How often is a random quantum state k-entangled?

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include

Non-additivity of Renyi entropy and Dvoretzky's Theorem

The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden-Winter disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).Comment: 8 pages, LaTeX; v2: added and updated references, minor editorial changes, no content change

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

Phase transitions for random states and a semi-circle law for the partial transpose

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k < k_0. By "random" we mean here "uniformly distributed on the sphere of the corresponding Hilbert space." The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.Comment: 5 pages, 2 figures. This short note contains a high-level overview of two long and technical papers, arXiv:1011.0275 and arXiv:1106.2264. Version 2: unchanged results, editorial changes, added reference, close to the published articl

On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.Comment: 10 pages, 1 figure; ver. 2 - minor changes, new proof of lemma

Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant ρ such that ∥⋅∥X⊗πY⩽ρ∥⋅∥X⊗εY holds for all Banach spaces of dimensions dimX=n and dimY=m, where X⊗πY and X⊗εY are the projective and injective tensor products. By requiring that X=Y, one obtains a symmetrised version of the above ratio, denoted by rs(n). We prove that r(n,m)⩾19/18 for all n,m⩾2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and rs(n), showing that, up to log factors: rs(n) is of the order n−−√ (which is sharp); r(n, n) is at least of the order n1/6; and r(n, m) grows at least as min{n,m}1/8. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ℓ1/ℓ2/ℓ∞ trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements