Leggett–Garg inequality
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The Leggett–Garg inequality, named for
Anthony James Leggett Sir Anthony James Leggett (born 26 March 1938) is a British-American theoretical physicist and professor emeritus at the University of Illinois at Urbana-Champaign. Leggett is widely recognised as a world leader in the theory of low-temperatur ...
and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism (macroscopic realism) is a classical
worldview A worldview or world-view or ''Weltanschauung'' is the fundamental cognitive orientation of an individual or society encompassing the whole of the individual's or society's knowledge, culture, and point of view. A worldview can include natural ...
defined by the conjunction of two postulates: # Macrorealism per se: "A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states." # Noninvasive measurability: "It is possible in principle to determine which of these states the system is in without any effect on the state itself, or on the subsequent system dynamics."


In quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the e ...
, which plays an important role in understanding the nature of the
Einstein–Podolsky–Rosen paradox The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen, with which they argued that the description of physical reality provided by quantum mechanics was incom ...
. Here
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
plays the central role.


Two-state example

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_ = 1 between times t_1 and t_3. That is to say, that for ''N'' realisations of the experiment, the temporal correlation reads : C_ = \frac \sum_^N Q_r(t_1) Q_r(t_3) = 1. We look at this case in some detail. What can be said about what happens at time t_2? Well, it is possible that C_ = C_ = 1, so that if the value of Q at t_1 is \pm 1, then it is also \pm 1 for both times t_2 and t_3. It is also quite possible that C_ = C_ = -1, so that the value of Q at t_1 is flipped twice, and so has the same value at t_3 as it did at t_1. So, we can have both Q(t_1) and Q(t_2) anti-correlated as long as we have Q(t_2) and Q(t_3) anti-correlated. Yet another possibility is that there is no correlation between Q(t_1) and Q(t_2). That is, we could have C_ = C_ = 0. So, although it is known that if Q = \pm 1 at t_1, it must also be \pm 1 at t_3; the value at t_2 may as well be determined by the toss of a coin. We define K as K = C_ + C_ - C_. In these three cases, we have K = 1, -3, -1 respectively. All that was for complete correlation between times t_1 and t_3. In fact, for any correlation between these times K = C_ + C_ - C_ \le 1. To see this, we note that : K = \frac \sum_^N \big(Q(t_1) Q(t_2) + Q(t_2) Q(t_3) - Q(t_1) Q(t_3)\big)_r. It is easily seen that for every realisation r, the term in the parentheses must be less than or equal to unity, so that the result for the average is also less than (or equal to) unity. If we have four distinct times rather than three, we have K = C_ + C_ + C_ - C_ \le 2, and so on. These are the Leggett–Garg inequalities. They express the relation between the temporal correlations of \langle Q(\text) Q(\text) \rangle and the correlations between successive times in going from the start to the end. In the derivations above, it has been assumed that the quantity ''Q'', representing the state of the system, always has a definite value (macrorealism per se) and that its measurement at a certain time does not change this value nor its subsequent evolution (noninvasive measurability). A violation of the Leggett–Garg inequality implies that at least one of these two assumptions fails.


Experimental violations

One of the first proposed experiments for demonstrating a violation of macroscopic realism employs superconducting quantum interference devices. There, using
Josephson junction In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mec ...
s, one should be able to prepare macroscopic superpositions of left and right rotating macroscopically large electronic currents in a superconducting ring. Under sufficient suppression of decoherence one should be able to demonstrate a violation of the Leggett–Garg inequality. However, some criticism has been raised concerning the nature of indistinguishable electrons in a Fermi sea. A criticism of some other proposed experiments on the Leggett–Garg inequality is that they do not really show a violation of macrorealism because they are essentially about measuring spins of individual particles.Foundations and Interpretation of Quantum Mechanics. Gennaro Auletta and
Giorgio Parisi Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics and complex systems. His best known contributions are the QCD evolution equations for parton densit ...
, World Scientific, 2001 ,
In 2015 Robens ''et al.'' demonstrated an experimental violation of the Leggett–Garg inequality using superpositions of positions instead of spin with a massive particle. At that time, and so far up until today, the Cesium atoms employed in their experiment represent the largest quantum objects which have been used to experimentally test the Leggett–Garg inequality. The experiments of Robens ''et al.'' as well as Knee ''et al.'', using ideal negative measurements, also avoid a second criticism (referred to as “clumsiness loophole”) that has been directed to previous experiments using measurement protocols that could be interpreted as invasive, thereby conflicting with postulate 2. Several other experimental violations have been reported, including in 2016 with neutrino particles using the
MINOS In Greek mythology, Minos (; grc-gre, Μίνως, ) was a King of Crete, son of Zeus and Europa. Every nine years, he made King Aegeus pick seven young boys and seven young girls to be sent to Daedalus's creation, the labyrinth, to be eaten ...
dataset. Brukner and Kofler have also demonstrated that quantum violations can be found for arbitrarily large ''macroscopic'' systems. As an alternative to
quantum decoherence Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave ...
, Brukner and Kofler are proposing a solution of the quantum-to-classical transition in terms of ''coarse-grained'' quantum measurements under which usually no violation of the Leggett–Garg inequality can be seen anymore. Experiments proposed by Mermin and Braunstein and Mann would be better for testing macroscopic realism, but warns that the experiments may be complex enough to admit unforeseen loopholes in the analysis. A detailed discussion of the subject can be found in the review by Emary et al.


Related inequalities

The four-term Leggett–Garg inequality can be seen to be similar to the
CHSH inequality In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics can not be reproduced by local hidden-variable theories. Experimental verification of the i ...
. Moreover, ''equalities'' were proposed by Jaeger ''et al.''


In popular culture

Leggett-Garg Inequalities is the title of a music album released in 2021 by the Japanese band, First Prequel


See also

*
Leggett inequality The Leggett inequalities, named for Anthony James Leggett, who derived them, are a related pair of mathematical expressions concerning the correlations of properties of entangled particles. (As published by Leggett, the inequalities were exemplifi ...


References

{{DEFAULTSORT:Leggett-Garg Inequality Quantum information science Interpretations of quantum mechanics Physics theorems Inequalities Quantum measurement Physics experiments Thought experiments in quantum mechanics