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arXiv:2607.11369

Moment-based PPT criteria for random bipartite states

Cécilia Lancien, Kieran McShane

quant-phmath-phmath.COmath.FAmath.PRSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on $\mathbb C^d\otimes\mathbb C^d$, obtained as the marginal over an environment $\mathbb C^s$ of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level $m\in\mathbb N$ in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension $s=λ_md^2$ at which the behavior of the associated random state switches from violating to satisfying it, with probability going to $1$ as $d$ grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension $d$ is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.

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Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd\mathbb C^d\otimes\mathbb C^dCd⊗Cd, obtained as the marginal over an environment Cs\mathbb C^sCs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈Nm\in\mathbb Nm∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2s=λ_md^2s=λm​d2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 111 as ddd grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension ddd is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
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