X-arability of mixed quantum states

• URL: http://arxiv.org/abs/2409.18948v1
• Date: Fri, 27 Sep 2024 17:49:59 GMT
• Title: X-arability of mixed quantum states
• Authors: Harm Derksen, Nathaniel Johnston, Benjamin Lovitz, Aravindan Vijayaraghavan,
• Abstract summary: We propose and study a unified notion of separability, which we call X-arability. For a subset of pure states X, we say that a mixed quantum state is X-arable if it lies in the convex hull of X. We develop unified tools and provable guarantees for X-arability, which already give new results for the standard separability problem.
• Score: 9.15858941222986
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The problem of determining when entanglement is present in a quantum system is one of the most active areas of research in quantum physics. Depending on the setting at hand, different notions of entanglement (or lack thereof) become relevant. Examples include separability (of bosons, fermions, and distinguishable particles), Schmidt number, biseparability, entanglement depth, and bond dimension. In this work, we propose and study a unified notion of separability, which we call X-arability, that captures a wide range of applications including these. For a subset (more specifically, an algebraic variety) of pure states X, we say that a mixed quantum state is X-arable if it lies in the convex hull of X. We develop unified tools and provable guarantees for X-arability, which already give new results for the standard separability problem. Our results include: -- An X-tension hierarchy of semidefinite programs for X-arability (generalizing the symmetric extensions hierarchy for separability), and a new de Finetti theorem for fermionic separability. -- A hierarchy of eigencomputations for optimizing a Hermitian operator over X, with applications to X-tanglement witnesses and polynomial optimization. -- A hierarchy of linear systems for the X-tangled subspace problem, with improved polynomial time guarantees even for the standard entangled subspace problem.

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