Celine Nadal

Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

Using a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropys probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachment of a single-charge drop from the continuum sea of all the other charges. These transitions, respectively, control the left and right tails of the entropys probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics.

Index Distribution of Gaussian Random Matrices

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N x N matrix belonging to Gaussian orthogonal (beta=1), unitary (beta=2) or symplectic (beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N+/N scales, for large N, as P(c,N) approximately = exp[-betaN(2)Phi(c)] where the rate function Phi(c), symmetric around c=1/2 and universal (independent of beta), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.

Statistical Distribution of Quantum Entanglement for a Random Bipartite State

We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachment of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of part of these results appeared recently in Nadal et al. (Phys. Rev. Lett. 104:110501, 2010).

How many eigenvalues of a Gaussian random matrix are positive

We study the probability distribution of the index N(+), i.e., the number of positive eigenvalues of an N×N Gaussian random matrix. We show analytically that, for large N and large N(+) with the fraction 0≤c=N(+)/N≤1 of positive eigenvalues fixed, the index distribution P(N(+)=cN,N)~exp[-βN(2)Φ(c)] where β is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function Φ(c) is computed explicitly for all 0≤c≤1. It is independent of β and displays a quadratic form modulated by a logarithmic singularity around c=1/2. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance Δ(N) of index fluctuations growing as Δ(N)~lnN/βπ(2) for large N. For β=2, this result is independently confirmed against an exact finite-N formula, yielding Δ(N)=lnN/2π(2)+C+O(N(-1)) for large N, where the constant C for even N has the nontrivial value C=(γ+1+3ln2)/2π(2)≃0.185 248… and γ=0.5772… is the Euler constant. We also determine for large N the probability that the interval [ζ(1),ζ(2)] is free of eigenvalues. Some of these results have been announced in a recent letter [Phys. Rev. Lett. 103, 220603 (2009)].

Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution.

We exploit a relation between the mean number N(m) of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behavior of N(m) in the simplest glasslike transition occuring in a toy model of a single particle in an N-dimensional random environment, with N>>1. Varying the control parameter μ through the critical value μ(c) we analyze in detail how N(m)(μ) drops from being exponentially large in the glassy phase to N(m)(μ)~1 on the other side of the transition. We also extract a subleading behavior of N(m)(μ) in both glassy and simple phases. The width δμ/μ(c) of the critical region is found to scale as N(-1/3) and inside that region N(m)(μ) converges to a limiting shape expressed in terms of the Tracy-Widom distribution.

RIGHT TAIL ASYMPTOTIC EXPANSION OF TRACY–WIDOM BETA LAWS

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian β matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail of Tracy–Widom β laws, for all β > 0, by studying the double scaling limit.

Purity distribution for generalized random Bures mixed states

We compute the distribution of the purity for random density matrices (i.e. random mixed states) in a large quantum system, distributed according to the Bures measure. The full distribution of the purity is computed using a mapping to random matrix theory and then a Coulomb gas method. We find three regimes that correspond to two phase transitions in the associated Coulomb gas. The first transition is characterized by an explosion of the third derivative on the left of the transition point. The second transition is of first order, it is characterized by the detachment of a single charge of the Coulomb gas. A key remark in this paper is that the random Bures states are closely related to the O(n) model for n = 1. This actually led us to study ?generalized Bures states? by keeping n general instead of specializing to n = 1.