67 research outputs found
The distillability problem revisited
An important open problem in quantum information theory is the question of
the existence of NPT bound entanglement. In the past years, little progress has
been made, mainly because of the lack of mathematical tools to address the
problem. (i) In an attempt to overcome this, we show how the distillability
problem can be reformulated as a special instance of the separability problem,
for which a large number of tools and techniques are available. (ii) Building
up to this we also show how the problem can be formulated as a Schmidt number
problem. (iii) A numerical method for detecting distillability is presented and
strong evidence is given that all 1-copy undistillable Werner states are also
4-copy undistillable. (iv) The same method is used to estimate the volume of
distillable states, and the results suggest that bound entanglement is
primarily a phenomenon found in low dimensional quantum systems. (v) Finally, a
set of one parameter states is presented which we conjecture to exhibit all
forms of distillability.Comment: Several corrections, main results unchange
On two-distillable Werner states
We consider bipartite mixed states in a quantum system. We say
that is PPT if its partial transpose is positive
semidefinite, and otherwise is NPT. The well-known Werner states are
divided into three types: (a) the separable states (the same as the PPT
states); (b) the one-distillable states (necessarily NPT); and (c) the NPT
states which are not one-distillable. We give several different formulations
and provide further evidence for validity of the conjecture that the Werner
states of type (c) are not two-distillable.Comment: 19 pages, expanded version containing new result
A few steps more towards NPT bound entanglement
We consider the problem of existence of bound entangled states with
non-positive partial transpose (NPT). As one knows, existence of such states
would in particular imply nonadditivity of distillable entanglement. Moreover
it would rule out a simple mathematical description of the set of distillable
states. Distillability is equivalent to so called n-copy distillability for
some n. We consider a particular state, known to be 1-copy nondistillable,
which is supposed to be bound entangled. We study the problem of its two-copy
distillability, which boils down to show that maximal overlap of some projector
Q with Schmidt rank two states does not exceed 1/2. Such property we call the
the half-property. We first show that the maximum overlap can be attained on
vectors that are not of the simple product form with respect to cut between two
copies. We then attack the problem in twofold way: a) prove the half-property
for some classes of Schmidt rank two states b) bound the required overlap from
above for all Schmidt rank two states. We have succeeded to prove the
half-property for wide classes of states, and to bound the overlap from above
by c<3/4. Moreover, we translate the problem into the following matrix analysis
problem: bound the sum of the squares of the two largest singular values of
matrix A \otimes I + I \otimes B with A,B traceless 4x4 matrices, and Tr
A^\dagger A + Tr B^\dagger B = 1/4.Comment: 15 pages, Final version for IEEE Trans. Inf. Theor
Quantum Correlations and Quantum Non-Locality: A Review and a Few New Ideas
In this paper we make an extensive description of quantum non-locality, one
of the most intriguing and fascinating facets of quantum mechanics. After a
general presentation of several studies on this subject, we consider if quantum
non-locality, and the friction it carries with special relativity, can
eventually find a "solution" by considering higher dimensional spaces.Comment: 1
Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory
PhD thesis (University of York). The thesis covers in a unified way the
material presented in quant-ph/0403073, quant-ph/0502040, quant-ph/0504160,
quant-ph/0510035, quant-ph/0512012 and quant-ph/0603283. It includes two large
review chapters on entanglement and distillation.Comment: 192 page
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