Dependencies

A linear map is \(k\)-positive if and only if its blockwise ampliation is positive on \(M_{D}(\mathbb {C})\otimes M_{k}(\mathbb {C})\).

Let \(K=(K_i)_{i=0}^{d-1}\) be a unital Kraus family, that is, \(\sum _i K_i K_i^\dagger =\mathbb {1}\). Then for all \(X\in M_{D}(\mathbb {C})\), \(\mathcal{K}(X^\dagger X)\ge \mathcal{K}(X)^\dagger \mathcal{K}(X)\), where \(\mathcal{K}(Y)=\sum _i K_i Y K_i^\dagger \).

For an orthogonal projection \(P\) and linear map \(E\), the corner \(PM_{D}(\mathbb {C})P\) is preserved if \(E(PXP)=PE(PXP)P\) for every \(X\).

The corner subspace associated with \(P\) is \(\{ PXP:X\in M_{D}(\mathbb {C})\} \).

Irreducibility of the corner-restricted map.

The rank of an orthogonal projection \(P\in M_{D}(\mathbb {C})\), equivalently the dimension of its range.

For \(t\ge 0\) and \(n\in \mathbb {N}\), one has \(T_{nt}=(T_t)^n\).

Every density matrix is nonzero.

If a corner compression is invariant under \(E\), it remains invariant under every power \(E^n\).

If \(L(X)=\mu X\), then \(e^{tL}(X)=e^{t\mu }X\) for all \(t\in \mathbb {R}\).

If \(\mu \) is in the spectrum of \(L\), then \(e^{t\mu }\) is in the spectrum of \(e^{tL}\).

If \(e^{it\theta }\) is a root of unity for every \(t{\gt}0\), then \(\theta =0\).

Peripheral spectral points of the generator have vanishing real part.

The number of peripheral eigenvalues is bounded by \(\dim (M_{D}(\mathbb {C}))\).

If all powers of a peripheral eigenvalue remain peripheral, then some positive power not exceeding \(\dim (M_{D}(\mathbb {C}))\) equals \(1\).

For an irreducible channel with faithful fixed point, every power of a peripheral eigenvalue is again peripheral.

Let \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be CP maps with \(T_1\le T_2\). Then there exist ancilla dimensions \(r_1,m\), Heisenberg-form Stinespring matrices \(V_1:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^{r_1}\) and \(V_2:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^m\) realizing \(T_1,T_2\) via \(T_i(A)=V_i^\dagger (A\otimes \mathbb {1})V_i\), and a rectangular contraction \(\widetilde C:\mathbb {C}^m\to \mathbb {C}^{r_1}\) with \(\widetilde C^\dagger \widetilde C\le \mathbb {1}_m\) such that

This is the explicit square corollary obtained by constructing both dilation matrices, rather than the supplied-dilation statement of Theorem 3.11.7.

Let \(I\) be a nonempty finite set, and let \(T_i,T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive linear maps such that \(\sum _{i\in I}T_i=T\). If \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) satisfies \(T(A)=V^\dagger (A\otimes \mathbb {1}_r)V\), then there are positive semidefinite operators \(P_i\in M_{r}(\mathbb {C})\) satisfying \(\sum _{i\in I}P_i=\mathbb {1}_r\) and \(T_i(A)=V^\dagger (A\otimes P_i)V\) for every \(i\in I\) and \(A\in M_{D}(\mathbb {C})\).

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 \(H\in M_{D}(\mathbb {C})\) be Hermitian and let \(A\in M_{D}(\mathbb {C})\) be invertible with \(HA=AH^{T}\) and \(A^{T}=-A\). Then every eigenvalue of \(H\) is at least two-fold degenerate.

Let \(H,V\in M_{D}(\mathbb {C})\) with \(H\) Hermitian, \(V\) unitary, \(HV^{\dagger }=V^{\dagger }H^{T}\), and \(V^{T}=-V\). Then every eigenvalue of \(H\) is at least two-fold degenerate.

The following conditions are equivalent:

1. \(\tau _{E^m}{\gt}0\) for every sufficiently large \(m\);

2. \(\mathcal S_m=M_{D}(\mathbb {C})\) for every sufficiently large \(m\).

For \(s\in [0,1]\), \(t\in [0,1]\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(A^s B^{1-s})\) is jointly concave:

Obtained from Theorem 7.7.13 by taking \(K=\mathbb {1}\).

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A_1,A_2,B\), the map \(A\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A,B_1,B_2\), the map \(B\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

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 \(T_\varphi \).

Let \(\{ \Phi _i\} \) be a quantum instrument on \(M_{d}(\mathbb {C})\) with \(d\geq 1\) whose total channel is the identity. Then for every outcome \(i\) the probability \(p_i(\rho )\) is a non-negative constant, the same for every \(\rho \) of unit trace.

For \(t \ge 0\), setting \(s=t\) in (5) gives

where \(M = \sup _{u \in [0,t]} \| e^{uL}\| \).

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).

For \(t \ge 0\), if \(\| e^{sL}\| \le 1\) and \(\| e^{sL'}\| \le 1\) for every \(s \in [0,t]\), then \(\| e^{tL'} - e^{tL}\| \le t\, \| L' - L\| \).

Let \(1\le k\) with \(k+1{\lt}D\). Then there is a map on \(M_{D}(\mathbb {C})\) that is \(k\)-positive but not \((k+1)\)-positive; the cone of \(k\)-positive maps strictly contains the cone of \((k+1)\)-positive maps. Every inclusion between consecutive amplification dimensions below the top one is therefore strict.

If \(P\in M_{D}(\mathbb {C})\) is positive semidefinite, then

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 Dyson–Phillips series \(\sum _n \widetilde{T}^{(n)}_t\) converges in operator norm for every \(t \ge 0\).

Every nonzero fixed point of an irreducible channel has nonzero trace.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map. Then for all density matrices \(\rho _1,\rho _2\in M_{D}(\mathbb {C})\), \(\lVert T(\rho _1)-T(\rho _2)\rVert _{\operatorname{tr}} \leq \lVert \rho _1-\rho _2\rVert _{\operatorname{tr}}\). See [ Wol12 , Chapter 8, Eq. (8.80) ] .

One has \(\sum _{n=0}^{\infty }\widetilde{T}^{(n)}_t = e^{tL'}\) in tsum form.

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.

Let \(T\) be a complex endomorphism on the same \(D\)-dimensional coordinate space as \(\Lambda ,N\in M_D(\mathbb C)\), and take \(\mu =\mu (T)\) from Definition 10.3. If \(\Lambda \) is diagonal, \(N\) is strictly upper triangular, and \(\lVert \Lambda \rVert _\infty =\mu \), then the coarse and refined estimates of Theorem 10.10 hold with this shared source-shaped \(\mu \).

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

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

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) ] .

For a Lindblad operator \(L_j\), the adjoint dissipator on observables is

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

The adjoint generator of a Lindblad form is

The adjoint kernel is the set of observables \(A\) satisfying \(L^*(A)=0\).

For a bipartite density operator \(\rho \in M_{d_A}(\mathbb {C})\otimes M_{d_B}(\mathbb {C})\), its operator-Schmidt rank is the least integer \(r\) for which there are matrices \(A_t\in M_{d_A}(\mathbb {C})\) and \(B_t\in M_{d_B}(\mathbb {C})\) satisfying

No Hermiticity or positivity condition is imposed on the factors. This is the definition in [ DlCDN19 , Equation (1) ] ; the same formula defines the rank of an arbitrary complex bipartite matrix.

For \(r,s\), let \(P_{\mathrm{top}}=C_{r,s}^\dagger C_{r,s}\) and \(P_{\mathrm{bot}}=\mathbb {1}-P_{\mathrm{top}}\). Both are PSD (for \(P_{\mathrm{bot}}\), by Lemma 3.11.10) and \(P_{\mathrm{top}}+P_{\mathrm{bot}}=\mathbb {1}\).

Let \(\{ R_i\} _{i\in I}\) be a finite family of simple rings. The \(i\)-th block ideal in \(\prod _{k\in I}R_k\) is

For natural numbers \(r,s\), let \(C=C_{r,s}\in \mathbb {C}^{r\times (r+s)}\) be the rectangular \(r\times (r+s)\) matrix whose rows are the first \(r\) rows of the identity on \(\mathbb {C}^{r+s}\): \(C_{ij}=1\) if \(j=i{\lt}r\) and \(0\) otherwise.

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

For a linear map \(E : M_{D_{\rm in}}(\mathbb {C})\to M_{D_{\rm out}}(\mathbb {C})\), its \(k\)-fold ampliation acts between the corresponding block-matrix algebras by

For a matrix \(U\) on \(M_{D}(\mathbb {C})\), the Breuer–Hall map on \(M_{D}(\mathbb {C})\) is

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 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the channel determinant \(\det T\) is the determinant of the matrix of \(T\) with respect to the standard matrix-unit basis of \(M_{D}(\mathbb {C})\). This is the quantity studied in [ Wol12 , Section 6.1 ] .

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

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 ] .

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 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) with Choi–Jamiołkowski operator \(\tau \), the purity of \(\tau \) is \(\operatorname{tr}[\tau ^\dagger \tau ]\), the sum

of the squared moduli of the entries of \(\tau \); in particular it is non-negative.

