Dependencies

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

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

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

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

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

This is [ Wol12 , Equation (5.2) ] .

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

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.

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

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

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

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

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).

For a Kraus map \(E\), define

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

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

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

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

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

Consider the linear map \(T : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\) defined by

This is the map from [ Wol12 , Example 5.3 ] .

If \(P\) and \(R\) are Hermitian, then the block matrix \(\begin{pmatrix} P & Q \\ Q^\dagger & R \end{pmatrix}\) is Hermitian.

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.

\(B \cdot B^+ = P_{\mathrm{supp}}(B B^\dagger )\), where \(P_{\mathrm{supp}}(X)\) denotes the support projection of \(X\) (the orthogonal projection onto the range of \(X\)).

If \(A {\gt} 0\), \(X \ge 0\), and \(\operatorname{tr}(A X) = 0\), then \(X = 0\).

Let \(A\in M_{D}(\mathbb {C})\) be positive semidefinite, let \(c\in \mathbb {C}\), and let \(\psi \in \mathbb {C}^D\). If \(A\leq c|\psi \rangle \! \langle \psi |\), then there is a non-negative scalar \(a\) such that \(A=a|\psi \rangle \! \langle \psi |\).

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

Let \(E:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a \(2\)-positive complex-linear map. For every \(A,B\in M_{D_{\rm in}}(\mathbb {C})\) and all \(v,w\in \mathbb C^{D_{\rm out}}\) such that \(E(B^\dagger B)\, w = E(B^\dagger A)\, v\), we have

This is the pseudoinverse-free form of [ Wol12 , Eq. (5.4) ] : the usual statement \(E(A^\dagger B)\, E(B^\dagger B)^{-1}E(B^\dagger A)\le E(A^\dagger A)\) (with inverse taken on the range) follows by taking \(w = E(B^\dagger B)^{+}E(B^\dagger A)\, v\).

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) ] .

If \(\operatorname{ran}(A) \subseteq \operatorname{ran}(B)\), then there exists \(C\) such that \(A = B C\).

Let \(A, B \in M_{D}(\mathbb {C})\). The following are equivalent:

1. \(\operatorname{range}(A) \subseteq \operatorname{range}(B)\),

2. \(A A^\dagger \le \mu B B^\dagger \) for some \(\mu \ge 0\),

3. \(A = B C\) for some \(C \in M_{D}(\mathbb {C})\).

If \(A v \in \operatorname{ran}(B)\) for every vector \(v\), then there exists \(C\) with \(A = B C\).

Let \(E:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a \(2\)-positive complex-linear map. For every \(A,B\in M_{D_{\rm in}}(\mathbb {C})\),

Let \(m\) be finite and \(B,D\in M_{m}(\mathbb {C})\). If \(tB+t^2D\) is positive semidefinite for every \(t\in \mathbb R\), then \(B=0\).

If \(\operatorname{range}(A) \subseteq \operatorname{range}(B)\), then \(A = B \cdot (B^+ A)\).

If \(A = B C_1 = B C_2\) and \(\operatorname{range}(C_1), \operatorname{range}(C_2) \subseteq \operatorname{range}(B^\dagger )\), then \(C_1 = C_2\). In particular, \(C = B^+ A\) is the unique such factor.

If a Kraus map is both unital and trace-preserving, then \(\operatorname{tr}(E(X)^\dagger E(X)) \le \operatorname{tr}(X^\dagger X)\) for all \(X \in M_{D}(\mathbb {C})\).

If \(E\) is \(k\)-positive, then \(E^*\) is \(k\)-positive.

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) ] .

If \(\sum _i K_i^\dagger K_i = \mathbb {1}\), then the adjoint Kraus map satisfies, for every \(X \in M_{D}(\mathbb {C})\),

Let \(E:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a \(2\)-positive complex-linear map. For every \(A,B\in M_{D_{\rm in}}(\mathbb {C})\), \(\ker (E(B^\dagger B))\subseteq \ker (E(A^\dagger B))\) as subspaces of \(\mathbb C^{D_{\rm out}}\).

