Local Indistinguishability Research Articles

The construction and weakly local indistinguishability of multiqubit unextendible product bases

The construction and weakly local indistinguishability of multiqubit unextendible product bases

A New Quantum Sealed-Bid Auction Protocol with a Set of Local Indistinguishable Orthogonal Product States

Quantum sealed-bid auction (QSA) is a special form of transaction with significant applications in the economic and financial fields. Using a unique set of locally indistinguishable orthogonal product (LIOP) states, we propose a new QSA protocol in this paper. In the protocol, the bid message is encoded as a quantum sequence of LIOP states, and the different particles of LIOP states are transmitted separately. Even though an attacker obtains a portion of the particles, they cannot recover the entire bid message because of the local indistinguishability of LIOP states. Once the auctioneer announces the winner’s bid, all bidders are able to confirm the authenticity of their bid. With the help of a semi-honest third party, collusion between the auctioneer and a malicious bidder can be discovered. Finally, our protocol is capable of meeting all requirements for secure sealed-bid auctions through security and completeness analysis. Additionally, the proposed protocol does not require any entangled resources and complicated operations, so it can be easily implemented in practice.

Hierarchical activation of quantum nonlocality: Stronger than local indistinguishability

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.

Unextendible product operator basis

Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality, which is the powerful resource for quantum information processing. In this work, we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB), and strongly UPUOB. We construct their examples and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB, which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs, and the distinguishability of them requires entanglement resources.

Quantum Concentration Inequalities

We establish Transportation Cost Inequalities (TCIs) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: First, we generalize the Dobrushin uniqueness condition to prove that Gibbs states of 1D commuting Hamiltonians satisfy a TCI at any positive temperature and provide conditions under which this first result can be extended to non-commuting Hamiltonians. Next, using a non-commutative version of Ollivier’s coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs H=(V,E) satisfy a TCI with constant scaling as O(|V|). Third, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Fourth, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.

Strong quantum nonlocality for multipartite entangled states

Recently, Halder et al. (in Phys Rev Lett 122:040403, 2019) present two sets of strong nonlocality of orthogonal product states based on the local irreducibility. However, for a set of locally indistinguishable orthogonal entangled states, the remaining question is whether the states can reveal strong quantum nonlocality. Here we present a general definition of strong quantum nonlocality based on the local indistinguishability. Then, in \(2 \otimes 2 \otimes 2\) quantum system, we show that a set of orthogonal entangled states is locally reducible but locally indistinguishable in all bipartitions, which means the states have strong nonlocality. Furthermore, we generalize the result in N-qubit quantum system with \(N\geqslant 3\). Finally, we also construct a class of strong nonlocality of orthogonal entangled states in \(d\otimes d\otimes \cdots \otimes d, d\geqslant 3\). Our results extend the concept of strong nonlocality for entangled states.

Local indistinguishability and incompleteness of entangled orthogonal bases: Method to generate two-element locally indistinguishable ensembles

We relate the phenomenon of local indistinguishability of orthogonal states with the properties of unextendibility and uncompletability of entangled bases for bipartite and multipartite quantum systems. We prove that all two-qubit unextendible entangled bases are of size three and they cannot be perfectly distinguished by separable measurements. We identify a method of constructing two-element orthogonal ensembles, based on the concept of unextendible entangled bases, that can potentially lead to information sharing applications. Two-element ensembles form the fundamental unit of ensembles, and yet does not offer locally indistinguishable ensembles for pure state elements. Going over to mixed states does open this possibility, but can be difficult to identify. The method provided using unextendible entangled bases can be used for their systematic generation. In multipartite systems, we find a class of unextendible entangled bases for which the unextendibility property remains conserved across all bipartitions. We also identify nonlocal operations, local implementation of which require entangled resource states from a higher-dimensional quantum system.

Nonlocal sets of orthogonal product states in an arbitrary multipartite quantum system

Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we firstly give a simple method to construct a nonlocal set of orthogonal product states in $\otimes_{j=1}^{n}\mathbb{C}^{d}$ for $d\geq 2$. Then we give an ingenious proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer states in the same quantum system. Furthermore, we generalize these two results to a more general $\otimes_{i=1}^{n}\mathbb{C}^{d_{j}}$ quantum system for $d_{j}\geq 2$. Compared with the existing results, the nonlocal set of multipartite orthogonal product states constructed by our method has fewer elements and is more simpler.