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) have Choi matrix \(\tau \) on \(\mathbb {C}^{d'}\otimes \mathbb {C}^d\). For \(X\in M_{d\times k}(\mathbb {C})\), the right-factor compression is the matrix on \(\mathbb {C}^{d'}\otimes \mathbb {C}^k\) with entries

The associated vector on \(\mathbb {C}^d\otimes \mathbb {C}^k\) has coefficients \(d^{-1/2}X_{a,p}\).

For \(X\in M_{d\times k}(\mathbb {C})\), let \(R_X\) be the matrix from \(\mathbb {C}^{d'}\otimes \mathbb {C}^d\) to \(\mathbb {C}^{d'}\otimes \mathbb {C}^k\) with entries

On the cyclic index set \(\mathbb {Z}/d\mathbb {Z}\), the Choi-type map is

where \(D\) keeps the diagonal of a matrix and \(U_{k0}\) is the cyclic shift by \(k\).

For a vector \(v\) on \(\mathbb {Z}/d\mathbb {Z}\), set

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

The commutant of a Lindblad form is the set of matrices commuting with \(H\), with every Lindblad operator \(L_j\), and with every adjoint \(L_j^\dagger \).

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 \(M_D(\mathbb {C})\).

The commutator form of the Lindblad equation writes the generator as

where \([A,B] = AB - BA\). This is [ Wol12 , Equation (7.22) ] .

Let \(V\) and \(V'\) be finite-dimensional real Hilbert spaces, let \(K\subseteq V\) be a closed pointed convex cone with nonempty interior, let \(T:V\to V'\) be linear, and fix \(c\in V\) and \(b\in V'\). As in [ Wol12 , Chapter 4, equations (4.1)–(4.2) ] , define the dual cone, the primal and dual feasible sets, and their values by

The values belong to the extended real line. The conventions are \(\inf \varnothing =+\infty \) and \(\sup \varnothing =-\infty \); an objective unbounded below has infimum \(-\infty \), and one unbounded above has supremum \(+\infty \). The nonempty-interior assumption is Wolf’s convention for a conic program; the definitions and weak duality do not require it.

Following [ Wol12 , Chapter 4, lines 72–78 ] , the primal problem is strictly feasible when there is an \(x\in \operatorname {int}K\) with \(T(x)=b\). The dual problem is strictly feasible when there is a \(y\in V'\) such that \(c-T^*(y)\in \operatorname {int}K^*\). A primal optimizer is an \(x^0\in \mathcal F_p\) satisfying \(\langle c|x^0\rangle \leq \langle c|x\rangle \) for every \(x\in \mathcal F_p\); a dual optimizer is a \(y^0\in \mathcal F_d\) satisfying \(\langle b|y\rangle \leq \langle b|y^0\rangle \) for every \(y\in \mathcal F_d\).

A dynamical semigroup \(T\) is norm-continuous if the map \(t \mapsto T_t\) is continuous in the operator norm topology. In finite dimension this coincides with strong continuity.

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 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 \(\lambda _{i_1}=c\), \(\rho _{i_1}=\rho _1\).

The restriction of \(E\) to an invariant corner \(PM_{D}(\mathbb {C})P\).

For an idempotent \(P\in M_{D}(\mathbb {C})\), the subspace \(\{ X\mid PXP=X\} \) and the corner \(\{ PXP\mid X\in M_{D}(\mathbb {C})\} \) are identified as \(\mathbb {C}\)-linear spaces by the identity on underlying matrices.

For linear maps \(S,T:M_{d_{\rm in}}(\mathbb {C})\to M_{d_{\rm out}}(\mathbb {C})\), write \(T\le S\) (in the CP order) when \(S-T\) is completely positive.

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 \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\). A block-permutation structure for \(T\) consists of a finite index set \(\iota \), a family of orthogonal projections \(P_k\in M_{D}(\mathbb {C})\) for \(k\in \iota \), and a permutation \(\sigma \in \mathrm{Sym}(\iota )\), not necessarily a single cycle, such that, for every \(k\in \iota \) and \(X\in M_{D}(\mathbb {C})\),

In general, \(\sigma \) may have multiple disjoint cycles.

A map \(\Phi \colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is completely copositive if \(\Phi \circ \theta \) is completely positive, where \(\theta \) is transposition. It is decomposable if \(\Phi =\Phi _{\mathrm{cp}}+\Phi _{\mathrm{ccp}}\) with \(\Phi _{\mathrm{cp}}\) completely positive and \(\Phi _{\mathrm{ccp}}\) completely copositive. It is indecomposable if it is positive and not decomposable.

A witness \(W\) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) is decomposable when

This is the explicit cone in Wolf’s Equation (3.15), with partial transposition on the first factor.

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

In the existing matrix-family coordinates for \(\mathcal A=\bigoplus _{k\in I}M_{d_k}(\mathbb {C})\), its density states are the families \(A=(A_k)_{k\in I}\) such that every \(A_k\succeq 0\) and \(\sum _k\operatorname{tr}(A_k)=1\).

Let

and let \(\mathcal H_A=\bigoplus _{k\in I}\mathbb {C}^{n_k}\) and \(\mathcal H_B=\bigoplus _{l\in J}\mathbb {C}^{m_l}\). Write \(\iota _A,\iota _B\) for the block-diagonal embeddings and \(\pi _A,\pi _B\) for diagonal-block compression. For a linear map \(T:\mathcal A\to \mathcal B\), its canonical full-matrix extension is

For \(X\in \mathcal A\) and \(Y\in \mathcal B\), the trace adjoint \(T^*:\mathcal B\to \mathcal A\) is defined by

This is the block-diagonal realization needed for the fixed-point and classification arguments in [ CPGSV16 , Appendix C.4, lines 1980–2003 ] . It is only an auxiliary construction: it does not assert the classification conclusion of that passage.

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and the reduced density matrix \(\operatorname{tr}_{1}[\tau ]\) of its Choi matrix \(\tau =(T\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)\) is proportional to the identity. This is the normal form in [ Wol12 , Proposition 2.9 ] .

A linear map \(T : M_{d_1}(\mathbb {C}) \to M_{d_2}(\mathbb {C})\) between matrix algebras of possibly different dimensions is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and \(T^{*}(\mathbb {1})\propto \mathbb {1}\), where \(T^{*}\) is the trace-pairing adjoint. This is the normal-form condition of [ Wol12 , Proposition 2.9 ] ; by the rectangular Choi–Jamiolkowski correspondence it is equivalent to both partial traces of the Choi matrix being proportional to the identity.

A family of linear maps \(T : \mathbb {R}\to (M_{D}(\mathbb {C}) \to _{\ell } M_{D}(\mathbb {C}))\) is a dynamical semigroup if

1. (semigroup law) \(T_{t+s} = T_t \circ T_s\) for all \(t,s \ge 0\); and

2. (initial condition) \(T_0 = \operatorname{id}\).

This is [ Wol12 , Equation (7.1) ] .

Let \(d_A{\gt}0\) and let \(\rho _{BC}\) be positive semidefinite. Define

Under the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), the right partial trace removes \(C\).

This formulation has the same value \(I(A{:}B) = S(\rho _A) + S(\rho _B) - S(\rho _{AB})\) as in Definition 13.7.1.

Let \(\rho _A\) and \(\rho _{BC}\) be positive semidefinite, and define

We use the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), so that the right partial trace removes \(C\).

This is the entropy formulation of the Hayashi Markov decomposition from Definition 13.6.55.

If \(P_A\) is the support projection of \(\rho _A\), define

This formulation has the same value \(S(\rho ) = -\sum _i \lambda _i \log \lambda _i\) as in Definition 13.4.1.

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:

Given a generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\), the exponential semigroup is

In \(\mathbb {C}^2 \otimes \mathbb {C}^2\), let \(v = |01\rangle - |10\rangle \). Equivalently, \(v_{(0,1)}=1\), \(v_{(1,0)}=-1\), and all other coefficients are zero.

For \(t{\gt}0\), set \(h(t)=-t\log t\), and set \(h(0)=0\). The entropy of a probability distribution \(a=(a_z)_{z\in Z}\) on a finite set is \(H(a)=\sum _{z\in Z}h(a_z)\). For a joint probability distribution \(P\) with row and column marginals \(p\) and \(q\), put

Let \(X\) and \(Y\) be finite sets. A matrix \(P=(P_{x,y})_{x\in X,y\in Y}\) is a joint probability distribution if \(P_{x,y}\geq 0\) for every \(x\in X\) and \(y\in Y\), and

Its row and column marginals are respectively

For \(r\geq 1\), let \(W_0:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) be the isometry \(W_0x=x\otimes e_0\).

A Fitting decomposition of a linear endomorphism \(f:V\to V\) on a finite-dimensional vector space over an algebraically closed field consists of:

1. \(f\) is nilpotent on the generalized \(0\)-eigenspace,

2. \(f\) is invertible on each generalized \(\mu \)-eigenspace for \(\mu \neq 0\),

3. the generalized eigenspaces span \(V\),

4. the generalized eigenspaces are linearly independent.

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.

The flip operator \(F\) on \(\mathbb {C}^d \otimes \mathbb {C}^d\) is

Its matrix entries are

Let

A real matrix \(C\in M_4(\mathbb R)\) is \(M\)-selfadjoint when \(C^{\mathsf T}M=MC\).

