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arXiv:math/0607175 · Operators and Matrices

Extremal marginal tracial states in coupled systems

Geoffrey L. Price, Shôichirô Sakai

Published Jan 1, 2007• 8 citations
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Abstract

Let be the convex set consisting of all states on the tensor product B B of the algebra B = M n (C) of all n n matrices over the complex numbers C with the property that the restrictions BI and IB are the unique tracial states on B I and I B . We find necessary and sufficient conditions for such a state, called a marginal tracial state, to be extremal in . We also give a characterization of those extreme points in which are pure states. We conjecture that all extremal marginal tracial states are pure states.

DOI
10.7153/oam-01-10

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Let be the convex set consisting of all states on the tensor product B B of the algebra B = M n (C) of all n n matrices over the complex numbers C with the property that the restrictions BI and IB are the unique tracial states on B I and I B . We find necessary and sufficient conditions for such a state, called a marginal tracial state, to be extremal in . We also give a characterization of those extreme points in which are pure states. We conjecture that all extremal marginal tracial states are pure states.
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Extremal Marginal States of Maximal Rank in (d,d+m)(d, d+m)(d,d+m)
We study the extreme points of the convex set C(ρ1,ρ2)\mathcal{C}(ρ_1,ρ_2)C(ρ1​,ρ2​) of bipartite quantum states with fixed marginals ρ1ρ_1ρ1​ and ρ2ρ_2ρ2​. We construct extreme points in (d, d+m)(d,\,d+m)(d,d+m) dimension, of rank d+md+md+m, matching the highest possible value, for all d≥3d\geq 3d≥3, m>d2−2d−22m > \frac{d^2-2d-2}{2}m>2d2−2d−2​ (when d=2d=2d=2, m≥1m\geq 1m≥1). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of C(ρ1,ρ2)\mathcal{C}(ρ_1,ρ_2)C(ρ1​,ρ2​), it is sufficient to study the special case C(D1,D2)\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)C(D1​,D2​), where the marginals are diagonal. Additionally, we observe that it is sufficient to consider d1≤d2d_1\leq d_2d1​≤d2​. Thus, our results show that apart from possibly a few finite cases, for each d1d_1d1​, the maximal rank is achieved almost all times.
We study the convex set C(ρ1,ρ2) of all bipartite quantum states with fixed marginal states ρ1 and ρ2. The extremal states in this set have recently been characterized by Parthasarathy [Ann. Henri Poincaré (to appear), quant-ph/0307182]. Here we present an alternative necessary and sufficient condition for a state in C(ρ1,ρ2) to be extremal. Our approach is based on a canonical duality between bipartite states and a certain class of completely positive maps and has the advantage that it is easier to check and to construct explicit examples of extremal states. In dimension 2×2 we give a simple new proof for the fact that all extremal states in C(121,121) are precisely the projectors onto maximally entangled wave functions. We also prove that in higher dimension this does not hold and construct an explicit example of an extremal state in C(131,131) that is not maximally entangled. Generalizations of this result to higher dimensions are also discussed.
We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.