Dependencies

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

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

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

Let

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

A 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

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

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

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

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

A 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.

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

No trace-preservation normalization is imposed.

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

The 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 \(i \mapsto (A^i)^\dagger \).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 linear map \(T : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is trace-preserving if \(\operatorname{tr}(T(X)) = \operatorname{tr}(X)\) for all \(X\). For \(D' = D\) this is the notion of Definition 2.1.2.

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

Write \(\tau _T\) for the Choi matrix of \(T\). For all linear maps \(T,S:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\), all \(c\in \mathbb {C}\), all finite index sets \(s\) and all families \((T_i)_{i\in s}\),

Let \(s\subseteq \mathbb {C}\) be a finite set. Assume that for every \(\mu \in s\) there exists an exponent \(p_\mu {\gt}0\) with \(\mu ^{p_\mu }=1\). Then there exists \(p{\gt}0\) such that \(\mu ^p=1\) for all \(\mu \in s\).

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

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

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

When the input and output matrix algebras agree, rectangular Kraus complete positivity is equivalent to the square-map notion of complete positivity.

If \(\mathcal S\) is trace-preserving completely positive and \(X\geq 0\), then \(\mathcal S(X)\geq 0\).

Let \((Q_k)_k\) be a finite family of orthogonal projections satisfying \(\sum _k Q_k=\mathbb {1}\). If \(k\ne \ell \), then \(Q_kQ_\ell =0\).

Let \(E(X)=\sum _iK_iXK_i^\dagger \) be a Kraus map that is unital (\(\sum _iK_iK_i^\dagger =\mathbb {1}\)), and assume:

1. the adjoint Kraus map \(E^*(X)=\sum _iK_i^\dagger XK_i\) has a positive definite fixed point \(\rho {\gt}0\), and

2. \(E\) is irreducible.

Then \(\mathrm{peripheral}(E)\) is closed under powers: if \(\mu \in \mathrm{peripheral}(E)\), then \(\mu ^n\in \mathrm{peripheral}(E)\) for every \(n\in \mathbb {N}\).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a linear map, and assume that \(E\) has a nonzero fixed point \(\rho \neq 0\). Let \(p{\gt}0\). If every peripheral eigenvalue \(\mu \) of \(E\) satisfies \(\mu ^p=1\), then \(\mathrm{peripheral}(E^p)=\{ 1\} \).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a bounded linear map. If its spectral radius is one, then \(D\geq 1\).

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

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

For all \(X, Y \in M_{D}(\mathbb {C})\), \(\operatorname{tr}\! (X\, E(Y)) = \operatorname{tr}\! (E^*(X)\, Y)\).

The trace-adjoint identity of Theorem 2.10.2 holds when the Kraus family is the family of matrices of an MPS tensor and the two Kraus maps are written as transfer maps.

If \(A\) is left canonical, then, for every bond matrix \(X\),

The adjoint Kraus map associated with \((K_i^\dagger )_i\) is the Kraus map associated with \((K_i)_i\).

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 \(E(\rho )=\rho \).

Every quantum channel is positive.

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

Every linear map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is a complex linear combination of four completely positive maps. If \(T\) is Hermitian, that is \(T(B^\dagger )=T(B)^\dagger \) for every \(B\in M_{d}(\mathbb {C})\), then \(T\) is a real linear combination of two of them. This is [ Wol12 , Proposition 2.2 ] .

The conditions \(T(\mathbb {1}_d)\propto \mathbb {1}_{d'}\) and \(T^*(\mathbb {1}_{d'})\propto \mathbb {1}_d\) hold if and only if \(\operatorname{tr}_B(\tau )\propto \mathbb {1}_{d'}\) and \(\operatorname{tr}_A(\tau )\propto \mathbb {1}_d\). This is the doubly-stochastic clause of [ Wol12 , Proposition 2.1 ] .

The Choi matrix is Hermitian, \(\tau =\tau ^\dagger \), if and only if \(T(B^\dagger )=T(B)^\dagger \) for every \(B\in M_{d}(\mathbb {C})\). This is the Hermiticity clause of [ Wol12 , Proposition 2.1 ] .

The assignments \(T\mapsto \tau \) and \(\tau \mapsto T\), the latter defined entrywise by

are mutual inverses: every linear map is recovered from its Choi matrix, and every operator on \(\mathbb {C}^{d'}\otimes \mathbb {C}^{d}\) is the Choi matrix of the map it defines. This is [ Wol12 , Proposition 2.1, Equation (2.4) ] .

The full trace of the Choi matrix is \(\operatorname{tr}(\tau )=\operatorname{tr}(T^*(\mathbb {1}_{d'}))/d\). This is the normalization clause of [ Wol12 , Proposition 2.1 ] .

