Dependencies

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

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

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

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

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

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

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

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

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

and \(T\) is completely positive on \(S\) if entrywise positivity \((\forall k{\lt}r,\ X_k\ge 0)\) always implies \((T\otimes \operatorname{id}_j)(X)\ge 0\), for every \(j\). This is Definition 1.6.10 applied block by block, one summand at a time.

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

Define the corresponding Hermitian matrix by

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

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

A linear map \(\mathcal S\) 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.

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

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

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

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

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

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

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

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

Let \(\rho \) be a matrix on a finite-dimensional space \(B\). The state-preparation map from matrices on \(A\) to matrices on \(A\otimes B\) is

This is the elementary preparation operation used in the maps \(\mathcal T_1\) and \(\mathcal S_1\) of [ CPGSV16 , Appendix C.2, lines 1527–1533 and 1551–1555 ] . It is relocated here from the MPDO renormalization chapter; its Kraus-action and conditional-expectation consequences for the density-operator setting appear in Section 1.8 below.

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

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

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

For an operator \(\rho \) on \(H_L\otimes H_R\), define \((\operatorname{tr}_L\rho )_{r,s}:=\sum _l\rho _{(l,r),(l,s)}\).

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

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

A vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) purifies a density operator \(\rho \in M_{d_A}(\mathbb {C})\) if \(\operatorname{tr}_B[|\psi \rangle \! \langle \psi |]=\rho \).

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

A vector \(\psi \in \mathbb {C}^{D}\otimes \mathbb {C}^{k}\) is identified with its coefficient matrix \(C_\psi \in M_{D,k}(\mathbb {C})(\mathbb {C})\). Its Schmidt rank is \(\operatorname{SR}(\psi )=\operatorname{rank}(C_\psi )\). We write \(\operatorname{SR}(\psi )\le r\) for the corresponding bounded-rank condition.

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

Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map and let \([a_{ij}]_{i,j=1}^n\) be a positive semidefinite block matrix with entries in \(M_D(\mathbb {C})\) such that the images \(T(a_{ij})\) pairwise commute. Then

for every family of vectors \(\{ \psi _i\} _{i=1}^n\subset \mathbb {C}^D\).

Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map with commutative range. Then \(T\) is completely positive.

Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map whose trace-pairing adjoint \(T^*\) has commutative range. Then \(T\) is completely positive.

A linear map \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) is completely positive if and only if \(T^*\) is.

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.

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.

The composition of two completely positive rectangular Kraus maps is completely positive. If the two families are \((A_i)_i\) and \((B_j)_j\), the composite family is \((A_iB_j)_{i,j}\).

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

The composition of two trace-preserving completely positive maps is again trace-preserving completely positive. If \(\mathcal{S}\) has Kraus operators \(A_i\) and \(\mathcal{T}\) has Kraus operators \(B_j\), then \(\mathcal{S}\circ \mathcal{T}\) has Kraus operators \(A_iB_j\).

The identity map \(\operatorname{id}(X)=X\) is trace-preserving completely positive, with single Kraus operator \(A_0=I\).

If \(\rho \) has a convex decomposition \(\rho =\sum _i\lambda _i\rho _i\) giving \(\rho _1\) a positive weight \(c\), then \(\ker (\rho )\subseteq \ker (\rho _1)\).

For rectangular matrices \(L\) and \(Q\) over a commutative semiring and every integer \(N\geq 1\),

Every finite rectangular Kraus family defines a completely positive map, without a resolution-of-identity assumption.

The map \(\operatorname{tr}_B\) is trace-preserving and completely positive for arbitrary finite-dimensional spaces \(A\) and \(B\).

For every matrix \(X\), one has \(\Phi _V(X)=VXV^\dagger \).

For every rectangular operator \(V:H\to K\), the map \(X\mapsto VXV^\dagger \) is completely positive.

If \(\rho \geq 0\), then the map \(X\mapsto X\otimes \rho \) is completely positive. No trace normalization is required.

