .

Let \(T\) be an endomorphism of a complex normed space, let \(S\subseteq \mathbb {C}\) be finite, and let \((n_k)_{k\geq 0}\) be a sequence of nonnegative integers such that \(\mu ^{n_k}\to 1\) for every \(\mu \in S\). If \(x\) belongs to the sum of the \(\mu \)-eigenspaces for \(\mu \in S\), then \(T^{n_k}(x)\to x\).

If \(\rho \geq 0\) and its support projection is the identity, then \(\rho \) is positive definite.

Let \(E\) be an irreducible positive map on \(M_{D}(\mathbb {C})\) and let \(A\geq 0\) be nonzero. Then

This is the implication (1)\(\Rightarrow \)(2) in [ Wol12 , Theorem 6.2 ] . Complete positivity is not assumed.

Every nonzero irreducible completely positive map on \(M_{D}(\mathbb {C})\) has the restricted CP spectral properties of Definition 8.15.1.

If \(A\) is an injective MPS tensor, then its transfer map \(\mathcal{E}_A\) is irreducible.

If a completely positive map on \(M_{D}(\mathbb {C})\) has the restricted CP spectral properties of Definition 8.15.1, then it is irreducible.

Let \(E\) be a nonzero completely positive map on \(M_{D}(\mathbb {C})\). Then \(E\) is irreducible if and only if it has the restricted CP spectral properties of Definition 8.15.1. This is a CP-map variant of [ Wol12 , Theorem 6.4 ] : uniqueness is required only among positive semidefinite Perron eigenvectors, not on the full eigenspace. Those restricted properties are nevertheless sufficient for the irreducibility equivalence.

If \(E\) is an irreducible map and \(c \neq 0\), then \(cE\) is also irreducible. For positive real \(c\), this is the scalar completely positive case of [ Wol12 , Proposition 6.6 ] ; the abstract irreducibility statement above also allows any nonzero complex \(c\).

Let \(E\) be an irreducible positive map on \(M_{D}(\mathbb {C})\). If \(\rho \geq 0\), \(\rho \neq 0\), and

then \(\rho \) is positive definite.

If the Kraus map of \(\{ K_i\} _{i=0}^{d-1}\) is irreducible, then so is the Kraus map of the conjugate-transposed family \(\{ K_i^\dagger \} \).

Let \(\rho \geq 0\) be fixed by the Kraus map of a finite matrix family \(K\), and let \(Q\) be the support projection of \(\rho \). Then \(Q\) is an orthogonal projection and

for every \(i\).

Let \(K\) be a finite matrix family with Kraus map \(E\), and let \(Q\) be an orthogonal projection. If

for every \(X\in M_{D}(\mathbb {C})\), then \((\mathbb {1}-Q)K_iQ=0\) for every \(i\).

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), if \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \), then the Kraus map \(\mathcal{E}_K\) is irreducible.

Let \(\tau _{E^m}\) be the normalized Choi matrix of the \(m\)-fold iterate of \(E\). Then

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

The Kraus map of a finite matrix family \(\{ K_i\} _{i=0}^{d-1}\) is completely positive.

Let \(\{ K_i\} _{i=0}^{d-1}\) be a family with \(D {\gt} 0\) and some \(K_i \neq 0\), whose Kraus map \(\mathcal K_K\) is irreducible. Then there exist a positive definite matrix \(\sigma \) and a positive real \(r {\gt} 0\) such that

This combines the eigenvector-existence part of [ Wol12 , Theorem 6.5 ] with the irreducible upgrade to positive definiteness ( [ Wol12 , Theorem 6.3(2) ] ).

Let \(D{\gt}0\), and let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices satisfying

Suppose there is a common length \(q\ge 0\) such that \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Then the Kraus map \(\mathcal{E}_K\) is primitive: it has a nonzero fixed point and \(1\) is its only peripheral eigenvalue.