Every unital Kraus map satisfies the abstract Schwarz inequality of Definition 6.8.1.1.

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

Let \(E\) be unital and let \(E^*\) be trace-preserving with a positive definite fixed point \(\rho {\gt} 0\). If \(E(X) = \mu X\) with \(|\mu | = 1\), then \(E(X^\dagger X) = E(X)^\dagger E(X)\).

For a unital Kraus map \(E\), the Kadison–Schwarz gap decomposes as

Let \(E\) be a Kraus map that is both unital and trace-preserving. If \(E(X) = \mu X\) with \(|\mu | = 1\), then the Kadison–Schwarz gap vanishes:

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 \(E:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a \(2\)-positive complex-linear map. For every \(A,B\in M_{D_{\rm in}}(\mathbb {C})\),

Let \(E\) be a unital Kraus map. Then

Together these are the Kraus-map specialization of Theorem 6.8.1.3.

If \(E\) is unital, then the restricted map \(E|_{\mathcal{A}(E)}:\mathcal{A}(E)\to M_{D}(\mathbb {C})\) is a \(*\)-algebra homomorphism. This is the homomorphism conclusion of [ Wol12 , Theorem 5.7 ] .

For a unital Kraus map \(E\), the full multiplicative domain \(\mathcal{A}(E)\) is a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). This is the algebraic content of [ Wol12 , Theorem 5.7 ] .

For a unital Kraus map \(E\), both \(\mathcal{A}_R(E)\) and \(\mathcal{A}_L(E)\) are unital subalgebras of \(M_{D}(\mathbb {C})\).

For every factorisation \(A = B C\), the pseudoinverse witness attains the minimal operator norm: \(\| B^+ A\| \le \| C\| \).

Let \(f : M_{D}(\mathbb {C}) \to \mathbb {C}\) be a complex-linear functional, where \(D{\gt}0\). If \(f(X) \geq 0\) for every \(X \geq 0\) and \(f(\mathbb {1})=1\), then there is a density matrix \(\rho \in M_{D}(\mathbb {C})\) such that \(f(X) = \operatorname{tr}(\rho X)\) for every \(X \in M_{D}(\mathbb {C})\).

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 with \(T(\mathbb {1})\le \mathbb {1}\). If \(a\le 0\le b\) and \(a\mathbb {1}\le A\le b\mathbb {1}\), then

This is the matrix form of [ Wol12 , Equation (5.21) ] .

If \(T:M_{n}(\mathbb {C})\to M_{m}(\mathbb {C})\) is positive and \(A,B\in M_{n}(\mathbb {C})\) satisfy \(A\le B\), then \(T(A)\le T(B)\) in \(M_{m}(\mathbb {C})\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive with \(T(\mathbb {1})\le \mathbb {1}\), and let \(A\in M_{D}(\mathbb {C})\) be Hermitian. If \(\operatorname{spec}(A)\subseteq [a,b]\) with \(a\le 0\le b\), then

This is [ Wol12 , Equation (5.21) ] .

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 \(m,d{\gt}0\), and let \(E:M_{md}(\mathbb {C})\to M_{md}(\mathbb {C})\) be a positive complex-linear map. Suppose that the image of \(E\) is contained in \(\mathbb {1}_m\otimes M_{d}(\mathbb {C})\) and that \(E(\mathbb {1}_m\otimes X)=\mathbb {1}_m\otimes X\) for every \(X\in M_{d}(\mathbb {C})\). Then there is a density matrix \(\rho \in M_{m}(\mathbb {C})\) such that, for every \(A\in M_{md}(\mathbb {C})\),

This is the one-factor case of [ Wol12 , Proposition 1.5 and Equation (1.40) ] .

