Leggett–Garg inequality

The Leggett–Garg inequality, named for Anthony James Leggett and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism (macroscopic realism) is a classical worldview defined by the conjunction of two postulates:

In quantum mechanics, the Leggett–Garg inequality is violated, meaning that the time evolution of a system cannot be understood classically. The situation is similar to the violation of Bell's inequalities in Bell test experiments which plays an important role in understanding the nature of the Einstein–Podolsky–Rosen paradox. Here quantum entanglement plays the central role. The violation of Bell's inequalities rules out local hidden variable theories which attempt to restore the realism in the sense that definiteness of the outcome in a single measurement can be ensured by using a supplementary variable along with the wave function which can not be obtained in the standard Copenhagen Interpretation of quantum mechanics in its various formulations.

As well as Einstein's famous "God does not play dice" objection to quantum mechanics, there was Einstein's still more fundamental objection that the Moon is still there when nobody looks. If the violation of the Leggett–Garg inequality can be demonstrated on the macroscopic scale, this would challenge even this notion of realism.

The simplest form of the Leggett–Garg inequality derives from examining a system that has only two possible states. These states have corresponding measurement values $Q=\pm 1$ . The key here is that we have measurements at two different times, and one or more times between the first and last measurement. The simplest example is where the system is measured at three successive times $t_{1}<t_{2}<t_{3}$ . Now suppose, for instance, that there is a perfect correlation $C_{13}$ of 1 between times $t_{1}$ and $t_{3}$ . That is to say, that for N realisations of the experiment, the temporal correlation reads

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