Suppose that \(K\) is trace preserving and that its Kraus map \(E\) is irreducible with peripheral spectrum \(\{ 1\} \). This is the implication from item 1 to items 3 and 4 of [ Wol12 , Theorem 6.8(1,3,4) ] . Then, for every sufficiently large \(m\),

Moreover,

Let \(\{ K_i\} _{i=0}^{d-1}\) be a finite Kraus family and let \(\rho \) be positive definite with \(\sum _i K_i^\dagger \rho K_i = \rho \). Define the gauged family \(B_i = \rho ^{1/2} K_i \rho ^{-1/2}\). Then \(B\) is trace-preserving: \(\sum _i B_i^\dagger B_i = \mathbb {1}\).

If \(\mathcal S_m=M_{D}(\mathbb {C})\), then \(H_m(K,\psi )=\mathbb {C}^D\) for every nonzero \(\psi \in \mathbb {C}^D\).

Let \(\sigma \) be a fixed point of the adjoint transfer map \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \), and let \(S\) be invertible with \(S^\dagger S = \sigma \). Define \(A'^i = SA^iS^{-1}\). Then \(\sum _i (A'^i)^\dagger A'^i = \mathbb {1}\), i.e. the gauged Kraus map is trace-preserving. This is the left-canonical normalization.

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.

If \(\rho \geq 0\) and \(\rho \neq 0\), then its support projection is nonzero.

Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\), and let \(A, B \ge 0\) be nonzero positive semidefinite matrices with \(\operatorname{tr}(BA) = 0\). Then there exists \(t\) with \(1 \le t \le D - 1\) such that \(\operatorname{tr}(BE^t(A)) {\gt} 0\).

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 a positive trace-preserving or unital linear map with \(D{\gt}0\). If \(\lambda \) is an eigenvalue of \(T\) with \(|\lambda |=1\), then \(\ker (T-\lambda )^k=\ker (T-\lambda )\) for all \(k\ge 1\). This is [ Wol12 , Proposition 6.2 ] .

If \(T_\infty \) denotes the mean-ergodic projection of a positive trace-preserving map, then

This is the absorption identity adjacent to [ Wol12 , Equation (6.14) ] .

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) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, with \(D{\gt}0\), and let \(\lambda _{1},\dots ,\lambda _{m}\) denote the distinct eigenvalues of \(T\) of modulus one. Then

both pointwise and in operator norm. This is [ Wol12 , Equation (6.15) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, with \(D{\gt}0\). Then \(X\in T_\phi (M_{D}(\mathbb {C}))\) if and only if, for every \(\varepsilon {\gt}0\), there is a positive integer \(n\) such that \(\lVert T^n(X)-X\rVert \leq \varepsilon \). This is the recurrent-vector characterization following [ Wol12 , Equation (6.15) ] . Here Wolf’s \(\mathbb {N}\) is read as the positive integers; allowing \(n=0\) would make the recurrence condition hold for every \(X\).

The peripheral and non-peripheral spectral subspaces are complementary.

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 \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(D{\gt}0\), and assume that \(E(\sigma )\neq 0\) for every nonzero positive semidefinite matrix \(\sigma \). Then there are a nonzero positive semidefinite matrix \(\rho \) and a real number \(r{\gt}0\) such that \(E(\rho )=r\rho \).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(D{\gt}0\). Then there are a nonzero positive semidefinite matrix \(\rho \) and a real number \(r\geq 0\) such that

This is the eigenvector-existence part of [ Wol12 , Theorem 6.5 ] . It does not by itself identify \(r\) with the spectral radius.

