Local discrimination of quantum states is a fundamental task in distributed quantum information processing and underlies applications such as quantum communication, data hiding, and secret sharing. Here we investigate a weak form of local indistinguishability by asking how easily it can disappear when the candidate set is reduced or an additional copy of the unknown state is supplied. We introduce inclusion-minimal locally indistinguishable sets, namely, locally indistinguishable sets for which every proper subset is perfectly distinguishable by local operations and classical communication (LOCC), and show that every finite locally indistinguishable set contains such a subset. We further find that any inclusion-minimal locally indistinguishable sets becomes perfectly distinguishable by LOCC when two identical copies are available, although a single copy is insufficient. Fininally, We construct explicit inclusion-minimal locally indistinguishable product-state sets in $(\mathbb C^d)^{\otimes n}$ for every odd $d=2k+1$ and $n\ge2$. These results show that local indistinguishability can be nontrivial at the single-copy level yet fragile under either the removal of candidate states or a modest increase in copy resources, providing a complementary perspective on the structure of quantum nonlocality.
Local discrimination of quantum states is a fundamental task in distributed quantum information processing and underlies applications such as quantum communication, data hiding, and secret sharing. Here we investigate a weak form of local indistinguishability by asking how easily it can disappear when the candidate set is reduced or an additional copy of the unknown state is supplied. We introduce inclusion-minimal locally indistinguishable sets, namely, locally indistinguishable sets for which every proper subset is perfectly distinguishable by local operations and classical communication (LOCC), and show that every finite locally indistinguishable set contains such a subset. We further find that any inclusion-minimal locally indistinguishable sets becomes perfectly distinguishable by LOCC when two identical copies are available, although a single copy is insufficient. Fininally, We construct explicit inclusion-minimal locally indistinguishable product-state sets in (Cd)⊗n for every odd d=2k+1 and n≥2. These results show that local indistinguishability can be nontrivial at the single-copy level yet fragile under either the removal of candidate states or a modest increase in copy resources, providing a complementary perspective on the structure of quantum nonlocality.
Quantum nonlocality has different manifestations that, in general, are revealed by local measurements of the parts of a composite system. In this paper, we study nonlocality arising from a set of orthogonal states that cannot be perfectly distinguished by local operations and classical communication (LOCC). Such a set is deemed nonlocal, for a joint measurement on the whole system is necessary for perfect discrimination of the states with certainty. On the other hand, a set of orthogonal states that can be perfectly distinguished by LOCC is believed to be devoid of nonlocal properties. Here, we show that there exist orthogonal sets that are locally distinguishable but without local redundancy (i.e., they become nonorthogonal on discarding one or more subsystems) whose nonlocality can be activated by local measurements. In particular, a state chosen from such a set can be locally converted, with certainty, into another state, the identity of which can now only be ascertained by global measurement and no longer by LOCC. In other words, a locally distinguishable set without local redundancy may be locally converted into a locally indistinguishable set with certainty. We also suggest an application, namely, local hiding of information, that allows us to locally hide locally available information without losing any part. Once hidden, the information in its entirety can only be retrieved using entanglement.
We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, where d is odd, Zhang et al. [Z.-C. Zhang et al., Phys. Rev. A 90, 022313 (2014)] have constructed d2 orthogonal product basis quantum states that are locally indistinguishable. We find a subset that contains $6d\ensuremath{-}9$ orthogonal product states that are still locally indistinguishable. We generalize our method to an arbitrary bipartite quantum system ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We present a small set with only $3(m+n)\ensuremath{-}9$ orthogonal product states and prove that these states are local operations and classical communication (LOCC) indistinguishable. Even though these $3(m+n)\ensuremath{-}9$ product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication.
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.
Quantum nonlocality, rooted in the quantum state discrimination task, contributes to the development of the theory of local distinguishability of quantum states. Recently, Zhu et al. [Phys. A (Amsterdam, Neth.) 624, 128956 (2023)] found that a nonlocal set of orthogonal product states (OPSs) can be perfectly identified by local operations and classical communication after removing a specific quantum state in a bipartite quantum system. They called this phenomenon minimal nonlocality. It is obvious that minimal nonlocality effectively reflects the lower boundary of the elements contained in a nonlocal set of OPSs. In this paper, we generalize the methods of constructing nonlocal sets with minimal nonlocality from bipartite systems to multipartite systems and from orthogonal product states to entangled states. First, we give a method to construct a set of OPSs with minimal nonlocality in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and then we extend it to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system for $d\ensuremath{\ge}4$ and $n\ensuremath{\ge}3$. In addition, we give a method to construct a nonlocal set of orthogonal states with entanglement relations in the ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ quantum system, and we prove that this set has minimal nonlocality, which can also be extended to the ${\ensuremath{\bigotimes}}_{i=1}^{n}{\mathbb{C}}^{d}$ quantum system.
Quantum nonlocality admits several operational manifestations, one of which emerges from sets of orthogonal quantum states that cannot be perfectly distinguished by local operations and classical communication (LOCC). Such sets are regarded as nonlocal because their perfect discrimination requires global measurements. In contrast, sets that are perfectly distinguishable by LOCC are generally considered locally accessible and operationally classical. In this work, we investigate the role of incompletability in local state discrimination and introduce the notion of \emph{activation of incompletability}. Specifically, we demonstrate the existence of orthogonal sets that are initially perfectly distinguishable by LOCC and free from local redundancy, but which can be transformed via LOCC into strictly incompletable sets. We prove that activation of incompletability necessarily implies activation of nonlocality, whereas the converse fails in general, thereby establishing a hierarchy between the two activation phenomena. Furthermore, within the framework of local incoherent operations and classical communication (LICC), we show that any set whose incompletability can be activated can nevertheless be extended to a complete orthonormal basis of the Hilbert space, although the resulting completed basis is no longer perfectly distinguishable by LOCC. Our results uncover a fundamental interplay among local distinguishability, incompletability, coherence, and nonlocality, and provide new insight into the structure of locally accessible quantum information.
A set of orthogonal quantum states is said to be locally indistinguishable if they cannot be perfectly distinguished by local operations and classical communication (LOCC). Otherwise, the states are locally distinguishable. Interestingly, locally indistinguishable states can have productive applications in quantum information processing protocols. In this sense, locally indistinguishable states are useful. On the other hand, it is usual to consider that locally distinguishable states are useless. Nevertheless, recent works suggest that locally distinguishable states should be given due consideration as in certain situations these states can be converted to locally indistinguishable states under orthogonality-preserving LOCC (OP-LOCC). Such a counterintuitive phenomenon motivates us to ask when the aforesaid conversion is possible and when it is not. In this work, we provide different structures of locally distinguishable product and entangled states which do not allow the aforesaid conversion. We also provide certain structures of locally distinguishable states which allow the aforesaid conversion. In this way, we classify the locally distinguishable sets by introducing hierarchies among them. In a multipartite system, this study becomes more involved as there exist multipartite locally distinguishable sets which cannot be converted to locally indistinguishable sets by OP-LOCC across any bipartition. We say this as ``no activation across bi-partitions".