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arXiv:1508.02271 · Physical Review A

Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach

J. J. Halliwell

Published Feb 26, 2016• 87 citations
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Abstract

The Leggett-Garg (LG) inequalities were proposed in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying notion of macrorealism. Here, the LG inequalities are explored using a simple quasiprobability linear in the projection operators to describe the properties of the system at two times. We show that this quasiprobability is measurable, has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the later time are independent of whether an earlier measurement was made, a generalization of the no-signaling in time condition of Kofler and Brukner. We argue that this quasiprobability, appropriately measured, provides a noninvasive measure of macrorealism per se at the two-time level. This measure, when combined with the LG inequalities, provides a characterization of macrorealism more detailed than that provided by the LG inequalities alone. When the quasiprobability is non-negative, the LG system has a natural parallel with the Einstein-Podolsky-Rosen-Bohm system and Fine's theorem. A simple spin model illustrating key features of the approach is exhibited.

DOI
10.1103/physreva.93.022123

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The Leggett-Garg (LG) inequalities were proposed in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying notion of macrorealism. Here, the LG inequalities are explored using a simple quasiprobability linear in the projection operators to describe the properties of the system at two times. We show that this quasiprobability is measurable, has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the later time are independent of whether an earlier measurement was made, a generalization of the no-signaling in time condition of Kofler and Brukner. We argue that this quasiprobability, appropriately measured, provides a noninvasive measure of macrorealism per se at the two-time level. This measure, when combined with the LG inequalities, provides a characterization of macrorealism more detailed than that provided by the LG inequalities alone. When the quasiprobability is non-negative, the LG system has a natural parallel with the Einstein-Podolsky-Rosen-Bohm system and Fine's theorem. A simple spin model illustrating key features of the approach is exhibited.
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable QQQ. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly studied test, are only necessary conditions for macrorealism, but, building on earlier work J. J. Halliwell, Phys. Rev. A 93, 022123 (2016), it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with a very different set of necessary and sufficient conditions for macrorealism, namely the no-signaling in time (NSIT) conditions proposed by J. Kofler and C. Brukner, Phys. Rev. A 87, 052115 (2013) and L. Clemente and J. Kofler, Phys. Rev. A 91, 062103 (2015), which ensure that all probabilities for QQQ at one and two times are independent of whether earlier or intermediate measurements are made in a given run, and do not require (but imply) the LG inequalities. We argue that tests based on the LG inequalities have the form of very weak classicality conditions and can be satisfied in the face of moderate interference effects, but those based on NSIT conditions have the form of much stronger coherence witness conditions, satisfied only for zero interference. The two tests differ in their implementation of noninvasive measurability and so are testing different notions of macrorealism: the augmented LG tests are indirect, entailing a combination of the results of different experiments with only compatible quantities measured in each experimental run, in close analogy with Bell tests, and are primarily tests for macrorealism per se; in contrast, the NSIT tests entail sequential measurements of incompatible quantities and are primarily tests for noninvasiveness.
The Leggett-Garg inequalities were proposed in order to assess whether sets of pairs of measurements on a quantum system at different times can be consistent with a notion of realism for the set of all variables measured. However, their interpretation is severely complicated by the difficulty of distinguishing violations of realism from the invasive effects of sequential measurements. The typically imposed requirement, of non-invasive measurability (NIM), is often too stringent to be implemented experimentally. Here, we propose a weaker alternative which is to replace the standard quantum-mechanical formula for two-time sequential measurements with a closely related quasi-probability which is linear in the projection operators. This quasi-probability is indirectly measurable (for two times), has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the later time are independent of whether an earlier measurement was made, i.e. satisfies the analogue of NIM for quasi-probabilities. Attention is restricted only to those parameter ranges for which the quasi-probability is non-negative, a condition named linear positivity by Goldstein and Page, and the quasi-probability may then be regarded as a probability for the alternatives at two times. Violations of the Leggett-Garg inequalities within this restricted parameter range are then clear refutations of realism and not consequences of the invasiveness of measurement. We exhibit a simple spin model in which linear positivity holds for the two-time quasi-probabilities but the Leggett-Garg inequalities are violated.
Abstract The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism – the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. We argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine’s theorem applies and these augmented conditions are then both necessary and sufficient. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. We argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.
Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which the usual LGI is satisfied by QM, but Wigner's form of the LGI (WLGI) is violated. This implies that for such unsharp measurements, the QM violation of macrorealism cannot be tested using the usual LGI, but can be tested using WLGI. In showing this, we take into account the general form of the usual LGI involving an arbitrary number of pairs of two-time correlation functions. Another recently proposed necessary condition for macrorealism, called ``no-signaling in time,'' is also probed, showing that its QM violation persists for arbitrarily unsharp measurements.
Macroscopic Realism (MR) says that a macroscopic system is always determinately in one or other of the macroscopically distinguishable states available to it. The Leggett-Garg (LG) inequality was derived to allow experimental test of whether or not this doctrine is true; it is also often thought of as a temporal version of a Bell-inequality. Despite recent interest in the inequality, controversy remains regarding what would be shown by its violation. Here we resolve this controversy, which arises due to an insufficiently general and model-independent approach to the question so far. We argue that LG's initial characterisation of MR does not pick out a particularly natural realist position, so we articulate an operationally well-defined and well-motivated position in its place. We show that much weaker conditions than LG's are sufficient to derive the inequality: in the first instance, its violation only demonstrates that certain measurements fail to be non-disturbing at the operational level. We articulate three distinct species of MR-ist position, and argue that it is only the first of these which can be refuted by LG inequality violation. This first position is an attractive one, so ruling it out remains of interest, however. A crucial role is played in LG's argument by the assumption of noninvasive measurability. We show that this notion is ambiguous between the weaker notion of disturbance at the operational level, and the stronger notion of invasiveness at the ontic level of properties of the system. Ontic noninvasiveness would be required to rule out MR per se but this property is not entailed by MR, and its presence cannot be established in a model-independent way. It follows that despite the formal parallels, Bell's and LG's inequalities are not methodologically on a par. We close with some reflections on the implications of our analysis for the pedagogy of quantum superposition.
Leggett-Garg (LG) tests for macrorealism were originally designed to explore quantum coherence on the macroscopic scale. Interference experiments and systems modeled by harmonic oscillators provide useful examples of situations in which macroscopicity has been approached experimentally and are readily turned into LG tests for a single dichotomic variable QQQ. Applying this approach to the double-slit experiment in which a noninvasive measurement at the slits is included, we exhibit LG violations. We find that these violations are always accompanied by destructive interference. The converse is not true in general and we find that there are nontrivial regimes in which there is destructive interference but the two-time LG inequalities are satisfied, which implies that it is in fact often possible to assign (indirectly determined) probabilities for the interferometer paths. Similar features have been observed in recent work involving a LG analysis of a Mach-Zehnder interferometer and we compare with those results. We extend the analysis to the triple-slit experiment again finding LG violations, and we also exhibit examples of some surprising relationships between LG inequalities and NSIT conditions that do not exist for dichotomic variables. For the simple harmonic oscillator, we find an analytically tractable example showing a two-time LG violation with a gaussian initial state, echoing in simpler form recent results of Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)].