LOCAL REALISM
a.k.a.
LOCAL HIDDEN VARIABLES

Local reality can be tested experimentally. Consider two observers, usuallly named Alice and Bob (shortly A and B), affecting the common system by their free choices and then measuring faster than the communication speed (in particular speed of light). If the results of measurements cannot be desribed by any local and real process, then the local reality is violated. The test can be performed analogously using also more observers, e.g. A(lice), B(ob), C(hris), D(avid), ... but it is usually more challenging.

Bell inequality

In 1964 John Stewart Bell invented a test of local reality, later reformulated by Clauser, Horne, Shimony and Holt (CHSH) [1], involving two observers (Alice and Bob or A and B). Each observer is free to apply one of two types of influences, 1 or 2 (if there are more types, they are binned into the two classes). Then they quickly make observations (quicker that the distance between them divided by the communication speed). The possible results for each observer are dichotomic, only +1 or −1 (if there are more, they are binned). These results, for each observer, A and B, and each influence, 1 and 2 will be denoted by A₁, B₁,A₂, B₂. If values of the results are ±1 then the inequality −2 ≤ A₁B₁ + A₁B₂ + A₂B₁ − A₂B₂ ≤ +2 holds. Since the results can be random, the final Bell-CHSH inequality involves statistical correlations (denoted by <>)

−2 ≤ <A₁B₁> + <A₁B₂> + <A₂B₁> − <A₂B₂> ≤ +2.

The correlations are calculated as follows. Alice and Bob perform pairs influence-measurement, write down the type of influence, the result of the measurement, and the order number of the pair (different numbers for different pairs). The time between influence and measurement in a single pair must be smaller than the communication time (Alice-Bob distance by the communication speed). After sufficiently many pairs are ready, an arbiter takes the protocols of Alice and Bob and calculates the correlations, averaging products of results with the same order number and types of influences. If the local realism holds and the experiment is performed as described above then the Bell-CHSH inequality must be statisfied, up to statistical error.

Quantum violation

Historical note: The first suggestion that quantum mechanics may violate local realism has been put forward by Einstein, Podolsky and Rosen in 1935[2]. However, it was Bell the first to propose the concrete test presented below [1].
In quantum mechanics the observed quantities are represented by Hermitian operators (algebraic matrices) O = O^†, while the (pure) states by vectors |ψ>, with Hermitian-conjugate vectors <ψ| = (|ψ>)^†. The scalar product is defined <φ|ψ>. The states should be normalized, <ψ|ψ> = 1. The states recognized by Alice and Bob are linear combinations of product states |ab>, where a,b denote states recognized by Alice and Bob, respectively. In particular, we distinguish two ortonormal states |±> for each obsrever. Alice and Bob are able to measure observables O = X cos(φ) + Z sin(φ) where X = |+><−| + |−><+| and Z = |+><+| − |−><−|. The observables measured by Alice and Bob are given by A,B = O in the appropriate subspace of states (a or b) with angles α and β in place of φ, respectively. Let us consider the special, Bell singlet state |ψ> given by √2|ψ> = |+−> − |−+>. In this case <AB> = <ψ|AB|ψ> = − cos(α-β). For the particular choice α₁ = 0, α₂ = π/2, β₁ = 3π/4, β₂ = π/4, we get <A₁B₁> = <A₁B₂> = <A₂B₁> = − <A₂B₂> = −1/√2. As a result, we get

<A₁B₁> + <A₁B₂> + <A₂B₁> − <A₂B₂> = −2√2 < −2.

The Bell-CHSH inequality is therefore violated along with local realism.
Improtant note: The above example relies on strongly applied standard Copenhagen interpretation of quantum mechanics. Under this interpretation the measurements can be performed instantly. In reality, the time of the free choice, the measurement is always a matter of interpretation, because the choice and finish of the measurement can be in principle pushed arbitrarily in time. Moreover, not always fast influences and measurements are feasible. There is also no perfect dichotomic observable, which is often circumvented by proper binning.