For a complex-linear map \(E:\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define its Frobenius transport by

For finite index sets \(I\) and \(J\), define

This is a complex-linear isometric equivalence when the matrix space is equipped with the Frobenius norm.

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

A generator decomposition consists of a completely positive map \(\phi : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) and a matrix \(\kappa \in M_{d}(\mathbb {C})\), defining the linear map

This is [ Wol12 , Equation (7.14) ] .

For a projection \(P\), the generator preserves the compression \(PM_{D}(\mathbb {C})P\) if, for every \(X\in M_{D}(\mathbb {C})\),

Following Ha’s displayed decomposition [ Ha98 , pp. 594–595 ] , continue to use zero-based cyclic indices, and let \(\mathcal B_d\) be the set of pairs

This is the zero-based form of Ha’s two displayed ranges for the vectors \(\beta _{1j}\) and \(\beta _{ij}\); it is empty when \(d=3\). Define

and let

Let \(\sigma (x\otimes y)=y\otimes x\), and write \(A^\sigma \) for the corresponding exchange of both tensor factors of a bipartite matrix. This exchange is an involution. If \(A=\sum _{i,j}a_{ij}\otimes e_{ij}\) and \(\Phi :M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) is linear, define the Eom–Kye bilinear pairing by

For \(J_d=\sum _{i,j}e_{ij}\otimes e_{ij}\), its associated matrix pairing is \(\langle X,J_d\rangle =\sum _{i,j}X_{(i,i),(j,j)}\).

The Choi-type map may be transported from the cyclic basis \(\mathbb {Z}/d\mathbb {Z}\) to the standard basis indexed by \(0,\ldots ,d-1\), and \(A_\gamma \) may be transported in the opposite direction on both tensor factors. These simultaneous changes of basis identify the two forms of the pairing in Theorem 4.8.1.36; they do not add a positivity hypothesis.

Fix \(d\ge 3\) and \(\gamma {\gt}0\). Following Ha’s proof of [ Ha98 , Theorem 2.1 ] , put \(m_k=\frac32(3^k-1)\) for \(0\le k{\lt}d\), choose \(\zeta =\exp (2\pi i/3^d)\), and enumerate the \(3^d\)-th roots of unity by \(\omega _i=\zeta ^i\) for \(0\le i{\lt}3^d\). Define

If \(S e_p=e_{p+1}\) with cyclic indices, set

With \(\circ \) denoting coordinatewise multiplication, let

Condition (4): there exists a nontrivial projector \(P\) and a Lindblad form \((H,\{ L_j\} )\) for \(L\) such that \((\mathbb {1}-P)L_jP=0\) and \((\mathbb {1}-P)\kappa P=0\) for all \(j\), where \(\kappa =iH+\frac{1}{2}\sum _jL_j^\dagger L_j\).

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 generator \(L\) has a faithful stationary state if there exists a positive definite density matrix \(\rho _0\) with \(L(\rho _0)=0\).

Condition (3): there exists a nontrivial orthogonal projector \(P\) such that \(e^{tL}(PM_{D}(\mathbb {C})P)\subseteq PM_{D}(\mathbb {C})P\) for all \(t\ge 0\).

The commutant of the Lindblad family \(\{ L_j\} '\) equals \(\mathbb {C}\cdot \mathbb {1}\), i.e. the Lindblad operators act irreducibly on \(M_{D}(\mathbb {C})\).

Condition (1): there exists a density matrix \(\rho _0\) with nontrivial kernel such that \(e^{tL}(\rho _0)=\rho _0\) for all \(t\ge 0\).

Condition (2): there exists a density matrix \(\rho _0\) with nontrivial kernel satisfying \(L(\rho _0)=0\).

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) has Schmidt number at most \(n\) when it is a finite sum of pure-state projectors of Schmidt rank at most \(n\),

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).

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 Lindblad form has traceless Kraus operators if \(\operatorname{tr}(L_j) = 0\) for every \(j\).

A Hayashi Markov decomposition of a tripartite state \(\rho _{ABC}\) consists of a finite direct-sum decomposition

together with a unitary change of basis on \(B\), a probability vector \((p_j)_j\), and density matrices \(\rho _{A B_j^L}\) and \(\rho _{B_j^R C}\) such that, in the adapted basis, the state becomes

The terminology follows Hayashi’s presentation of quantum Markov structure  [ Hay06 ] ; the block decomposition used by the MPDO argument is the structure theorem of [ HJPW04 ] .

The Hermitian \(n\times n\) matrices form a finite-dimensional real Hilbert space for Wolf’s trace pairing \(\langle A|B\rangle =\operatorname {Re}\operatorname{tr}(AB)\). Its positive-semidefinite matrices form a closed proper cone \(K_{\mathrm{psd}}\) satisfying \(K_{\mathrm{psd}}^*=K_{\mathrm{psd}}\), and \(\operatorname {int}K_{\mathrm{psd}}=\{ A:A{\gt}0\} \). The last identity also covers the zero-dimensional matrix space, where positive definiteness is vacuous and the cone is the whole space.

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

Let \(A\) be Hermitian, with spectral decomposition \(A=U\operatorname{diag}(\lambda _i)U^\dagger \). Its support projection is

A family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) is Hilbert–Schmidt orthonormal when

This is the pairing \(\operatorname{tr}(PA^\dagger B)\) of [ Wol12 , Equation (2.18) ] in the case \(P=\mathbb {1}\).

For a finite-dimensional system \(A\), there is a finite family of effects \((M_s)_s\), with \(0\leq M_s\leq \mathbf1_A\), whose complex linear span is the full matrix algebra on \(A\). One member is the identity. For an operator \(X\) on \(A\otimes B\), define its conditional slice by

The family may be chosen from the four rank-one effects occurring in the polarization identity, scaled so that every member is bounded by the identity.

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 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.

Associated to an instrument are the total channel \(\sum _i\Phi _i\), the unnormalized update \(\rho \mapsto \Phi _i(\rho )\) for each outcome \(i\), the outcome probability \(p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))\), and the normalized posterior state \(\Phi _i(\rho )/p_i(\rho )\) whenever \(p_i(\rho )\neq 0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a bijective complex-linear map. Write \(T^{-1}\) for the inverse linear map supplied by the associated linear equivalence; it is both a left and a right inverse of \(T\).

An invertible filtering operation on \(M_{D}(\mathbb {C})\) is a completely positive map

If \(T:M_{d_1}(\mathbb {C})\to M_{d_2}(\mathbb {C})\), pre- and postfiltering are written in Wolf’s order as \(\Phi _2\circ T\circ \Phi _1\). The matrices \(X\) are retained as part of the data; in particular, matrices which differ by a phase are not identified.

Let \(I\) be a finite index set and let \(C\in \mathbb C^{I\times I}\) be invertible. Congruence by \(C\) is the complex-linear equivalence

whose inverse is

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 \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is conditionally completely positive (CCP) if it admits a generator decomposition, i.e. there exist a CP map \(\phi \) and a matrix \(\kappa \) such that \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \). This is condition 1 of [ Wol12 , Proposition 7.2 ] .

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 \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is a GKSL generator if \(e^{tL}\) is a quantum channel (CPTP) for all \(t \ge 0\).

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 span of the Lindblad operators is closed under Hermitian conjugation, i.e. it forms a \(*\)-subspace of \(M_{D}(\mathbb {C})\).

A Naimark dilation of a POVM \(\{ E_i\} \) consists of an isometry \(V\) into a larger Hilbert space together with a projective measurement \(\{ P_i\} \) there such that \(E_i=V^\dagger P_iV\) for all \(i\).

A projection is nontrivial if it is orthogonal and neither \(0\) nor \(\mathbb {1}\).

A bipartite matrix \(\rho \) has the positive-partial-transpose (PPT) property when its first-factor partial transpose is positive semidefinite, \(\rho ^{T_1}\ge 0\).

A quantum dynamical semigroup is a norm-continuous dynamical semigroup whose time slices are quantum channels for every \(t \ge 0\).

A quantum dynamical semigroup is reducible if it has a nontrivial invariant compression.

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) is separable when it is a finite sum of Kronecker products of positive semidefinite matrices,

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is trace-annihilating if \(\operatorname{tr}(L(\rho )) = 0\) for all \(\rho \in M_{d}(\mathbb {C})\). This is the infinitesimal version of trace preservation.

A generator decomposition \((\phi , \kappa )\) satisfies the trace constraint if \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), where \(\phi ^*\) is the Hilbert–Schmidt adjoint. This is the condition in [ Wol12 , Equation (7.20) ] .

Let \(V:\mathbb {C}^k\to \mathbb {C}^D\) be an isometry, so that \(V^\dagger V=\mathbb {1}_k\). Define \(\iota _V:M_{k}(\mathbb {C})\to M_{D}(\mathbb {C})\) by

The map \(\iota _V\) is complex-linear, multiplicative, and \(*\)-preserving. In general it is not unital: \(\iota _V(\mathbb {1}_k)=VV^\dagger \) is the projection onto the range of \(V\).

