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 \(\mathcal{K}(Y)=\sum _i K_i Y K_i^\dagger \).

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.

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

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

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

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a quantum channel if it is both completely positive and trace-preserving (CPTP), following  [ Wol12 ] .

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

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

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

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 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.

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.

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

Its matrix entries are

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

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

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 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 ] .

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 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 \(i,j\).

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

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) 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.

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

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

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 \(\mathbb {C}^k\to \mathbb {C}^D\).

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 \(\eta {\gt}0\).

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.

The trace-pairing adjoint \(E^* : M_{D'}(\mathbb {C}) \to M_{D}(\mathbb {C})\) of a linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is the adjoint for the bilinear pairing \((A,B)\mapsto \operatorname{tr}(AB)\).

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is trace-preserving if \(\operatorname{tr}(E(X)) = \operatorname{tr}(X)\) for all \(X\).

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

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.

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

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.

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.

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

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

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

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 \(\operatorname{diag}(a_i)-|v\rangle \! \langle v|\ge 0\).

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.

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

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 \(|\psi \rangle \):

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 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 \((X,Y)\mapsto \Re \operatorname{tr}(XY)\).

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

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.

Let \(P,A\in M_{D}(\mathbb {C})\) with \(P\ge 0\), and let \(Q=\operatorname{supp}(P)\) be the support projection of \(P\). Then

Let \(P\in M_{D,D}(\mathbb {C})(\mathbb {C})\) be Hermitian, idempotent, and of rank \(k\). Then there exists \(V\in M_{D,k}(\mathbb {C})(\mathbb {C})\) such that \(P=VV^\dagger \).

Let \(X\in M_{d\times k}(\mathbb {C})\) and suppose \(k\le d\). Then there exists a Hermitian projection \(P\in M_{d}(\mathbb {C})\) of rank \(k\) such that \(PX=X\).

Applying the \(k\)-fold ampliation of \(T\) to the rank-one matrix \(|\psi \rangle \! \langle \psi |\), where \(\psi _{(a,p)}=d^{-1/2}X_{a,p}\), is exactly the right-factor compression of the Choi matrix of \(T\) by \(X\).

If \(p\) is a vector with \(\sum _i |p_i|^2\le 1\), then \(\mathbb {1}-|p\rangle \! \langle p|\ge 0\).

Writing \(C_i\) for the coefficient matrix of \(\phi _i\), the reduced density operator factors as \(\rho _i=C_iC_i^\dagger \).

Each reduced eigenvector density has unit trace, and for \(n\ge D'\) its Ky-Fan \(n\)-norm is that trace:

The reduced density operator \(\rho _i\) is positive semidefinite.

Let \(X\in M_{d\times k}(\mathbb {C})\) and let \(\eta \in \mathbb {C}^{d'}\otimes \mathbb {C}^k\). Then \(R_X^\dagger \eta \in \mathbb {C}^{d'}\otimes \mathbb {C}^d\) has Schmidt rank at most \(k\).

If \(\psi \in \mathbb {C}^{d'}\otimes \mathbb {C}^d\) has Schmidt rank at most \(k\), then there exist \(X\in M_{d\times k}(\mathbb {C})\) and \(\eta \in \mathbb {C}^{d'}\otimes \mathbb {C}^k\) such that \(\psi =R_X^\dagger \eta \).

For every \(V\in M_{d\times k}(\mathbb {C})\), the right-tensor factor satisfies \(R_{VV^\dagger }=R_V^\dagger R_V\).

Let \(X\in M_{d\times k}(\mathbb {C})\) and \(P\in M_{d}(\mathbb {C})\). Then the right-tensor factors satisfy \(R_XR_P=R_{PX}\).

Assume \(D',D\ge 1\). For \(n\ge 1\) the set \(E_n(\tau )\) is nonempty, and for Hermitian \(\tau \) it is bounded below by the least eigenvalue \(\nu _{\min }\) of \(\tau \); the expectation of \(\tau \) in any normalized vector of Schmidt rank at most \(n\) is one of its elements. Hence \(\inf E_n(\tau )\) exists.

If \(\operatorname{SR}(\psi )\le k\) and \(k\le l\), then \(\operatorname{SR}(\psi )\le l\).

Assume \(d{\gt}0\). A vector \(\psi \in \mathbb {C}^d\otimes \mathbb {C}^d\) has Schmidt rank at most \(k\) if and only if there is a matrix \(X\in M_{d}(\mathbb {C})\) of rank at most \(k\) such that \(\psi _{(i,j)}=d^{-1/2}X_{i,j}\).

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) have Choi matrix \(\tau \). Suppose \(P\in M_{d}(\mathbb {C})\) and \(X\in M_{d\times k}(\mathbb {C})\) satisfy \(PX=X\). If \(R_P\tau R_P^\dagger \ge 0\), then the rectangular right compression of \(\tau \) by \(X\) is positive semidefinite.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a linear map and let \(X\in M_{D}(\mathbb {C})\) satisfy \(X^\dagger T^*(\mathbb {1})X=\mathbb {1}\). Then, for every \(\rho \in M_{D}(\mathbb {C})\),

Suppose that \(N\ge 1\) and \(m{\lt}s\). If \(f_{N,m,s}(y)\le B\) for every strictly positive vector \(y\), then the same inequality holds for every nonnegative vector when a literal \(0/0\) summand is assigned Lean’s totalized value zero. In particular, the conclusion holds with \(B=N/s\) once the strictly positive inequality has been proved.

