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International Conference on Quantum, Nano and Micro Technologies (ICQNM)

Strict Hierarchy of Bell Theorems

Gilles Brassard A, André Allan Méthot

Published Feb 1, 2008• 1 citation

Abstract

As proved by John Bell, quantum mechanics exhibits correlations in spacelike separated bipartite systems that are impossible to reproduce by classical means. There are three levels of "Bell theorems", depending on which aspects of the quantum correlations can or cannot be reproduced classically. The original "Bell inequalities" (Bl) require a perfect classical simulation of all quantum probabilities. With "Bell theorems without inequalities" (BTWI), we ask the classical simulation to be able to produce precisely the outputs that could occur according to quantum mechanics, but we do not worry about their exact probabilities. With "pseudo-telepathy" (PT), we are satisfied if the classical simulation produces only outputs allowed by quantum mechanics, but not necessarily all of them. Bell's original proof of Bl involved a maximally entangled 2times2 bipartite state such as the singlet state. Hardy proved that BTWI are possible in dimension 2times2, but his construction used a non-maximally entangled state. Here, we prove that no 2times2 maximally entangled state can serve to produce BTWI. Combining this with the fact that 2times2 entangled states cannot be used at all for the purpose of PT, it follows a strict hierarchy on the quantum resources that are required to exhibit the various levels of Bell theorems.

DOI: 10.1109/icqnm.2008.13

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As proved by John Bell, quantum mechanics exhibits correlations in spacelike separated bipartite systems that are impossible to reproduce by classical means. There are three levels of "Bell theorems", depending on which aspects of the quantum correlations can or cannot be reproduced classically. The original "Bell inequalities" (Bl) require a perfect classical simulation of all quantum probabilities. With "Bell theorems without inequalities" (BTWI), we ask the classical simulation to be able to produce precisely the outputs that could occur according to quantum mechanics, but we do not worry about their exact probabilities. With "pseudo-telepathy" (PT), we are satisfied if the classical simulation produces only outputs allowed by quantum mechanics, but not necessarily all of them. Bell's original proof of Bl involved a maximally entangled 2times2 bipartite state such as the singlet state. Hardy proved that BTWI are possible in dimension 2times2, but his construction used a non-maximally entangled state. Here, we prove that no 2times2 maximally entangled state can serve to produce BTWI. Combining this with the fact that 2times2 entangled states cannot be used at all for the purpose of PT, it follows a strict hierarchy on the quantum resources that are required to exhibit the various levels of Bell theorems.

Ever since the work of Bell, it has been known that entangled quantum states can produce non-local correlations between the outcomes of separate measurements. However, for almost forty years, it has been assumed that the most non-local states would be the maximally entangled ones. Surprisingly it is not the case: non-maximally entangled states are generally more non-local than maximally entangled states for all the measures of non-locality proposed to date: Bell inequalities, the Kullback-Leibler distance, entanglement simulation with communication or with non-local boxes, the detection loophole and efficiency of cryptography. In fact, one can even find simple examples in low dimensions, confirming that it is not an artefact of a specifically constructed Hilbert space or topology. This anomaly shows that entanglement and non-locality are not only different concepts, but also truly different resources. We review the present knowledge on this anomaly, point out that Hardy's theorem has the same feature, and discuss the perspectives opened by these discoveries.

It is a well-established fact that some quantum correlations can be nonlocal, meaning that they cannot be described by a local hidden variable model. Certain quantum correlations have a form of nonlocality so strong that they cannot be reproduced even by models having an arbitrarily small local hidden variable component. These correlations are called fully nonlocal and lead to Bell inequalities in which the maximum quantum value saturates the non-signaling bound. A well-known example of this effect, which is also referred to as quantum pseudo-telepathy or all-versus-nothing proofs of nonlocality, is the quantum distribution fulfilling the Peres-Mermin square, in which the underlying state is a

4\times4

dimensional maximally entangled state. Other examples of full nonlocality are known but, so far, all of them are for maximally entangled states and it is an open question whether maximal entanglement is necessary for full nonlocality. In this work, we first establish a link between full nonlocality and the concept of antidistinguishability of quantum states. We use this connection to show that in every bipartite

d\times d

Hilbert space, with

d\geq3

, there are non-maximally entangled states that are fully nonlocal. In fact, we derive simple sufficient conditions for full nonlocality that are only based on the smallest and largest Schmidt coefficients. We also show that in every dimension there exist pure entangled states that do not exhibit full nonlocality. Finally, we show that all pure entangled states can be activated to show full nonlocality in the many-copy scenario.

The Bell theorem can take the form of several inequalities, as we have seen in §4.2. Moreover, since it was discovered, the theorem has stimulated the discovery of several other mathematical contradictions between the predictions of quantum mechanics and those of local realism. We review a few of them in this chapter: GHZ contradictions (§5.1) and their generalization (§5.2), Cabello's inequality (§5.3), and Hardy's impossibilities (§5.4). Finally, in §5.5, we discuss the notion of contextuality and introduce the BKS theorem. GHZ contradiction For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism [188, 189]. They involve a sign contradiction (100% violation) for perfect correlations, while the BCHSH inequalities are violated by about 40% (Cirelson bound) and deal with situations where the results of measurements are not completely correlated. In this section, we discuss three-particle systems, but generalizations to N particles are possible (§5.2). Derivation GHZ contradictions may occur in various systems, not necessarily involving spins. Initially, they were introduced in the context of entanglement swapping (§6.3.2) for four particles [188] or entanglement of three spinless particles [189].

The paper G. Brassard, R. Cleve, and A. Tapp, Phys. Rev. Lett. 83, 1874 (1999) begins thus: ``Bell's celebrated theorem shows that certain scenarios involving bipartite quantum measurements result in correlations that are impossible to simulate with a classical system if the measurement events are spacelike separated. If the measurement events are timelike separated, then classical simulation is possible, at the expense of some communication. Our goal is to quantify the required amount of communication.'' In this note we tighten the bounds on the amount of communication required to simulate a von Neumann measurement on a Bell pair.