A linear map \(E : M_{D_{\rm in}}(\mathbb {C}) \to M_{D_{\rm out}}(\mathbb {C})\) is \(k\)-positive if \(E \otimes \operatorname{id}_{k}\) is positive from \(M_{D_{\rm in}}(\mathbb {C}) \otimes M_{k}(\mathbb {C})\) to \(M_{D_{\rm out}}(\mathbb {C}) \otimes M_{k}(\mathbb {C})\).

Assume in addition that the common average \(\bar\rho \) of Definition 13.6.56 is positive definite. Hayden–Jozsa–Petz–Winter instead reduce to this case by shrinking to the joint support of \(\rho _1,\ldots ,\rho _K\); that reduction is not re-derived here. By Lemma 13.6.57, \(\bar\rho \) is a positive definite fixed point of every \(F\in \mathbf F\), so Theorem 10.2.10 makes the fixed-point set of each adjoint map,

a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). The common invariant algebra is

and \(X\in A_0\) if and only if \(F^*(X)=X\) for every \(F\in \mathbf F\).

Fix \(F_0\in \mathbf F\) with \(A_0=A_{F_0}\) as in Theorem 13.6.61. Let \(P_0\) be the finite-dimensional mean-ergodic projection of the Schrödinger map \(F_0\), and define \(P_0^*\) as its trace-pairing adjoint.

Let \(\rho _1,\ldots ,\rho _K\) be density matrices in \(M_{D}(\mathbb {C})\), \(K\ge 1\). A trace-preserving completely positive Kraus family \(F\) preserves \(\rho _1,\ldots ,\rho _K\) if \(F\rho _k=\rho _k\) for every \(k\). Write

for the set of such operations – non-empty since the identity operation belongs to it – and

for their common average.

A Kossakowski form consists of \(H=H^\dagger \), a finite family of matrices \(\{ F_k\} _{k=0}^{n-1}\), and a positive semidefinite matrix \(C \in M_{n}(\mathbb {C})\), defining

In Wolf’s formula one takes \(\{ F_k\} \) to be a basis of traceless matrices; here we keep only the algebraic data needed for the conversion to Lindblad form.

The minimal number of Lindblad operators across all GKSL representations of \(L\), which equals the rank of the Kossakowski matrix in the orthonormal-basis representation.

The adjoint Kraus map is

When the \(\{ K_i\} \) are the matrices of an MPS tensor \(A\), this is the transfer map of the conjugate-transposed family \(i \mapsto (A^i)^\dagger \).

The Kraus commutant is

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

Let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices. For a word \(w=(i_1,\ldots ,i_n)\), its word evaluation is

with the empty word evaluated as the identity matrix.

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.

The fixed-point set of a Kraus map \(E(X)=\sum _iK_iXK_i^\dagger \) is \(\operatorname{Fix}(E)=\{ X\in M_{D}(\mathbb {C}):E(X)=X\} \).

Given operators \(\{ K_i\} _{i=0}^{d-1}\) with \(K_i \in M_{D}(\mathbb {C})\), the Kraus map is

A linear map is completely positive (Definition 2.1.5) if and only if it can be written in this form, as in (1).

Let

Write \(\iota _{\mathcal A},\pi _{\mathcal A}\) and \(\iota _{\mathcal B},\pi _{\mathcal B}\) for the corresponding block-diagonal embeddings and diagonal-block compressions. The canonical full-matrix extension of \(T:\mathcal A\to \mathcal B\) is

The map \(T\) is a direct-sum Kraus map when \(\widehat T\) has a Kraus representation.

Let \(\{ A_i\} _{i=0}^{d-1}\subseteq M_{D_1}(\mathbb {C})\) 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.

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian with eigenvalues \(\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _D\) listed in decreasing order. For \(0\le k\le D\), set

This is the sum of the \(k\) largest signed eigenvalues. Indices beyond \(D\) contribute zero, so the value stabilizes at \(\operatorname{Re}\operatorname{tr}(A)\) once \(k\) reaches \(D\).

The Ky-Fan \(k\)-norm \(\| A\| _{(k)}\) is by definition the sum of the \(k\) largest singular values of \(A\). It agrees with \(S_k(A)\) exactly when \(A\) is positive semidefinite, where eigenvalues and singular values coincide; for an indefinite Hermitian \(A\) the two differ. For \(A=\operatorname{diag}(1,-2)\), one has \(S_1(A)=1\) but \(\| A\| _{(1)}=2\). The results below concern \(S_k(A)\) for arbitrary Hermitian \(A\), so on the positive-semidefinite cone they are statements about the Ky-Fan norm.

A Lindblad form consists of a Hermitian matrix \(H = H^\dagger \) (the Hamiltonian) and a family of matrices \(\{ L_j\} _{j=0}^{r-1}\) (the Lindblad operators), defining the linear map

where \([A,B] = AB - BA\) and \(\{ A,B\} _+ = AB + BA\). This is [ Wol12 , Equation (7.21) ] .

The \(\mathbb {C}\)-linear span of the Lindblad operators \(\{ L_j\} \).

Let \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) be complex-linear. Its coefficient matrix \(C(\mathcal L)\in M_{e^2\times d^2}(\mathbb {C})\) in the matrix-unit bases is defined by

If a Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\) acts on the first factor while a finite-dimensional space \(R\) is left unchanged, the corresponding local Stinespring matrix is \(W=V\otimes \mathbb {1}_R\).

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) for its matrix in the normalized Pauli basis \(\{ \sigma _{0}/\sqrt{2},\sigma _{1}/\sqrt{2}, \sigma _{2}/\sqrt{2},\sigma _{3}/\sqrt{2}\} \). The channel is in diagonal Lorentz normal form when it is unital and every off-diagonal entry of \(\widehat{T'}\) is zero.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in non-diagonal Lorentz normal form when, for some \(x \in [0,1]\),

Trace preservation supplies the first row of the displayed matrix.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in singular Lorentz normal form when

equivalently, every input state is mapped to \((1+\sigma _{3})/2\). Trace preservation supplies the first row of the displayed matrix.

A linear map \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) maps the positive semidefinite cone onto itself if \(T\) is positive and every positive semidefinite matrix is the image under \(T\) of a positive semidefinite matrix. This is condition (1) of [ Wol12 , Proposition 3.6 ] .

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 \(\operatorname{tr}(|\Omega \rangle \! \langle \Omega |)=1\).

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

Let \(H=\bigoplus _k H_k\) 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.

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

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

A multi-cycle decomposition of \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) consists of a finite cycle index set \(\iota \), a per-cycle period \(m:\iota \to \mathbb {N}_{{\gt}0}\), and a family of orthogonal projections \(P_{c,k}\in M_{D}(\mathbb {C})\) for \(c\in \iota \) and \(k\in \{ 0,\ldots ,m(c){-}1\} \). For every \(c\in \iota \) and \(k\in \{ 0,\ldots ,m(c){-}1\} \), the per-cycle cyclic action is \(T(P_{c,k+1})=P_{c,k}\). The multiplicative-domain factorizations are \(T(P_{c,k}X)=T(P_{c,k})T(X)\) and \(T(XP_{c,k})=T(X)T(P_{c,k})\).

A multi-cycle decomposition gives a single-index block-permutation structure on the disjoint-union index \(\Sigma _{c\in \iota }\{ 0,\ldots ,m(c){-}1\} \): the permutation is the product of the per-cycle cyclic shifts \(k\mapsto k+1\), whose cycle decomposition has one cycle per \(c\in \iota \); the projections are \((c,k)\mapsto P_{c,k}\); and the multiplicative-domain factorizations descend componentwise. The resulting map forgets the explicit cycle indexing.

The quantum mutual information of a bipartite state \(\rho _{AB}\) is

The Naimark isometry \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n\) of a POVM is the Stinespring-type construction

For a POVM \(\{ E_i\} \), each effect \(E_i\ge 0\) admits a square-root factorisation \(E_i=M_i^\dagger M_i\) with \(M_i\in M_{D}(\mathbb {C})\). The operators \(M_i\) are the Naimark Kraus square roots.

The Naimark projectors on \(\mathbb {C}^D\otimes \mathbb {C}^n\) are

Following Lewenstein–Kraus–Cirac–Horodecki [ LKCH00 ] , a normalized decomposable witness is an operator of the form

The transpose remains on Wolf’s first tensor factor.

Let \(I\) be finite and let \(A=\mathbb {R}^I\), with coordinatewise multiplication and the faithful positive functional \(\varphi (x)=\sum _{i\in I}x_i\). Its strictly positive invertible cone is

and the induced Hilbert norm and Nowosad functional are

For \(w\in A\), write \(P(w)=\operatorname {alg}_{\mathbb {R}}\{ w,w^{-1}\} \) for its Laurent-polynomial subalgebra.

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.

For a finite family of matrices \(\{ C_i\} _{i\in \iota }\) and a defect block \(S\), one forms an isometric dilation whose rows are the adjoints of the matrices \(C_i\) together with the adjoint of \(S\). One also forms the scalar block-diagonal matrix whose \(\iota \)-blocks carry prescribed weights and whose defect block carries a single scalar.

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 \((X_k^\dagger )_{k{\lt}r}\).

Let \(V:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^r\) be a supplied Stinespring matrix, and let \(|\Omega \rangle =d'^{-1/2}\sum _{a=1}^{d'}|a,a\rangle \) be normalized. Wolf’s auxiliary operator is

or, in coordinates, \(W_{j,(i,a)}=d'^{-1/2}V_{(a,j),i}\).