Suppose that \(m{\lt}N\), \(m{\lt}s\), and \(N\le s\). Let strictly positive vectors \(x^{(r)}\) converge coordinatewise to a nonzero nonnegative vector \(a\) with at least one vanishing forward denominator. Then

Suppose that \(N\ge 1\) and \(m{\lt}s\). Let \(a\colon \mathbb {Z}/N\mathbb {Z}\to \mathbb {R}\) be nonnegative and nonzero. If one forward denominator vanishes, there is an index \(j\) such that

If \(m{\lt}N\), the \(m\) indices \(j-1,\ldots ,j-m\) are distinct. Each has zero numerator, and its forward denominator contains the positive coordinate \(a_j\).

If \(d\ge 3\), \(1\le m\le d-2\), \(s\ge d\), and \(j\ne 0\) in \(\mathbb {Z}/d\mathbb {Z}\), then

Suppose \(D{\gt}2\). It is false that the Breuer–Hall map is indecomposable for every antisymmetric contraction \(U\). Indeed, \(U=0\) satisfies \(U^{\mathsf T}=-U\) and \(U^{\dagger }U\le \mathbb {1}\), but its Breuer–Hall map is

which is completely copositive and decomposable.

Suppose \(D{\gt}2\) and \(U\) is an antisymmetric unitary on \(\mathbb {C}^{D}\), \(U^{\mathsf T}=-U\) and \(U^{\dagger }U=\mathbb {1}\). Then the Breuer–Hall map \(T_{\mathrm{BH}}\) is an indecomposable positive map.

If \(U\) is antisymmetric, \(U^{\mathsf T}=-U\), and a contraction, \(U^{\dagger }U\le \mathbb {1}\), then the Breuer–Hall map \(T_{\mathrm{BH}}\) sends every positive semidefinite matrix to a positive semidefinite matrix.

For \(D\ge 1\), the Choi matrix of the reduction map is

For \(D\ge 1\) the Choi matrix of \(T_n\) is the reduction witness:

For \(D\ge 1\), the Choi matrix of \(T_\eta \) is

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

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

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

For \(X\in M_{d\times k}(\mathbb {C})\) and \(\eta \in \mathbb {C}^{d'}\otimes \mathbb {C}^k\),

If \(\tau \) is the Choi matrix of \(T\), then

where \(C_T(X)\) is the right-factor Choi compression by \(X\).

Let \(\theta (X)=X^T\) be matrix transposition on \(M_{D}(\mathbb {C})\), with \(D \ge 1\). Then

This is [ Wol12 , Equation 3.1 ] .

Let \(d\ge 3\), \(2\le n\le d-3\), and put \(x_i=|v_i|^2\). Then

with zero-denominator summands interpreted as zero. Consequently, \(T_C(|v\rangle \! \langle v|)\ge 0\).

For every \(d\ge 3\) and \(1\le n\le d-2\), the Choi-type map \(T_C\) sends every positive semidefinite matrix to a positive semidefinite matrix. This is the positivity assertion of Wolf’s Example 3.1.

For every \(d\ge 3\) and \(n=1\), the Choi-type map \(T_C\) sends every positive semidefinite matrix to a positive semidefinite matrix. This is the bottom slice of the positivity assertion of Wolf’s Example 3.1.

For every \(d\ge 3\) and \(n=d-2\), the Choi-type map \(T_C\) sends every positive semidefinite matrix to a positive semidefinite matrix. This is the case \(n=d-2\) of the positivity assertion of Wolf’s Example 3.1 and subsumes the case \(d=3\), \(n=1\).

For each \(n\) the set \(S_n\) of trace-one bipartite states of Schmidt number at most \(n\) is compact.

Assume \(d{\gt}0\). For \(X\in M_{d\times k}(\mathbb {C})\), the vector \(\psi _X\) in (83) satisfies \(\operatorname{SR}(\psi _X)=\operatorname{rank}X\).

For every \(X\in M_{D}(\mathbb {C})\), the conjugation filter \(\rho \mapsto X\rho X^\dagger \) is completely positive, hence positive. Therefore, if \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is positive, then \(\rho \mapsto T(X\rho X^\dagger )\) is positive. If in addition \(X^\dagger X=\mathbb {1}\), then the conjugation filter is trace-preserving and hence is a quantum channel.

Let \(Y\in M_{D}(\mathbb {C})\) be invertible. Then the map \(X\mapsto YXY^\dagger \) and the map \(X\mapsto YX^TY^\dagger \) each map the positive semidefinite cone onto itself. This is the implication (3) \(\Rightarrow \) (1) of [ Wol12 , Proposition 3.6 ] .

For each \(n\) the set \(S_n\) of bipartite states of Schmidt number at most \(n\) is convex.

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

Every completely positive map is \(k\)-positive for all \(k \ge 1\).

Every completely positive map is two-positive.

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

For maps \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) with \(D\ge 1\), complete positivity is equivalent to \(D\)-positivity.

Let \(d'\geq 1\) and let \(T\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\). Then \(T\) is decomposable if and only if the output-first Choi matrix of its trace adjoint \(T^*\colon M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) is a decomposable witness:

If \(\Phi \colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is decomposable and \(\rho \) is a positive semidefinite matrix on \(\mathbb {C}^{d}\otimes \mathbb {C}^{k}\) with the PPT property, then \((\Phi \otimes \operatorname{id})(\rho )\ge 0\).

The set of decomposable witnesses in Equation (3.15) is convex.

Assume \(D{\gt}0\). Every bipartite matrix \(W\) on \(\mathbb {C}^D\otimes \mathbb {C}^D\) is the Choi matrix \(\tau _T=(T\otimes \mathbb {1})(|\Omega \rangle \langle \Omega |)\) of some linear map \(T\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\).

