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arXiv:2412.20979

Entanglement in bipartite X-states: Analytical results for the volume of states with positive partial transpose

Yaqing Xy Wang, József Zsolt Bernád

Published Dec 30, 2024• 0 citations
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Abstract

We provide an analytical formula for the volume ratio between bipartite X-states with positive partial transpose and all bipartite X-states. The result applies to arbitrary $m \times n$-bipartite systems and the volume expressions are derived with respect to the Hilbert-Schmidt measure.

DOI
10.34734/fzj-2025-02565

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We provide an analytical formula for the volume ratio between bipartite X-states with positive partial transpose and all bipartite X-states. The result applies to arbitrary m×nm \times nm×n-bipartite systems and the volume expressions are derived with respect to the Hilbert-Schmidt measure.
Abstract Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres–Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes’ rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit-qutrit, and qubit-four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit–qutrit states. Based on this numerical approach with the help of the Clauser–Horne–Shimony–Holt inequality and the Collins–Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
While entanglement is believed to be an important ingredient in the understanding of quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive an analytic upper-bound on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split that rapidly outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.