arXiv:2403.07229 · Linear Algebra and its Applications

Characterizations of homomorphisms among unital completely positive maps

Andre Kornell

Published Jan 15, 2025• 0 citations

Abstract

We prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if it is completely entropy-nonincreasing, where the relevant notion of entropy is a variant of von Neumann entropy. This adjusted von Neumann entropy is the negative of the relative entropy with respect to the uniform state on the C ⁎ -algebra, up to an additive constant. As an intermediate step, we prove that a unital completely positive map between finite-dimensional C ⁎ -algebras is a homomorphism if and only if its adjusted Choi operator is a projection. Both equivalences generalize familiar facts about stochastic maps between finite sets.

DOI: 10.1016/j.laa.2025.01.014

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arXiv:2403.07229 · Linear Algebra and its Applications

Characterizations of homomorphisms among unital completely positive maps

Andre Kornell

Published Jan 15, 2025• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.laa.2025.01.014

Related papers

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Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$ , or equivalently with a commutant; returning to $\mathcal{N}$ itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is $*$ -anti-isomorphic to itself. Naturality cannot be dropped: whenever $\mathcal{M}$ is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$, or equivalently with a commutant; returning to $\mathcal{N}$ itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is $*$-anti-isomorphic to itself. Naturality cannot be dropped: whenever $\mathcal{M}$ is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.

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We prove recovery theorems for

α

z

Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and

L^p

-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.

Abstract For a state

\omega

on a C

^{*}

-algebra

A

, we characterize all states

\rho

in the weak* closure of the set of all states of the form

\omega \circ \varphi

, where

\varphi

is a map on

A

of the form

\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,

\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1

(

a_i\in A

n\in \mathbb {N}

). These are precisely the states

\rho

that satisfy

\|\rho |J\|\leq \|\omega |J\|

for each ideal

J

A

. The corresponding question for normal states on a von Neumann algebra

\mathcal {R}

(with the weak* closure replaced by the norm closure) is also considered. All normal states of the form

\omega \circ \psi

, where

\psi

is a quantum channel on

\mathcal {R}

(that is, a map of the form

\psi (x)=\sum \nolimits _ja_j^{*}xa_j

, where

a_j\in \mathcal {R}

are such that the sum

\sum \nolimits _ja_j^{*}a_j

converge to

1

in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C

^{*}

-algebra and for properly infinite von Neumann algebras the converse also holds.

Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on

\mathcal{M}\bar{\otimes}\mathcal{N}

but on the tensor product of

\mathcal{M}

with the \emph{opposite} algebra of

\mathcal{N}

, or equivalently with a commutant; returning to

\mathcal{N}

itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is

*

-anti-isomorphic to itself. Naturality cannot be dropped: whenever

\mathcal{M}

is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.