Experimental status

Four experiments violated either Bell-CHSH inequality with preselection or related Eberhard inequality [3] assumning locality in the sense of special relativity - no signalling faster than light. In Delft [4] the states in 1.3km separated nitrogen-vacancy centers entangled with photons combining midway have been measured by microwave radiation and laser light deflections in about 1 microsecond. B-CHSH inequality has been violated by 2 standard deviations by 245 events. In NIST [5], 10⁵ entangled photons have been measured by polarization changing Pockels cells, polarizing splitters and photodetectors, separated by 180m. In Vienna [6], 10⁶ photons have been measured similarly as in NIST, with separation 60m. In NIST and Vienna Eberhard inequality has been violated with 5 and 11.5 standard deviations, respectively. In Munich [7], 10⁴ rubidium atoms have been entangled with pthotons (like in Delft) which combined to preselect a valid state, measured by polarized lased light. B-CHSH inequality has been violated by 6.7 standard deviations. Important note: a new Vienna experiment [8] is invalid because it relies on fair sampling assumption (detection loophole). By deep data analysis [9] I have found that Delft and NIST experiment show also violation of no-signaling at the level of 2 standard deviations. Data from Vienna and Munich did not violate no-signaling, but are inconsisent with the simple 2x2 Bell model. The possible reason, additional influence of Pockels cells, has not been yet confirmed experimentally. In addition, I have recently shown [10] that local realism is in direct conflict with relativity not even relying on Bell-type inequalities. Therefore, the current status is: LOCAL REALISM IS HARDLY VALID WITH RELATIVITY BUT RELATIVITY (NO-SIGNALING) IS AT STAKE. Next experiments will show if violation of no-signaling and relativity can be confirmed with higher confidence level. In addition, experiments are continued to increase trust in freedom of choice and outcome objectivity (actual time moment of the measurement) which is always assumed to some extent.

Weak positivity

It is tempting to relax the assumption that the results are dichotomic (±1) and allow arbitrary real numbers. However, the violation of Bell-CHSH inequality then does not imply violation of local realism. Moreover, every test based solely on second (and first) order correlations, of type <XY> will not violate local realism[11] because the results can be reproduced by local and real Gausssian probability density ρ ∝ exp(−∑ C⁻¹_XY XY/2), where C is the correlation matrix with elements 2C_XY = <XY+YX> and C⁻¹ is its inverse. Since C is positive definite, the probability is well defined (first order averages can be either included via identity 1 or subtracted X → X − <X>.
Note: So why does Bell-CHSH inequallity contain only 2. order correlations? Because the assumption of dichotomic results ±1 is equivalent to <(A² − 1)²> = 0, which involves fourth order correlations.

Fourth order inequality

Knowing that we cannot stop at the second order, the lowest possible order to find a Bell-CHSH-type inequality for unbounded results is 4. Such an inequality reads[11]

|<A₁B₁(A₁² + B₁²)> + <A₁B₂(A₁² + B₂²)> + <A₂B₁(A₂² + B₁²)> − <A₂B₂(A₂² + B₂²)>| ≤
≤ (<A₁⁴> + <A₂⁴> + <B₁⁴> + <B₂⁴>)/2 + ∑_{D ≠ C; E ≠ C,D,D'} <C⁴>^1/4<D⁴>^1/4 <D² − E²>^1/2/4

where the last sum is over C,D,E = A,B_1,2 and X_n' = X_3−n. The inequality reduces to the standard Bell-CHSH inequality for dichotomic results.

References

[1] J.S. Bell, Physics (Long Island City, NY) 1, 195 (1964) [hard to find]; J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969); review article by A. Shimony in Stanford Encyclopedia of Philosophy, see plato.stanford.edu/entries/bell-theorem/
[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935)
[3] P.H. Eberhard, Phys. Rev. A 47, R747 (1993)
[4] B. Hensen et al., Nature 526, 682 (2015)
[5] L.K. Shalm et al., Phys. Rev. Lett. 115, 250402 (2015)
[6] M. Giustina et al., Phys. Rev. Lett. 115, 250401 (2015)
[7] W. Rosenfeld et al., Phys. Rev. Lett. 119, 010402 (2017)
[8] J. Handsteiner et al., Phys. Rev. Lett. 118, 060401 (2017)
[9] A. Bednorz, Phys. Rev. A 95, 042118 (2017); PDF (APS Copyright); arXiv:1511.03509
[10] A. Bednorz, Phys. Rev. D 94, 085032 (2016); PDF (APS Copyright); arXiv:1605.09129
[11] A. Bednorz and W. Belzig, Phys. Rev. B 83, 125304 (2011) ; PDF (APS Copyright); arXiv:1006.4991