The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations, as has been experimentally observed, brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up a fraction of just 14% of measurement independence. The underlying model is deterministic and no-signalling. It may thus be favourably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signalling. A local deterministic model is also given that achieves the maximum possible violation of the well known Bell-CHSH inequality, at a cost of only 1/3 of measurement independence.
A hidden variable model reproducing the quantum mechanical probabilities for a spin singlet is presented. The model violates only the hypothesis of independence of the distribution for the hidden variables from the detectors settings and vice versa (Measurement Independence). It otherwise satisfies the hypotheses of Setting Independence, Outcome Independence -- made in the derivation of Bell inequality -- and that of compliance with Malus's Law -- made in the derivation of Leggett inequality. It is shown that the violation of the Measurement Independence hypothesis may be explained alternative…
Most quantum information tasks based on Bell tests relie on the assumption of measurement independence. However, it is difficult to ensure that the assumption of measurement independence is always met in experimental operations, so it is crucial to explore the effects of relaxing this assumption on Bell tests. In this paper, we discuss the effects of relaxing the assumption of measurement independence on 1-parameter family of Bell (1-PFB) tests. For both general and factorizable input distributions, we establish the relationship among measurement dependence, guessing probability, and the maxim…
Bell's theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell's inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribu…
Bell inequalities or Bell-like experiments are supposed to test hidden variable theories based on three intuitive assumptions: determinism, locality and measurement independence. If one of the assumptions of Bell inequality is properly relaxed, the probability distribution of the singlet state, for example, can be reproduced by a hidden variable model. Models that deal with the relaxation of some condition above, with more than one hidden variable, have been studied in the literature nowadays. In this work the relation between the number of hidden variables and the degree of relaxation necessa…
Three classes of local hidden-variable models that violate both Bell and Leggett inequalities are presented. The models, however, do not reproduce the quantum mechanical predictions, hence they are experimentally testable. It is concluded that on one hand neither Bell or Leggett inequality fully captures the essential counterintuitiveness of quantum mechanics, while on the other hand the hypothesis of outcome independence and that of locality are uncorrelated.
We in this Letter derive analytic formulas of Bell correlations in terms of quantum probability statistics under the assumption of measuring outcome-independence. For a spin-1/2 singlet state we find analytically that the violations of Bell-type inequalities are really related to the quantum non-local correlations. However, the Bell and Clauser-Horne-Shimony-Holt (CHSH) inequalities are always satisfied for the spin-1 singlet states. More generally the quantum non-locality does not lead to the violation of Bell and CHSH inequalities for the integer-spin singlet since the non-local interference…
We give a sufficient condition of nonlocality in order to reproduce singlet spin correlations. For a given pair of hidden variables and measurement directions this condition determines only the product of the outcomes and reproduces statistical correlations for all measurement directions; but fails to give a complete description of sub-systems, provides no means to calculate all joint probabilities and puts no constraints on signaling. In order to complete the model we introduce an additional condition. In this case we observe that the character of the nonlocality changes (outcome independence…
The experimental violation of Bell inequality establishes necessary but not sufficient conditions that any theory must obey. Namely, a theory compatible with the experimental observations can satisfy at most two of the three hypotheses at the basis of Bell's theorem: free will, no-signaling, and outcome-Independence. Quantum mechanics satisfies the first two hypotheses but not the latter. Experiments not only violate Bell inequality, but show an excellent agreement with quantum mechanics. This fact restricts further the class of admissible theories. In this work, the author determines the form…
If nonlocality is to be inferred from a violation of Bell's inequality, an important assumption is that the measurement settings are freely chosen by the observers, or alternatively, that they are random and uncorrelated with the hypothetical local variables. We study the case where this assumption is weakened, so that measurement settings and local variables are at least partially correlated. As we show, there is a connection between this type of model and models which reproduce nonlocal correlations by allowing classical communication between the distant parties, and a connection with models…
Experimental free-will or measurement independence is one of the crucial assumptions in derivation of any nonlocal theorem. Any nonlocal correlation obtained in quantum world can have a local deterministic explanation if there is no experimental free-will in choosing the measurement settings. Recently, in [Phys. Rev. Lett. {\bf105}, 250404 (2010)] it has been shown that to obtain a local deterministic description for singlet state correlation one does not need to give up measurement independence completely, but a partial measurement dependence suffices. In three party scenario considering GHZ …
Device independent protocols rely on the violation of Bell inequalities to certify properties of the resources available. The violation of the inequalities are meaningless without a few well-known assumptions. One of these is measurement independence, the property that the source of the states measured in an inequality is uncorrelated from the measurements selected. Since this assumption cannot be confirmed, we consider the consequences of relaxing it and find that the definition chosen is critically important to the observed behavior. Considering a definition that is a bound on the min-entrop…
A local and deterministic model of quantum correlations is always possible, as shown explicitly by Brans in 1988: one simply needs the physical systems being measured to have a suitable statistical correlation with the physical systems performing the measurement, via some common cause. Hence, to derive no-go results such as Bell inequalities, an assumption of measurement independence is crucial. It is a surprisingly strong assumption -- less than 1/15 bits of prior correlation suffice for a local model of the singlet state of two qubits -- with ramifications for the security of quantum communi…
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 f…
Device independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully `causal' simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables an…
The combination of various physically plausible properties, such as no signaling, determinism, and experimental free will, is known to be incompatible with quantum correlations. Hence, these properties must be individually or jointly relaxed in any model of such correlations. The necessary degrees of relaxation are quantified here, via natural distance and information-theoretic measures. This allows quantitative comparisons between different models in terms of the resources, such as the number of bits, of randomness, communication, and/or correlation, that they require. For example, measuremen…
