MOMENT-BASED PPT CRITERIA FOR RANDOM BIPARTITE STATES

Characterizing entanglement in bipartite quantum systems is a computationally difficult task. While the Positive Partial Transpose (PPT) criterion is a standard tool, it requires full state tomography, which is experimentally demanding. This paper investigates a hierarchy of relaxations of the PPT criterion based on the low-order moments of the partially transposed state, which are more accessible experimentally. The authors aim to determine the typical performance of these criteria on high-dimensional random bipartite mixed states.

The authors analyze random mixed states on C^d ⊗ C^d, generated as the marginal of a uniformly distributed pure state in a larger environment C^s. They define the m-th level of the hierarchy, the p_{2m+1}-PPT criterion, which involves checking the positive semi-definiteness of a Hankel matrix constructed from the first 2m+1 moments of the partially transposed state. Using tools from random matrix theory, combinatorics of permutations, and the theory of orthogonal polynomials, the authors compute the asymptotic average and variance of these moments as d grows, and use concentration of measure arguments to establish the threshold behavior.

The study identifies a sharp threshold for the environment dimension s = λ_m d^2 at which the behavior of a random state switches from typically violating to typically satisfying the m-th level of the moment-based PPT hierarchy. The threshold parameter is found to be λ_m = 4 cos^2(π / (m + 2)). As the level m increases, this threshold approaches 4, which is the known threshold for the standard PPT criterion. This provides a quantitative comparison of how these implementable criteria perform relative to the full PPT criterion in high-dimensional settings.