If choices between different possible measurement settings for a correlation experiment are not made quite as ‘freely’ as we might believe, then one can explain the resulting correlations in a local and deterministic manner. This includes those quantum correlations which violate Bell inequalities.

In particular, to derive and to test any standard Bell inequality, at least one observer must be able to select between two or more incompatible measurement settings. These settings can correspond, for example, to different spin directions or to different angles of light polarisation, depending on the physical system under investigation.

Crucially, the selection of measurement setting must be assumed to be completely independent of the preparation of the system – that is, having no correlation between the choice of measurement settings and any variables describing the system. Without this assumption – called “measurement setting independence” – no standard Bell inequality can be derived.

But what if one does not assume measurement setting independence? By allowing a sufficient degree of correlation between the settings and the system variables, one can in fact write down a local and deterministic model for any quantum correlation – whether or not this correlation violates a standard Bell inequality. Naturally enough, this possibility of ‘saving’ locality and determinism merits serious consideration.

It might immediately be objected, and indeed often is, that if measurement settings cannot be freely selected, then experimenters cannot have ‘free will’. Yet such experimental free will certainly appears to exist – and might even be considered necessary for making meaningful physical investigations.

However, in addition to the well-known possibility put forward by various philosophers, that free will is merely an illusion, there are at least three strong physical counterarguments to this objection:

First, in experimental tests of Bell inequalities to date, and in applications such as quantum cryptography and randomness extraction, measurement settings are in fact selected by random number generators, rather than by an experimenter. Thus, measurement setting independence should not be strictly interpreted in terms of ‘free will’ of human beings.

Second, the assumption of measurement setting independence is equivalent to assuming there is no common past cause that correlates the system variables with the measurement settings. But since the system and any physical mechanism for selecting measurement settings (such as an experimenter or a random number generator) do share a common past, there is no physically compelling reason for ruling out such a common cause.

Third and finally, a further assumption typically used in deriving Bell inequalities is *determinism*, i.e., all measurement outcomes are predetermined. The motivation for this assumption is that some fundamental deterministic theory underlies nature. However, such a deterministic theory should reasonably be expected to predict *all* physical phenomena – including the measurement settings in particular. Thus measurement setting independence is difficult to reconcile with the assumption of determinism.

Surprisingly, only a small degree of measurement setting dependence is required for a local and deterministic model of quantum correlations. For example, just 1 one bit of information shared between the system variables and measurement settings every 15 measurements is sufficient for such a model of a pair of maximally entangled qubits. This possibility is highly relevant to quantum cryptographic and randomness extraction protocols based on the violation of a Bell inequality by such qubits.

For example, if the random number generators used in such a protocol have been subtly compromised by an adversary, it is possible for the local measurement outcomes to be fully predetermined and known to the adversary, despite the observed violation of a standard Bell inequality. However, it is worth noting that ‘relaxed’ Bell inequalities can be derived that allow modifications of such protocols to be safely used, providing the degree of measurement setting dependence is sufficiently small. For instance, the bound of the CHSH inequality can be adapted from 2 to 2.2 when measurement settings appear to be predetermined 10% of the time.

Michael J. W. Hall

published online on 4 Feb 2014