Uncertainty Relations Represented by Tracial or Non-Tracial Positive Linear Maps (Theory of function spaces and related topics)

In this paper, we explore equivalent conditions for the positivity of a quadratic equation P(x)=Ax2+Bx+C with operator coefficients. Specifically, we show that Ax2+2Bx+C≥0 for all x∈R if and only if A,C≥0, B=B∗, and there exists a self-adjoint operator H such that [A+C+HB+i(A−C)B+i(A−C)A+C−H]≥0.Also, we show that if A and B are two unital C∗-algebras, Φ:A→B is a unital tracial positive linear map, and ϕ:A→A is a linear map satisfying ϕ(X)∗ϕ(X)≤ϕ(X∗X) for all X∈A, then VΨ(A)♯VΨ(B)≥±12Ψ([A,B]),where A and B are self-adjoint elements in A, Ψ=Φ∘ϕ, and VΨ(X)=Ψ(X∗X)−Ψ(X∗)Ψ(X) denotes the generalized variance of an observable X.In this paper, we explore equivalent conditions for the positivity of a quadratic equation P(x)=Ax2+Bx+C with operator coefficients. Specifically, we show that Ax2+2Bx+C≥0 for all x∈R if and only if A,C≥0, B=B∗, and there exists a self-adjoint operator H such that [A+C+HB+i(A−C)B+i(A−C)A+C−H]≥0.Also, we show that if A and B are two unital C∗-algebras, Φ:A→B is a unital tracial positive linear map, and ϕ:A→A is a linear map satisfying ϕ(X)∗ϕ(X)≤ϕ(X∗X) for all X∈A, then VΨ(A)♯VΨ(B)≥±12Ψ([A,B]),where A and B are self-adjoint elements in A, Ψ=Φ∘ϕ, and VΨ(X)=Ψ(X∗X)−Ψ(X∗)Ψ(X) denotes the generalized variance of an observable X.

Linear and Multilinear AlgebraJan 24, 20250 citations

Uncertainty Relations and Their Applications

Frontiers research topicsJan 1, 20220 citations

Relative entropy representations via tracial joint spectral measures

Lukas Schmitt

We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)$ , where $μ_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\mapsto \mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)$, where $μ_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\mapsto \mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.

quant-pharXiv:2609.12547Sep 14, 20260 citations

We show that for positive semidefinite

A

and

B

the relative entropy can be written as

D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)

, where

μ_{A,B}

denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve

t\mapsto \mathrm{tr}[A-tB]_+

is piecewise affine exactly when

A

and

B

commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.