Let \(\rho \) be a trace-one Hermitian bipartite state whose Schmidt number exceeds \(n\). Then there is a Hermitian operator \(W\) such that, for every \(\psi \) of Schmidt rank at most \(n\),

This is the only-if direction of Wolf’s Proposition 3.3 [ Wol12 , Chapter 3, Proposition 3.3 ] : a state of Schmidt number larger than \(n\) is detected by an entanglement witness for \(S_n\).

Every real-linear functional \(g\) on the space of square complex matrices is the trace form of a Hermitian matrix: there is a Hermitian \(H_0\) with \(g(X)=\operatorname{Re}\operatorname{tr}(X H_0)\) for every Hermitian \(X\).

Assume \(d'{\gt}0\). Let \(W\) be a Hermitian operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) with \(\operatorname{Re}\langle \psi |W|\psi \rangle \ge 0\) for every \(\psi \) of Schmidt rank at most \(n\). Then \(W\) is the Choi matrix of an \(n\)-positive map \(P\colon M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\). This is the Choi–Jamiołkowski translation of the entanglement witness of Wolf’s Proposition 3.3 into the \(n\)-positive map of Wolf’s Proposition 3.4 [ Wol12 , Chapter 3, Proposition 3.4 ] .

Let \(\rho \) be a trace-one Hermitian bipartite state on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) whose Schmidt number exceeds \(n\). Then there is an \(n\)-positive map \(T\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) such that \((T\otimes \mathbb {1}_{d'})(\rho )\) is not positive semidefinite. This is the if direction of Wolf’s Proposition 3.4 [ Wol12 , Chapter 3, Proposition 3.4, Equations (3.13)–(3.14) ] .

Assume \(d{\gt}0\). Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be \(k\)-positive, with Choi matrix \(\tau \). If a right-factor matrix has the form \(P=VV^\dagger \) with \(V\in M_{d\times k}(\mathbb {C})\), then \(R_P\tau R_P^\dagger \geq 0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a positive map between matrix algebras of possibly different dimensions, and let \(X\in M_{D}(\mathbb {C})\) satisfy \(X^\dagger T^*(\mathbb {1})X=\mathbb {1}\), where \(T^*\) is the trace-pairing adjoint. Then \(\rho \mapsto T(X\rho X^\dagger )\) is positive and trace-preserving.

This is the algebraic trace-normalization step in the proof of [ Wol12 , Chapter 3, Lemma “Making positive maps trace preserving” ] . The inverse-square-root choice is recorded in the next theorem.

If \(d\ge 3\) and \(\gamma {\gt}0\), then Ha’s displayed block transpose is precisely

Here \(T_2\) denotes transposition in the second tensor factor, exactly as in Ha’s block notation.

The orientation obtained from [ EK00 , equations (12)–(13), p. 137; Theorem 3.3 and the concluding shuffled pairing, pp. 138–139 ] is

If \(|\Omega _d\rangle =d^{-1/2}\sum _i e_i\otimes e_i\), then \(J_d=d|\Omega _d\rangle \! \langle \Omega _d|\), and hence

The pairing is unchanged under a simultaneous reindexing of the finite coordinates of \(A\) and of the domain and range coordinates of \(\Phi \).

Let \(d\ge 3\), \(1\le n\le d-2\), and \(\gamma {\gt}0\). For the Choi-type map \(T_C\) and Ha’s matrix \(A_\gamma \),

In particular, if \(0{\lt}\gamma {\lt}1\), the real part of each expression in (72) is strictly negative.

Put \(\rho _\gamma =A_\gamma ^\sigma \). For every \(d\ge 3\) and \(\gamma {\gt}0\),

Every vector \(z_{r,i}\) in (60) has Schmidt rank at most two. Consequently \(A_r\) and \(A_\gamma \) belong to the cone \(V_2\) generated by projectors onto vectors of Schmidt rank at most two; in particular, \(A_\gamma \ge 0\).

A matrix of Schmidt number at most \(n\) is positive semidefinite. States of Schmidt number at most \(n\) are closed under addition and under multiplication by a non-negative real scalar, and a bound on the Schmidt number relaxes to any larger bound.

A bipartite matrix has Schmidt number at most one if and only if it is separable.

A bipartite state on \(M_{d}(\mathbb {C})\otimes M_{k}(\mathbb {C})\) of Schmidt number at most \(n\) satisfies \((T\otimes \operatorname{id}_k)(\rho )\ge 0\) for every \(n\)-positive map \(T\colon M_{d}(\mathbb {C})\to M_{r}(\mathbb {C})\). This is the only-if direction of Wolf’s Proposition 3.4 [ Wol12 , Chapter 3, Proposition 3.4 ] , with the map output and innocent-bystander dimensions allowed to vary independently.

On \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), with no relation imposed between the two factors, a state of Schmidt number at most \(n\) satisfies \((T\otimes \operatorname{id})(\rho )\ge 0\) for every \(n\)-positive endomorphism \(T\) of \(M_{d}(\mathbb {C})\). This compatibility declaration is the specialization of Theorem 4.13.5 used by the reduction criterion below.

For \(1\le n{\lt}D\), a bipartite state on the square system \(M_{D}(\mathbb {C})\otimes M_{D}(\mathbb {C})\) of Schmidt number at most \(n\) satisfies \((T_n\otimes \operatorname{id})(\rho )\ge 0\).

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

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

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

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

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

Let \(d,d'\geq 1\). If \(T\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is an indecomposable positive map, then there is a PPT entangled density operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\).

Let \(\Phi \colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be a positive map. If some positive semidefinite matrix \(\rho \) on \(\mathbb {C}^{d}\otimes \mathbb {C}^{k}\) with the PPT property has \((\Phi \otimes \operatorname{id})(\rho )\) not positive semidefinite, then \(\Phi \) is indecomposable. The detection hypothesis alone, without positivity of \(\Phi \), still implies that \(\Phi \) is not decomposable.

