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arXiv:2201.12832 · Physical Review A

Activating strong nonlocality from local sets: An elimination paradigm

Subhendu B. Ghosh, Tathagata Gupta, A. V. Ardra, Anandamay Das Bhowmik, Sutapa Saha, Tamal Guha, Amit Mukherjee

Published Jul 25, 2022• 13 citations
arXivHTMLPDF

Abstract

Apart from the Bell nonlocality, which deals with the correlations generated from the local input-output statistics, quantum theory exhibits another kind of nonlocality that involves the indistinguishability of the locally preparable set of multipartite states. While Bell-type nonlocality cannot be activated from a given local correlation via local operations and shared randomness, it is already reported that the latter kind of nonlocality can be activated from a ``local,'' i.e., locally distinguishable set of states. However, recently it is shown that a stronger notion of such a nonlocality, which deals with elimination instead of discrimination, can be activated from locally preparable bipartite states of dimension $7\ifmmode\times\else\texttimes\fi{}8$. The present work observes that the same notion can be demonstrated even in lower dimensional multipartite systems. Importantly, the possibly strongest version of such an activation is further depicted here, where none of the transformed product states can be eliminated, even if all but one of the parties come together.

DOI
10.1103/physreva.106.l010202

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Apart from the Bell nonlocality, which deals with the correlations generated from the local input-output statistics, quantum theory exhibits another kind of nonlocality that involves the indistinguishability of the locally preparable set of multipartite states. While Bell-type nonlocality cannot be activated from a given local correlation via local operations and shared randomness, it is already reported that the latter kind of nonlocality can be activated from a ``local,'' i.e., locally distinguishable set of states. However, recently it is shown that a stronger notion of such a nonlocality, which deals with elimination instead of discrimination, can be activated from locally preparable bipartite states of dimension 7\ifmmode×\else\texttimes\fi87\ifmmode\times\else\texttimes\fi{}87\ifmmode×\else\texttimes\fi8. The present work observes that the same notion can be demonstrated even in lower dimensional multipartite systems. Importantly, the possibly strongest version of such an activation is further depicted here, where none of the transformed product states can be eliminated, even if all but one of the parties come together.
The quantum state discrimination primitive becomes highly nontrivial in the limited measurement setting and leads to different classes of impossibility, viz., indistinguishability, unmarkability, irreducibility, etc. These phenomena, often referred to as another nonlocal aspect of quantum theory, have utmost importance in the domain of data hiding, secret sharing, etc. Motivated by this, recently a significant effort has been devoted to activate local indistinguishability from locally distinguishable nontrivial sets of quantum states. In the present work, we introduce other stronger notions of quantum nonlocality and depict a series of hierarchical nonlocality activation from nontrivial sets of locally distinguishable entangled states. Moreover, we appropriately moderate the strict condition of nontriviality, introduced in earlier literature, and come up with interesting examples in support of our claim.
A set of orthogonal quantum states in multipartite systems is of genuine nonlocality if it is locally indistinguishable in every bipartition. If it is locally reducible when the parties are separated, we say that it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 1}; otherwise, it has genuine nonlocality of type~\uppercase\expandafter{\romannumeral 2}. For a locally distinguishable set without local redundancy, if there exist some orthogonality preserving local measurements such that each outcome leads to a locally indistinguishable set, then we say that it exhibits the activation of nonlocality. We activate type-\uppercase\expandafter{\romannumeral 1} and type-\uppercase\expandafter{\romannumeral 2} genuine nonlocality of orthogonal product state sets in tripartite systems. In particular, we tackle the local irredundancy problem with partial trace operation and ppp-ary numeral systems to significantly simplify the proofs. Our results also address the open question raised by S. Bandyopadhyay \textit{et al.}[\href{https://link.aps.org/doi/10.1103/PhysRevA.104.L050201}{Phys. Rev. A \textbf{104}, L050201 (2021)}]. Furthermore, we observe the activation of hidden genuine nonlocality in multipartite systems, which highlights the applications of nonlocality based on state discrimination in different practical scenarios.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.
Based on the conviction of switching information from locally accessible to locally hidden environs, the concept of hidden nonlocality activation has recently been highlighted by Bandyopadhyay et al. [Phys. Rev. A 104, L050201 (2021)]. They have demonstrated that a certain locally distinguishable set of pure quantum states can be transformed into a locally indistinguishable set with certainty through orthogonality preserving local measurements. This transformation makes the set locally inaccessible, despite being locally accessible before. This phenomenon is defined as the activation of hidden nonlocality. In this paper, we present two classes of locally distinguishable sets within $(2m+1)\ensuremath{\bigotimes}2\ensuremath{\bigotimes}(2m+1)$ systems. One class reveals nonlocality through local operations, whereas the other requires joint measurements for it. As the latter class depends on nonlocal operations to exhibit nonlocality, it arguably has a lower degree of nonlocality, and accordingly, can be considered as more local compared to the first class. This analysis exhibits a stronger manifestation of locality by elucidating the nuanced interplay between these distinct local phenomena within the framework of quantum state discrimination. Furthermore, we also explore their significant applications in the context of data hiding. Additionally, we introduce the concept of strong local set and compare it with various activable sets, highlighting differences in terms of locality.
Abstract Recently, Halder et al (2019 Phys. Rev. Lett. 122 040403) proposed the concept of strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of its subsystems. Due to the complexity of the problem, most results are limited to tripartite systems. Here we consider a weaker form of nonlocality which is called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for every bipartition of the subsystems. In this work, we study how to construct sets of orthogonal product states which are genuinely nonlocal. Firstly, we present a set of product states with simple structure in bipartite systems that is locally indistinguishable. After that, based on a simple observation, we present a general method to construct genuinely nonlocal sets of multipartite product states by using those sets that are genuinely nonlocal but with less parties. As a consequence, we obtain that genuinely nonlocal sets of fully product states exist in any L parties systems ⨂ i = 1 L C d i provided L ⩾ 3 and d i ⩾ 3 for all i .
A set of quantum states is said to be antidistinguishable if, upon being given a randomly chosen state from the set, it is possible to exclude with certainty a state that was definitely not prepared. We study quantum nonlocality within the framework of local state exclusion and prove that any set of multipartite pure states containing a state that is orthogonal to all the others is locally antidistinguishable. Next, we introduce the task of local state antimarking\textit{local state antimarking}local state antimarking, where, instead of a single state, a non-repetitive sequence of states from a known set of multipartite states is randomly selected and distributed to spatially separated parties, whose objective is to identify, using LOCC, at least one sequence that was not supplied. We present an ensemble of product states that is not globally antidistinguishable; yet, by choosing states from it without replacement, one can construct sequences that can be perfectly excluded globally but, importantly, not locally. This reveals a hitherto unexplored form of `nonlocality without entanglement'. Finally, for a given ensemble, we compare its local antidistinguishability and antimarkability with its conclusive local distinguishability and markability. We demonstrate that no strict hierarchy exists between these paradigms: there exist product-state ensembles that permit one task while strictly forbidding the other, and vice versa.