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

For a matrix \(V:\mathbb C^K\to \mathbb C^J\) and a complex-linear map \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define \(\Psi (X)=V^\dagger \Phi (X)V\).

Let \(\{ \psi _k\} _{k {\lt} m}\) be an ensemble and let \(n\) be a natural number. The padded ensemble \(\{ \psi _k^{\mathrm{pad}}\} _{k {\lt} n}\) is \(\psi _k\) for \(k {\lt} m\) and the zero vector for \(m \le k {\lt} n\). For \(m \le n\) the padded family agrees with \(\psi \) on every index \(k {\lt} m\) and retains the whole ensemble; for \(n {\lt} m\) only the first \(n\) vectors survive. The intended use is \(m \le n\).

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

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

The completed Petz map is

Put \(Q_\tau =\mathbf1_L-P_\tau \) and \(\omega _R=\operatorname{tr}_L\sigma \). The complementary term is

Let \(\sigma \) be positive semidefinite on \(H_L\otimes H_R\), and set \(\tau =\operatorname{tr}_R\sigma \). The Petz transpose formula on the support of \(\tau \) is

This is the support formula of [ HJPW04 , Theorem 3, equation (8) ] . It is not asserted to be trace preserving on operators outside the support of \(\tau \).

For a bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), the partial transpose over the first factor is the matrix \(\rho ^{T_1}\) with entries

transposing the first index pair \(i,j\) and fixing the second pair \(k,l\).

For a bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), the partial transpose over the second factor is the matrix \(\rho ^{T_2}\) with entries

transposing the second index pair \(k,l\) and fixing the first pair \(i,j\).

Define Pauli time reversal by \(\Theta (X)=\operatorname{tr}(X)\mathbb {1}-X\). Its Pauli transfer matrix is

For a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\), define its Pauli-block diagonal truncation by

This is the truncation in [ Wol12 , Proposition 2.10, Section 2.4 ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 984–998.

Every Hermitian matrix \(M\in M_2^\dagger (\mathbb {C})\) has unique real Pauli coordinates \(x=(x_0,x_1,x_2,x_3)\) such that

This identifies \(M_2^\dagger (\mathbb {C})\) with \(\mathbb {R}^4\). Write

for the Minkowski quadratic form, its matrix, and the closed future cone. These are the coordinates used immediately before [ Wol12 , Eq. (2.41) ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 1040–1044.

Let \(\sigma _0=\mathbb {1},\sigma _1,\sigma _2,\sigma _3\) be the Pauli matrices. The Pauli transfer matrix of a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\) is \(\widehat T\in M_4(\mathbb {C})\) with entries

We write

where \(\Delta =(\widehat T_{ij})_{i,j=1}^{3}\). Thus \(r^{\mathsf T}\) and \(v\) are exactly the two off-diagonal Pauli blocks. This is the representation used in [ Wol12 , Section 2.4, Eq. (2.39) ] .

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 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) ] .

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

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

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 positive operator-valued measure with \(n\) outcomes on \(\mathbb {C}^D\) is a family \(\{ E_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^D\) satisfying the resolution of identity

Given an isometry \(V:\mathbb {C}^D\to \mathbb {C}^{d'}\) with \(V^\dagger V=\mathbb {1}_D\) and a family \(\{ P_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^{d'}\) summing to the identity, the pulled-back operators \(E_i:=V^\dagger P_iV\) form a POVM. In particular, any projective measurement on the dilation (a special case of a PSD resolution of identity) pulls back to a POVM.

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

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

For \(P\in M_{D}(\mathbb {C})\), define

Given a finite family \(\{ \psi _i\} _{i \in \iota }\) of (unnormalized) vectors in \(\mathbb {C}^D\), its pure-ensemble density is \(\rho = \sum _i |\psi _i\rangle \! \langle \psi _i|\). Weights \(p_i \geq 0\) with \(\sum p_i = 1\) can be absorbed by replacing \(\psi _i \mapsto \sqrt{p_i} \psi _i\).

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 matrices \(\rho ,\sigma \in M_{D}(\mathbb {C})\), define the trace-log expression

On the physical domain where \(\rho \) is a density matrix and \(\sigma \) is positive definite, this is the Umegaki relative entropy.

A real-eigenvalue block is the ordinary real Jordan block \(J_n(a)\) with eigenvalue \(a\) and ones on its first superdiagonal. If \(a\pm \tau i\) is a non-real conjugate pair, put

Its real Jordan chain of length \(m\) has \(B(a,\tau )\) on the block diagonal and \(I_2\) on the first block superdiagonal. On the product indexing of this \(2m\)-dimensional real block, its unsigned metric reverses both the chain coordinate and the two-dimensional realification coordinate. No sign characteristic is attached to a non-real block.

For a complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\), its unnormalized rectangular Choi matrix \(J(\mathcal L)\in M_{de}(\mathbb {C})\) is the reshaping

When \(d=e\), this is \(d\) times the normalized Choi matrix convention in [ Wol12 , Proposition 2.1 ] .

A complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) is a Hilbert–Schmidt contraction if, for every \(X\in M_{d}(\mathbb {C})\),

Let \(\tau \) be a Hermitian operator on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) with normalized eigenvectors \(\phi _i\). The reduced density operator of the \(i\)-th eigenvector on the first factor is \(\rho _i=\operatorname{tr}_2|\phi _i\rangle \! \langle \phi _i|\).

The reduction map on \(M_{D}(\mathbb {C})\), for \(k\in \mathbb {N}\), is

The reduction witness on \(M_{D}(\mathbb {C})\otimes M_{D}(\mathbb {C})\), for \(n\in \mathbb {N}\), is

For \(u\ge 0\) and \(s{\gt}0\), the residual slice index is \(m_n=\left\lfloor nu/s\right\rfloor \).

For \(u\ge 0\) and \(s{\gt}0\), the residual slice time is the remainder \(r_n=s\, \mathrm{fract}\! \left(nu/s\right)\).

For square complex matrices \(\rho \) and \(\omega \) and every \(\alpha \in \mathbb {R}\), set

Real powers are defined by continuous functional calculus, with negative powers equal to zero on the zero eigenspace. Thus \(\widetilde Q_\alpha \) is total even when \(\alpha =0\) or \(\omega \) is singular. For \(\alpha {\gt}1\), on positive semidefinite inputs satisfying \(\ker \omega \subseteq \ker \rho \), this convention gives the finite sandwiched Rényi trace term. In this regime, if the support inclusion fails, the totalized value remains finite but is not the divergence trace term.

For square matrices \(\rho \) and \(\omega \) of the same size, define

The negative power is defined by functional calculus and vanishes on the kernel of \(\omega \). On trace-one positive semidefinite matrices satisfying \(\ker \omega \subseteq \ker \rho \), its logarithm is the order-two sandwiched Rényi divergence [ MLDS\(^{+}\)13 , Definition 2 ] .

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

For a matrix \(A\in M_{D}(\mathbb {C})\), the singular values \(s_0(A),s_1(A),\ldots \) form a finitely supported family: only finitely many are nonzero. The Schatten one-norm of \(A\) is their sum, that is, the sum of the finitely many nonzero singular values:

This is the \(p=1\) case of the Schatten \(p\)-norm [ Wol12 , Chapter 8, Section 8.1 ] . The equivalent closed formula \(\lVert A\rVert _1=\sum _{i=0}^{D-1}s_i(A)\), summing over all \(D\) singular values including trailing zeros, is part of Theorem 13.1.4.

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.

For a matrix \(\tau \) on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) and a bound \(n\), the set

collects the real parts of the quadratic forms of \(\tau \) in the normalized vectors of Schmidt rank at most \(n\). For Hermitian \(\tau \), the setting of every statement below, the quadratic form is real and \(E_n(\tau )\) is the set of expectations of \(\tau \) in those vectors.

The Schmidt singular values of a vector \(\psi \in \mathbb {C}^{D}\otimes \mathbb {C}^k\) are the singular values of the coefficient matrix \(C_\psi \), viewed as a linear map \(\mathbb {C}^k\to \mathbb {C}^D\).

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

For Hermitian data \(F_i\), the traces \(\operatorname{tr}(F_iX)\) are real; define \(T(X)_i=\operatorname{tr}(F_iX)\). Then \(T^*(y)=\sum _i y_iF_i\). Consequently the conic primal constraint is precisely \(X\geq 0\) and \(\operatorname{tr}(F_iX)=b_i\), while conic dual feasibility is precisely \(F_0-\sum _i y_iF_i\geq 0\). The two conic strict-feasibility predicates become, respectively, \(X{\gt}0\) with the trace constraints and \(F_0-\sum _i y_iF_i{\gt}0\), exactly as in [ Wol12 , Chapter 4, lines 85–105 ] .

Let \(d\ge 1\). A symmetric informationally complete family in dimension \(d\) consists of \(d^2\) rank-one orthogonal projections \(P_i\in M_{d}(\mathbb {C})\) satisfying

A matrix \(P\in M_{D}(\mathbb {C})\) is a rank-one orthogonal projection if \(P=P^\dagger \), \(P^2=P\), and \(\operatorname{rank}P=1\).

