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 $n$ such that $\mu^{*n}$ is stochastically dominated by $\nu^{*n}$ for two given probability measures $\mu$ and $\nu$. As a consequence we obtain a similar theorem on the majorization order for vectors in $\R^d$. 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 ℓp\ell_p 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 $y$, the set $T(y)$ of vectors $x$ such that we have $x \otimes z \prec y \otimes z$ for some $z$, where $\prec$ denotes the standard majorization relation. Our main result is that the closure of $T(y)$ in the $\ell_1$ norm can be fully described by inequalities on the $\ell_p$ norms: $\|x\|_p \leq \|y\|_p$ for all $p \geq 1$. This is a first step towards a complete description of $T(y)$ itself. It can also be seen as a $\ell_p$-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 ($x^{\otimes n} \prec y^{\otimes n}$ for given $n$). The main new tool is a variant of Cram\'er$ theorem on large deviations for sums of i.i.d. random variables