The output-factor partial trace of the Choi matrix is \(\operatorname{tr}_A(\tau )=(T^*(\mathbb {1}_{d'}))^{T}/d\), where \(T^*\) is the trace-pairing adjoint; hence \(T^*(\mathbb {1}_{d'})=\mathbb {1}_d\), equivalently \(T\) preserves the trace, if and only if \(\operatorname{tr}_A(\tau )=\mathbb {1}_d/d\). This is the trace-preservation clause of [ Wol12 , Proposition 2.1 ] .

For every \(A\in M_{d'}(\mathbb {C})\) and \(B\in M_{d}(\mathbb {C})\),

This is the second line of [ Wol12 , Proposition 2.1, Equation (2.1) ] .

The input-factor partial trace of the Choi matrix is \(\operatorname{tr}_B(\tau )=T(\mathbb {1}_d)/d\); hence \(T(\mathbb {1}_d)=\mathbb {1}_{d'}\) if and only if \(\operatorname{tr}_B(\tau )=\mathbb {1}_{d'}/d\). This is the unitality clause of [ Wol12 , Proposition 2.1 ] .

The controlled dependent partial trace is trace-preserving and completely positive.

If \(X_{ii}\) denotes the \(i\)th diagonal block of \(X\), then \([\mathcal C_{\operatorname{tr}}(X)]_{ii}=\operatorname{tr}_{B_i}(X_{ii})\).

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

Let \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) be completely positive in the Kraus sense. For every \(k\) and every positive block matrix \(M \in M_k(M_{D}(\mathbb {C}))\), the entrywise image \((E(M_{ab}))_{a,b}\) is again positive in \(M_k(M_{D}(\mathbb {C}))\).

Every Kraus-represented completely positive map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) determines a completely positive map between the matrix \(C^*\)-algebras in the entrywise positivity sense.

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 ] .

Every quantum channel \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) admits a finite family \((K_i)_i\) in \(M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

and

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.

Let \((v_i)_{i\in I}\) be a finite family in \(\mathbb {C}^n\). Then

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

If the Kraus map associated with \((K_i)_i\) is irreducible, then so is the Kraus map associated with \((K_i^\dagger )_i\).

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

If \(\mathcal S:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) is trace-preserving and completely positive, then its trace-pairing adjoint satisfies

This is the Schrödinger–Heisenberg duality of [ Wol12 , Section 1.2 ] .

Every trace-preserving completely positive map in rectangular Kraus form preserves the matrix trace.

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) ] .

Let \(E\) be a unital Kraus map. If \(E(X^\dagger X) = E(X)^\dagger E(X)\) for some \(X\), then, for every \(i\),

Suppose that

Every eigenvalue \(\mu \) of \(\mathcal M_{A,B}\) satisfies \(|\mu |\leq 1\).

Suppose that

Then \(\varrho (\mathcal M_{A,B})\leq 1\).

Two sets of Kraus operators \(\{ K_j\} \) and \(\{ \widetilde{K}_\ell \} \) represent the same completely positive map if and only if there is a unitary \(U\) with \(K_j=\sum _\ell U_{j\ell }\widetilde{K}_\ell \), where the smaller set is padded with zero operators; equivalently, after padding, the families are related by an isometry \(V\) with \(V^\dagger V=\mathbb {1}\). This is item 4 of [ Wol12 , Theorem 2.1 ] .

Let \(T(X)=\sum _{j=1}^{r}K_jXK_j^\dagger \) with \(K_j\in M_{d'\times d}(\mathbb {C})\). Then \(T\) is trace preserving if and only if \(\sum _j K_j^\dagger K_j=\mathbb {1}_d\), and unital if and only if \(\sum _j K_jK_j^\dagger =\mathbb {1}_{d'}\). This is item 1 of [ Wol12 , Theorem 2.1 ] .

Every completely positive map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) admits a Kraus representation with \(r=\operatorname {rank}(\tau )\) Hilbert–Schmidt orthogonal Kraus operators, \(\operatorname{tr}[K_i^\dagger K_j]\propto \delta _{ij}\); the off-diagonal traces vanish and the diagonal traces are nonzero. This is item 3 of [ Wol12 , Theorem 2.1 ] .

The minimal number of Kraus operators of a completely positive map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(r=\operatorname {rank}(\tau )\leq dd'\). This is item 2 of [ Wol12 , Theorem 2.1 ] .

Let \(\{ K_i\} _{i=0}^{d-1}\) be matrices in \(M_{D}(\mathbb {C})\) with \(\sum _{i=0}^{d-1}K_i^\dagger K_i=\mathbb {1}\), and let

be the associated Kraus map. Every eigenvalue \(\mu \) of \(\mathcal K_K\) satisfies \(|\mu | \le 1\). This is the trace-preserving specialization of [ Wol12 , Proposition 6.1 ] .

Let \(\{ K_i\} _{i=0}^{d-1}\) be a trace-preserving Kraus family on \(M_{D}(\mathbb {C})\). The spectral radius of its Kraus map \(\mathcal K_K\) is at most \(1\).