The sip matrix of size \(n\) is \(P_n\). A signed sip block is \(\varepsilon P_n\), where \(\varepsilon =1\) or \(\varepsilon =-1\). Each such block is symmetric, invertible, and its own inverse.

For \(w\in H_L\) and a distinguished basis vector \(e_r\in H_R\), define the lift \(J_r w=w\otimes e_r\in H_L\otimes H_R\).

An SL-filtering for \(D \times D\) matrices is a completely positive map of the form \(\Phi (X)=SXS^{\dagger }\), where \(S \in M_{D}(\mathbb {C})\) satisfies \(\det S=1\). Such maps are invertible; when \(D\geq 1\), they have Kraus rank 1.

For \(X\in \operatorname{SL}(2,\mathbb {C})\), let \(L(X)\) be the real linear transformation of \(\mathbb {R}^4\) determined by Wolf’s Hermitian congruence

Its Pauli-basis entries are

The trace is real. This is the four-dimensional action in [ Wol12 , Eq. (2.41) ] ; it is not the three-dimensional adjoint action \(M\mapsto UMU^{-1}\) on traceless Pauli matrices. The special orthochronous Lorentz group is described, in the row-action convention of the source, by

Regard \(\mathrm{SO}^+(1,3)\) from (95) as a matrix group. The spinor action defines the homomorphism

A tripartite density matrix \(\rho _{ABC}\) satisfies SSA equality if

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\). A linear operator \(f:\mathbb {C}^D\to \mathbb {C}^D\) is an intertwiner from \(W\) into \(W'\) when \(f(W)\subseteq W'\) and \(f(Ax)=A(fx)\) for every \(A\in S\) and \(x\in W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\). A subspace \(W\subseteq \mathbb {C}^D\) is irreducible under \(S\) when it is nonzero, invariant under every member of \(S\), and its only invariant subspaces are \(\{ 0\} \) and \(W\).

Let \(S\) be a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\), and let \(W,W'\subseteq \mathbb {C}^D\) be invariant under \(S\). The pieces \(W\) and \(W'\) are of the same type when there is an intertwiner from \(W\) into \(W'\) that is nonzero on \(W\).

For an irreducible channel \(E\) on \(M_{D}(\mathbb {C})\) with \(D\geq 1\), the stationary state is the unique density-matrix fixed point \(\rho _\infty \) satisfying \(E(\rho _\infty )=\rho _\infty \), \(\rho _\infty {\gt}0\), and \(\operatorname{tr}(\rho _\infty )=1\). Existence, uniqueness, and positive definiteness follow from Theorem 8.14.1.

The stationary support of an irreducible channel \(E\) is the support projection of its stationary state (Definition 10.4.2).

Given possibly rectangular Kraus operators \(K_j:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\), the Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\otimes \mathbb {C}^r\) is defined by

Thus \(V=\sum _j K_j\otimes |j\rangle \). It is an isometry precisely when the Kraus family satisfies the trace-preserving normalization.

For a Kraus family \((K_j)_{j\in J}\) indexed by an arbitrary finite set \(J\), define \((V_J)_{(i,j),k}=(K_j)_{ik}\).

Let \(\tau =\sum _i\lambda _i|i\rangle \! \langle i|\) be positive semidefinite. Its inverse square root on the support is

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.

Let \(\tau \geq 0\). Its support-restricted negative quarter power \(\tau ^{-1/4}_{\operatorname {supp}}\) acts by \(x^{-1/4}\) on every positive eigenspace and vanishes on the kernel. For a complex-linear map \(\Phi \), define

For a real parameter \(\eta \), the map \(T_\eta \) on \(M_{D}(\mathbb {C})\) is

The map is defined for every real \(\eta \); at \(\eta =0\) it reduces to \(\rho \mapsto \operatorname{tr}(\rho ) \mathbb {1}\), and the positivity threshold below assumes \(\eta {\gt}0\).

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 norm of \(A\in M_{D}(\mathbb {C})\) is its Schatten one-norm:

For \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(T_\phi \) be its peripheral spectral projection and define

This is Wolf’s notation in [ Wol12 , Chapter 8, Proposition “Convergence towards asymptotic states”, Eq. (8.112) ] .

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

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) and a trace-self-dual basis \(\{ \sigma _i\} \), define coefficients \(t_{ij}:=\operatorname{tr}(\sigma _i T(\sigma _j))\).

A basis \(\{ \sigma _i\} _i\) of \(M_{D}(\mathbb {C})\) is trace-self-dual when its coordinate functionals are given by trace pairing:

For any matrix \(Y\in M_{D'}(\mathbb {C})\), define \(T'_Y:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) by

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.

For a tripartite matrix \(\rho _{ABC}\) on \(\mathbb {C}^{d_A} \otimes \mathbb {C}^{d_B} \otimes \mathbb {C}^{d_C}\), the partial trace over \(A\) is

The partial trace over \(A\) and \(C\) is

The partial trace over \(C\) is

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

For a linear map \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) and a family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) used on both sides, the transfer matrix of \(T\) in \((\sigma _\alpha )\) is

which is [ Wol12 , Equation (2.20) ] with \(F_\alpha =G_\alpha \).

For matrices \(A,B\) on \(M_{d}(\mathbb {C})\) and the SWAP operator \(F\) on \(M_{d}(\mathbb {C})\otimes M_{d}(\mathbb {C})\), the transposition witness is \(W=(A\otimes B)\, F\, (A\otimes B)^\dagger \). These are the entanglement witnesses associated with the transposition map.

For Pauli directions \(a,b\in \{ x,y,z\} \) and real rates \(\gamma _a,\gamma _b\), define

In particular, \(\mathcal L_{a,b}(\mathbb {1})=0\).

The maximally mixed qubit state is \(\tau _1=\frac12\mathbb {1}_2\).

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

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

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

The four source-indexed patterns are: four real one-dimensional blocks; one two-dimensional non-real conjugate-pair block and two real one-dimensional blocks; one real Jordan block of size two and two real one-dimensional blocks; or one real Jordan block of size three and one real one-dimensional block.

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

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

For a Hermitian matrix \(\rho \in M_{D}(\mathbb {C})\) with eigenvalues \(\lambda _0,\ldots ,\lambda _{D-1}\), the von Neumann entropy is

where \(0\log 0:=0\).

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) ] .

For matrices \(\sigma \in \mathbb C^{I\times I}\) and \(\tau \in \mathbb C^{J\times J}\) and a complex-linear map \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define

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) ] .

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

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

Let \(S\) be invertible and suppose that \(S\mathbf1=S^{\mathsf T}\mathbf1=s\mathbf1\), where \(s{\gt}0\). For

a second positive point \(a\) is transformed to

where powers are taken coordinatewise.

For \(x\colon \mathbb {Z}/d\mathbb {Z}\to \mathbb {R}\), define the forward denominator and homogeneous functional

For the standard character \(\chi _j\) of \(\mathbb {Z}/d\mathbb {Z}\), the corresponding cyclic matrix eigenvalue is

These are Yamagami’s parameters with stride \(l=1\). Negating the cyclic index converts this forward window to the backward window in the Choi-type rank-one weight.

For a non-negative family \(f_{0},\ldots ,f_{D-1}\) of real numbers, or more generally for a non-negative family indexed by a finite set with \(D\) elements,

For \(A\in M_{D}(\mathbb {C})\) and a complex-linear map \(S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),

If \(\psi \) and \(\phi \) are orthogonal unit vectors, then \(\lVert |\psi \rangle \! \langle \psi |-|\phi \rangle \! \langle \phi |\rVert _2=\sqrt2\). Consequently the orthogonal-pure-state supremum for \(S\) is at most \(\sqrt D\, \lVert S\rVert _{2\to 2}\sqrt2\). These are precisely the estimates in Wolf Equations (8.115)–(8.116).

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 \(\{ \psi _i\} _{i=1}^n\subset \mathbb {C}^D\).

The block-top matrix satisfies \(CC^\dagger =\mathbb {1}_r\).

The block-top matrix satisfies \(C^\dagger C\le \mathbb {1}_{r+s}\).

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

If \(U\) is antisymmetric, \(U^{\mathsf T}=-U\), and a contraction, \(U^{\dagger }U\le \mathbb {1}\), then for every vector \(v\) the image \(T_{\mathrm{BH}}(|v\rangle \langle v|)\) of the rank-one operator \(|v\rangle \langle v|\) is positive semidefinite.

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 \(E(\sigma )=\sigma \).

For a Hermitian matrix \(A\), a real function \(f\), and a unitary \(U\), the continuous functional calculus satisfies

Let \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) be a linear map, \(\widehat{T}\) its matrix with respect to the matrix-unit basis and \(\tau \) its Choi–Jamiołkowski operator. Then

Equivalently \(\widehat{T}=D\, \tau ^{\Gamma }\), where the involution \(\tau \mapsto \tau ^{\Gamma }\) is defined by \(\langle m,n|\tau ^{\Gamma }|k,\ell \rangle =\langle m,k|\tau |n,\ell \rangle \). This is [ Wol12 , Eq. (2.22) ] .

