Leggett–Garg inequalities

Abstract

In contrast to the spatial Bell's inequalities which probe entanglement between spatially-separated systems, the Leggett-Garg inequalities test the correlations of a single system measured at different times. Violation of a genuine Leggett-Garg test implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. Quantum mechanics violates the inequalities on both accounts and the original motivation for these inequalities was as a test for quantum coherence in macroscopic systems. The last few years has seen a number of experimental tests and violations of these inequalities in a variety of microscopic systems such as superconducting qubits, nuclear spins, and photons. In this article, we provide an introduction to the Leggett-Garg inequalities and review these latest experimental developments. We discuss important topics such as the significance of the non-invasive measurability assumption, the clumsiness loophole, and the role of weak measurements. Also covered are some recent theoretical proposals for the application of Leggett-Garg inequalities in quantum transport, quantum biology and nano-mechanical systems.

Leggett–Garg inequalities for a quantum top affected by classical noise

“…Presence of entanglement can be detected [15] by the violation of various forms of Bell inequalities [5]. Recent studies on quantum correlations in time show that such correlations are no less interesting as these in space [12]. Non-classical time correlations between outcomes of measurements realized at different time instants are indicated by the violation of the Leggett-Garg inequalities (LGIs) [21].…”

“…In Ref. [12], there is an up-to-date review of both basic theoretical and experimental achievements related to the LGI.…”

“…The correlator K 3 ≡ K 3 (t) is a combination of three time correlation functions given by an expected value of an anti-commutator [12] C ji = 1 2 {Q(t j ), Q(t i )} of a dichotomous observable Q(t) in the Heisenberg picture with corresponding measurement values q(t) = ±1 at three successive time instants t 1 < t 2 < t 3 . Due to the existence of superposition states, quantum mechanics should violate the above inequality.…”

“…Due to the existence of superposition states, quantum mechanics should violate the above inequality. Experimental testing of LGIs and search for their possible violation are subtle problems due to required non-invasiveness of measurement of the quantity Q [12] and 'macroscopicity' of quantum states. The LGIs are interesting by their own also in microscopic systems [6] so the second requirement is often abandoned.…”

“…The LGIs are interesting by their own also in microscopic systems [6] so the second requirement is often abandoned. There are also certain conceptual difficulties related to the problem of the 'clumsiness loophole' [12]. In order to avoid confusion, we simply claim that the violation of the LGI in (1) is a hallmark of the violation of the Leggett-Garg realism in the considered system.…”