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Characterizing locally distinguishable orthogonal product states
Yuan Feng, Yaoyun Shi
TL;DR
A complete, computationally verifiable characterization of LOCC-indistinguishable orthogonal product sets was unknown; this paper supplies one for 3⊗3. It shows these sets fall into three classes, with membership computationally determinable from pairwise inner products.
Problem
A complete, computationally verifiable characterization of LOCC-indistinguishable OPSs was unknown, with 3⊗3 the smallest dimension where such a characterization was not known.
Source: p. 2Method
The proof characterizes irreducible OPBs through R9 rectangular representations, then uses this structure to establish the 3⊗3 classification.
Source: pp. 2–3Results
In 3⊗3, LOCC-indistinguishable OPSs are exactly the three stated classes, with 5, 8, or 9 states corresponding to classes (2), (3), and (1).
Source: p. 2Takeaways & Limitations
Membership in the three classes can be determined computationally from pairwise inner products of the state components.
Source: p. 2Extending the characterization to higher dimensions remains open, and the rectangular representation lemma does not hold in all dimensions.
Source: p. 4
Abstract
from arXiv · showhide
Bennett et al. \cite{BDF+99} identified a set of orthogonal {\em product} states in the $3\otimes 3$ Hilbert space such that reliably distinguishing those states requires non-local quantum operations. While more examples have been found for this counter-intuitive ``nonlocality without entanglement'' phenomenon, a complete and computationally verifiable characterization for all such sets of states remains unknown. In this Letter, we give such a characterization for the $3\otimes 3$ space.