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The Leggett-Garg Inequalities and Linear Positivity: an Alternative to Non-Invasive Measurability

J. J. Halliwell

Published Aug 10, 2015• 0 citations
OpenAlex

Abstract

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.

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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.

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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.
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.
The Leggett-Garg inequalities are a set of inequalities obeyed by classical systems but violated in quantum theory. Their violation has been taken as evidence that quantum theory lacks a `realistic' formulation. However in addition to realism the derivation of the Leggett-Garg inequalities relies on another, more obscure assumption of `non-invasive measurability.' The significance of this assumption and the consequences for the interpretation of violations of the Leggett-Garg inequalities have been hotly debated. In this paper we present a pedagogical introduction to the issues, focussing on the significance of non-invasive measurability. We give a simple derivation of the Leggett-Garg inequalities paying particular attention to where and when the two assumptions are used, and we give an example of a realist but not non-invasively measurable hidden variable theory which violates them. We also discuss recent attempts to experimentally implement non-invasive measurements via so called `undetectable measurements' and we show that even undetectable measurements are not necessarily non-invasive.
The Leggett-Garg inequality is a widely used test of the "quantumness" of a system, and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Legget-Garg inequality disproves macroscopic realism and non-invasiveness. Nevertheless, recent results point out that macroscopic realism is a model dependent notion and that one should always be able to attribute to invasiveness a violation of a Legget-Garg inequality. This opens some natural questions: how to provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the measurability of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors or to back action.
The Leggett-Garg inequality is a widely used test of the ``quantumness'' of a system and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Leggett-Garg inequality disproves macroscopic realism and noninvasiveness. Nevertheless, recent results point out that macroscopic realism is a model-dependent notion and that one should always be able to attribute to invasiveness a violation of a Leggett-Garg inequality. This opens some natural questions: How do we provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional-independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the measurability of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors, or to back action.
The arrival-time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the xxx axis at two different times. We explore this problem using a two-time quasiprobability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or, more generally, ambiguous). We argue that when positive, it corresponds to a measurement-independent arrival-time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasiprobability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of ``quantumness,'' in which the arrival-time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasiprobability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.
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.