Erratum: Feasibility of Bell tests with theWstate [Phys. Rev. A81, 033823 (2010)]

Abstract

The three-particle Bell inequality, Eq. (11) in [1], as used in our paper reads 1 + P (z i = −1,z j = −1,z k = 1) + P (z i = −1,z j = 1,z k = −1) + P (z i = 1,z j = −1,z k = −1)This interpretation of Eq. (11) in [1] is unfortunately not a true Bell inequality because, for a given mapping of the particle number {1,2,3} onto {i,j,k}, one can find a local hidden variable (LHV) model that violates the inequality. At the time of writing, we believed that while one can find LHV models that violate any specific mapping, one could not, with the same model, violate more than one of the three ensuing inequalities due to different mappings. Thus, if the choice of mapping (and hence the choice of measurement performed) was done outside the light cone (to close the communication loophole), the inequalities would discriminate between LHV and nonlocal models. We have, however, found that this is not true. Denote a LHV preparation of the three particles ), where z 1 = −1 means that if particle 1 is measured in the Z basis, the result is always −1, and if it is instead measured in the X basis, the result x 1 is 1 and so on. Assume the mapping i = 1, j = 2, and k = 3. This preparation has P (z 1 = −1,z 2 = 1,z 3 = −1) = 1, and the rest of the terms in the inequality (1) are zero. Hence, the left-hand side of (1) is 2, a clear violation of the inequality. Due to the i ↔ j symmetry of the inequality, the mapping i = 2, j = 1, and k = 3 gives the same violation of 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Quantum efficiency η 2 Middle term of Bell's inequality FIG. 4. The sum of the middle terms in (2) as a function of the detector efficiency η 2 for β = 0.71 and measuring the x basis using unbalanced homodyning.the inequality. One also finds that, using this preparation but any of the four remaining mappings {1,2,3} onto {i,j,k}, the left-hand side of the inequality equals unity.Similarly, using the mappings i = 1, j = 3, and k = 2 or i = 3, j = 1, and k = 2 and the preparation ) or the mappings i = 2, j = 3, and k = 1 or i = 3, j = 2, and k = 1 and the preparation

Feasibility of loophole-free nonlocality tests with a single photon

“…, 0). We note that the first term on the lefthand side was not present in previous works [21,24,25]. However, it is easy to check that the inequality is still valid, and since the term is always positive it can only increase any violation of the inequality.…”

“…In Ref. [24] an alternative method was proposed for the implementation of measurements of σx . In this setup the incoming field is displaced by mixing with a coherent state on a highly transmitting beam splitter and subsequently measured with a single-photon detector.…”

“…However, this choice of α suggested in Ref. [24] is not necessarily optimal. When the single-photon detector has a non-unit efficiency η spd , (4) becomes…”

Extreme nonlocality with one photon

“…Another method, utilising unbalanced homodyne detection, has recently been put forward to test our proposal [34]. Within this scheme, a quantum efficiency of 69% is required for a single photon, three-mode W state to exhibit a detectable violation of local realism.…”

Algebraic and algorithmic frameworks for optimized quantum measurements

“…µ were discussed at length in Refs. [39,46,47].) By implementing generalized measurements, all figures of merit improve.…”