For every square complex matrix \(X\),

If \(E\) has a nonzero fixed point \(\rho \neq 0\) with \(E(\rho )=\rho \), then \(1\) is a peripheral eigenvalue of \(E\).

If \(P \in M_{D}(\mathbb {C})\) is an orthogonal projection, then \(\mathbb {1}-P\) is an orthogonal projection.

For a complex square matrix, the two descriptions \(P=P^\dagger \), \(P^2=P\) and \(P^*=P\), \(P^2=P\) are equivalent.

The peripheral spectrum of the Kraus map associated with \((K_i^\dagger )_i\) is the complex conjugate of the peripheral spectrum associated with \((K_i)_i\).

On a finite-dimensional space, the set of peripheral eigenvalues is finite.

If the set of peripheral eigenvalues of a linear endomorphism \(E\) on a finite-dimensional space is closed under powers (\(\mu \in \mathrm{peripheral}(E)\) and \(n\geq 1\) imply \(\mu ^n\in \mathrm{peripheral}(E)\)), then every peripheral eigenvalue is a root of unity.

Let \(E(X)=\sum _iK_iXK_i^\dagger \) be a Kraus map that is unital, and assume that the adjoint Kraus map \(E^*(X)=\sum _iK_i^\dagger XK_i\) has a positive definite fixed point and that \(E\) is irreducible. Then every peripheral eigenvalue of \(E\) is a root of unity.

Let \(D\geq 1\), and let \(E\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be an irreducible quantum channel. Then every peripheral eigenvalue of \(E\) is a root of unity.

Let \(E\) be a linear endomorphism on a finite-dimensional space. If \(\mu \) is a peripheral eigenvalue of \(E\) and \(\mu ^n\) is an eigenvalue of \(E\) for every \(n\geq 1\), then \(\mu \) is a root of unity.

If \(E:M_{n}(\mathbb {C})\to M_{m}(\mathbb {C})\) is positive and \(X\in M_{n}(\mathbb {C})\) satisfies \(X = X^\dagger \), then \(E(X) = E(X)^\dagger \) in \(M_{m}(\mathbb {C})\).

Let \(D\geq 1\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive. Then, for every \(X\in M_{D}(\mathbb {C})\),

This is the Russo–Dye estimate invoked in the proof of [ Wol12 , Proposition 6.1 ] , at local source line 84. Here \(\| \cdot \| _\infty \) is the C\(^*\)-operator norm, rather than the Frobenius norm or a row-sum norm.

Let \(D\geq 1\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive. Every eigenvalue \(\mu \) of \(T\) satisfies

and consequently

This is [ Wol12 , Proposition 6.1 ] .

Let \(D{\gt}0\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. If \(T(X)=\mu X\) for some nonzero \(X\), then \(|\mu |\leq 1\). This is the unit-disk conclusion of [ Wol12 , Proposition 6.1 ] .

Let \(D\geq 1\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. Then \(1\) is an eigenvalue with a nonzero positive semidefinite eigenvector, every eigenvalue belongs to the closed unit disk, and \(\varrho (T)=1\). No complete-positivity assumption is made.

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.

If \(E\) is trace-preserving and \(E(\rho )=\rho \), then, for \(n\geq 1\),

where \(P\) is the fixed-point projection.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be trace-preserving, and let \(\rho \neq 0\) satisfy \(E(\rho )=\rho \) and \(\operatorname{tr}(\rho )\neq 0\). Let \(P\) be the fixed-point projection associated to \(\rho \), as defined in (15). Assume that:

1. \(E\) is primitive;

2. every eigenvalue \(\mu \) of \(E\) satisfies \(|\mu |\le 1\);

3. whenever \(E(X)=X\) and \(\operatorname{tr}(X)=0\), one has \(X=0\).

Then every eigenvalue \(\nu \) of \(E-P\) satisfies \(|\nu |{\lt}1\).

A linear map with a nonzero fixed point is primitive if and only if its period is \(1\).

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

Let \(E\) be a unital Kraus map. If \(E(X^\dagger X) = E(X)^\dagger E(X)\), then, for every \(Y \in M_{D}(\mathbb {C})\),

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\ge 0\) with \(A^\dagger A\le B\) and \(B\) commuting with \(A\) (\(BA=AB\)), then

This is [ Wol12 , Theorem 5.6 ] .

Let \(Q\) be a self-adjoint idempotent on a finite-dimensional space, and let \(U\) be a coisometry onto that space, so that \(UU^\dagger =\mathbb {1}\). Then \(U^\dagger Q U\) is self-adjoint and idempotent.

For matrices \(\rho \) and \(X\) and a finite family \((K_i)_i\),

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 \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), 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.

If a Kraus family \((K_i)_i\) satisfies \(\sum _i K_i^\dagger K_i=\mathbb {1}\), then the family \((K_i^\dagger )_i\) is unital.