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Journal of Quantum Information Science

Incomparability Through Superposition of Many States under LOCC

Amit Bhar, Santi Prasad Sinha

Published Jan 1, 2011• 0 citations
Publisher

Abstract

In this paper we investigate the global property of the states constructed through superposition of many states by using the concept of incomparability under LOCC as the inherent property of the states.In our work we are able to form a bridge between comparable and incomparable classes of states through linear superposition of a states and the concept of strong incomparability criterian under LOCC. PACS number(s): 03.67.Hk, 03.65.Ud, 03.65.Ta, 03.67.-a, 89.70+c.

DOI
10.4236/jqis.2011.11001

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In this paper we investigate the global property of the states constructed through superposition of many states by using the concept of incomparability under LOCC as the inherent property of the states.In our work we are able to form a bridge between comparable and incomparable classes of states through linear superposition of a states and the concept of strong incomparability criterian under LOCC. PACS number(s): 03.67.Hk, 03.65.Ud, 03.65.Ta, 03.67.-a, 89.70+c.
Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmidt rank from a finite number of 2x2 pure entangled states only by deterministic LOCC. We provide general theory for the case of 3x3 system of incomparable states by the above processes where incomparability seems to be the most hardest one.
Since the canonical commutation relations for a finite number of degrees of freedom have many inequivalent irreducible representations, the states of a physical system may span more than one such represen-tation. Superseparability is defined to be the case in which no meaning can be attached to a superposition of vectors belonging to inequivalent representations. In this report, which is basically nonmathematical, we trace the origins of superseparability and suggest two experiments that may establish the existence of the phenomenon. We then dis-cuss which-path experiments and interaction-free measurements in the light of superseparability, and conclude with stating some open prob-lems. A mathematical appendix provides the essential mathematical background. ∗Preliminary version 1