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arXiv:quant-ph/0406021 · Journal of Mathematical Physics

On extremal quantum states of composite systems with fixed marginals

Oliver Rudolph

Published Oct 18, 2004• 25 citations

Abstract

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.

DOI: 10.1063/1.1776642

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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.

Extremal Marginal States of Maximal Rank in

(d, d+m)

We study the extreme points of the convex set

\mathcal{C}(ρ_1,ρ_2)

of bipartite quantum states with fixed marginals

ρ_1

and

ρ_2

. We construct extreme points in

(d,\,d+m)

dimension, of rank

d+m

, matching the highest possible value, for all

d\geq 3

,

m > \frac{d^2-2d-2}{2}

(when

d=2

,

m\geq 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

\mathcal{C}(ρ_1,ρ_2)

, it is sufficient to study the special case

\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)

, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider

d_1\leq d_2

. Thus, our results show that apart from possibly a few finite cases, for each

d_1

, the maximal rank is achieved almost all times.

We discuss the problem of characterizing upper bounds on entanglement in a bipartite quantum system when only the reduced density matrices (marginals) are known. In particular, starting from the known two-qubit case, we propose a family of candidates for maximally entangled mixed states with respect to fixed marginals for two qutrits. These states are extremal in the convex set of two-qutrit states with fixed marginals. Moreover, it is shown that they are always quasidistillable. As a by-product we prove that any maximally correlated state that is quasidistillable must be pure. Our observations for two qutrits are supported by numerical analysis.

Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.

An extremal element of the convex set of composite quantum states in

M_2\otimes M_3

, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.

An extremal element of the convex set of composite quantum states in

M_2\otimes M_3

, whose marginals are all normalised identities has been constructed. It is found to be a mixed state and is entangled as well.