Let \(E\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be linear, and let \(\rho \) and \(\sigma \) be positive-definite fixed points of \(E\). Suppose every nonzero positive-semidefinite fixed point of \(E\) is positive definite. Then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(E\colon M_D(\mathbb C)\to M_D(\mathbb C)\) be a positive, trace-preserving linear map and \(X=E(X)\) a fixed point. Decompose \(X\) into Hermitian and anti-Hermitian parts and then each into orthogonal positive and negative parts, yielding four positive semidefinite operators \(P_1,\dots ,P_4\). Then \(E(P_j)=P_j\) for \(j=1,\dots ,4\); in other words, every fixed point is a \(\mathbb C\)-linear combination of four positive-semidefinite fixed points. In particular, if \(E\) is a channel and \(H=H^\dagger \) is fixed by \(E\), then there are positive-semidefinite fixed points \(Q_1,Q_2\) such that \(H=Q_1-Q_2\).

This is Wolf Proposition 6.8.

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 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 \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and unital. Then \(1\) is an eigenvalue of \(T\), every eigenvalue belongs to the closed unit disk, and \(\varrho (T)=1\). No complete-positivity assumption is made.

Let \(T\) be an endomorphism of a vector space over a field. If \(T(x)=\mu x\), then

for every \(n\geq 0\).

Let \(E\) be an irreducible positive map on \(M_{D}(\mathbb {C})\). If \(\rho \geq 0\), \(\rho \neq 0\), and \(E(\rho )=\lambda \rho \) for some \(\lambda \in \mathbb {C}\), then \(\rho \) is positive definite.

Let \(E\) be an irreducible completely positive map on \(M_{D}(\mathbb {C})\). Suppose \(\rho \neq 0\) and \(\rho ,\sigma \geq 0\) satisfy \(E(\rho )=r\rho \) and \(E(\sigma )=r\sigma \) for a real number \(r{\gt}0\). Then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\). This is the completely positive specialization of the positive-eigenvector uniqueness in [ Wol12 , Theorem 6.3(2–3) ] .

Assume \(D \ge 1\). Let \(A\) be an MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then there exists \(\rho \ge 0\), \(\rho \neq 0\), with \(\mathcal{E}_A(\rho ) = \rho \).

Let \(A\) be an injective MPS tensor and let \(\rho \ge 0\), \(\rho \neq 0\), be a fixed point of \(\mathcal{E}_A\). Then \(\rho \) is positive definite.

If \(E\) is completely positive and irreducible and \(\rho \ge 0\), \(\rho \neq 0\) is a fixed point of \(E\), then \(\rho \) is positive definite.

Let \(A\) be an injective MPS tensor. If \(\rho ,\sigma \geq 0\) are nonzero fixed points of \(\mathcal{E}_A\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(A\) be an MPS tensor whose transfer map is irreducible. If \(\rho ,\sigma \geq 0\) are fixed points of \(\mathcal{E}_A\) and \(\rho \neq 0\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(E\) be a completely positive irreducible map on \(M_{D}(\mathbb {C})\). If \(\rho ,\sigma \geq 0\) are fixed points of \(E\) and \(\rho \neq 0\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

Assume \(D\ge 1\). Let \(A\) be an MPS tensor such that its transfer map is irreducible and \(\sum _i (A^i)^\dagger A^i=\mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then \(\mathcal{E}_A\) has a unique positive definite fixed point, up to scalar multiple.

Let \(\mathcal{E}_A(\rho ) = \rho \) and let \(S\) be an invertible matrix with \(SS^\dagger = \rho \). Define the gauged operators \(A'^i = S^{-1}A^iS\). Then \(\sum _i A'^i(A'^i)^\dagger = \mathbb {1}\), i.e. the gauged Kraus map is unital. This is the right-canonical normalization.

Let \(D\geq 1\), let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive, and suppose \(X{\gt}0\) and \(T(X)=rX\) with \(r{\gt}0\). Then \(\varrho (T)=r\). Irreducibility and complete positivity are not needed once this Perron pair is supplied. This is the final similarity argument in [ Wol12 , Theorem 6.3(4) ] .

Under the hypotheses of Theorem 8.9.7, the real-valued spectral radius of \(E\) is also equal to \(r\).

Let \(D\ge 1\) and let \(T\colon M_D(\mathbb C)\to M_D(\mathbb C)\) be a continuous, trace-preserving, positive (not necessarily linear) map. Then \(T\) has at least one stationary state: a density matrix \(\rho \) such that \(T(\rho )=\rho \).

