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arXiv:1903.03957 · Europhysics Letters (EPL)

Local-hidden-state models for T-states using finite shared randomness

Yuan-Yuan Zhang, Fu-Lin Zhang

Published Sep 2, 2019• 1 citation
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Abstract

The study of local models using finite shared randomness originates from the consideration about the cost of classically simulating entanglement in composite quantum systems. We construct explicitly two families of local-hidden-state (LHS) models for T-states, by mapping the problem to the Werner state. The continuous decreasing of shared randomness along with entanglement, as the anisotropy increases, can be observed in the one from the most economical model for the Werner state. The construction of the one for separable states shows that the separable boundary of T-states can be generated from the one of the Werner state, and the cost is 2 classical bits.

DOI
10.1209/0295-5075/127/20007

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The study of local models using finite shared randomness originates from the consideration about the cost of classically simulating entanglement in composite quantum systems. We construct explicitly two families of local-hidden-state (LHS) models for T-states, by mapping the problem to the Werner state. The continuous decreasing of shared randomness along with entanglement, as the anisotropy increases, can be observed in the one from the most economical model for the Werner state. The construction of the one for separable states shows that the separable boundary of T-states can be generated from the one of the Werner state, and the cost is 2 classical bits.
The statistics of local measurements performed on certain entangled states can be reproduced using a local hidden variable (LHV) model. While all known models make use of an infinite amount of shared randomness, we show that essentially all entangled states admitting a LHV model can be simulated with finite shared randomness. Our most economical model simulates noisy two-qubit Werner states using only log_{2}(12)≃3.58 bits of shared randomness. We also discuss the case of positive operator valued measures, and the simulation of nonlocal states with finite shared randomness and finite communication. Our work represents a first step towards quantifying the cost of LHV models for entangled quantum states.
We present a general criterion for deciding the existence of local-hidden-state (LHS) models and consequently local-hidden-variable (LHV) models for arbitrary states that can be implemented by semidefinite programming. This method can be applied to any given state and general (POVM) measurements and can also be adapted to generate random states with LHS models. As applications we provide an estimate of the fraction of two-qubit entangled states that have LHS models, and present new explicit examples of such states, including Bell-diagonal states, and noisy GHZ and W states. We also generate a list of 750000 new entangled two-qubit states with LHS model for projective measurements, 1500 for POVM measurements, and 150 genuine multipartite entangled three-qubit local states.
This Master's thesis has two central subjects: the simulation of correlations generated by local measurements on entangled quantum states by local hidden-variables models and the revelation of hidden nonlocality. We present and detail the Werner's local model and the hidden nonlocality of some Werner states of dimension d≥5d\geq5d≥5, the Gisin-Degorre's local model for a Werner state of dimension d=2d=2d=2 and the local model of Hirsch et al. for mixtures of the singlet state and noise, all of them for projective measurements. Finally, we introduce the local model for POVMs of Hirsch et al. for a state constructed upon the singlet with noise, that still violates the CHSH inequality after local filters are applied, hence presenting the so-called genuine hidden nonlocality.
Quantum entanglement is a fundamental feature of quantum mechanics, yet certain entangled states that are unsteerable can be classically simulated in steering scenarios, making them unable to exhibit quantum steering. Despite their significance, a systematic comparison of such entangled states has not been explored. In this work, we quantify the resource content of unsteerable quantum states in terms of the amount of shared randomness required to simulate the assemblages they generate in the steering scenario. We rigorously demonstrate that the simulation cost is unbounded even for certain unsteerable two-qubit states. Moreover, the simulation cost of entangled two-qubit states is always strictly larger than that for any separable state. A significant portion of our results rests on the relationship between the simulation cost of two-qubit Werner states and that of noisy spin measurements. Using noisy spin measurements as our central example, we also investigate the minimum number of outcomes a parent measurement requires to simulate a given set of compatible measurements. Although certain continuous measurement families admit a finite-outcome parent measurement, we identify scenarios where the simulation cost is unbounded. Our results establish previously unknown lower bounds and upper bounds on the shared randomness simulation cost, supported by connections between the simulation cost of noisy spin measurements and various geometric inequalities, including ones from the zonotope approximation problem in Banach space theory.
The problem of sharing quantum correlations is an interesting problem in the study of quantum information theory. Silva et al. proposed the study of sharing quantum nonlocality at first. They studied the fundamental limits on nonlocality, asking whether a single pair of entangled qubits could generate a long sequence of nonlocal correlations. At the same time, the sequential scenario was also introduced first, in which Alice and Bob each have half of a pair of entangled qubit states. The first Bob measures his half and then passes his part to a second Bob who measures again and so on. Obviously, even partial preservation of entanglement in a shared state in spite of a few sequences of local operations performed by the sharing parties can be important for information processing schemes in which entanglement is utilized as a resource. Thus, the problem of sharing quantum entanglement has also been extensively investigated. Recently, Srivastava et al. proved that there exist a class of T-states whose entanglement can be shared by arbitrarily many independent observers in [ Phys. Rev. A 2022 105 062413 ]. Here, we want to find whether there are other entangled states that can be shared entanglement arbitrarily many times. In this paper, we consider the problem of sharing quantum entanglement when the initial shared state is a two-qubit entangled Werner state. The goal is to maximize the number of Bobs that can share entanglement with a single Alice. By suitably choosing the entanglement witness operator and the unsharp measurement settings by the Bobs, we prove that there exist two-qubit entangled initial shared Werner states whose entanglement can be detected by arbitrarily many sequential observers Bobs with a single Alice. Then, we also consider the special case of the Werner state, that is, the maximally entangled state as the initial shared state. In this case, its entanglement can also be witnessed arbitrarily many times, and the number of Bobs increases with the decrease of parameter.
The correlations of certain entangled quantum states can be fully reproduced via a local model. We discuss in detail the practical implementation of an algorithm for constructing local models for entangled states, recently introduced by Hirsch et al. [Phys. Rev. Lett. 117, 190402 (2016)] and Cavalcanti et al. [Phys. Rev. Lett. 117, 190401 (2016)]. The method allows one to construct both local hidden state (LHS) and local hidden variable (LHV) models, and can be applied to arbitrary entangled states in principle. Here, we develop a systematic implementation of the algorithm, discussing the choice of the free parameters. For the case of two-qubit states, we design a ready-to-use procedure. This allows us to construct LHS models (for projective measurements) that are almost optimal, as we show for Bell diagonal states, for which the optimal model has recently been derived. Finally, we show how to construct fully analytical local models, based on the output of the convex optimization procedure.