Let \(U\) be any matrix on the first factor. If \(\rho \) has the PPT property then

is positive semidefinite; no invertibility or unitarity of \(U\) is required. Specializing to a unitary \(U\) this is the partial transpose taken in the basis changed by \(U\), so the positivity of the partial transpose is independent of the local basis, recovering Wolf’s basis-independence statement [ Wol12 , Equation (3.17) ] .

A bipartite matrix has the PPT property if and only if its second-factor partial transpose is positive semidefinite: \(\rho ^{T_1}\ge 0\iff \rho ^{T_2}\ge 0\).

If \(\rho \) and \(\sigma \) are separable then so is \(\rho +\sigma \).

Every separable matrix has the positive-partial-transpose property: if \(\rho \) is separable then \(\rho ^{T_1}\ge 0\). A negative partial transpose therefore certifies that a state is entangled.

For positive semidefinite \(A\) on \(M_{d}(\mathbb {C})\) and \(B\) on \(M_{d'}(\mathbb {C})\), the Kronecker product \(A\otimes B\) is separable.

Every separable matrix is positive semidefinite.

For \(d\ge 1\) and a separable matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), applying the reduction map \(T_1(X)=\operatorname{tr}(X)\mathbb {1}-X\) to the first factor keeps the result positive semidefinite: \((T_1\otimes \operatorname{id})(\rho )\ge 0\). This is the separability instance of the operator inequality underlying the reduction criterion at the first level.

If \(\rho \) is separable and \(c\ge 0\) is real then \(c\, \rho \) is separable.

If \(E\) and \(F\) are \(k\)-positive, then \(E+F\) is \(k\)-positive.

For each \(k\), the set of \(k\)-positive linear maps is closed in the finite-dimensional space of linear maps.

Assume \(d{\gt}0\). A linear map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive if and only if, for every \(X\in M_{d\times k}(\mathbb {C})\), the right-factor compression of the Choi matrix of \(T\) by \(X\) is positive semidefinite.

If a linear map is \(k\)-positive and \(m \le k\), then it is \(m\)-positive.

Positivity under amplification is monotone in \(k\).

A linear map \(E:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive if and only if, for every vector \(\phi \in \mathbb {C}^d\otimes \mathbb {C}^k\), \(E^{(k)}(|\phi \rangle \! \langle \phi |)\ge 0\). This is the pure-state reduction used in [ Wol12 , Chapter 3, Proposition 3.1 ] .

Assume \(d{\gt}0\). If \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive, then, for every \(X\in M_{d\times k}(\mathbb {C})\),

where \(\tau \) is the Choi matrix of \(T\).

Assume \(d{\gt}0\). If \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive, then \(\langle \psi ,\tau _T\psi \rangle \ge 0\) for every \(\psi \in \mathbb {C}^{d'}\otimes \mathbb {C}^d\) with \(\operatorname{SR}(\psi )\le k\).

If \(E\) is \(k\)-positive and \(c \ge 0\), then \(cE\) is \(k\)-positive.

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

A linear map is \(k\)-positive if and only if its trace-pairing adjoint is \(k\)-positive.

The zero map is \(k\)-positive.

If \(E\) is unital and two-positive, then for all \(X\in M_{D}(\mathbb {C})\), \(E(X^\dagger X)\ge E(X)^\dagger E(X)\).

Let \(H\in M_{D}(\mathbb {C})\) be Hermitian and let \(T\) be an anti-unitary operator with \([H,T]=0\) and \(T^2=-\mathbb {1}\). Then every eigenvalue of \(H\) is at least two-fold degenerate [ Wol12 , Chapter 3, Theorem 3.1 ] .

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 \(|\psi \rangle \neq 0\) satisfies \(H|\psi \rangle =\lambda |\psi \rangle \) and the partner does not vanish, \(A|\overline\psi \rangle \neq 0\), then the eigenspace of \(\lambda \) has dimension at least two.

For every Hermitian \(A\in M_{D}(\mathbb {C})\) and every \(k\ge 0\), there is an orthogonal projection \(P\) of rank \(\min (k,D)\) with \(\operatorname{Re}\operatorname{tr}(PA)=S_k(A)\).

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian and \(0\le k{\lt}D\). Then the sum of its \(k\) largest eigenvalues is the largest value of \(\operatorname{Re}\operatorname{tr}(PA)\) attained by an orthogonal projection \(P\) of rank \(k\):

This is Ky Fan’s maximum principle [ Fan49 ] ; see also [ Bha97 , Problem I.6.15, Exercise II.1.13 ] . For a positive semidefinite \(A\), where \(S_k(A)\) coincides with the Ky-Fan norm \(\| A\| _{(k)}\), it gives the extremal overlap of [ Wol12 , Lemma 3.1 ] .

For every Hermitian \(A\in M_{D}(\mathbb {C})\) and every orthogonal projection \(P\) of rank \(k{\lt}D\), one has \(\operatorname{Re}\operatorname{tr}(PA)\le S_k(A)\).

Let \(\phi \in \mathbb {C}^{D}\otimes \mathbb {C}^k\) be a normalized vector with reduced density matrix \(\rho =\operatorname{tr}_2|\phi \rangle \! \langle \phi |\) on the first factor, and let \(1\le n{\lt}D\). The largest squared overlap \(|\langle \phi | \psi \rangle |^2\) attained by a normalized vector \(\psi \) of Schmidt rank at most \(n\) is the Ky-Fan \(n\)-norm of \(\rho \):

Because \(\rho \) is positive semidefinite, this norm is the sum of its \(n\) largest eigenvalues, which coincide there with its singular values. This is [ Wol12 , Lemma 3.1 ] for \(1\le n{\lt}D\); the top index \(n=D\), where the value is \(\operatorname{tr}\rho =1\), is Theorem 4.2.2.4.