This matches Wolf Theorem 6.11 exactly: the hypotheses are continuity, positivity (\(X\geq 0\Rightarrow T(X)\geq 0\)), and trace preservation (\(\operatorname{tr}(T(X))=\operatorname{tr}(X)\)). Linearity is not assumed.

The proof is Wolf’s argument: positivity and trace preservation imply that \(T\) restricts to a continuous self-map of the compact convex set of density matrices; the density-matrix version of Brouwer (Theorem 8.17.6) then yields a fixed point.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive, and let \(T^*\) denote its adjoint for the trace pairing. Then \(T\) is irreducible if and only if \(T^*\) is irreducible. This is the observation at local source lines 604–606, immediately preceding [ Wol12 , Theorem 6.3 ] .

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.

If \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), then the transfer map \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) is a quantum channel (CPTP).

The transfer map \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) is completely positive.

Let \(A\) be an MPS tensor whose transfer map \(\mathcal{E}_A\) is irreducible. If \(\rho \geq 0\), \(\rho \neq 0\), and \(\mathcal{E}_A(\rho )=\rho \), then \(\rho \) is positive definite.

Let \(E\) be a completely positive map on \(M_{D}(\mathbb {C})\). Then \(E\) is irreducible if and only if, for every \(t{\gt}0\) and every nonzero \(A\geq 0\), one has \(\exp (tE)(A){\gt}0\). This is the completely positive specialization of the equivalence in [ Wol12 , Theorem 6.2(3) ] ; Theorem 8.13.1 is its forward implication.

Each assertion that the corresponding property holds for every sufficiently large \(m\) is equivalent to the existence of a threshold with Wolf’s explicit universal quantifier:

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive and trace preserving, with the Kraus decomposition above. Then the following are equivalent:

1. \(T\) is primitive: it is irreducible and its peripheral spectrum consists only of \(1\);

2. there is an \(n\in \mathbb N\) such that, for every \(m\geq n\) and every nonzero \(\psi \in \mathbb {C}^D\), \(K_m|\psi \rangle =\mathbb {C}^D\);

3. there is a \(q\in \mathbb N\) such that \(K_m=M_{D}(\mathbb {C})\) for every \(m\geq q\);

4. there is a \(q\in \mathbb N\) such that \(\tau _m{\gt}0\) for every \(m\geq q\).

The thresholds in items 3 and 4 can be chosen equal. If \(n\) and \(q\) are chosen minimal, then \(n\leq q\). This is [ Wol12 , Theorem 6.8 ] .

Let \(D\geq 1\) and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be irreducible and positive. There are a density matrix \(X{\gt}0\) and a real number \(r\geq 0\) such that \(T(X)=rX\) and the following assertions hold.

1. Every lower feasible value is at most \(r\), every upper feasible value is at least \(r\), and \((X,r)\) is feasible in both senses. Thus \(r\) is the global maximum of the lower values and the global minimum of the upper values.

2. The ordinary complex eigenspace at \(r\) is \(\mathbb {C}X\), and hence has dimension one. No assertion about the generalized eigenspace is made.

3. If \(Y\geq 0\) is nonzero, \(\lambda {\gt}0\), and \(T(Y)=\lambda Y\), then \(\lambda =r\).

4. The spectral radius satisfies \(\varrho (T)=r\).

If \(T\neq 0\), the same conclusions hold with \(r{\gt}0\). This is the source form of [ Wol12 , Theorem 6.3 ] .

The boundary \(r=0\) is necessary without the hypothesis \(T\neq 0\). On \(M_{1}(\mathbb {C})\) the zero map is positive and irreducible because the only orthogonal projections are \(0\) and \(\mathbb {1}\), but its spectral radius is zero. The printed claim that the distinguished eigenvalue is strictly positive therefore omits this one-dimensional case.