If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\mathcal P_\rho :X\mapsto X\otimes \rho \) is trace-preserving completely positive.

If \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) is positive, so is \(T^*\).

Let \(\tau '\) be as in Lemma 1.6.12 and let \(Y=\sum _kv_kv_k^\dagger \) be any rank-one decomposition of its Riesz matrix. Then the map \(T'(B)_{ij}:=n\cdot \tau '(B\otimes |i\rangle \langle j|)\) equals \(\sum _kK_kBK_k^\dagger \) for the reshaped operators \(K_k:\mathbb {C}^m\to \mathbb {C}^n\), \(K_k:=\sqrt n\, v_k^\dagger \) (viewing \(v_k\) as an \(m\times n\) matrix). Complete positivity of \(T'\) is assembled from this identity together with the positive semidefiniteness of \(Y\) at Theorem 1.6.17.

Let \(\tau '\) be a \(\mathbb {C}\)-linear functional on \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) built from a \(\mathbb {R}\)-linear functional by the Hermitian-decomposition construction of Lemma 1.6.12, and suppose \(\operatorname{Re}\tau '(X)\ge 0\) for every positive semidefinite \(X\in M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\). Let \(Y\) be the matrix representing \(\tau '\) through the trace pairing, \(\tau '(X)=\operatorname{tr}(YX)\). Then \(Y\) is positive semidefinite.

If \(d_A\le d_B\), every \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) admits a Schmidt decomposition.

Let \(A\), \(B\) be positive semidefinite operators on the same space with \(\ker (A)\subseteq \ker (B)\), and let \(c{\gt}0\). Then

where the inverse is taken on the range of \(A\).

For every Hermitian \(A\in S\otimes M_{n}(\mathbb {C})\),

There is a \(\mathbb {C}\)-linear functional \(\tau '\) on \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) agreeing with \(\tau \) on \(S\otimes M_{n}(\mathbb {C})\) and satisfying \(\operatorname{Re}\tau '(A)\le \| A\| _\infty \, \operatorname{Re}\tau '(\mathbb 1)\) for every Hermitian \(A\in M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\).

For every \(B\in S\) and every \(i,j\),

Let \(\varphi :M_{d}(\mathbb {C})\to \mathbb {C}\) be a \(\mathbb {C}\)-linear functional satisfying \(\operatorname{Re}\varphi (A)\le \| A\| _\infty \, \operatorname{Re}\varphi (\mathbb 1)\) for every Hermitian \(A\in M_{d}(\mathbb {C})\), with \(\operatorname{Re}\varphi (\mathbb 1)\ge 0\). Then \(\operatorname{Re}\varphi (A)\ge 0\) for every positive semidefinite \(A\in M_{d}(\mathbb {C})\).

For every operator \(M\) on \(H_L\) and every operator \(\rho \) on \(H_L\otimes H_R\), one has \(\operatorname{tr}((M\otimes \mathbb {1}_R)\rho )=\operatorname{tr}(M\operatorname{tr}_R\rho )\).

For every operator \(M\) on \(H_R\), one has \(\operatorname{tr}((\mathbb {1}_L\otimes M)\rho )=\operatorname{tr}(M\operatorname{tr}_L\rho )\).

Every vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) admits a Schmidt decomposition. This is [ Wol12 , Proposition 1.1 ] .

There are no projective rays when \(d=0\). When \(d=1\), every pure-state matrix is the identity and

If \(d\geq 2\), then

The identity channel has Choi matrix \(|\Omega \rangle \! \langle \Omega |\).

