Strong quantum nonlocality and unextendibility without entanglement in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-partite systems with odd <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>
Yiyun He, Fei Shi, Xiande Zhang
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-dimensional hypercubes, we present strongly nonlocal orthogonal product sets in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-partite systems for all odd <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>3</mml:mn></mml:math>. Based on this decomposition, we give explicit constructions of unextendible product bases in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-partite systems for odd <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&#x2265;</mml:mo><mml:mn>3</mml:mn></mml:math>. Furthermore, we apply our results to quantum secret sharing, uncompletable product bases, and PPT entangled states.