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arXiv:2202.09034 · Quantum

Bounds on the smallest sets of quantum states with special quantum nonlocality

Mao-Sheng Li, Yanling Wang

Published Sep 7, 2023• 21 citations
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Abstract

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in C d 1 ⊗ C d 1 ⊗ ⋯ ⊗ C d N for any d i ≥ 2 and 1 ≤ i ≤ N ?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.

DOI
10.22331/q-2023-09-07-1101

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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 .

arXiv:2011.10286•Journal of Physics A Mathematical and Theoretical•Sep 21, 2021•26 citations
An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in C d 1 ⊗ C d 1 ⊗ ⋯ ⊗ C d N for any d i ≥ 2 and 1 ≤ i ≤ N ?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.
The nonlocal set has received wide attention over recent years. Shortly before, Li and Wang arXiv:2202.09034 proposed the concept of a locally stable set: The only possible orthogonality preserving measurement on each subsystem is trivial. Locally stable sets present stronger nonlocality than those sets that are just locally indistinguishable. In this work, we focus on the constructions of locally stable sets in multipartite quantum systems. First, two lemmas are put forward to prove that an orthogonality-preserving local measurement must be trivial. Then we present the constructions of locally stable sets with minimum cardinality in bipartite quantum systems ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}3)$ and ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{2}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2})$. Moreover, for the multipartite quantum systems ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}n}\phantom{\rule{4pt}{0ex}}(d\ensuremath{\ge}2)$ and ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{{d}_{i}}\phantom{\rule{4pt}{0ex}}(3\ensuremath{\le}{d}_{1}\ensuremath{\le}{d}_{2}\ensuremath{\le}\ensuremath{\cdots}\ensuremath{\le}{d}_{n})$, we also obtain d+1d+1d+1 and dn+1{d}_{n}+1dn​+1 locally stable orthogonal states, respectively. Fortunately, our constructions reach the lower bound of the cardinality on the locally stable sets, which provides a positive and complete answer to an open problem raised in arXiv:2202.09034.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving measurement can be performed to this set. In general, it is difficult to show such a orthogonality-preserving measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully construct orthogonal product sets (OPSs) in any possible three, four and five-partite systems that are locally irreducible in every bipartition. Moreover, we also show the existence of unextendible product bases (UPBs) with such properties in d⊗d⊗dd\otimes d\otimes dd⊗d⊗d for any d≥3d\geq 3d≥3. Therefore, these OPSs and UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Both results solve the open questions given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. Our results shed a new light on the strong quantum nonlocality without entanglement.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unextendible product bases (UPBs) that are locally irreducible in every bipartition in d ⊗ d ⊗ d for any d ≥ 3 , and 3 ⊗ 3 ⊗ 3 achieves the minimum dimension for the existence of such UPBs. These UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Our result solves an open question given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.
Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.
Quantum nonlocality based on state discrimination describes the global property of the set of orthogonal states and has a wide range of applications in quantum cryptographic protocols. Strongest nonlocality is the strongest form of quantum nonlocality recently presented in multipartite quantum systems: a set of orthogonal multipartite quantum states is strongest nonlocal if the only orthogonality-preserving local measurements on the subsystems in every bipartition are trivial. In this work, we found a construction of strongest nonlocal sets in Cd1⊗Cd2⊗Cd3\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}Cd1​⊗Cd2​⊗Cd3​ (2≤d1≤d2≤d3)(2\leq d_{1}\leq d_{2}\leq d_{3})(2≤d1​≤d2​≤d3​) of size d2d3+1d_2d_3+1d2​d3​+1 without stopper states. Then we obtain the strongest nonlocal sets in four-partite systems with d3+1d^3+1d3+1 orthogonal states in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^d\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd (d≥2)(d\geq2)(d≥2) and d2d3d4+1d_{2}d_{3}d_{4}+1d2​d3​d4​+1 orthogonal states in Cd1⊗Cd2⊗Cd3⊗Cd4\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}Cd1​⊗Cd2​⊗Cd3​⊗Cd4​ (2≤d1≤d2≤d3≤d4)(2\leq d_{1}\leq d_{2}\leq d_{3}\leq d_{4})(2≤d1​≤d2​≤d3​≤d4​). Surprisingly, the number of the elements in all above constructions perfectly reaches the recent conjectured lower bound and reduces the size of the strongest nonlocal set in Cd⊗Cd⊗Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd⊗Cd⊗Cd of [\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108}, 062407 (2023)}] by d−2d-2d−2. In particular, the general optimal construction of the strongest nonlocal set in four-partite system is completely solved for the first time, which further highlights the theory of quantum nonlocality from the perspective of state discrimination.
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 .