On the representation of local indistinguishability operators

This paper studies local indistinguishability operators, i.e., symmetric and transitive fuzzy relations that do not need to be reflexive. This is an important generalization of global indistinguishability relations (fuzzy relations satisfying the reflexivity property in addition) because there are interesting families of fuzzy relations that are non-reflexive. One case are decomposable relations, that are generated by a fuzzy subset and contains the t-norms as an important subfamily. Also the relations associated naturally to fuzzy subgroups are local indistinguishability operators. In this paper these relations will be studied stressing the way they can be generated. A representation theorem will be proved and related to the concepts of extensionality and of fuzzy rough set. Decomposable local indistinguishability operators will also be studied and related with one-dimensional ones in the sense of the previous representation theorem. The presence of these relations in the study of fuzzy subgroups will also be analyzed.

Security analysis and improvement of an arbitrated quantum signature scheme

We analyze the security of Jiang et al.’s arbitrated quantum signature scheme based on local indistinguishability of orthogonal product states. First, Jiang et al.’s scheme cannot resist against the forgery attack. Second, it is insecure against disavowal attack. Then, an improved arbitrated quantum signature scheme is proposed. The new scheme overcomes the security drawbacks of Jiang et al.’s scheme and it is secure against forgery attack and disavowal attack. What is more, it is more efficient than the old scheme. On the other hand, our quantum signature scheme is based on product states, which can be obtained straightforwardly, thus our scheme is easier to implement than the schemes using cluster states.

Constructions of one-way LOCC indistinguishable sets of generalized Bell states

In this paper, we mainly consider the local indistinguishability of the set of bipartite generalized Bell states (GBSs). We systematically show constructions of small sets of GBSs with cardinalities greatly smaller than d which are not distinguishable by one-way local operations and classical communication (1-LOCC) in \(d\otimes d\). The constructions, based on linear system and Vandermonde matrix, are different for odd dimensions than for even dimensions. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs and greatly improve previous results in Zhang et al. (Phys Rev A 91:012329, 2015) and Wang et al. (Quantum Inf Process 15:1661, 2016). Among others, the case that d is odd of the results shows that the set of 4 GBSs in \(5\otimes 5\) in Fan (Phys Rev A 75:014305, 2007) is indeed a 1-LOCC indistinguishable set which cannot be distinguished by Fan’s method, and the minimum cardinality of 1-LOCC indistinguishable set of GBSs in \(7\otimes 7\) is 5. The case that d is even of the results shows that there exists a 1-LOCC indistinguishable set of 18 GBSs in \(100\otimes 100\).

Arbitrary Quantum Signature Based on Local Indistinguishability of Orthogonal Product States

Digital signature plays an important role in cryptography. Many quantum digital signature (QDS) schemes have been proposed up to now since the security of classic digital signature (CDS) schemes becomes more and more vulnerable with the development of quantum computing algorithms. Most of the existing quantum signature schemes are based on probabilistic comparison of quantum states, which makes the schemes very complicated. In this paper, we propose a new QDS scheme based on local indistinguishability of orthogonal product states. In the scheme, the receiver cooperates with the arbitrator to verify the valid of the signature. The analysis of security and efficiency shows that our scheme is secure and efficient.

Local indistinguishability of multipartite orthogonal product bases

So far, very little is known about local indistinguishability of multipartite orthogonal product bases except some special cases. We first give a method to construct an orthogonal product basis with n parties each holding a $$\frac{1}{2}(n+1)$$ -dimensional system, where $$n\ge 5$$ and n is odd. The proof of the local indistinguishability of the basis exhibits that it is a sufficient condition for the local indistinguishability of an orthogonal multipartite product basis that all the positive operator-valued measure elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then, we construct a set of n-partite product states, which contains only 2n members and cannot be perfectly distinguished by local operations and classic communication. All the results lead to a better understanding of the phenomenon of quantum nonlocality without entanglement in multipartite and high-dimensional quantum systems.

Local indistinguishability and edge modes revealed by the sub-system fidelity

We propose to use the sub-system fidelity, defined by comparing a pair of reduced density matrices, to identify and locate zero modes relating two degenerate ground states in one-dimensional interacting many-body systems, where the degeneracy arises from either symmetry protected topology (SPT), or from discrete symmetry breaking (DSB). A theorem is provided to construct locally indistinguishable (LI) states by linear recombination of the degenerate states of minimal bulk entanglement entropy. The theorem enables us to construct local operators that swap exactly the LI states and are therefore the zero energy modes in the many-body system. Interestingly, they can be located anywhere in the DSB case, but can only be accommodated near the edges in the SPT case. This can be used to identify or distinguish SPT states against the so-called cat states in the DSB case. We illustrate the results for the anisotropic Haldane chain and the interacting Kitaev model.

