Abstract The assumption of measurement independence is required for a local deterministic model to conduct a Bell test. So, the violation of a Bell inequality by such a model implies that this assumption must be relaxed. The degree to which this assumption needs to be relaxed to achieve violation of some bipartite Bell inequalities, has been investigated in [Phys. Rev. Lett. 105, 250404(2010), Phys. Rev. A 99, 012121(2019)]. In this work, we study the minimum degree of relaxation required to simulate violations of various well-known tripartite Bell inequalities. Local deterministic models are also provided to achieve the violations of these Bell inequalities. PACS numbers: 03.67.Mn, 03.65.Ud.;
Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters
M1 and
M2, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.
The demonstration and use of Bell-nonlocality, a concept that is fundamentally striking and is at the core of applications in device independent quantum information processing, relies heavily on the assumption of measurement independence, also called the assumption of free choice. The latter cannot be verified or guaranteed. In this paper, we consider a relaxation of the measurement independence assumption. We briefly review the results of Phys. Rev. Lett. 113, 190402 (2014), which show that with our relaxation, the set of so-called measurement dependent local (MDL) correlations is a polytope, i.e. it can be fully described using a finite set of linear inequalities. Here we analyze this polytope, first in the simplest case of 2 parties with binary inputs and outputs, for which we give a full characterization. We show that partially entangled states are preferable to the maximally entangled state when dealing with measurement dependence in this scenario. We further present a method which transforms any Bell-inequality into an MDL inequality and give valid inequalities for the case of arbitrary number of parties as well as one for arbitrary number of inputs. We introduce the assumption of independent sources in the measurement dependence scenario and give a full analysis for the bipartite scenario with binary inputs and outputs. Finally, we establish a link between measurement dependence and another strong hindrance in certifying nonlocal correlations: nondetection events.
It is well known that the effect of quantum nonlocality, as witnessed by violation of a Bell inequality, can be observed even when relaxing the assumption of measurement independence, i.e., allowing for the source to be partially correlated with the choices of measurement settings; however, what is the minimal amount of measurement independence needed to observe quantum nonlocality? Here we explore this question and consider models with strong measurement-dependent locality, where measurement choices can be perfectly determined in almost all rounds of the Bell test. Nevertheless, we show that quantum nonlocality can still be observed in this scenario, which we conjecture is minimal within the framework we use. We also discuss potential applications in randomness amplification.
A local deterministic model can reproduce Bell-inequality violations only if the hidden state depends on the measurement settings. Hall found the minimum such dependence for the singlet state [Phys. Rev. Lett. 105, 250404 (2010)] and posed the corresponding problem for the Mermin state [Phys. Rev. A 84, 022102 (2011)]. This paper solves it, for the
n-party GHZ-Mermin scenario at every
n≥3, with the marginals reproduced as well as the correlators. The minimum measurement-dependence fraction is
Fmin(n)=R/[2(R+1)], with
R=2⌊(n−1)/2⌋ the Mermin violation ratio:
1/3,2/5,4/9,8/17,…, always below
1/2. The lower bound follows from Mermin's inequality by a general overlap bound; the upper bound is an explicit model, uniform on the hidden-variable classes that violate the fewest constraints, with closed-form counts; a reduction theorem shows that for Mermin scenarios the marginals impose no additional dependence. The Mermin values are also certified by integer-arithmetic enumeration through
n=22. The formula was stated as a conjecture in the first version of this paper and was first proved, by a different route, by Wang, Yao and Chen; the proof here was obtained independently and the two agree. Every minimal contradiction of size
k among stabilizer constraints has correlator-only minimum
1/(k−1). With the marginals included the minimum is the same on every core of the
C5 ring cluster, on the size-4 and size-5 cores of
C4 and on every four-core of
C6, but larger on the size-6 and size-7 cores of
C4 and on 112 of the 1,920 five-cores of
C6, with exact certificates on both sides in every case.
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 defined by 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 optimized inequalities.