Let \(K\) be finite, let \(m_k\geq 1\), and set

For \(A\in \operatorname{End}(\mathcal H)\), write \(A_{kk}\) for its \(k\)-th diagonal block. The map

has zero off-diagonal output blocks. It is trace-preserving and completely positive, and it is a conditional expectation onto the coordinate subalgebra

Thus its range lies in this subalgebra, it fixes the subalgebra pointwise, and it is unital and idempotent. This is the normalized trace-preserving coordinate choice obtained by specializing the block densities in [ Wol12 , Proposition 1.5 and Equation (1.40) ] to maximally mixed densities. Relative to Wolf’s \(M_{d_k}(\mathbb {C})\otimes \mathbb 1_{m_k}\) convention, the displayed \(\mathbb 1_{m_k}\otimes M_{d_k}(\mathbb {C})\) order is obtained by interchanging the tensor factors. This theorem concerns only the displayed direct-sum coordinates; it does not conjugate the subalgebra by an arbitrary unitary or corestrict the codomain. This coordinate restriction is documented in [ con26j ] .

Let \(F:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) be completely positive and let \(E:M_{p}(\mathbb {C})\to M_{q}(\mathbb {C})\) have a Kraus representation. Then \(E\circ F\) is completely positive.

Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map. If \(T\) has commutative range, or \(T^*\) has commutative range, then \(T\) is completely positive. This is [ Wol12 , Proposition 1.6 ] ; the commutative-range hypothesis is the range-only reading of Wolf’s “\(\mathcal A\) or \(\mathcal B\) commutative” hypothesis that his own proof establishes and that his remark immediately after the proof licenses for an arbitrary operator system mapping into a commutative algebra. See [ con26k ] for the precise relationship between the two phrasings.

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.

Let \(\iota \) be the block-diagonal embedding of \(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) into \(M_M(\mathbb {C})\), \(M=\sum _kd_k\). If \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) is an operator system, then \(\iota (S)\) is an operator system in \(M_M(\mathbb {C})\); and if \(T:S\to M_{p}(\mathbb {C})\) is completely positive on \(S\), then the induced map \(T'':\iota (S)\to M_{p}(\mathbb {C})\), \(T''(\iota (X))=T(X)\), is completely positive on \(\iota (S)\).

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

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 \(\iota _j\) be the block-diagonal embedding

Then \(\iota _j(X)\) is positive semidefinite exactly when every \(X_k\) is positive semidefinite. Moreover, for all \(a,b{\lt}j\),

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.

Let \(m,n\ge 1\), let \(S\subseteq M_{m}(\mathbb {C})\) be an operator system, and let \(T:S\to M_{n}(\mathbb {C})\) be a completely positive linear map on \(S\). Then there is a completely positive map \(T':M_{m}(\mathbb {C})\to M_{n}(\mathbb {C})\), in rectangular Kraus form, that agrees with \(T\) on \(S\).

Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\ge 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system (Definition 1.7.1), and let \(T:S\to M_{p}(\mathbb {C})\) be completely positive on \(S\) (Definition 1.7.2). Then there is a completely positive map \(T':\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) (Definition 1.7.3) that agrees with \(T\) on \(S\).

Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\geq 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system, let \(\mathcal B\subseteq M_{p}(\mathbb {C})\) be a unital \(*\)-subalgebra, and let \(T:S\to \mathcal B\) be completely positive, where positivity in \(\mathcal B\) is read through its inclusion in \(M_{p}(\mathbb {C})\). Then there is a linear map

whose inclusion into \(M_{p}(\mathbb {C})\) is completely positive and which satisfies \(\widetilde T|_S=T\).

Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\geq 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system, and let \(\mathcal B\subseteq M_{p}(\mathbb {C})\) be a unital \(*\)-subalgebra. Suppose that \(T:S\to M_{p}(\mathbb {C})\) is completely positive and \(T(S)\subseteq \mathcal B\). Then there is a completely positive map \(G:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) such that \(G(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C}))\subseteq \mathcal B\) and \(G|_S=T\).

