Leggett–Garg inequality

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 simplest form of the Leggett–Garg inequality derives from examining a system that has only two possible states. These states have corresponding measurement values $Q=\backslash pm\; 1$. 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 . Now suppose, for instance, that there is a perfect correlation $C\_\{13\}\; =\; 1$ between times $t\_1$ and $t\_3$. That is to say, that for N realisations of the experiment, the temporal correlation reads

: $C\_\{13\}\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{r=1\}^N\; Q\_r(t\_1)\; Q\_r(t\_3)\; =\; 1.$

We look at this case in some detail. What can be said about what happens at time $t\_2$? Well, it is possible that $C\_\{12\}\; =\; C\_\{23\}\; =\; 1$, so that if the value of $Q$ at $t\_1$ is $\backslash pm\; 1$, then it is also $\backslash pm\; 1$ for both times $t\_2$ and $t\_3$.

It is also quite possible that $C\_\{12\}\; =\; C\_\{23\}\; =\; -1$, 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 $C\_\{12\}\; =\; C\_\{23\}\; =\; 0$.

So, although it is known that if $Q\; =\; \backslash pm\; 1$ at $t\_1$, it must also be $\backslash pm\; 1$ at $t\_3$; the value at $t\_2$ may as well be determined by the toss of a coin.

We define $K$ as $K\; =\; C\_\{12\}\; +\; C\_\{23\}\; -\; C\_\{13\}$.

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 $K\; =\; C\_\{12\}\; +\; C\_\{23\}\; -\; C\_\{13\}\; \backslash le\; 1$. To see this, we note that

: $K\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{r=1\}^N\; \backslash big(Q(t\_1)\; Q(t\_2)\; +\; Q(t\_2)\; Q(t\_3)\; -\; Q(t\_1)\; Q(t\_3)\backslash big)\_r.$

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 $K\; =\; C\_\{12\}\; +\; C\_\{23\}\; +\; C\_\{34\}\; -\; C\_\{14\}\; \backslash le\; 2$, and so on. These are the Leggett–Garg inequalities. They express the relation between the temporal correlations of $\backslash langle\; Q(\backslash text\{start\})\; Q(\backslash text\{end\})\; \backslash rangle$ 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.

One of the first proposed experiments for demonstrating a violation of macroscopic realism employs superconducting quantum interference devices. There, using Josephson junctions, 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.{{cite journal | last=Leggett | first=A J | title=Testing the limits of quantum mechanics: motivation, state of play, prospects | journal=Journal of Physics: Condensed Matter | volume=14 | issue=15 | date=2002-04-05 | issn=0953-8984 | doi=10.1088/0953-8984/14/15/201 | pages=R415–R451| s2cid=250911999 }} However, some criticism has been raised concerning the nature of indistinguishable electrons in a Fermi sea.{{Cite journal |doi = 10.1007/s10701-011-9598-4|arxiv = 1001.1777|bibcode = 2012FoPh...42..256W|title = Addressing the Clumsiness Loophole in a Leggett-Garg Test of Macrorealism|year = 2012|last1 = Wilde|first1 = Mark M.|last2 = Mizel|first2 = Ari|journal = Foundations of Physics|volume = 42|issue = 2|pages = 256–265| s2cid=73699503 }}{{cite thesis|title=Superconducting qubit in a resonator: test of the Leggett-Garg inequality and single-shot readout|author=A. Palacios-Laloy|type=PhD |year=2010|url=http://iramis.cea.fr/spec/Pres/Quantro/static/wp-content/uploads/2010/10/Palacios-Laloy-Thesis1.pdf}}

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, World Scientific, 2001 {{isbn|981-02-4614-5}}, {{isbn|978-981-02-4614-3}} In 2015 Robens et al.{{cite journal | last1=Robens | first1=Carsten | last2=Alt | first2=Wolfgang | last3=Meschede | first3=Dieter | last4=Emary | first4=Clive | last5=Alberti | first5=Andrea | title=Ideal Negative Measurements in Quantum Walks Disprove Theories Based on Classical Trajectories | journal=Physical Review X| volume=5 | issue=1 | date=2015-01-20 | issn=2160-3308 | doi=10.1103/physrevx.5.011003 | page=011003|doi-access=free| bibcode=2015PhRvX...5a1003R | arxiv=1404.3912 }} 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.{{cite journal |last=Knee |first=George C. |year=2015 |title=Viewpoint: Do Quantum Superpositions Have a Size Limit? |journal=Physics |volume=8 |issue=6 |page=6 |doi=10.1103/Physics.8.6|doi-access=free }}

