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arXiv:1509.05136

Are weak measurements really necessary for Leggett-Garg type measurements?

N. D. Hari Dass

Published Sep 17, 2015• 0 citations
arXivHTMLPDF

Abstract

Leggett-Garg inequalities are an important milestone in our quest to bridge the classical-quantum divide. An experimental investigation of these inequalities requires the so called \emph{non-invasive measurements}(NIM). It has become popular to invoke weak measurements as the means of realising NIMs to very good approximation, because of their allegedly low disturbance of systems under measurement. In this note, this is shown to be a myth; it is shown, by simple estimates of errors, that for comparable levels of statistical errors, even the strong or projective measurements can be used. In fact, it is shown that resource-wise, strong measurements are even preferable.

DOI
10.48550/arxiv.1509.05136

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Leggett-Garg inequalities are an important milestone in our quest to bridge the classical-quantum divide. An experimental investigation of these inequalities requires the so called \emph{non-invasive measurements}(NIM). It has become popular to invoke weak measurements as the means of realising NIMs to very good approximation, because of their allegedly low disturbance of systems under measurement. In this note, this is shown to be a myth; it is shown, by simple estimates of errors, that for comparable levels of statistical errors, even the strong or projective measurements can be used. In fact, it is shown that resource-wise, strong measurements are even preferable.
Leggett and Garg derived inequalities that probe the boundaries of classical and quantum physics by putting limits on the properties that classical objects can have. Historically, it has been suggested that Leggett-Garg inequalities are easily violated by quantum systems undergoing sequences of strong measurements, casting doubt on whether quantum mechanics correctly describes macroscopic objects. Here I show that Leggett-Garg inequalities cannot be violated by any projective measurement. The perceived violation of the inequalities found previously can be traced back to an inappropriate assumption of non-invasive measurability. Surprisingly, weak projective measurements cannot violate the Leggett-Garg inequalities either because even though the quantum system itself is not fully projected via weak measurements, the measurement devices are.
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 inequalities probe the classical-quantum boundary by putting limits on the sum of pairwise correlation functions between classical measurement devices that consecutively measured the same quantum system. The apparent violation of these inequalities by standard quantum measurements has cast doubt on quantum mechanics' ability to consistently describe classical objects. Recent work has concluded that these inequalities cannot be violated by either strong or weak projective measurements [1]. Here I consider an entropic version of the Leggett-Garg inequalities that are different from the standard inequalities yet similar in form, and can be defined without reference to any particular observable. I find that the entropic inequalities also cannot be be violated by strong quantum measurements. The entropic inequalities can be extended to describe weak quantum measurements, and I show that these weak entropic Leggett-Garg inequalities cannot be violated either even though the quantum system remains unprojected, because the inequalities describe the classical measurement devices, not the quantum system. I conclude that quantum mechanics adequately describes classical devices, and that we should be careful not to assume that the classical devices accurately describe the quantum system.
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.
Ambiguous measurements do not reveal complete information about the system under test. Their quantum-mechanical counterparts are semiweak (or in the limit, weak) measurements and here we discuss their role in tests of the Leggett-Garg inequalities. We show that, while ambiguous measurements allow one to forgo the usual noninvasive measurability assumption, to derive a Leggett-Garg inequality that may be violated, we are forced to introduce another assumption that equates the invasive influence of ambiguous and unambiguous detectors. Based on this assumption, we derive signaling conditions that should be fulfilled for the plausibility of the Leggett-Garg test. We then propose an experiment on a three-level system with a direct quantum-optics realization that satisfies all signaling constraints and violates a Leggett-Garg inequality.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Notion of weak value coordinates for state spaces are introduced and elaborated.