117 research outputs found
On the separability of unitarily invariant random quantum states - the unbalanced regime
We study entanglement-related properties of random quantum states which are
unitarily invariant, in the sense that their distribution is left unchanged by
conjugation with arbitrary unitary operators. In the large matrix size limit,
the distribution of these random quantum states is characterized by their
limiting spectrum, a compactly supported probability distribution. We prove
several results characterizing entanglement and the PPT property of random
bipartite unitarily invariant quantum states in terms of the limiting spectral
distribution, in the unbalanced asymptotical regime where one of the two
subsystems is fixed, while the other one grows in size.Comment: New section on PPT matrices with large Schmidt numbe
Asymptotics of random density matrices
We investigate random density matrices obtained by partial tracing larger
random pure states. We show that there is a strong connection between these
random density matrices and the Wishart ensemble of random matrix theory. We
provide asymptotic results on the behavior of the eigenvalues of random density
matrices, including convergence of the empirical spectral measure. We also
study the largest eigenvalue (almost sure convergence and fluctuations)
Gaussianization and eigenvalue statistics for random quantum channels (III)
In this paper, we present applications of the calculus developed in Collins
and Nechita [Comm. Math. Phys. 297 (2010) 345-370] and obtain an exact formula
for the moments of random quantum channels whose input is a pure state thanks
to Gaussianization methods. Our main application is an in-depth study of the
random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284
(2008) 263-280] and used recently by Brandao and Horodecki [Open Syst. Inf.
Dyn. 17 (2010) 31-52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to
refine the Hastings counterexample to the additivity conjecture in quantum
information theory. This model is exotic from the point of view of random
matrix theory as its eigenvalues obey two different scalings simultaneously. We
study its asymptotic behavior and obtain an asymptotic expansion for its von
Neumann entropy.Comment: Published in at http://dx.doi.org/10.1214/10-AAP722 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic domination for iterated convolutions and catalytic majorization
We study how iterated convolutions of probability measures compare under
stochastic domination. We give necessary and sufficient conditions for the
existence of an integer such that is stochastically dominated by
for two given probability measures and . As a consequence
we obtain a similar theorem on the majorization order for vectors in . In
particular we prove results about catalysis in quantum information theory
Compatibility of quantum measurements and inclusion constants for the matrix jewel
In this work, we establish the connection between the study of free
spectrahedra and the compatibility of quantum measurements with an arbitrary
number of outcomes. This generalizes previous results by the authors for
measurements with two outcomes. Free spectrahedra arise from matricial
relaxations of linear matrix inequalities. A particular free spectrahedron
which we define in this work is the matrix jewel. We find that the
compatibility of arbitrary measurements corresponds to the inclusion of the
matrix jewel into a free spectrahedron defined by the effect operators of the
measurements under study. We subsequently use this connection to bound the set
of (asymmetric) inclusion constants for the matrix jewel using results from
quantum information theory and symmetrization. The latter translate to new
lower bounds on the compatibility of quantum measurements. Among the techniques
we employ are approximate quantum cloning and mutually unbiased bases.Comment: v5: section 3.3 has been expanded significantly to incorporate the
generalization of the Cartesian product and the direct sum to matrix convex
sets. Many other minor modifications. Closed to the published versio
Catalytic majorization and norms
An important problem in quantum information theory is the mathematical
characterization of the phenomenon of quantum catalysis: when can the
surrounding entanglement be used to perform transformations of a jointly held
quantum state under LOCC (local operations and classical communication) ?
Mathematically, the question amounts to describe, for a fixed vector , the
set of vectors such that we have for
some , where denotes the standard majorization relation. Our main
result is that the closure of in the norm can be fully
described by inequalities on the norms: for all
. This is a first step towards a complete description of
itself. It can also be seen as a -norm analogue of Ky Fan dominance
theorem about unitarily invariant norms. The proofs exploits links with another
quantum phenomenon: the possibiliy of multiple-copy transformations
( for given ). The main new tool is a
variant of Cram\'er$ theorem on large deviations for sums of i.i.d. random
variables
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