At the top index \(n=D\), the Schmidt-rank constraint \(\operatorname{SR}(\psi )\le D\) is vacuous, so the largest squared overlap is the Ky-Fan \(D\)-norm of \(\rho \):

The maximum is attained at \(\psi =\phi \). The Ky-Fan \(D\)-norm sums all eigenvalues, so for a normalized \(\phi \) it reduces to \(\| \rho \| _{(D)}=\operatorname{tr}\rho =1\). Together with Theorem 4.2.2.3, this covers the full range \(1\le n\le D\) of [ Wol12 , Lemma 3.1 ] .

For \(d,d'\geq 1\), the normalized decomposable witnesses are precisely the trace-one section of the cone in Equation (3.15). This set is compact and convex.

Let \(d,d'\geq 1\), and let \(W\) be a Hermitian operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) such that \(\operatorname{tr}(W)=1\), \(W\) is not decomposable, and

Then there is a PPT entangled density operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\).

Let \(d,d'\geq 1\), and let \(W\) be a Hermitian operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) such that \(\operatorname{tr}(W)=1\) and \(W\) is not decomposable. Then there is an operator \(R\) such that

Let \(\rho \) be a trace-one Hermitian bipartite state. Then the Schmidt number of \(\rho \) exceeds \(n\) if and only if there is a Hermitian operator \(W\) such that, for every \(\psi \) of Schmidt rank at most \(n\),

This is the corrected form of Wolf’s Proposition 3.3 [ Wol12 , Chapter 3, Proposition 3.3 ] : a state has Schmidt number larger than \(n\) exactly when it is detected by an entanglement witness for \(S_n\). The source prints exact Schmidt rank \(n\), while the convex set and proof require Schmidt rank at most \(n\); the defect is recorded in [ con26a ] .

Every real linear operator \(T\colon A\to A\) is bounded for the norm induced by \(\varphi \): there is \(C\ge 0\) such that

An element \(q\in A\) belongs to \(P(w)\) exactly when it is constant on every value class of \(w\). If \(i\in I\) and a linear functional \(D\colon P(w)\to \mathbb {R}\) satisfies

then \(D=0\).

Let \(T\colon A\to A\) be real linear, and suppose that \(u,v\in A_{++}\) are local minima of \(\lambda _T\). Put \(w=u^{-1}v\) and \(\delta =u^{-1}T(u)\). Then

and \(\lambda _T\) is constant on \(uP(w)\cap A_{++}\). The same conclusion for two local maxima of \(\lambda _T\) is obtained by applying the local-minimum statement to \(-T\).

A linear map is \(1\)-positive if and only if it is positive.

For matrices \(U,V\) on the first factor and a bipartite matrix \(\rho \), conjugating inside by \(V\) and \(U\), taking the first-factor partial transpose, and conjugating outside by \(U\) and \(V\) produces the original partial transpose conjugated by \(U U^{\mathsf T}\) and \(V^{\mathsf T}V\):

If \(\rho \) is Hermitian then so is its first-factor partial transpose \(\rho ^{T_1}\).

Applying the first-factor partial transpose twice returns the original matrix: \((\rho ^{T_1})^{T_1}=\rho \).

For a finite family of bipartite matrices, \((\sum _i f_i)^{T_1}=\sum _i f_i^{T_1}\).

The second-factor partial transpose is the full transpose of the first-factor one: \(\rho ^{T_2}=(\rho ^{T_1})^{\mathsf T}\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive complex-linear map. Suppose that

for every Hermitian matrix \(H\in M_{D}(\mathbb {C})\). Then there is an invertible matrix \(Y\in M_{D}(\mathbb {C})\) such that either

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

for every \(X\in M_{D}(\mathbb {C})\). This is precisely the implication (2) \(\Rightarrow \) (3) of [ Wol12 , Proposition 3.6 ] .

Let \(\rho \) be a PPT entangled density operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\). Then there is an indecomposable positive map \(T\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\). This is the PPT-state-to-map direction of Wolf’s Proposition 3.5 [ Wol12 , Chapter 3, Proposition 3.5 ] .

Let \(d,d'\geq 1\). There exists an indecomposable positive map \(T\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) if and only if there exists an entangled density operator \(\rho \) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) with \(\rho ^{T_1}\geq 0\). This is Wolf’s Proposition 3.5 in its rectangular form [ Wol12 , Chapter 3, Proposition 3.5 ] .

The equivalence in Theorem 4.10.15 holds for all finite dimensions \(d,d'\). If \(d=0\) or \(d'=0\), both existence statements are false.

Let \(P\in M_{D}(\mathbb {C})\) be positive semidefinite and let \(Q=\operatorname{supp}(P)\). Then the complex linear span of \(C(P)\) is the full support corner:

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

Assume \(d{\gt}0\) and \(k\le d\). A linear map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive if and only if, for every Hermitian projection \(P\in M_{d}(\mathbb {C})\) of rank \(k\), \(R_P\tau R_P^\dagger \geq 0\), where \(\tau \) is the Choi matrix of \(T\). This is the rank-\(k\) projection formulation of [ Wol12 , Chapter 3, Proposition 3.1 ] .

