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arXiv:2410.05400 · Physical Review A

Separable ellipsoids around multipartite states

Robin Y. Wen, Gilles Parez, Liuke Lyu, William Witczak‐Krempa, Achim Kempf

Published Jul 15, 2025• 2 citations
arXivHTMLPDF

Abstract

We show that, in finite dimensions, around any $m$-partite product state $ρ_{\rm prod}=ρ_1\otimes...\otimesρ_m$, there exists an ellipsoid of separable states centered around $ρ_{\rm prod}$. This separable ellipsoid contains the separable ball proposed in previous works, and the volume of the ellipsoid is typically exponentially larger than that of the ball, due to the hierarchy of eigenvalues in typical states. We further generalize this ellipsoidal criterion to a trace formula that yields separable region around all separable states, and further study biseparability. Our criteria not only help numerical procedures to rigorously detect separability, but they also lead to a nested hierarchy of SLOCC-stable subsets that cover the separable set. We apply the procedure for separability detection to 3-qubit X states, genuinely entangled 4-qubit states mixed with noise, and the 1d transverse field Ising model at finite temperature to illustrate the power of our procedure for understanding entanglement in physical systems.

DOI
10.1103/tyhj-1wlq

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The quantum steering ellipsoid of a two-qubit state is the set of Bloch vectors that Bob can collapse Alice's qubit to, considering all possible measurements on his qubit. We provide an elementary construction of the ellipsoid for arbitrary states, calculate its volume, and explain how this geometric representation can be made faithful. The representation provides a range of new results, and uncovers new features, such as the existence of "incomplete steering" in separable states. We show that entanglement can be analyzed in terms of three geometric features of the ellipsoid and prove that a state is separable if and only if it obeys a "nested tetrahedron" condition.
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