Topological quantum error correction in the Kitaev honeycomb model

The Kitaev honeycomb model is an approximate topological quantum error correcting code in the same phase as the toric code, but requiring only a 2-body Hamiltonian. As a frustrated spin model, it is well outside the commuting models of topological quantum codes that are typically studied, but its exact solubility makes it more amenable to analysis of effects arising in this noncommutative setting than a generic topologically ordered Hamiltonian. Here we study quantum error correction in the honeycomb model using both analytic and numerical techniques. We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space. These bounds are tighter than can be achieved using known general properties of topological phases. Our proofs are specialized to the honeycomb model, but some of the methods may nonetheless be of broader interest. Following this, we numerically study noise caused by thermalization processes in the perturbative regime close to the toric code renormalization group fixed point. The appearance of non-topological excitations in this setting has no significant effect on the error correction properties of the honeycomb model in the regimes we study. Although the behavior of this model is found to be qualitatively similar to that of the standard toric code in most regimes, we find numerical evidence of an interesting effect in the low-temperature, finite-size regime where a preferred lattice direction emerges and anyon diffusion is geometrically constrained. We expect this effect to yield an improvement in the scaling of the lifetime with system size as compared to the standard toric code.

Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are $$\mathcal {C}^m\otimes \mathcal {C}^{n_1}$$Cm?Cn1 with $$5\le m\le n_1$$5≤m≤n1 and $$\mathcal {C}^m\otimes \mathcal {C}^{n_2}$$Cm?Cn2 with $$5\le m \le n_2$$5≤m≤n2, where $$n_1$$n1 is odd and $$n_2$$n2 is even. The second one is in $$\mathcal {C}^m\otimes \mathcal {C}^n$$Cm?Cn$$(m, n\ge 4)$$(m,n?4). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.

The local indistinguishability of multipartite product states

We study the perfectly local indistinguishability of multipartite product states. Firstly, we follow the method of Zhang et al. (Phys Rev A 93:012314, 2016) to give another more concise set of $$2n-1$$ orthogonal product states in $${\mathbb {C}}^m\otimes {\mathbb {C}}^n$$ $$(4\le m\le n)$$ which can not be distinguished by local operations and classical communication. Then we use the three-dimensional cubes to present some product states which give us an intuitive view on how to construct locally indistinguishable product states in tripartite quantum systems. At last, we give an explicit construction of locally indistinguishable orthogonal product states for general multipartite systems.

LOCC indistinguishable orthogonal product quantum states.

We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of 2k+i ⊗ 2l+j (i, j ∈ {0, 1} and i ≥ j ) and 3k+i ⊗ 3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of 3k+i ⊗ 3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only 2k ⊗ 2l and 2k+1 ⊗ 2l but also 2k ⊗ 2l+1 and 2k+1 ⊗ 2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of 2k ⊗ 2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.

Local indistinguishability of orthogonal product states

In the general bipartite quantum system $m \otimes n$, Wang \emph{et al.} [Y.-L Wang \emph{et al.}, Phys. Rev. A \textbf{92}, 032313 (2015)] presented $3(m+n)-9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\otimes n $. First, in $3\otimes n (3< n)$ quantum system, we construct $3n-2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\otimes n (4\leq m\leq n)$, we present $3n+m-4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\otimes n(3\leq m\leq n)$, we show a smaller set with $2n-1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.

Nonlocality of orthogonal product-basis quantum states

We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of $\mathbb{C}^d\otimes\mathbb{C}^d$, where $d$ is odd, Zhang \emph{et al} have constructed $d^2$ orthogonal product basis quantum states which are locally indistinguishable in [Phys. Rev. A. {\bf 90}, 022313(2014)]. We find a subset contains with $6d-9$ orthogonal product states which are still locally indistinguishable. Then we generalize our method to arbitrary bipartite quantum system $\mathbb{C}^m\otimes\mathbb{C}^n$. We present a small set with only $3(m+n)-9$ orthogonal product states and prove these states are LOCC indistinguishable. Even though these $3(m+n)-9$ 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.