arXiv:2002.12887 · Journal of Mathematical Physics

Positive maps and trace polynomials from the symmetric group

Felix Huber

Published Feb 1, 2021• 19 citations

Abstract

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.

DOI: 10.1063/5.0028856

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arXiv:2002.12887 · Journal of Mathematical Physics

Positive maps and trace polynomials from the symmetric group

Felix Huber

Published Feb 1, 2021• 19 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/5.0028856

Related papers

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M_m(\mathbb{C}) \otimes M_n(\mathbb{C})

. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.

Φ:\mathcal A \to \mathcal B

between

C^\ast

-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality

Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)

and form an intermediate class between positive and completely positive maps. We completely classify

U(n)

-equivariant on

M_n(\mathbb C)

and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker

DU(n)

-equivariance (diagonal unitary symmetry) and for tensor-product symmetries

U(n_1) \otimes U(n_2)

. In each case, the parameter regions where

Φ

is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the

U(n)

-equivariant family satisfies

\mathrm{PPT} \iff \mathrm{EB}

, while the

DU(2)

, symmetric

DU(3)

U(2) \otimes U(2)

and

U(2) \otimes U(3)

, families obey the

\mathrm{PPT}^2

conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the

\mathrm{PPT}^2

property holds.

We show the linearity and injectivity of two maps

\phi_1

and

\phi_2

between certain subsets of matrix spaces that satisfy

\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)

. We apply it to characterize maps

\phi_i

(

i=1, \ldots, m

) satisfying

\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)

on maps

\mathcal{S}\to \mathcal{S}

in which

\mathcal{S}

is the set of

n

-by-

n

general or Hermitian matrices for

m\ge 3

and positive definite matrices for

m\ge 2

, and on completely positive maps

M_n\to M_n

for

m\ge 2

. Then we give several equivalent characterizations of invertible completely positive maps, that is,

\phi: M_n\to M_k

and

\psi:M_k\to M_n

such that

\psi(\phi(A))=A

for

A\in M_n

, and provide a comprehensive list of such examples. These maps are connected to the trace-of-product preserving property and are useful in quantum information theory. In addition, we show that a linear map

\phi

between matrix spaces satisfies

\phi(I)=I

and

\phi(A^k)=\phi(A)^k

for a fixed integer

k\ge 2

if and only if

\phi(A^r)=\phi(A)^r

for all

r\in {\mathbb Z}_+

. We characterize maps

\phi, \psi: \mathcal{S} \to \mathcal{S}

satisfying

\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)

and

\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)

for a fixed

k\ge 2

, where

\mathcal{S}

is the set of general or Hermitian matrices.