Let \(A,B\in M_d(\mathbb {C})\) be Hermitian. If their characteristic polynomials agree, then there is a unitary matrix \(U\in M_d(\mathbb {C})\) such that

Let \(T:M_d(\mathbb {C})\to M_d(\mathbb {C})\) be complex linear. Suppose that \(T(A)\) is Hermitian whenever \(A\) is Hermitian and that

for every Hermitian \(A\). Then every Hermitian \(A\) satisfies

Let \(A,B\in M_d(\mathbb {C})\) be Hermitian. Suppose \(\sigma (B)=\sigma (A)\) and the roots of \(\chi _A\) have no repetitions. Then \(\operatorname{tr}(B^k)=\operatorname{tr}(A^k)\) for every \(k\in \mathbb {N}\).

If \(\varepsilon {\gt}0\), then \(\lambda _0(\varepsilon ){\gt}\cdots {\gt}\lambda _{d-1}(\varepsilon )\), so the roots of \(\chi _{A_\varepsilon }\) have no repetitions. Moreover, \(A_\varepsilon \to A\) as \(\varepsilon \to 0\). In particular, the positive sequence \(\varepsilon _n=1/(n+1)\) gives simple-spectrum matrices \(A_{\varepsilon _n}\to A\). All assertions remain valid for \(d=0\).

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be complex linear. Suppose that

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

for every Hermitian \(A\in M_{d}(\mathbb {C})\). Then there is a unitary \(U\) such that either

for every \(X\in M_{d}(\mathbb {C})\), or

for every \(X\in M_{d}(\mathbb {C})\). This is the spectrum-preserver classification in [ Wol12 , Chapter 1, Spectrum preserving maps ] .

If \(A\) is Hermitian with eigenvalues \(\lambda _0,\ldots ,\lambda _{d-1}\), listed with multiplicity, then for every \(k\in \mathbb {N}\),

This includes \(d=0\), when both sides are empty sums.

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

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

If \(\Phi (X)=\sum _iA_iXA_i^\dagger \) is a rectangular Kraus completely positive map, then \(\Phi \otimes \operatorname{id}_j\) is a Kraus completely positive map with Kraus operators \(A_i\otimes \mathbb 1_j\). In particular, \((\Phi \otimes \operatorname{id}_j)(X)\) is positive semidefinite whenever \(X\) is.

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 complex vector space and let \(T,S:M_{D}(\mathbb {C})\to E\) be complex-linear maps. If

for every \(v\in \mathbb {C}^D\), then \(T=S\).

Let \(\rho \) be positive semidefinite on \(H_L\otimes H_R\), and let \(P_L\) be the orthogonal projector onto the support of \(\operatorname{tr}_R\rho \). Then \((P_L\otimes \mathbb {1}_R)\rho =\rho \).

Under the same assumptions, \(\rho (P_L\otimes \mathbb {1}_R)=\rho \). For \(H_L=H_A\otimes H_X\) and \(H_R=H_B\), the two absorption identities are precisely the \(P_{AX}\) support-projector identities in [ CPGSV16 , Appendix D.2, lines 2228–2235 ] .

For every unital \(*\)-subalgebra \(\mathcal B\subseteq M_{n}(\mathbb {C})\), there is a trace-preserving completely positive map \(E:M_{n}(\mathbb {C})\to M_{n}(\mathbb {C})\) that is a conditional expectation onto \(\mathcal B\): it is unital and idempotent, its range lies in \(\mathcal B\), and it fixes every element of \(\mathcal B\). This is the normalized trace-preserving choice in the family described by [ Wol12 , Proposition 1.5 and Equation (1.40) ] .

For each \(k\in \mathbb {N}\), the function \(A\mapsto \operatorname{tr}(A^k)\) on \(M_d(\mathbb {C})\) is continuous. Thus \(A_n\to A\) implies \(\operatorname{tr}(A_n^k)\to \operatorname{tr}(A^k)\).

