Testing quantum correlations versus single-particle properties within Leggett’s model and beyond

Abstract

Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are nonlocal in the sense of violating a Bell inequality 1 . Nevertheless, Bell's strong sentence 'Correlations cry out for explanations' (ref. 2) remains relevant. The maturing of quantum information science and the discovery of the power of non-local correlations, for example for cryptographic key distribution beyond the standard quantum key distribution schemes 3-5 , strengthen Bell's wish and make it even more timely. In 2003, Leggett proposed an alternative model for non-local correlations 6 that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. These inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions; they are also stronger than previously published and experimentally tested Leggett-type inequalities 6-9 . The simplest of the new inequalities is experimentally violated. Then we go beyond Leggett's model, and show that we cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.Formally, a correlation is a conditional probability distribution P(α, β|a, b), where α, β are the outcomes observed by two partners, Alice and Bob, when they make measurements labelled by a and b, respectively. On the abstract level, a and b are merely inputs, freely and independently chosen by Alice and Bob. On a more physical level, Alice and Bob hold two subsystems of a quantum state; in the simple case of qubits, the inputs are naturally characterized by vectors on the Poincaré sphere, hence the notation a,b.How should we understand non-local correlations, in particular those corresponding to entangled quantum states? A natural approach consists in decomposing P(α, β|a, b) into a statistical mixture of hopefully simpler correlations:Bell's locality assumption is P l (α, β|a, b) = P A l (α|a)P B l (β|b), admittedly the simplest choice, but an inadequate one as it turns out: quantum correlations violate Bell's locality 1 . Setting out to explore other choices, it is natural to require first that the P l fulfil the so-called no-signalling condition, that is, that none of the correlations P l results from a communication between Alice and Bob. This can be guaranteed by ensuring spacelike separation between Alice and Bob. Non-signalling correlations happen without any time ordering: there is not a first event, let us say on Alice's side, that causes the second event via some spooky action at a distance. We may phrase it differently: nonsignalling correlations happen from outside space-time, in the sense that there is no story in space-time that tells us how they happen. This is the case in orthodox quantum physics, or in some illuminating toy models such as the non-local box of Popescu and Rohrlich (PR box) 10 . Mathematically, the no-signalling condition reads P l (α|a,b) = P l (α|a) and P l (β|a,b) = P l (β|b): Alice'...

Leggett-type nonlocal realistic inequalities without any constraint on the geometrical alignment of measurement settings

“…Note that the inequality derived and experimentally tested by Branciard et al [6] is a special case of the inequality (6) by assuming a specific geometrical constraint that requiresn i 's to be mutually orthogonal and |β 1 | = |β 2 | = |β 3 |. For a photon pair prepared in a pure singlet state, one can show that settings for observing the maximum violation of the inequality (6) are in which Bob's choices are such that the three directionsn i 's of (b i −b i ) are orthogonal where |β 1 | = |β 2 | = |β 3 | ≈ 36.9 o , withâ i 's chosen by Alice to be along the directions of b i +b i (see Appendix B for a proof).…”

“…As a consequence, the experimental violation of any relevant LNR inequality can be unambiguously concluded only if the supremum of such a range of LNR bounds is violated. However, an estimation of such a range of LNR bounds by taking into account all possible imprecisions that may occur in realizing the required configuration of settings is severely restricted for any of the LNR inequalities derived earlier [2][3][4][5][6]. This is essentially because the validity of any such inequality is in itself contingent upon certain geometrical constraints being strictly satisfied by the measurement settings.…”

“…This additional assumption was, however, not required in the subsequent works [4][5][6] that showed empirical violation of the suitably derived forms of LNR inequalities. Nevertheless, an undesirable feature besets all such studies since different forms of LNR inequalities that have been derived and tested to date hold good only if certain geometrical constraints are exactly satisfied by the spatial arrangement of the relevant measurement settings.…”

One sided indeterminism alone is not a useful resource to simulate any nonlocal correlation

“…As quantum mechanical predictions tally with the experimental data so only the second query is relevant in this regard. Recently lot of work has been done to simulate nonlocal correlations by violating these assumptions jointly or individually [4][5][6][7][8][9][10][11][12][13]. In particular, simulation protocols for singlet state correlation was given in [15][16][17][18].…”

“…In particular, simulation protocols for singlet state correlation was given in [15][16][17][18]. The main objective of all these works ( [4][5][6][7][8][9][11][12][13][14]) was to simulate nonlocal correlations assuming two sided relaxation of these constraints collectively or individually. The question that naturally arises in this context is whether joint or individual one sided relaxation of these constraints can achieve any Bell violation.…”