Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ^{\prime})$ be tracial von-Neumann algebras and let $φ:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $τ(B)=1$ , a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, τ)$ is said to be recoverable if $\mathcal{R}(φ(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $φ$ . In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circφ$ . We show that there exists a completely positive, trace preserving map $ψ:\mathcal{M}\to\mathcal{M}$ such that $ψ(A)$ is recoverable for all $A$ and $(\mathcal{R}\circφ)^n\toψ$ in norm as operators on $L^p(\mathcal{M},τ)$ for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in $L^1$ . Finally, we prove an interesting decomposition theorem for normal states on $\mathcal{M}$ .Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ^{\prime})$ be tracial von-Neumann algebras and let $φ:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $τ(B)=1$, a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, τ)$ is said to be recoverable if $\mathcal{R}(φ(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $φ$. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circφ$. We show that there exists a completely positive, trace preserving map $ψ:\mathcal{M}\to\mathcal{M}$ such that $ψ(A)$ is recoverable for all $A$ and $(\mathcal{R}\circφ)^n\toψ$ in norm as operators on $L^p(\mathcal{M},τ)$ for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in $L^1$. Finally, we prove an interesting decomposition theorem for normal states on $\mathcal{M}$.

quant-pharXiv:2605.08829May 12, 20260 citations

Entropy of Completely Positive Maps and Applications to Quantum Information Theory

Ichiro Fujimoto

Trends in mathematicsJan 1, 20160 citations

Absolute continuity for operator valued completely positive maps on C∗-algebras

Aurelian Gheondea, Ali S. Kavruk

Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.

Journal of Mathematical PhysicsFeb 1, 200915 citations

On Quantum Correlations and Positive Maps

Władysław A. Majewski

arXiv:math-ph/0403024Letters in Mathematical PhysicsFeb 1, 20043 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

Let

(\mathcal{M},τ)

and

(\mathcal{N},τ^{\prime})

be tracial von-Neumann algebras and let

φ:\mathcal{M}\to\mathcal{N}

be a strictly completely positive, trace preserving map. Given a positive, invertible

B\in\mathcal{M}

with

τ(B)=1

, a state on

\mathcal{M}

given by a positive

A\in L^1(\mathcal{M}, τ)

is said to be recoverable if

\mathcal{R}(φ(A))=A

where

\mathcal{R}

is the Petz recovery map corresponding to

B

and

φ

. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of

\mathcal{R}\circφ

. We show that there exists a completely positive, trace preserving map

ψ:\mathcal{M}\to\mathcal{M}

such that

ψ(A)

is recoverable for all

A

and

(\mathcal{R}\circφ)^n\toψ

in norm as operators on

L^p(\mathcal{M},τ)

for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in

L^1

. Finally, we prove an interesting decomposition theorem for normal states on

\mathcal{M}

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.