Let \(\rho \) and \(\rho _1\) be two density operators acting on the same space and let \(c{\gt}0\). There is a convex decomposition of the form \(\rho =\sum _i\lambda _i\rho _i\) with \(\rho _1\) carrying weight \(c\) iff \(\ker (\rho )\subseteq \ker (\rho _1)\) and

where the inverse is taken on the range of \(\rho \). Wolf’s Proposition “Maximal weight in convex decomposition” [ Wol12 ] states this with the inverse form \(c\le \| \rho ^{-1/2}\rho _1 \rho ^{-1/2}\| _\infty ^{-1}\); the two are equivalent here because the kernel inclusion together with \(\operatorname{tr}(\rho _1)=1\) (so \(\rho _1\ne 0\)) forces \(\| \rho ^{-1/2}\rho _1\rho ^{-1/2}\| _\infty {\gt}0\), so dividing the displayed multiplicative inequality through by that norm is valid.

For every natural number \(m\), the matrix \(m^{-1}\mathbb 1_m\) is positive semidefinite.

If \(m\geq 1\), then \(\operatorname{tr}(m^{-1}\mathbb 1_m)=1\).

Suppose that for every \(k\) the sectorwise Kraus family resolves the identity, \(\sum _a A_{k,a}^\dagger A_{k,a}=I_{H_k}\). Then the orthogonally controlled map \(\mathcal C\) is trace-preserving and completely positive.

If \(k\ne l\), then \(\mathcal C(X)_{kl}=0\).

Let \(X_{kk}:H_k\to H_k\) denote the \(k\)th diagonal block of \(X\). Then

For every equivalence \(e:I\simeq J\) between finite index sets, the reindexing map \(R_e\) is trace-preserving and completely positive.

For square matrices \(X\) on \(A\) and \(Y\) on \(B\), \(\operatorname{tr}_{A}(X\otimes Y)=\operatorname{tr}(X)Y\).

Let \((P_s)_{s\in I}\) be a finite family of orthogonal projections on \(H\) such that \(\sum _sP_s=I_H\). For each \(s\), let \(\Phi _s:\operatorname{End}_{\mathbb {C}}(H)\to \operatorname{End}_{\mathbb {C}}(K)\) be trace-preserving and completely positive. Then

is trace-preserving and completely positive.

Let \((A_a:H\to K)_a\) be a finite family of rectangular operators. If \(\sum _a A_a^\dagger A_a=I_H\), then \(X\mapsto \sum _a A_aXA_a^\dagger \) is trace-preserving and completely positive.

If \(e:I\simeq I'\) and \(A'_b=A_{e^{-1}(b)}\), then \(\Phi _{A'}=\Phi _A\).

For square matrices \(X\) on \(A\) and \(Y\) on \(B\), \(\operatorname{tr}_B(X\otimes Y)=\operatorname{tr}(Y)X\).

Let \(H\) and \(K\) be finite-dimensional complex vector spaces, and let \(V:H\to K\) satisfy \(V^\dagger V=I_H\). Then the map

is trace-preserving and completely positive. This is the general one-isometry form of the local basis change used in [ CPGSV16 , Appendix C.2, lines 1439 and 1520 ] .

If \(V^\dagger V=I_H\), then \(\Phi _V\) is trace-preserving and completely positive.

If \(\rho \succeq 0\) and \(A_j(e_a)=e_a\otimes \sqrt\rho \, e_j\), then

If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\sum _j A_j^\dagger A_j=I_A\).

If \(A, B \in M_{n}(\mathbb {C})\) satisfy \(\operatorname{tr}(A^k) = \operatorname{tr}(B^k)\) for all \(k \ge 1\), then \(A\) and \(B\) have the same characteristic polynomial. It is enough to assume this equality for \(1 \le k \le n\).

For finite-dimensional \(H_L\) and \(H_R\), if \(\rho \geq 0\), then \(\operatorname{tr}_L\rho \geq 0\).