Assume \(d{\gt}0\) and \(k\le d\). Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) have Choi matrix \(\tau \). Suppose that \(R_P\tau R_P^\dagger \geq 0\) for every Hermitian projection \(P\in M_{d}(\mathbb {C})\) of rank \(k\). Then \(T\) is \(k\)-positive.

Assume \(d{\gt}0\). Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be \(k\)-positive, with Choi matrix \(\tau \). If \(P\in M_{d}(\mathbb {C})\) is Hermitian, satisfies \(P^2=P\), and has rank \(k\), then \(R_P\tau R_P^\dagger \geq 0\). This is the forward rank-\(k\) projection-compression direction of [ Wol12 , Chapter 3, Proposition 3.1 ] .

Let \(n\ge 1\) and let \(\rho \) be a bipartite matrix, with \(\rho ^{F}\) its image under the factor swap \(\rho ^{F}_{(i_2,i_1),(j_2,j_1)}=\rho _{(i_1,i_2),(j_1,j_2)}\). If \((T_n\otimes \operatorname{id})(\rho )\ge 0\) then \(n\, \mathbb {1}\otimes \rho _2\ge \rho \), and if \((T_n\otimes \operatorname{id})(\rho ^{F})\ge 0\) then \(n\, \rho _1\otimes \mathbb {1}\ge \rho \), where \(\rho _1\) and \(\rho _2\) are the reduced densities of \(\rho \) on the first and second factors. Because the factor swap is a unitary reindexing, \((\operatorname{id}\otimes T_n)(\rho )\ge 0 \iff (T_n\otimes \operatorname{id})(\rho ^{F})\ge 0\), so the second hypothesis is Wolf’s symmetric condition \((\operatorname{id}\otimes T_n)(\rho )\ge 0\) expressed through the swap.

This is the second step of Wolf’s two-step argument for [ Wol12 , Equation (3.18) ] . The source states the inequalities under the premise that \(\rho \) has Schmidt number at most \(n\), and reaches \((T_n\otimes \operatorname{id})(\rho )\ge 0\) from that premise by the \(n\)-positivity of \(T_n\); that first step is taken here as the hypothesis rather than derived. The first step and the composed full criterion with the Schmidt-number premise are Theorem 4.13.8.

On a general bipartite system \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\), a state of Schmidt number at most \(n\) satisfies the reduction criterion

with \(\rho _1\) and \(\rho _2\) the reduced densities on the first and second factors. No relation is imposed between \(d\) and \(d'\); each inequality carries only the \(n\)-positivity threshold of \(T_n\) on its acting factor.

For every \(k\in \mathbb {N}\) the reduction map \(T_k\) equals the family map \(T_\eta \) at \(\eta =k\): \(T_k=T_\eta \big|_{\eta =k}\).

For \(1\le k\le D\) the reduction map \(T_k\) on \(M_{D}(\mathbb {C})\) is \(k\)-positive.

For \(1\le k{\lt}D\) the reduction map \(T_k\) on \(M_{D}(\mathbb {C})\) is not \((k+1)\)-positive.

For \(D\ge 1\), the map \(T_1(X)=\operatorname{tr}(X)\mathbb {1}-X\) is completely copositive and hence decomposable. More precisely, the Choi matrix of \(T_1\circ \theta \), where \(\theta \) is transposition, is

with \(F\) the swap operator on \(\mathbb {C}^D\otimes \mathbb {C}^D\).

For \(D\ge 1\), the map \(T_1(X)=\operatorname{tr}(X)\mathbb {1}-X\) on \(M_{D}(\mathbb {C})\) is positive.

The reduction map is self-dual for the trace pairing:

Let \(d,d'{\gt}0\) and let \(\rho \) be a density operator on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\). Then \(\rho \) has Schmidt number at most \(n\) if and only if

This is Wolf’s rectangular Proposition 3.4 [ Wol12 , Chapter 3, Proposition 3.4 ] .

The zero vector has Schmidt rank zero, every vector satisfies \(\operatorname{SR}(\psi )\le \min (D,k)\), and every product vector \(u\otimes v\) has Schmidt rank at most one.

Assume \(d{\gt}0\). Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\). If \(\langle \psi ,\tau _T\psi \rangle \ge 0\) for every \(\psi \in \mathbb {C}^{d'}\otimes \mathbb {C}^d\) with \(\operatorname{SR}(\psi )\le k\), then \(T\) is \(k\)-positive. This includes \(k=0\) and imposes no nonemptiness assumption on the output factor.

Assume \(d{\gt}0\). A map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(k\)-positive if and only if \(\langle \psi ,\tau _T\psi \rangle \ge 0\) for every \(\psi \in \mathbb {C}^{d'}\otimes \mathbb {C}^d\) with \(\operatorname{SR}(\psi )\le k\).

Assume \(d{\gt}0\). For every \(\psi \in \mathbb {C}^d\otimes \mathbb {C}^k\), there is \(X\in M_{d\times k}(\mathbb {C})\) such that

In particular, if \(\operatorname{SR}(\psi )\le r\), then \(X\) may be chosen with \(\operatorname{rank}X\le r\).

The Schmidt singular values are indexed from zero. The nonzero Schmidt singular values of \(\psi \) are exactly those with index below \(\operatorname{SR}(\psi )\). Equivalently, for zero-based indexing, \(s_k(\psi )=0\) if and only if \(\operatorname{SR}(\psi )\le k\).

