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New Journal of Physics

Generalised versions of separable decompositions applicable to bipartite entangled quantum states

Hussain Anwar, Sania Jevtic, Oliver Rudolph, S. Virmani

Published Aug 13, 2019• 1 citation
Publisher

Abstract

Abstract The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the ‘smallest’ possible sets of non-physical local operators. We consider a few ways to define the word ‘smallest’ and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states.

DOI
10.1088/1367-2630/ab3adc

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Abstract The investigation of separable states in quantum theory has been driven by the notion that they are highly classical, in that they do not demonstrate nonlocality, and are in some contexts unable to support non-classical computation. The converse question, the extent to which entangled states do or do not support non-classical information processing, is less well understood. Motivated by this question we extend the notion of quantum separability into the entangled quantum states, by constructing separable decompositions that describe them with the ‘smallest’ possible sets of non-physical local operators. We consider a few ways to define the word ‘smallest’ and present techniques for obtaining them. The methods involve calculating certain forms of cross norm. The results generalise significantly the results obtained in our previous work on this topic (2015 New J. Phys. 17 093047), and can be be used to construct classical simulation methods and local hidden variable models for subsets of local measurements on entangled quantum states.
Entangled quantum states can be given a separable decomposition if we relax the restriction that the local operators be quantum states. Motivated by the construction of classical simulations and local hidden variable models, we construct `smallest' local sets of operators that achieve this. In other words, given an arbitrary bipartite quantum state we construct convex sets of local operators that allow for a separable decomposition, but that cannot be made smaller while continuing to do so. We then consider two further variants of the problem where the local state spaces are required to contain the local quantum states, and obtain solutions for a variety of cases including a region of pure states around the maximally entangled state. The methods involve calculating certain forms of cross norm. Two of the variants of the problem have a strong relationship to theorems on ensemble decompositions of positive operators, and our results thereby give those theorems an added interpretation. The results generalise those obtained in our previous work on this topic [New J. Phys. 17, 093047 (2015)].
According to usual definitions, entangled states cannot be given a separable decomposition in terms of products of local density operators.If we relax the requirement that the local operators be positive, then an entangled quantum state may admit a separable decomposition in terms of more general sets of single-system operators.This form of separability can be used to construct classical models and simulation methods when only a restricted set of measurements is available.With these motivations in mind, we ask what are the smallest sets of local operators such that a pure bipartite entangled quantum state becomes separable?We find that in the case of maximally entangled states there are many inequivalent solutions, including for example the sets of phase point operators that arise in the study of discrete Wigner functions.We therefore provide a new way of interpreting these operators, and more generally, provide an alternative method for constructing local hidden variable models for entangled quantum states under subsets of quantum measurements.
Nonlocality and its connections to entanglement are fundamental features of quantum mechanics that have found numerous applications in quantum information science. A set of correlations is said to be nonlocal if it cannot be reproduced by spacelike-separated parties sharing randomness and performing local operations. An important practical consideration is that the runtime of the parties has to be shorter than the time it takes light to travel between them. One way to model this restriction is to assume that the parties are computationally bounded. We therefore initiate the study of nonlocality under computational assumptions and derive the following results: (a) We define the set NeL\mathsf{NeL}NeL (not-efficiently-local) as consisting of all bipartite states whose correlations arising from local measurements cannot be reproduced with shared randomness and \emph{polynomial-time} local operations. (b) Under the assumption that the Learning With Errors problem cannot be solved in \emph{quantum} polynomial-time, we show that NeL=ENT\mathsf{NeL}=\mathsf{ENT}NeL=ENT, where ENT\mathsf{ENT}ENT is the set of \emph{all} bipartite entangled states (pure and mixed). This is in contrast to the standard notion of nonlocality where it is known that some entangled states, e.g. Werner states, are local. In essence, we show that there exist (efficient) local measurements producing correlations that cannot be reproduced through shared randomness and quantum polynomial-time computation. (c) We prove that if NeL=ENT\mathsf{NeL}=\mathsf{ENT}NeL=ENT unconditionally, then BQP≠PP\mathsf{BQP}\neq\mathsf{PP}BQP=PP. In other words, the ability to certify all bipartite entangled states against computationally bounded adversaries gives a non-trivial separation of complexity classes. (d) Using (c), we show that a certain natural class of 1-round delegated quantum computation protocols that are sound against PP\mathsf{PP}PP provers cannot exist.
Entanglement is not an intrinsic property of a quantum state: it depends on how the total system is factorized into subsystems via a tensor product structure. Just as a coherent quantum state can be made to appear classical by representing it in its eigenbasis, an entangled state can always be made to appear separable in a suitably chosen factorization. However, the same is not true for a set of states. It is not always possible to make a set of states simultaneously separable in a common factorization. Following recent work that defined a basis-independent notion of classicality for sets of states, we propose an extension of this idea to sets of bipartite states. This leads us to a factorization-independent notion of separability we call simulation separability. This notion is weaker than previously studied forms of separability for sets of states. It can be physically understood via an operational interpretation based on unitary gates and state-preparation devices that emit separable states. By showing that the set of all simulation-separable sets is convex, we are able to derive necessary and sufficient conditions for simulation separability. We also propose an experimentally implementable method for detecting sets that are not simulation separable. Finally, finite sets of two-qubit states are investigated numerically with respect to simulation separability.
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.
Entanglement allows for the nonlocality of quantum theory, which is the resource behind device-independent quantum information protocols. However, not all entangled quantum states display nonlocality. A central question is to determine the precise relation between entanglement and nonlocality. Here we present the first general test to decide whether a quantum state is local, and show that the test can be implemented by semidefinite programing. This method can be applied to any given state and for the construction of new examples of states with local hidden variable models for both projective and general measurements. As applications, we provide a lower-bound estimate of the fraction of two-qubit local entangled states and present new explicit examples of such states, including those that arise from physical noise models, Bell-diagonal states, and noisy Greenberger-Horne-Zeilinger and W states.