The matrix \(P_p\) is an orthogonal projection with trace one, and

The map \(p\mapsto P_p\) is injective. Moreover, a unitary \(U\) and coordinatewise conjugation act by

These are the pure-state matrix identities used in [ Wol12 , Chapter 1, Wigner’s theorem ] .

For arbitrary nonzero representatives \(v,w\in \mathbb {C}^d\),

Let \(f\) be a map on the rays of \(\mathbb {C}^d\) such that

for all rays \(p,q\). Then there is a unitary \(U\) such that either \(f(p)=U\cdot p\) for every \(p\), or \(f(p)=U\cdot \overline p\) for every \(p\).

Let \(A\in M_d(\mathbb {C})\) be Hermitian, and let \(p\) be a projective ray. If

then there is a projective ray \(q\) such that

Let \(\rho \in M_{d_A}(\mathbb {C})\) be a density operator with purification \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\). For every convex decomposition \(\rho =\sum _i\lambda _i\rho _i\) there is an instrument \(\{ T_i:M_{d_B}(\mathbb {C})\to M_{d_B}(\mathbb {C})\} \) acting on Bob’s system such that

for every \(i\). This is [ Wol12 , Proposition (Quantum steering) ] .

Let \(m\geq 1\). The linear map \(E:M_{m}(\mathbb {C})\otimes M_{d}(\mathbb {C})\to M_{m}(\mathbb {C})\otimes M_{d}(\mathbb {C})\) defined by

is trace-preserving and completely positive. It is a conditional expectation onto the unital \(*\)-subalgebra \(\mathbb 1_m\otimes M_{d}(\mathbb {C})\): it is unital and idempotent, its range lies in that subalgebra, and it fixes the subalgebra pointwise. This is the one-block trace-preserving choice in the family described by [ Wol12 , Proposition 1.5 and Equation (1.40) ] ; it supplies the corresponding one-block codomain retraction for the extension theorem, not the general finite-dimensional codomain construction.

Let \(\rho \) be positive semidefinite. If \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \), then \((\mathbb {1}_L\otimes P_R)\rho =\rho \).

Under the same assumptions, \(\rho (\mathbb {1}_L\otimes P_R)=\rho \).

For any Schmidt decomposition of \(\psi \) with coefficients \(\{ \lambda _j\} \), the number of nonzero \(\lambda _j\) equals \(\operatorname{SR}(\psi )\). This is [ Wol12 , Proposition 1.1 ] .

Let \(A,B\in M_d(\mathbb {C})\). Suppose \(\sigma (B)=\sigma (A)\) and the roots of \(\chi _A\) have no repetitions. Then the root multisets of \(\chi _A\) and \(\chi _B\) are equal.

For every nonzero \(v\in \mathbb {C}^d\),

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.

For projective rays \(p,q,r,s\) in \(\mathbb {C}^d\), if

then

For a bipartite vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) with coefficient matrix \(C\) and any linear map \(T\) with domain \(M_{d_B}(\mathbb {C})\),

Let \(e:H\simeq I\) be an equivalence of finite index sets and let \(U\) be a unitary matrix indexed by \(I\). Define

and

Both \(\Phi \) and \(\Phi ^{-1}\) are trace-preserving and completely positive.

Let \(e:H\simeq I\) be an equivalence of finite index sets and let \(U\) be a unitary matrix indexed by \(I\). The mutually inverse coordinate changes

and

both preserve positive semidefiniteness.

Let \(F\) be a bijection of the normalized pure states \(P_p\) such that \(\operatorname{tr}(F(P)F(Q))=\operatorname{tr}(PQ)\) for all normalized pure states \(P,Q\). Then there is a unitary \(U\) such that either \(F(P)=UPU^\dagger \) for every \(P\), or \(F(P)=UP^{\mathsf T}U^\dagger \) for every \(P\). This is a disjunction, not an exclusive alternative. For \(d=0\) the set of pure states is empty, while for \(d=1\) it is a singleton and the two alternatives coincide.