The experiments of Robens et al. as well as Knee et al.,{{cite journal | last1=Knee | first1=George C. | last2=Simmons | first2=Stephanie | last3=Gauger | first3=Erik M. | last4=Morton | first4=John J.L. | last5=Riemann | first5=Helge | last6=Abrosimov | first6=Nikolai V. | last7=Becker | first7=Peter | last8=Pohl | first8=Hans-Joachim | last9=Itoh | first9=Kohei M. | last10=Thewalt | first10=Mike L.W. | last11=Briggs | first11=G. Andrew D. | last12=Benjamin | first12=Simon C. | title=Violation of a Leggett–Garg inequality with ideal non-invasive measurements | journal=Nature Communications| volume=3 | issue=1 | year=2012 | issn=2041-1723 | doi=10.1038/ncomms1614 | pmid=22215081 | pmc=3272582 | page=606|display-authors=5| bibcode=2012NatCo...3..606K | arxiv=1104.0238 }} using ideal negative measurements, also avoid a second criticism (referred to as “clumsiness loophole”{{cite journal | last1=Wilde | first1=Mark M. | last2=Mizel | first2=Ari | title=Addressing the Clumsiness Loophole in a Leggett-Garg Test of Macrorealism | journal=Foundations of Physics| volume=42 | issue=2 | date=2011-09-13 | issn=0015-9018 | doi=10.1007/s10701-011-9598-4 | pages=256–265| arxiv=1001.1777 | bibcode=2012FoPh...42..256W | s2cid=73699503 }}) 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 dataset.{{cite journal | last1=Formaggio | first1=J. A. | last2=Kaiser | first2=D. I. | last3=Murskyj | first3=M. M. | last4=Weiss | first4=T. E. | title=Violation of the Leggett-Garg Inequality in Neutrino Oscillations | journal=Physical Review Letters | volume=117 | issue=5 | date=2016-07-26 | issn=0031-9007 | doi=10.1103/physrevlett.117.050402 | pmid=27517759 | page=050402| bibcode=2016PhRvL.117e0402F | arxiv=1602.00041 | s2cid=6127630 }}

Brukner and Kofler have also demonstrated that quantum violations can be found for arbitrarily large macroscopic systems. As an alternative to quantum decoherence, 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.{{cite journal | last1=Kofler | first1=Johannes | last2=Brukner | first2=Časlav | title=Classical World Arising out of Quantum Physics under the Restriction of Coarse-Grained Measurements | journal=Physical Review Letters | volume=99 | issue=18 | date=2007-11-02 | issn=0031-9007 | doi=10.1103/physrevlett.99.180403 | pmid=17995385 | page=180403|arxiv=quant-ph/0609079| bibcode=2007PhRvL..99r0403K | s2cid=34702806 }}{{cite journal | last1=Kofler | first1=Johannes | last2=Brukner | first2=Časlav | title=Conditions for Quantum Violation of Macroscopic Realism | journal=Physical Review Letters | volume=101 | issue=9 | date=2008-08-28 | issn=0031-9007 | doi=10.1103/physrevlett.101.090403 | pmid=18851590 | page=090403|arxiv=0706.0668| bibcode=2008PhRvL.101i0403K | s2cid=6060566 }}

Experiments proposed by Mermin{{cite journal | last=Mermin | first=N. David | title=Extreme quantum entanglement in a superposition of macroscopically distinct states | journal=Physical Review Letters| volume=65 | issue=15 | year=1990 | issn=0031-9007 | doi=10.1103/physrevlett.65.1838 | pmid=10042377 | pages=1838–1840| bibcode=1990PhRvL..65.1838M }} and Braunstein and Mann{{cite journal | last1=Braunstein | first1=Samuel L. | last2=Mann | first2=A. | title=Noise in Mermin'sn-particle Bell inequality | journal=Physical Review A| volume=47 | issue=4 | date=1993-04-01 | issn=1050-2947 | doi=10.1103/physreva.47.r2427 | pmid=9909338 | pages=R2427–R2430| bibcode=1993PhRvA..47.2427B }} 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.{{cite journal | last1=Emary | first1=Clive | last2=Lambert | first2=Neill | last3=Nori | first3=Franco | title=Leggett–Garg inequalities | journal=Reports on Progress in Physics | volume=77 | issue=1 | year=2014 | issn=0034-4885 | doi=10.1088/0034-4885/77/1/016001 | page=016001|arxiv=1304.5133| bibcode=2014RPPh...77a6001E | s2cid=9794268 }}

The four-term Leggett–Garg inequality can be seen to be similar to the CHSH inequality. Moreover, equalities were proposed by Jaeger et al.{{cite journal | last1=Jaeger | first1=Gregg | last2=Viger | first2=Chris | last3=Sarkar | first3=Sahotra | title=Bell-type equalities for SQUIDs on the assumptions of macroscopic realism and non-invasive measurability | journal=Physics Letters A | volume=210 | issue=1–2 | year=1996 | issn=0375-9601 | doi=10.1016/0375-9601(95)00821-7 | pages=5–10| bibcode=1996PhLA..210....5J }}

Leggett-Garg Inequalities is the title of a 2021 music album by the Japanese band First Prequel. [https://linkco.re/NEu3S4Nr?lang=en]

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