Measurement-dependent locality beyond independent and identically distributed runs

Abstract

When conducting a Bell test, it is normal to assume that the preparation of the quantum state is independent of the measurements performed on it. Remarkably, the violation of local realism by entangled quantum systems can be certified even if this assumption is partially relaxed. Here, we allow such measurement dependence to correlate multiple runs of the experiment, going beyond previous studies that considered independent and identically distributed (i.i.d.) runs. To do so, we study the polytope that defines block-i.i.d. measurement-dependent local models. We prove that non-i.i.d. models are strictly more powerful than i.i.d. ones, and comment on the relevance of this work for the study of randomness amplification in simple Bell scenarios with suitably optimised inequalities. INTRODUCTIONSince their introduction, by John Bell in 1964 [1], Bell inequalities have been a subject of extensive study, as they highlight the fact that quantum theory is incompatible with local realism. Numerous experimental tests of Bell inequalities have been carried out, with the results being overwhelmingly in favour of the quantum predictions. Of particular note are the recent loophole-free Bell tests [2][3][4], which simultaneously addressed several loopholes that had been raised regarding previous experiments. All these tests were conducted under the assumption that the choice of measurements and the state of the source are independent in each run. This observation should not be taken as a reservation: such measurement independence is an essential piece of the scientific method, and its negation would be rightly considered conspiratorial. This makes it all the more remarkable that quantum theory can be proved incompatible with local realism even if this assumption is relaxed to some extent.Indeed, while unrestricted measurement dependence would lead to an unfalsifiable superdeterminism [5], it was noted by Hall [6] that the violation of Bell inequalities keeps its meaning if some restrictions are made. This led to the study of measurement-dependent local (MDL) scenarios, where some correlation is allowed between the measurement choices and the source. A few subsequent works refined our understanding of measurement dependence [7][8][9][10], all sticking to known inequalities. A significant breakthrough was achieved when Pütz and coworkers noticed that the traditional Bell inequalities are no longer optimal: other linear constraints, suitably named MDL inequalities, more tightly define the conditions under which local realism holds in the MDL scenario. Their works [11,12] developed the mathematical framework to study these inequalities. Their most celebrated discovery is the following: there exist quantum correlations that violate local realism with "arbitrarily low measurement independence", that is, as long as the MDL model does not trivially allow us to reproduce all no-signalling correlations. The corresponding inequality has been tested in an experiment [13].Measurement dependence in Bell-type tests is also central in the tas...

Relaxed Bell inequalities with arbitrary measurement dependence for each observer

“…which is identical to the value ofĨ G (V T ) in Eq. (62). I 4 (z) grows monotonically with z, to a maximum value of ≈ 0.1423 bits (dark blue circle in Fig.…”

“…These include device-independent quantum key distribution [47][48][49] along with random-number generation and randomness expansion [50][51][52][53][54][55][56][57][58]. In particular, a malicious adversary with knowledge of an opponent's devices could conceivably undermine a variety of quantum information schemes by exploiting the measurement-independence loophole [29,36,[59][60][61][62][63][64].…”

Bell Nonlocality

“…The systematic study of non-i.i.d. strategies has been attempted only for the (2, 2; 2, 2) scenario under the min-entropy constraint (11.9); even then, rather partial results were obtained (Tan et al, 2016). One of them is worth mentioning because it shows, as claimed previously, that the MDL polytope changes significantly.…”

Quantum nonlocality in the presence of strong measurement dependence

“…In the second line, the condition (8) is not satisfied; nevertheless, p(a, b, x, y) 0 and therefore the second term is upper bounded by 0. The first term can also be bounded by noting that a,b,x,y ξω + a,b,x,y (1 − q)p(a, b, x, y| ) ξ (1 − q)ω + max , (16) where…”

“…This question has attracted broad attention in recent years and has been discussed following various approaches (see, e.g., [10][11][12][13][14][15][16][17][18]). A notable approach is that of Pütz et al [19,20], who presented a general framework for addressing these questions.…”