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arXiv:2105.08778

Characterization of preorders induced by positive maps in the set of Hermitian matrices

Julio I. de Vicente

Published May 18, 2021• 0 citations
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Abstract

Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix $A$ into another $B$ if and only if the vector of eigenvalues of $A$ majorizes that of $B$. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out that the measures in each of the two formalisms are essentially unique.

DOI
10.48550/arxiv.2105.08778

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Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix AAA into another BBB if and only if the vector of eigenvalues of AAA majorizes that of BBB. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out that the measures in each of the two formalisms are essentially unique.
We show the linearity and injectivity of two maps ϕ1\phi_1ϕ1​ and ϕ2\phi_2ϕ2​ between certain subsets of matrix spaces that satisfy tr⁡(ϕ1(A)ϕ2(B))=tr⁡(AB)\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)tr(ϕ1​(A)ϕ2​(B))=tr(AB). We apply it to characterize maps ϕi\phi_iϕi​ (i=1,…,mi=1, \ldots, mi=1,…,m) satisfying tr⁡(ϕ1(A1)⋯ϕm(Am))=tr⁡(A1⋯Am)\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)tr(ϕ1​(A1​)⋯ϕm​(Am​))=tr(A1​⋯Am​) on maps S→S\mathcal{S}\to \mathcal{S}S→S in which S\mathcal{S}S is the set of nnn-by-nnn general or Hermitian matrices for m≥3m\ge 3m≥3 and positive definite matrices for m≥2m\ge 2m≥2, and on completely positive maps Mn→MnM_n\to M_nMn​→Mn​ for m≥2m\ge 2m≥2. Then we give several equivalent characterizations of invertible completely positive maps, that is, ϕ:Mn→Mk\phi: M_n\to M_kϕ:Mn​→Mk​ and ψ:Mk→Mn\psi:M_k\to M_nψ:Mk​→Mn​ such that ψ(ϕ(A))=A\psi(\phi(A))=Aψ(ϕ(A))=A for A∈MnA\in M_nA∈Mn​, 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 ϕ\phiϕ between matrix spaces satisfies ϕ(I)=I\phi(I)=Iϕ(I)=I and ϕ(Ak)=ϕ(A)k\phi(A^k)=\phi(A)^kϕ(Ak)=ϕ(A)k for a fixed integer k≥2k\ge 2k≥2 if and only if ϕ(Ar)=ϕ(A)r\phi(A^r)=\phi(A)^rϕ(Ar)=ϕ(A)r for all r∈Z+r\in {\mathbb Z}_+r∈Z+​. We characterize maps ϕ,ψ:S→S\phi, \psi: \mathcal{S} \to \mathcal{S}ϕ,ψ:S→S satisfying tr⁡(ϕ(A)ψ(B))=tr⁡(AB)\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)tr(ϕ(A)ψ(B))=tr(AB) and tr⁡(ϕ(A)ψ(B)k)=tr⁡(ABk)\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)tr(ϕ(A)ψ(B)k)=tr(ABk) for a fixed k≥2k\ge 2k≥2, where S\mathcal{S}S is the set of general or Hermitian matrices.
We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.
Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one.
In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels Λ1,Λ2Λ_1, Λ_2Λ1​,Λ2​ and an arbitrary unknown input state, we show that the output state of the quantum channel Λ2Λ_2Λ2​ from the pair can be uniquely identified from the output statistics of the other channel Λ1Λ_1Λ1​ from the pair, using some POVM, iff Λ2Λ_2Λ2​ can be obtained from Λ1Λ_1Λ1​ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, Λ2Λ_2Λ2​ may not always be obtained from Λ1Λ_1Λ1​ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing Λ2Λ_2Λ2​ given that Λ1Λ_1Λ1​ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.
Quantum monotone metric was introduced by Petz [Linear Algebra Appl. 244, 81–96 (1996)], and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always the one based on completely positive, trace preserving maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics that have monotonicity under completely positive, trace non-increasing maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity, and monotonicity with respect to positive operators.