Let \(A \in M_{m \times n}(\mathbb {C})\) and \(B \in M_{n \times m}(\mathbb {C})\). Then the charpoly-root entropy sum is invariant under the cyclic swap \(AB \mapsto BA\):

with roots counted with algebraic multiplicity.

Let \(D\geq 1\). Two linear maps \(T,S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) with the same Choi matrix are equal.

For arbitrary \(Y\in M_{D'}(\mathbb {C})\), the rectangular (output-factor-first) Choi matrix of \(T'_Y\) is

See [ Wol12 , Chapter 8, Eq. (8.86) ] .

With the notation of Lemma 3.17.2,

For \(D\geq 1\) this is [ Wol12 , Eq. (6.28) ] , which states it in the divided form \(\operatorname{tr}[\tau ^\dagger \tau ]=D^{-2}\operatorname{tr}[\widehat{T}^\dagger \widehat{T}]\).

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 \(d\ge 2\) and let \(x_i\ge 0\) be indexed by \(\mathbb {Z}/d\mathbb {Z}\). Then, with zero-denominator summands read as \(0\),

Consequently, for \(n=1\) and \(x_i=|v_i|^2\), the reciprocal sum of the rank-one weights \(a_i=(d-1)x_i+x_{i-1}\) is at most \(1\).

For a vector \(v\) on \(\mathbb {Z}/d\mathbb {Z}\),

Let \(x_i\ge 0\) be indexed by a finite set, let \(T=\sum _j x_j\), and let \(\sigma \) be a permutation of the index set. Then, with zero-denominator summands read as \(0\),

For \(d\ge 3\) and \(n=1\), every vector \(v\) satisfies \(T_C(|v\rangle \! \langle v|)\ge 0\).

Assume \(n\le d-2\). If, with the convention that the summand is \(0\) at indices where \(a_i=0\), \(\sum _i |v_i|^2/a_i\le 1\), where \(a_i\) is the weight of Definition 4.8.1.17, then \(T_C(|v\rangle \! \langle v|)\ge 0\).

For \(d\ge 3\) and \(n=d-2\), every vector \(v\) satisfies \(T_C(|v\rangle \! \langle v|)\ge 0\).

Let \(d\ge 3\) and \(n=d-2\). With \(x_i=|v_i|^2\) and \(T=\sum _j x_j\), the Choi rank-one diagonal weight equals

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 \(\Phi \) is completely copositive, then \(\Phi \) is positive.

For \(X\in M_{d\times k}(\mathbb {C})\), the vector

has Schmidt rank at most \(k\).

Let \(A\in M_{D}(\mathbb {C})\) be positive definite and set \(S=A^{1/2}\). Then

The channel raised to the period preserves every cyclic corner.

Appropriate powers of the channel preserve each sector corner.

Let \(T\) be irreducible, let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by \(T\), and suppose that every orthogonal projection \(Q\) with \(QP_k=P_kQ=Q\) that is invariant under the corner restriction of \(T^m\) to \(P_kM_{D}(\mathbb {C})P_k\) admits an orthogonal projection \(R\) invariant under \(T\) on the full algebra with \(Q=0\iff R=0\) and \(Q=P_k\iff R=\mathbb {1}\). Then, for every \(k\), the restriction of \(T^m\) to the corner \(P_kM_{D}(\mathbb {C})P_k\) is irreducible.

Each sector restriction is primitive.

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.

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.

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

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.

A primitive \(m\)-th peripheral root yields \(m\) orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by the channel.

If \(\Phi \) is decomposable, then \(\Phi \) is positive.

The eigenvalues of a density matrix sum to \(1\).

Let \(A\in M_{D}(\mathbb {C})\) with \(D\geq 1\). If \(|\det A|=1\), then \(D\leq \operatorname{tr}(A^{\dagger }A)\).

For a square complex matrix \(A\) whose index set has \(n\) elements,

Let \(f:\mathbb {R}\to \mathbb {R}\) be convex on \([0,\infty )\), let \(A\) be a positive semidefinite matrix on a finite-dimensional space indexed by \(n\), and let \(v\in \mathbb {C}^n\) satisfy \(\langle v,v\rangle =1\). Then

where \(f(A)\) is defined by the Hermitian continuous functional calculus.

For positive diagonal matrices \(M\) and \(N\) and \(s\in (0,1)\),

Let \(a_i\ge 0\) and suppose \(v_i=0\) whenever \(a_i=0\). If, with the convention that \(|v_i|^2/a_i\) is read as \(0\) when \(a_i=0\), \(\sum _i |v_i|^2/a_i\le 1\), then \(\operatorname{diag}(a_i)-|v\rangle \! \langle v|\ge 0\).

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

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

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

in every summand.

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

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

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

Let \(x_1,\dots ,x_m\) 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 ] .

For a Hermitian operator \(\tau \) with normalized eigenvectors \(\phi _i\), the eigenvector overlaps recover the squared norm of \(\psi \):

For a Hermitian operator \(\tau \) with eigenvalues \(\nu _i\) and normalized eigenvectors \(\phi _i\), the expectation in a vector \(\psi \) expands as

If two linearly independent vectors are eigenvectors of \(M\in M_{D}(\mathbb {C})\) for the same eigenvalue \(\lambda \), then the eigenspace of \(\lambda \) has dimension at least two.

The von Neumann entropy of a Hermitian matrix is the \(-x\log x\) sum over the real parts of the roots of its characteristic polynomial \(\chi _\rho \):

If two Hermitian matrices are equal, then their von Neumann entropies are equal.

The zero matrix has zero von Neumann entropy: \(S(0)=0\).

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

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) have Choi matrix \(\tau \). For every \(V\in M_{d\times k}(\mathbb {C})\),

The first-environment embedding satisfies \(W_0^\dagger W_0=\mathbb {1}\).

The fixed-point projection \(P\) has operator trace \(\operatorname{Tr}(P)=1\).

For finite matrices \(A\) and \(B\) of the same shape,

Suppose \(T(X)=\sum _j K_j X K_j^\dagger \) and also \(T(X)=\sum _j K_j^\dagger X K_j\). Then \(T\) has a family of Hermitian Kraus operators, namely \(\tfrac 12(K_j+K_j^\dagger )\) and \(\tfrac {i}{2}(K_j-K_j^\dagger )\).

Let \((\sigma _\alpha )\) be a Hilbert–Schmidt orthonormal basis of \(M_{d}(\mathbb {C})\). The coordinates of \(X\in M_{d}(\mathbb {C})\) in this basis are \(\operatorname{tr}(\sigma _\alpha ^\dagger X)\); consequently \(X=0\) as soon as \(\operatorname{tr}(\sigma _\alpha ^\dagger X)=0\) for every \(\alpha \). If in addition every \(\sigma _\alpha \) is Hermitian, the coordinates are \(\operatorname{tr}(\sigma _\alpha X)\), so the basis is self-dual for the bilinear trace pairing.

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

For a positive-definite operator \(\tau \) on \(\mathbb {C}^{d_2}\otimes \mathbb {C}^{d_1}\), the infimum of

over \(S_{1}\in M_{d_1}(\mathbb {C})\) and \(S_{2}\in M_{d_2}(\mathbb {C})\) with \(\det S_{1}=\det S_{2}=1\) is attained. The two tensor factors may have different dimensions.

Let \(D\geq 1\) and let \(X\in \mathrm{GL}(D,\mathbb {C})\). There are a nonzero \(c\in \mathbb {C}\) and \(S\in \operatorname{SL}(D,\mathbb {C})\) such that

Moreover, \(\Phi _X\) has Kraus rank exactly one. The matrix scalar \(c\) is complex and need not be positive real; positivity applies only to the map scalar \(|c|^2\).

Suppose \(X_i=c_iS_i\), where \(c_i\neq 0\) and \(S_i\in \operatorname{SL}(d_i,\mathbb {C})\). For every linear map \(T:M_{d_1}(\mathbb {C})\to M_{d_2}(\mathbb {C})\),

If \(E\) is irreducible and unital, then every fixed point of \(E\) is a scalar multiple of \(\mathbb {1}\).

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.

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

Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(X \ne 0\) satisfies \(E(X) = \mu X\) with \(|\mu | = 1\), then \(X\) is invertible.

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

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

Every \(F\in \mathbf F\) satisfies \(F\bar\rho =\bar\rho \).

Let \(H\in M_{D}(\mathbb {C})\) be Hermitian and let \(A\in M_{D}(\mathbb {C})\) satisfy \(HA=AH^{T}\) and \(A^{T}=-A\). If \(H|\psi \rangle =\lambda |\psi \rangle \), then the partner vector \(A|\overline\psi \rangle \) is again a \(\lambda \)-eigenvector of \(H\) and is orthogonal to \(|\psi \rangle \):

Suppose \(K^{i_0}\varphi =\mu \varphi \) with \(\mu \neq 0\). For every \(n\), the span of \(K^w\varphi \) over words with \(|w|\leq n\) equals \(H_n(K,\varphi )\).

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

Let \(\rho \) be a positive semidefinite operator on \(A \otimes R\) with reduced state \(\rho _R = \operatorname{tr}_A \rho \). Then the singular reference \((\mathbb {1}_A / d_A) \otimes \rho _R\) satisfies the support condition \(\ker ((\mathbb {1}_A / d_A) \otimes \rho _R) \subseteq \ker \rho \).

