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arXiv:1612.05680 · Physical Review Letters

Universal Bell Correlations Do Not Exist

Cole Graham, William M. Hoza

Published Aug 1, 2017• 0 citations
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Abstract

We prove that there is no finite-alphabet nonlocal box that generates exactly those correlations that can be generated using a maximally entangled pair of qubits. More generally, we prove that if some finite-alphabet nonlocal box is strong enough to simulate arbitrary local projective measurements of a maximally entangled pair of qubits, then that nonlocal box cannot itself be simulated using any finite amount of entanglement. We also give a quantitative version of this theorem for approximate simulations, along with a corresponding positive result.

DOI
10.1103/physrevlett.119.050402

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We prove that there is no finite-alphabet nonlocal box that generates exactly those correlations that can be generated using a maximally entangled pair of qubits. More generally, we prove that if some finite-alphabet nonlocal box is strong enough to simulate arbitrary local projective measurements of a maximally entangled pair of qubits, then that nonlocal box cannot itself be simulated using any finite amount of entanglement. We also give a quantitative version of this theorem for approximate simulations, along with a corresponding positive result.
A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most 128cot⁡4θ128\cot^4θ128cot4θ calls for the Schmidt state cos⁡θ∣00⟩+sin⁡θ∣11⟩\cosθ|00\rangle+\sinθ|11\ranglecosθ∣00⟩+sinθ∣11⟩, 0<θ<π/40<θ<π/40<θ<π/4. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.
Bell's theorem states that, to simulate the correlations created by measurement on pure entangled quantum states, shared randomness is not enough: some 'non-local' resources are required. It has been demonstrated recently that all projective measurements on the maximally entangled state of two qubits can be simulated with a single use of a 'non-local machine'. We prove that a strictly larger amount of this non-local resource is required for the simulation of pure non-maximally entangled states of two qubits |ψ(α)⟩ = cosα|00⟩ + sinα|11⟩ with .
Bell's theorem states that, to simulate the correlations created by measurement on pure entangled quantum states, shared randomness is not enough: some `non-local' resources are required. It has been demonstrated recently that all projective measurements on the maximally entangled state of two qubits can be simulated with a single use of a `non-local machine'. We prove that a strictly larger amount of this non-local resource is required for the simulation of pure non-maximally entangled states of two qubits |ψ(α)⟩ = cosα|00⟩ + sinα|11⟩ with . 0<α≤π/7.8 .
We introduce a version of the chained Bell inequality for an arbitrary number of measurement outcomes and use it to give a simple proof that the maximally entangled state of two d-dimensional quantum systems has no local component. That is, if we write its quantum correlations as a mixture of local correlations and general (not necessarily quantum) correlations, the coefficient of the local correlations must be zero. This suggests an experimental program to obtain as good an upper bound as possible on the fraction of local states and provides a lower bound on the amount of classical communication needed to simulate a maximally entangled state in dxd dimensions. We also prove that the quantum correlations violating the inequality are monogamous among nonsignaling correlations and, hence, can be used for quantum key distribution secure against postquantum (but nonsignaling) eavesdroppers.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities – entanglement, non-locality and randomness – relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: Given a single entangled system, how many nonlocal correlations in a sequence can we obtain? And from this single entangled system, how many certified random numbers is it possible to generate? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from a single entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.