Let \(A \in M_{D}(\mathbb {C})\) be Hermitian. If \(\operatorname{tr}(AB)\geq 0\) for every positive semidefinite matrix \(B\), then \(A\geq 0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be complex-linear. Suppose that for every \(v\in \mathbb {C}^D\) there is a scalar \(c_v\in \mathbb {C}\) such that \(T(|v\rangle \! \langle v|)=c_v|v\rangle \! \langle v|\). Then there is a scalar \(c\in \mathbb {C}\) such that \(T=c\, \operatorname{id}\).

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 \(P \in M_{D_1}(\mathbb {C})\) and \(R \in M_{D_2}(\mathbb {C})\) be positive semidefinite, and let \(Q \in M_{D_1,D_2}(\mathbb {C})\). The following are equivalent:

1. The block matrix \(\begin{pmatrix} P & Q \\ Q^\dagger & R \end{pmatrix}\) is positive semidefinite.

2. One has \(\ker (R)\subseteq \ker (Q)\) and \(P\geq QR^+Q^\dagger \).

3. One has \(\ker (R)\subseteq \ker (Q)\), \(\ker (P)\subseteq \ker (Q^\dagger )\), and \(Q=P^{1/2}KR^{1/2}\) for the canonical support contraction \(K\), with \(\lVert K\rVert _\infty \leq 1\).

Here \(R^+\) denotes the pseudoinverse. The second support condition in the third clause is necessary in the singular case and is absent from the printed statement of Wolf’s Theorem 5.2.

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

For every linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\), the trace-pairing adjoint of \(E^*\) is \(E\).

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

A linear map is positive if and only if its trace-pairing adjoint is positive.

Let \(E:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a \(2\)-positive complex-linear map. For every \(A,B\in M_{D_{\rm in}}(\mathbb {C})\),

where \((\cdot )^+\) is the Moore–Penrose pseudoinverse, i.e. the inverse taken on the range. No vector witness is assumed to exist.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map such that, for a given \(B\in M_{D}(\mathbb {C})\), the per-\(B\) Schwarz hypothesis holds (Definition 6.9.2.7). Then for every \(A\in \mathcal{A}_B\) (Definition 6.9.2.8) and every \(X\in M_{D}(\mathbb {C})\),

This is the source’s equality theorem, local source Notes/WolfNoteTexSource/ch05_schwarz_inequalities.tex, line 198.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a Kraus map satisfying \(E(\mathbb {1})=\mathbb {1}\). Every eigenvalue \(\mu \) of \(E\) satisfies

This is the unital Kraus specialization of [ Wol12 , Proposition 6.1 ] .

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a Kraus map satisfying \(E(\mathbb {1})=\mathbb {1}\). Then, for every \(X\in M_{D}(\mathbb {C})\),

Let \(A,B\in M_{d}(\mathbb {C})\) be Hermitian. If \(A\leq B\), then their decreasingly ordered eigenvalues satisfy

This is the consequence of [ Wol12 , Equation (5.56) ] used immediately before Proposition 5.3.

The map from Example 6.9.1.1 is not completely positive.

The map from Example 6.9.1.1 is positive.

For every \(A \in M_{2}(\mathbb {C})\), \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\), where \(T\) is the map from Example 6.9.1.1.

Let \(E=T^*:M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\) be a positive complex-linear map. If \(T:M_{D_{\rm out}}(\mathbb {C})\to M_{D_{\rm in}}(\mathbb {C})\) is \(2\)-positive, then for all \(A,B\in M_{D_{\rm in}}(\mathbb {C})\),

This is exactly [ Wol12 , Theorem 5.3 and Eq. (5.4) ] , with the inverse taken on the range.

Let \(A,B\in M_{d}(\mathbb {C})\) be Hermitian. The corrected form of [ Wol12 , Proposition 5.3 and Equation (5.57) ] is, with \(\mathcal U(d)\) denoting the unitary matrices,

In particular, the unitary is chosen before the interval and the function; it depends only on \(A\) and \(B\). No continuity hypothesis on \(f\) is required, since both spectra are finite.