Dependencies

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:

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

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 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 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.

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 \(s\in (0,1)\),

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 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

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 \(Y_\theta =\theta Y_1+(1-\theta )Y_2\).

For \(s\in (0,1)\),

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

Fix \(t{\gt}0\) and positive-definite matrices with \(A_1\le A_2\) and \(B_1\le B_2\). The resolvent of the Kronecker model \(A\otimes \mathbb {1}+t\, (\mathbb {1}\otimes B^\top )\) of the commuting left- and right-multiplication superoperators is antitone:

For \(t{\gt}0\), positive-definite \(A_1,A_2,B_1,B_2\), 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 \(B_\theta =\theta B_1+(1-\theta )B_2\):

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

Let \(E^*\) be the adjoint of a trace-preserving Kraus map. If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\) commuting with \(A\) and satisfying \(A^\dagger A\le B\), then

Let \(E^*(X)=\sum _i K_i^\dagger X K_i\) be the adjoint Kraus map of a trace-preserving Kraus family, and let \(A\in M_{D}(\mathbb {C})\) be normal. Then the Schwarz gap

is positive semidefinite. Equivalently,

This is the completely positive case of [ Wol12 , Proposition 5.1 ] .

For \(s\in [0,1]\), any matrix \(K\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the function \((A,B)\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is jointly concave, satisfying (8) of Theorem 7.7.10.

For \(s\in [0,1]\), any matrix \(K\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), write \(F_s(A,B):=\Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then

See [ Lie73 ; And79 ] . This statement is the positive-definite, boundary-exponent (\(x+y=1\)) case of the Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] , which holds for positive semidefinite \(A,B\) and all exponents \(x,y\ge 0\) with \(x+y\le 1\).

For \(s\in [0,1]\), any matrix \(K\), and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is jointly concave, satisfying (8) of Theorem 7.7.10. This is the full Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] on the boundary line \(x+y=1\) (with \(x=s\), \(y=1-s\)): the positive-definiteness restriction of Theorem 7.7.10 is lifted. The general two-exponent region is obtained in Theorem 7.7.12.

Let \(x,y\ge 0\) with \(x+y\le 1\). For any matrix \(K\) and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), write \(F_{x,y}(A,B):=\Re \operatorname{tr}(K^\dagger A^x K B^y)\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then

This is Wolf’s Theorem 5.15 in the finite-dimensional matrix setting.

For a positive unital map \(T\) and a positive-definite matrix \(A\), the inequality \(T(\log A)\le \log (T(A))\) holds.

For elements \(a,b\) of a unital C\({}^\ast \)-algebra with \(0\le a,b\), an exponent \(p\in [1,2]\), and \(t\in [0,1]\), the map \(a\mapsto a^p\) is convex for the Loewner order on the positive cone:

This is the convex counterpart of the operator concavity of \(x\mapsto x^p\) for exponents \(p\in [0,1]\).

For a positive unital map \(T\) and positive-definite \(A\), one has \(T(\log A)\le \log (T(A))\). This is the corrected logarithmic specialization associated with [ Wol12 , Corollary 5.2(3) ] . The printed common hypothesis \(T(\mathbb {1})\le \mathbb {1}\) is insufficient: in dimension one, for \(0{\lt}c{\lt}1\), the positive subunital map \(T(x)=cx\) and \(A=1\) would give \(0=T(\log A)\le \log (T(A))=\log c\). The corrected theorem instead assumes \(T(\mathbb {1})=\mathbb {1}\); see [ con26i ] .

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), one has \(T(A^p)\le (T(A))^p\). This is [ Wol12 , Theorem 5.13 ] for concave \(f(x)=x^p\).

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), one has \((T(A))^p\le T(A^p)\). This is [ Wol12 , Theorem 5.11 ] for the subunital operator-convex case \(f(x)=x^p\).

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), the inequality \(T(A^p)\le (T(A))^p\) holds.

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), the inequality \((T(A))^p\le T(A^p)\) holds.

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

This is [ Wol12 , Theorem 5.6 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is normal, then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Proposition 5.1 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is subnormal (i.e. there exists a normal operator on a larger space whose compression to \(\mathbb {C}^D\) is \(A\)), then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Theorem 5.5 ] .

For \(p\in [0,1]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),

This follows from operator concavity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.

For \(p\in [1,2]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),

This follows from operator convexity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.