For complex matrices \(A\) and \(B\) of compatible rectangular sizes,

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

Let \(A\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) be positive semidefinite with \(A\leq |\Omega \rangle \! \langle \Omega |\). Then \(A=a|\Omega \rangle \! \langle \Omega |\) for a non-negative scalar \(a\).

For \(a,b{\gt}0\) and \(s\in (0,1)\),

For \(a,b{\gt}0\) and \(s\in (0,1)\), the function \(t\mapsto t^{s-1} \dfrac {ab}{a+tb}\) is integrable on \((0,\infty )\).

For positive-definite matrices \(A\) and \(B\) with \(A\le B\) in the Loewner order, the inverses satisfy \(B^{-1}\le A^{-1}\).

For a Hermitian matrix \(A\) and a unitary \(U\),

For positive definite matrices \(\rho \) and \(\tau \),

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a positive map for which \(T^*(\mathbb {1})\) is positive definite. Then there exists an invertible \(X\in M_{D}(\mathbb {C})\) such that \(\rho \mapsto T(X\rho X^\dagger )\) is a trace-preserving positive map [ Wol12 , Chapter 3, Lemma “Making positive maps trace preserving” ] .

If \(A\in M_{m,n}(\mathbb {C})(\mathbb {C})\) has rank at most \(k\), then there are matrices \(B\in M_{m,k}(\mathbb {C})(\mathbb {C})\) and \(C\in M_{k,n}(\mathbb {C})(\mathbb {C})\) such that \(BC=A\).

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

Let \(\tau \) be positive semidefinite on \(\mathcal H_S\), let \(X\) be a matrix on \(\mathcal H_S\), and let \(\overline\tau =\tau \otimes d_C^{-1}\mathbf1_C\). Then

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

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

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.

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

For a POVM \(\{ E_i\} \) with square roots \(E_i=M_i^\dagger M_i\), one has \(\sum _iM_i^\dagger M_i=\mathbb {1}\).

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:

Peripheral unitary eigenvectors can be normalized compatibly with the faithful fixed-point state.

For positive-definite matrices \(A_1,A_2,B_1,B_2\), \(s\in (0,1)\), and \(\theta \in [0,1]\), with \(\hat{A}=A\otimes \mathbb {1}\) and \(\hat{B}=\mathbb {1}\otimes B^\top \), setting \(A_\theta =\theta A_1+(1-\theta )A_2\) and \(B_\theta =\theta B_1+(1-\theta )B_2\), the fractional product \(\hat A^s\hat B^{1-s}\) is jointly concave in the Loewner order:

For positive-definite matrices \(A\), \(B\) and \(s\in (0,1)\),

where \(A\otimes \mathbb {1}\) and \(\mathbb {1}\otimes B^\top \) are regarded as operators on the Kronecker model space \(\mathbb {C}^{D\times D}\).

Let \(f\) be a function from a measure space to a finite matrix algebra. If \(f(x)\ge 0\) for almost every \(x\), then \(\int f(x)\, d\mu (x)\ge 0\).

Let \(Y\) be positive definite and let \(W\) be an isometry. Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), and let \(t{\gt}0\). Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (0,1)\), and let \(t{\gt}0\). Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (1,2)\), and let \(t{\gt}0\). Then

The inequality is reversed relative to the concave integrand, since \(g_{p,t}\) is operator convex.

If the defect block satisfies

then the dilation is an isometry, and compressing the weighted scalar block-diagonal matrix gives \(\sum _{i\in \iota }w_iC_iC_i^\dagger +tSS^\dagger \). The defect relation can also be written as \(\sum _{i\in \iota }C_iC_i^\dagger +SS^\dagger =\mathbb {1}\). Strict positivity of all weights together with \(t{\gt}0\) implies positive definiteness of the scalar block-diagonal matrix. If all weights are nonzero and the defect scalar is nonzero, the inverse diagonal has reciprocal weights, and compressing this inverse gives \(\sum _{i\in \iota }w_i^{-1}C_iC_i^\dagger +t^{-1}SS^\dagger \).

Let \(w_i\ge 0\) and \(t{\gt}0\). If the defect block satisfies \(SS^\dagger =\mathbb {1}-\sum _{i\in \iota }C_iC_i^\dagger \), then

Let \(A\ge 0\) be a finite matrix. There are positive semidefinite matrices \(P_i\) and non-negative real numbers \(\lambda _i\) such that

If the supplied dominating dilation is minimal, \(r_2=\operatorname{rank}(\tau _2)\), then \(W_2\) is surjective. Consequently a factorization \(W_1=CW_2\) determines \(C\) uniquely.

Let \(W_i:H\to R_i\) be finite-dimensional complex matrices. If

then there is a contraction \(C:R_2\to R_1\) such that \(W_1=CW_2\).

Suppose that \(T:M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) has the supplied representation \(T(A)=V^\dagger (A\otimes \mathbb {1}_r)V\). If \(\tau \) is the normalized Choi matrix of \(T\), then

If \(d'{\gt}0\), the assignment \(V\mapsto W\) is one-to-one. Moreover, for every \(C:\mathbb {C}^{r_2}\to \mathbb {C}^{r_1}\),

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

For positive-definite matrices \(X_1,X_2,Y_1,Y_2\) and \(\theta \in [0,1]\), the parallel sum \((X,Y)\mapsto X(X+Y)^{-1}Y\) is jointly concave in the Loewner order:

where \(X_\theta =\theta X_1+(1-\theta )X_2\) and \(Y_\theta =\theta Y_1+(1-\theta )Y_2\).

If \(\sigma \) has trace one, then

Let \(e_L:H_L\to H_L'\) and \(e_R:H_R\to H_R'\) be bijections, and write \(Z^{e_L\otimes e_R}\) for the corresponding simultaneous relabelling of the rows and columns of \(Z\). Then

Let \(A:H_A\to K_A\) and \(B:H_B\to K_B\) be linear maps between finite-dimensional complex spaces, and let \(X\) be an operator on \(H_A\otimes H_B\). If \(B^\dagger B=1\), then

If \(A^\dagger A=1\), then, symmetrically,

The identities include zero-dimensional spaces.

For every matrix \(X\), the support Petz map is given by (64).

Let \(P_\tau \) be the orthogonal projector onto the support of \(\tau =\operatorname{tr}_R\sigma \). Then, for every matrix \(X\),

For square operators \(X,Y\) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\),

Consequently, first-factor partial transpose is self-adjoint for the real trace pairing \((X,Y)\mapsto \Re \operatorname{tr}(XY)\).

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 \(n\in \mathbb {N}\).

The inverse of a peripheral eigenvalue is peripheral.

The product of two peripheral eigenvalues is peripheral.

Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(\mu , \nu \) are peripheral eigenvalues of \(E\), then so is \(\mu \nu \).

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 \(\mathrm{peripheral}(E^p)=\{ 1\} \).

For an irreducible unital Schwarz map with faithful adjoint fixed point, each peripheral eigenvalue admits a unitary eigenvector.

If \(E(U)=\mu U\) with \(|\mu |=1\) and \(U\) unitary, then \(E(U^n)=\mu ^nU^n\) for every \(n\).

If \(A\in M_{D}(\mathbb {C})\) is positive definite, then \((A^{1/2})^{-1}\) is invertible.

Let \(D\geq 1\), let \(M\in M_{D}(\mathbb {C})\) be positive-definite, and let \(\lambda _{\min }(M)\) be its smallest Hermitian eigenvalue. For every \(X\in M_{D}(\mathbb {C})\),

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

A linear map on matrices that sends every rank-one projector \(|w\rangle \! \langle w|\) to a positive semidefinite matrix sends every positive semidefinite matrix to a positive semidefinite matrix.

If \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) is a positive linear map and \(H\in M_{D}(\mathbb {C})\) is Hermitian, then \(T(H)\) is Hermitian.

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

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian with positive part \(A^+\). There is a matrix \(\Pi \) with \(0\leq \Pi \leq \mathbb {1}\), \(\Pi ^2=\Pi \), and \(\Pi A=A^+\), namely the orthogonal projection onto the support space of \(A^+\).

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map and let \(H\in M_{D}(\mathbb {C})\) be Hermitian, with Jordan decompositions \(H=P_+-P_-\) and \(T(H)=Q_+-Q_-\). Then \(\operatorname{tr}[Q_+]\leq \operatorname{tr}[P_+]\).

Let \(S,T\) be positive semidefinite matrices and let \(x\) be a vector. If \((t\mathbf1+S)^{-1}x=(t\mathbf1+T)^{-1}x\) for every \(t{\gt}0\), then \(\sqrt S\, x=\sqrt T\, x\). More generally, for any fixed matrix \(Q\), if \(Q(t\mathbf1+S)^{-1}x=Q(t\mathbf1+T)^{-1}x\) for every \(t{\gt}0\), then \(Q\sqrt S\, x=Q\sqrt T\, x\).

In particular, for positive definite \(A,B\) and every \(t{\gt}0\), put \(\Delta _{A,B}=A\otimes (B^{-1})^{\mathsf T}\). The source-\(B\) left–right resolvent satisfies

and

These are the positive-square-root specializations of the passage from relative modular resolvents to analytic functions of the relative modular operator in Jenčová–Ruskai, arXiv:0903.2895v4, lines 658–680.