At the top index \(n=D'\), the Schmidt-rank constraint is vacuous and each Ky-Fan \(D'\)-norm reduces to \(\| \rho _i\| _{(D')}=\operatorname{tr}\rho _i=1\). The Rayleigh characterization of the least eigenvalue then applies: every normalized \(\psi \) satisfies \(\nu _{\min }\le \langle \psi |\tau |\psi \rangle \), and a least-eigenvalue eigenvector attains \(\langle \psi |\tau |\psi \rangle =\nu _{\min }\). Together they give

There is no Schmidt-rank restriction. At \(n=D'\) the right-hand side of the upper bound of [ Wol12 , Chapter 3, Proposition 3.2 ] reads \(\nu +(\nu _--\nu )\| \rho _-\| _{(D')}=\nu _-=\nu _{\min }\), so the display is that bound at that index. The positivity index of [ Wol12 , Chapter 3, Proposition 3.1 ] runs over \(1\le n\le D\) with \(D\) the dimension of the second factor, so \(n=D'\) is an index the source speaks about when \(D'\le D\), and lies beyond its range when \(D'{\gt}D\). The lower bound of the same proposition reads, at that index, \(\nu _0+\sum _{i:\nu _i\le 0}(\nu _i-\nu _0)\), an inequality no stronger than (14), and Theorem 4.3.1.6 establishes it. With Theorems 4.2.2.6 and 4.2.2.7, both bounds then hold at every \(n\ge 1\).

Let \(\tau \) be a Hermitian operator on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) with eigenvalues \(\nu _i\), normalized eigenvectors \(\phi _i\) and reduced densities \(\rho _i=\operatorname{tr}_2|\phi _i\rangle \! \langle \phi _i|\). Assume \(D',D\ge 1\), let \(1\le n{\lt}D'\) and let \(\nu _0\ge 0\) be a lower bound for the positive eigenvalues, the smallest positive eigenvalue in [ Wol12 , Chapter 3 ] . Then

This is [ Wol12 , Chapter 3, equation (3.7) ] for \(n{\lt}D'\). The indices \(n\ge D'\) are Theorem 4.3.1.6, and the two together give the equation at every \(n\ge 1\).

Assume \(D',D\ge 1\). Once the Schmidt-rank bound reaches the dimension \(D'\) of the first tensor factor the constraint is vacuous, and each Ky-Fan norm collapses to \(\| \rho _i\| _{(n)}=\operatorname{tr}\rho _i=1\). For \(n\ge D'\),

which is the right-hand side of (3.8) there, since \(\nu +(\nu _--\nu )\| \rho _-\| _{(n)}=\nu _-=\nu _{\min }\) when \(\nu _-\) is the only non-positive eigenvalue. The right-hand side of (3.7) reads \(\nu _0+\sum _{i:\nu _i\le 0}(\nu _i-\nu _0)\), which can be strictly weaker once \(\tau \) has two non-positive eigenvalues: for the spectrum \((-2,-1,3)\) with \(\nu _0=3\) it is \(-6\) while \(\nu _{\min }=-2\). It coincides with \(\nu _{\min }\) when \(\nu _0=0\) and only one eigenvalue is strictly negative. It is still a lower bound: for those same \(n\), and for \(\nu _0\ge 0\) a lower bound for the positive eigenvalues as in Theorem 4.3.1.3,

and (3.7) holds at those indices as well. With Theorems 4.3.1.3 and 4.3.1.4 this covers every \(n\ge 1\), hence the range \(1\le n\le D\) of [ Wol12 , Chapter 3, Proposition 3.1 ] , where \(D\) is the dimension of the second tensor factor.

With \(\tau \), \(\phi _i\) and \(\rho _i\) as above and \(D',D\ge 1\), let \(1\le n{\lt}D'\) and suppose every eigenvalue of \(\tau \) is at most \(\nu \). Then, for every eigenvector index \(j\),

Taking \(\nu \) to be the largest positive eigenvalue and \(j\) the index of the unique non-positive eigenvalue \(\nu _-\), which is the situation in which all other eigenvalues are strictly positive, gives \(\inf E_n(\tau )\le \nu +(\nu _--\nu )\| \rho _-\| _{(n)}\), that is, [ Wol12 , Chapter 3, equation (3.8) ] for \(n{\lt}D'\). The indices \(n\ge D'\) are Theorem 4.3.1.6, and the two together give the equation at every \(n\ge 1\).

Let \(\tau \) be a Hermitian operator on \(\mathbb {C}^{D'}\otimes \mathbb {C}^{D}\) with eigenvalues \(\nu _i\) and normalized eigenvectors \(\phi _i\), and write \(\rho _i=\operatorname{tr}_2|\phi _i\rangle \! \langle \phi _i|\) for the reduced density operator of \(\phi _i\) on the first factor. Let \(1\le n{\lt}D'\) and let \(\nu _0\ge 0\) be a lower bound for the positive eigenvalues (the smallest positive eigenvalue). Then every normalized vector \(\psi \) of Schmidt rank at most \(n\) satisfies

Hence the same bound holds for the infimum over such vectors. This is the lower bound of [ Wol12 , Chapter 3, Proposition 3.2 ] for \(n{\lt}D'\); the top index \(n=D'\) is Theorem 4.2.2.8.

With \(\tau \), \(\phi _i\), and \(\rho _i\) as above, let \(1\le n{\lt}D'\) and suppose every eigenvalue of \(\tau \) is at most \(\nu \). For any eigenvector index \(j\), there is a normalized vector \(\psi \) of Schmidt rank at most \(n\) with

Hence the infimum over such vectors lies below that value. Taking \(\nu \) to be the largest positive eigenvalue and \(j\) the index of the unique non-positive eigenvalue \(\nu _-\) recovers the upper bound \(\nu +(\nu _--\nu )\| \rho _-\| _{(n)}\) of [ Wol12 , Chapter 3, Proposition 3.2 ] . The statement is for \(n{\lt}D'\); the top index \(n=D'\) is Theorem 4.2.2.8.

