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arXiv:1411.0216 · International Journal of Theoretical Physics

The Local Orthogonality Between Quantum States and Entanglement Decomposition

Sunho Kim, Junde Wu, Lin Zhang, Minhyung Cho

Published Jan 19, 2016• 0 citations
arXivHTMLPDF
DOI
10.1007/s10773-016-2918-x

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We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at every subsystem. In the bi-partite (N=2) case, this decomposition reduces to the (not necessarily unique) Schmidt decomposition and the entropy reduces to the entropy of entanglement
The Schmidt decomposition of quantum many-body states, which is the basis of many powerful quantum many-body techniques, relies on a partition of the whole system in terms of orthogonal local basis functions. By carefully investigating the spectrum of the truncated density matrices in a non-orthogonal basis, we propose in this work a non-orthogonal decomposition of Slater determinants, which reduces to the conventional Schmidt decomposition in the orthogonal limit. The new decomposition provides a natural tool for building local correlation spaces when the orbitals are overlapping, and we are able to extend some existing embedding methods based on the Schmidt decomposition like density matrix embedding theory (DMET) to overlapping subsystem partitions to achieve greater flexibility and efficiency. We also propose a new quantum embedding strategy as another utility of the non-orthogonal decomposition that bridges the \textit{ab initio} model potential (AIMP) theory and DMET, which does not require a mean-field calculation of the whole system like AIMP, and is able to capture quantum entanglement like DMET.
A set of orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. Although such a phenomenon has been shown to any three-, four-, and five-partite systems, the existence of strongly nonlocal orthogonal product sets in multipartite systems remains unknown. In this paper, by using a general decomposition of the N -dimensional hypercubes, we present strongly nonlocal orthogonal product sets in N -partite systems for all odd N ≥ 3 . Based on this decomposition, we give explicit constructions of unextendible product bases in N -partite systems for odd N ≥ 3 . Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.