Loopholes in Bell inequality tests of local realism

Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or “loopholes” in available experiment tests, local realist theories can reproduce the data from these experiments. This paper reviews such loopholes, what effect they have on Bell inequality tests, and how to avoid them in experiment. Avoiding all these simultaneously in one experiment, usually called a “loophole-free” or “definitive” Bell test, remains an open task, but is very important for technological tasks such as device-independent security of quantum cryptography, and ultimately for our understanding of the world. Submitted to: J. Phys. A: Math. Gen. Loopholes in Bell Inequality Tests of Local Realism 2 In a Bell inequality test of local realism, the word “loophole” refers to circumstances in an experiment that force us to make extra assumptions for the test to apply. For comparison, in the English language the word refers to an ambiguity in the description of a system, that can be used to circumvent the intent of the system. One example is a loophole in a system of law, meaning some unintended and/or unexpected circumstances where the law does not apply, or situations that it does not cover. Such a loophole in a law can be used to avoid the law without technically breaking it, one popular example of this is taxation law. An older meaning of the word is a narrow arrow slit in a castle wall where defenders can shoot arrows at attacking forces. Such loopholes are narrow because it should be impossible to enter the castle through them, the intent being to force attackers to enter through the well-defended main gate. In our case, the well-defended gate is the Bell theorem (Bell, 1964): that local realist models cannot give the predictions obtained from quantum mechanics. The Bell inequality is derived under the assumptions of local realism and is violated by quantum-mechanical predictions, and therefore local realist models cannot give quantum-mechanical predictions. However, when testing this in experiment, we are no longer in the simple, clean, ideal setting of the Bell theorem. There are unintended and/or unexpected circumstances that opens possibilities for local realism to give the output of the experiment, circumstances that constitute loopholes in Bell inequality tests of local realism. The two most well-known loopholes are the “locality” loophole (Bell, 1964) and the “efficiency” loophole (Pearle, 1970). There are a number of issues that fall roughly under these two labels, but before we discuss these a brief introduction into Bell inequality tests is needed, to review the explicit assumptions made in the inequalities (see Section 1 below), and to address some experimental circumstances that cannot be categorized as locality or efficiency problems. After this, we will look into the two mentioned loopholes and see that they actually are the base of two classes of loopholes with slightly different effects and scope, locality in Section 2 and efficiency in Section 3. A brief conclusion will follow in Section 4 with recommendations for an experimenter that wants to perform a loophole-free experimental test. 1. Violation of Local Realism The seminal paper by Einstein, Podolsky, and Rosen (EPR, 1935), asks the question “Can [the] Quantum-Mechanical Description of Physical Reality Be Considered Complete?” In the quantum-mechanical description of a physical system, the quantities momentum (P ) and position (Q) are not explicitly included, other than as (generalized) eigenvalues of non-commuting measurement operators. EPR argue that these physical quantities must correspond to an element of physical reality, and that a complete theory should include them in the description. Therefore, they argue, the quantum-mechanical description of physical reality cannot be considered complete. Bell (1964) enhances the argument by adding a statistical test that essentially shows that this completeness Loopholes in Bell Inequality Tests of Local Realism 3