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

Characterization of equivariant maps and application to entanglement detection

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced by Bhat. These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat and as a generalization of the equivariant maps studied by Collins et al. Using representation theory, we fully compute them and study their graphical representation, and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all k-entangled density matrices.Comment: 16 pages, 4 figure

Product of random projections, Jacobi ensembles and universality problems arising from free probability

We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.Comment: 28 pages, no figure, pdfLaTe

Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels

Using the graphical calculus and integration techniques introduced by the authors, we study the statistical properties of outputs of products of random quantum channels for entangled inputs. In particular, we revisit and generalize models of relevance for the recent counterexamples to the minimum output entropy additivity problems. Our main result is a classification of regimes for which the von Neumann entropy is lower on average than the elementary bounds that can be obtained with linear algebra techniques

Convex set of quantum states with positive partial transpose analysed by hit and run algorithm

The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi--circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0,3], which describes well the level density obtained numerically for PPT states.Comment: 11 pages, 4 figure