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arXiv:1703.09547 · Physical Review A

Ambiguous measurements, signaling, and violations of Leggett-Garg inequalities

Clive Emary

Published Oct 4, 2017• 23 citations
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Abstract

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.

DOI
10.1103/physreva.96.042102

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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.
© 2018 American Physical Society. We realize a quantum three-level system with photons distributed among three different spatial and polarization modes. Ambiguous measurements of the state of the qutrit are realized by blocking one out for the three modes at any one time. Using these measurements we construct a test of a Leggett-Garg inequality as well as tests of no-signaling-in-time for the measurements. We observe violations of the Leggett-Garg inequality that cannot be accounted for in terms of signaling. Moreover, we tailor the qutrit dynamics such that both ambiguous and unambiguous measurements are simultaneously nonsignaling, which is an essential step for the justification of the use of ambiguous measurements in Leggett-Garg tests.
We realize a quantum three-level system with photons distributed among three different spatial and polarization modes. Ambiguous measurements of the state of the qutrit are realized by blocking one out for the three modes at any one time. Using these measurements we construct a test of a Leggett-Garg inequality as well as tests of no-signaling-in-time for the measurements. We observe violations of the Leggett-Garg inequality that cannot be accounted for in terms of signaling. Moreover, we tailor the qutrit dynamics such that both ambiguous and unambiguous measurements are simultaneously nonsignaling, which is an essential step for the justification of the use of ambiguous measurements in Leggett-Garg tests.
We generalize the derivation of Leggett-Garg inequalities to systematically treat a larger class of experimental situations by allowing multiparticle correlations, invasive detection, and ambiguous detector results. Furthermore, we show how many such inequalities may be tested simultaneously with a single setup. As a proof of principle, we violate several such two-particle inequalities with data obtained from a polarization-entangled biphoton state and a semiweak polarization measurement based on Fresnel reflection. We also point out a nontrivial connection between specific two-party Leggett-Garg inequality violations and convex sums of strange weak values.
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.
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.
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.