Positive Linear Maps Research Articles

Asymptotic lifting for completely positive maps

Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

Variations on the Choi–Jamiołkowski isomorphism

The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗ -algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism (Choi 1975 Lin. Alg. Appl. 10 285) further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical structure in C∗ -algebras. We explain in detail how this dependence gives rise to the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiołkowski (1972 Rep. Math. Phys. 3 275). We then overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi’s theorem for Jamiołkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗ -algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.

Ergodic quantum processes on finite von Neumann algebras

Let (M,τ) be a tracial von Neumann algebra with a separable predual and let (Ω,P) be a probability space. A bounded positive random linear operator on L1(M,τ) is a map γ:Ω×L1(M,τ)→L1(M,τ) so that τ(γω(x)a) is measurable for all x∈L1(M,τ) and a∈M, and x↦γω(x) is bounded, positive, and linear almost surely. Given an ergodic T∈Aut(Ω,P), we study quantum processes of the form γTnω∘γTn−1ω∘⋯∘γTmω for m,n∈Z. Using the Hennion metric introduced in [21], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when γω is the predual of a normal positive linear map on M. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [21].

Generalizing Choi map in M3 beyond circulant scenario

We introduce a family of positive linear maps in the algebra of 3 × 3 complex matrices, which generalizes the seminal positive non-decomposable map originally proposed by Choi. Necessary and sufficient conditions for decomposability are derived and demonstrated. Specifically, the proposed maps evolve the circulant structure with three parameters into a doubly stochastic structure with five parameters.

Exposedness of elementary positive maps between matrix algebras

ABSTRACT The positive linear maps Ad s which send matrices x to s ∗ x s play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ad s generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

Generalized Refinements of Reversed AM-GM Operator Inequalities for Positive Linear Maps

We shall present some more generalized and further refinements of reversed AM-GM operator inequalities for positive linear maps due to Xue’s and Ali’s publications.

Integral formula for quantum relative entropy implies data processing inequality

Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.

Profitable entanglement for channel discrimination

We investigate the usefulness of side entanglement in discriminating between two generic qubit channels, up to unitary pre- and post-processing, and determine exact conditions under which it does enhance (as well as conditions under which it does not) the success probability. This is done in a constructive way by first analysing the problem for channels that are extremal in the set of completely positive and trace-preserving qubit linear maps and then for channels that are inside such a set.

Some inequalities involving eigenvalues and positive linear maps

Some inequalities involving eigenvalues and positive linear maps

Parameterized Wasserstein means

A new barycenter of positive definite matrices minimizing the quantum divergences associated with sandwiched quasi-relative entropy has been recently introduced. It generalizes the well-known Wasserstein mean for covariance matrices of Gaussian distributions with mean zero, so we call it the parameterized Wasserstein mean. In this article we prove a norm inequality of the parameterized Wasserstein mean, give bounds to the parameterized Wasserstein mean with respect to the Loewner order and several inequalities with positive unital linear maps, and show the extended version of the Lie-Trotter-Kato formula for the parameterized Wasserstein mean. Finally, we show the log-majorization properties of the parameterized Wasserstein mean by using the unique characterization of the Cartan mean.

Inequalities for products of singular values of matrices

Let A and B be positive semidefinite matrices, be an increasing convex function with , and let be a positive unital linear map. It is shown that for , where and are the singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given.

Morita Equivalences for Completely Positive Linear Maps and GNS-C*-Correspondences

Morita Equivalences for Completely Positive Linear Maps and GNS-C*-Correspondences

Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products

<abstract><p>We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.</p></abstract>

Some inequalities for eigenvalues and positive linear maps

Some inequalities for eigenvalues and positive linear maps

Mapping cone of k-entanglement breaking maps

In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce Schmidt number on taking composition with another completely positive map.

Two-moment characterization of spectral measures on the real line

Abstract In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$ , $k=p, q$ , imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$ -algebras is also provided.

Strong Morita Equivalence for Completely Positive Linear Maps on $$C^*$$-Algebras

We will introduce the notion of strong Morita equivalence for completely positive linear maps and study its basic properties. Also, we will discuss the relation between strong Morita equivalence for bounded \(C^*\)-bimodule linear maps and strong Morita equivalence for completely positive linear maps. Furthermore, we will show that if two unital \(C^*\)-algebras are strongly Morita equivalent, then there is a \(1-1\) correspondence between the two sets of all strong Morita equivalence classes of completely positive linear maps on the two unital \(C^*\)-algebras and we will show that the corresponding two classes of the completely positive linear maps are also strongly Morita equivalent.

Purity of the embeddings of operator systems into their C∗- and injective envelopes

We study the issue of issue of purity (as a completely positive linear map) for identity maps on operators systems and for their completely isometric embeddings into their C$^*$-envelopes and injective envelopes. Our most general result states that the canonical embedding of an operator system $\mathcal R$ into its injective envelope $\mathcal I(\mathcal R)$ is pure if and only if the C$^*$-envelope of $\mathcal R$ is a prime C$^*$-algebra. To prove this, we also show that the identity map on any AW$^*$-factor is a pure completely positive linear map. For embeddings of operator systems into their C$^*$-envelopes, the issue of purity is seemingly harder to describe in full generality, and so we focus here on operator systems arising from the generators of discrete groups. The question of purity of the identity is quite subtle for operator system that are not C$^*$-algebras, and we have results only for two universal operator systems arising from discrete groups. Lastly, a previously unrecorded feature of pure completely positive linear maps is presented: every pure completely positive linear map on an operator system $\mathcal R$ into an injective von Neumann algebra $\mathcal M$ has a pure completely positive extension to any operator system $\mathcal T$ that contains $\mathcal R$ as an operator subsystem, thereby generalising a result of Arveson for the injective type I factor $\mathcal B(\mathcal H)$.

Positive linear maps and spreads of normal matrices

We obtain some inequalities involving positive linear maps on matrix algebra. The special cases provide bounds for the spreads of normal matrices.

Mean inequalities for sector matrices involving positive linear maps

Mean inequalities for sector matrices involving positive linear maps