Let \(\eta {\gt}0\) and \(1\le k{\lt}D\). Then \(T_\eta \) is \(k\)-positive if and only if \(\eta \ge k\). This is [ Wol12 , Equation (3.11) ] for \(1\le k{\lt}D\); the top index \(k=D\), where \(k\)-positivity is complete positivity, is Theorem 4.5.1.4.

Let \(\eta {\gt}0\) and \(D\ge 1\). Then \(T_\eta \) is \(D\)-positive if and only if \(\eta \ge D\). Together with Theorem 4.5.1.3 this is [ Wol12 , Equation (3.11) ] over the full range \(1\le k\le D\), where the top index \(k=D\) is complete positivity.

For a pure state \(|\psi \rangle \langle \psi |\) on \(M_{d}(\mathbb {C})\otimes M_{k}(\mathbb {C})\) with \(\operatorname{SR}(\psi )\le n\) and any \(n\)-positive map \(T\colon M_{d}(\mathbb {C})\to M_{r}(\mathbb {C})\), applying \(T\) to the first factor keeps the result positive semidefinite: \((T\otimes \operatorname{id}_k)(|\psi \rangle \langle \psi |)\ge 0\).

For a linear map \(T\) on the first factor and a finite family of bipartite matrices, \((T\otimes \operatorname{id})(\sum _i f_i)=\sum _i (T\otimes \operatorname{id})(f_i)\).

Applying \(T_\eta \) to the first factor of a bipartite matrix \(\rho \) gives

where \(\rho _2\) is the reduced density of \(\rho \) on the second factor.

For a pure state \(|\psi \rangle \langle \psi |\) on the square bipartite system \(M_{D}(\mathbb {C})\otimes M_{D}(\mathbb {C})\) with \(\operatorname{SR}(\psi )\le n\) and \(1\le n{\lt}D\), applying \(T_n\) to the first factor keeps the result positive semidefinite: \((T_n\otimes \operatorname{id})(|\psi \rangle \langle \psi |)\ge 0\).

For a linear map \(P\colon M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) with Choi matrix \(\tau _P\) and any bipartite matrix \(\rho \) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\),

where \(P^{*}\colon M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is the trace-pairing adjoint of \(P\).

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian and suppose that \(\operatorname{Re}\operatorname{tr}(A)\ge 0\). If

then \(A\) is positive semidefinite.

This is the trace-non-negative form of the converse in [ Wol12 , Proposition 3.7 ] . The unrestricted squared implication printed there is false without a trace-sign condition: for \(A=-\mathbb {1}\), one has \(\operatorname{tr}(A)^2=D^2\ge D(D-1)=(D-1)\operatorname{tr}(A^2)\), although \(A\) is not positive semidefinite.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a positive map such that \(T^*(\mathbb {1})\) is positive definite. Put \(X=(T^*(\mathbb {1}))^{-1/2}\). Then \(\rho \mapsto T(X\rho X^\dagger )\) is positive and trace-preserving.

For every linear map \(E : M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), the trace-pairing adjoint commutes with ampliation:

For every linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\), every \(\rho \in M_{D'}(\mathbb {C})\), and every \(X \in M_{D}(\mathbb {C})\), one has

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

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

The first-factor partial transpose preserves the trace: \(\operatorname{tr}(\rho ^{T_1})=\operatorname{tr}(\rho )\).

If \(D \ge 2\), the matrix transposition map \(\theta (X)=X^T\) on \(M_{D}(\mathbb {C})\) is not completely positive.

The matrix transposition map \(\theta (X)=X^T\) on \(M_{D}(\mathbb {C})\) is positive and trace-preserving.

Every transposition witness \(W=(A\otimes B)\, F\, (A\otimes B)^\dagger \) is Hermitian.

Every two-positive map is positive.

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a complex-linear map. The following are equivalent:

1. \(T\) maps the cone of positive semidefinite matrices onto itself;

2. \(T\) is positive and preserves the rank of Hermitian matrices;

3. there is an invertible \(Y\in M_{D}(\mathbb {C})\) such that either \(T(X)=YXY^\dagger \) for every \(X\), or \(T(X)=YX^{\mathsf T}Y^\dagger \) for every \(X\).

This is [ Wol12 , Proposition 3.6 ] .

Let \(N\ge 3\), \(1\le m\le N-2\), and \(x\colon \mathbb {Z}/N\mathbb {Z}\to \mathbb {R}\) be nonnegative. With zero-denominator summands interpreted as zero,

On \(\mathbb {Z}/d\mathbb {Z}\), let

In the range \(d\ge 3\), \(1\le m\le d-2\), and \(s\ge d\), the matrix \(S\) has nonnegative entries, is circulant and invertible, and satisfies

More explicitly, for the unnormalized discrete Fourier transform,

Here \(q_0=0\) and \(q_j{\gt}0\) for \(j\ne 0\). Finally, \(Sx\) is the forward denominator and

Suppose in addition that \(S\) has nonnegative entries, that \(H\) is positive semidefinite, and that \(\ker H=\mathbb {R}\mathbf1\). Then every local maximum of \(f_S\) on \(A_{++}\) is projectively scalar. More precisely, a non-scalar local maximum \(a\) would produce the non-scalar vector

Let \(N\ge 3\), \(2\le m\le N-2\), and \(s\ge N\). Suppose that \(x\colon \mathbb {Z}/N\mathbb {Z}\to \mathbb {R}\) is nonnegative and \(x_{N-1}=0\), and let \(x'\colon \mathbb {Z}/(N-1)\mathbb {Z}\to \mathbb {R}\) be given in the standard cyclic coordinates by \(x'_i=x_i\) for \(0\le i{\lt}N-1\). Then