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Violations of a Leggett-Garg inequality without signalling for a photonic qutrit probed with ambiguous measurements
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Abstract
We realise a quantum three-level system with photons distributed among three different spatial and polarization modes. Ambiguous measurement of the state of the qutrit are realised 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-signalling-in-time for the measurements. We observe violations of the Leggett-Garg inequality that can not be accounted for in terms of signalling. Moreover, we tailor the qutrit dynamics such that both ambiguous and unambiguous measurements are simultaneously non-signalling, which is an essential step for the justification of the use of ambiguous measurements in Leggett-Garg tests.
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Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions, but have never identified which specific observables must fail such assignment. Using neutron interferometry, we remedy this shortcoming by showing that BKS contextuality can be confined to particular observables in the form of anomalous weak values, which can be directly witnessed through weak measurements. We construct a confined contextuality witness from weak values, which we measure experimentally to obtain a average violation of the noncontextual bound, with one contributing term violating an independent bound by more than . This experimentally measured confined contextuality confirms the quantum pigeonhole effect, wherein eigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle.
Weak measurement has provided new insight into the nature of quantum measurement, by demonstrating the ability to extract average state information without fully projecting the system. For single-qubit measurements, this partial projection has been demonstrated with violations of the Leggett–Garg inequality. Here we investigate the effects of weak measurement on a maximally entangled Bell state through application of the Hybrid Bell–Leggett–Garg inequality (BLGI) on a linear chain of four transmon qubits. By correlating the results of weak ancilla measurements with subsequent projective readout, we achieve a violation of the BLGI with 27 s.d.s. of certainty.
Macroscopic realism is the name for a class of modifications to quantum
theory that allow macroscopic objects to be described in a
measurement-independent fashion, while largely preserving a fully quantum
mechanical description of the microscopic world. Objective collapse theories
are examples which attempt to provide a physical mechanism for wavefunction
collapse, and thereby aim to solve the quantum measurement problem. Here we
describe and implement an experimental protocol capable of constraining
theories of this class, and show how it is related to the original Leggett-Garg
test, yet more noise tolerant and conceptually transparent. By conducting a set
of simple 'prepare, shuffle, measure' tests in a superconducting flux qubit, we
rule out (by over 77 standard deviations) those theories which would deny
coherent superpositions of 170 nA currents over a 9 ns timescale. Further, we
address the 'clumsiness loophole' by determining classical disturbance in a set
of control experiments.
We report on a stringent test of the nonclassicality of the motion of a massive quantum particle, which propagates on a discrete lattice. Measuring temporal correlations of the position of single atoms performing a quantum walk, we observe a 6σ violation of the Leggett-Garg inequality. Our results rigorously excludes (i.e., falsifies) any explanation of quantum transport based on classical, well-defined trajectories. We use so-called ideal negative measurements—an essential requisite for any genuine Leggett-Garg test—to acquire information about the atom’s position, yet avoiding any direct interaction with it. The interaction-free measurement is based on a novel atom transport system, which allows us to directly probe the absence rather than the presence of atoms at a chosen lattice site. Beyond the fundamental aspect of this test, we demonstrate the application of the Leggett-Garg correlation function as a witness of quantum superposition. Here, we employ the witness to discriminate different types of walks spanning from merely classical to wholly quantum dynamics.
Weak values arise experimentally as conditioned averages of weak (noisy) observable measurements that minimally disturb an initial quantum state, and also as dynamical variables for reduced quantum state evolution even in the absence of measurement. These averages can exceed the eigenvalue range of the observable ostensibly being estimated, which has prompted considerable debate regarding their interpretation. Classical conditioned averages of noisy signals only show such anomalies if the quantity being measured is also disturbed prior to conditioning. This fact has recently been rediscovered, along with the question whether anomalous weak values are merely classical disturbance effects. Here we carefully review the role of the weak value as both a conditioned observable estimation and a dynamical variable, and clarify why classical disturbance models will be insufficient to explain the weak value unless they can also simulate other quantum interference phenomena.
We discuss the use of inequalities of the Leggett-Garg type (LGtI) to witness
quantum coherence and present the first experimental violation of this type of
inequalities using a light-matter interfaced system. By separately benchmarking
the Markovian character of the evolution and the translational invariance of
the conditional probabilities, the observed violation of a LGtI is attributed
to the quantum coherent character of the process. These results provide a
general method to benchmark `quantumness' when the absence of memory effects
can be independently certified and confirm the persistence of quantum coherent
features within systems of increasing complexity.
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.
By combining the postulates of macrorealism with Bell-locality, we derive a qualitatively different hybrid inequality that avoids two loopholes that commonly appear in Leggett-Garg and Bell inequalities. First, locally-invasive measurements can be used, which avoids the "clumsiness" Leggett-Garg inequality loophole. Second, a single experimental ensemble with fixed analyzer settings is sampled, which avoids the "disjoint sampling" Bell inequality loophole. The derived hybrid inequality has the same form as the Clauser-Horne-Shimony-Holt Bell inequality; however, its quantum violation intriguingly requires weak measurements. A realistic explanation of an observed violation requires either the failure of Bell-locality, or a preparation-conspiracy of finely tuned and nonlocally-correlated noise. Modern superconducting and optical systems are poised to implement this test.
The violation of a Bell inequality is an experimental observation that forces the abandonment of a local realistic viewpoint-namely, one in which physical properties are (probabilistically) defined before and independently of measurement, and in which no physical influence can propagate faster than the speed of light. All such experimental violations require additional assumptions depending on their specific construction, making them vulnerable to so-called loopholes. Here we use entangled photons to violate a Bell inequality while closing the fair-sampling loophole, that is, without assuming that the sample of measured photons accurately represents the entire ensemble. To do this, we use the Eberhard form of Bell's inequality, which is not vulnerable to the fair-sampling assumption and which allows a lower collection efficiency than other forms. Technical improvements of the photon source and high-efficiency transition-edge sensors were crucial for achieving a sufficiently high collection efficiency. Our experiment makes the photon the first physical system for which each of the main loopholes has been closed, albeit in different experiments.
One of the most striking features of quantum mechanics is the profound effect exerted by measurements alone. Sophisticated quantum control is now available in several experimental systems, exposing discrepancies between quantum and classical mechanics whenever measurement induces disturbance of the interrogated system. In practice, such discrepancies may frequently be explained as the back-action required by quantum mechanics adding quantum noise to a classical signal. Here, we implement the "three-box" quantum game [Aharonov Y, et al. (1991) J Phys A Math Gen 24(10):2315-2328] by using state-of-the-art control and measurement of the nitrogen vacancy center in diamond. In this protocol, the back-action of quantum measurements adds no detectable disturbance to the classical description of the game. Quantum and classical mechanics then make contradictory predictions for the same experimental procedure; however, classical observers are unable to invoke measurement-induced disturbance to explain the discrepancy. We quantify the residual disturbance of our measurements and obtain data that rule out any classical model by ≳7.8 standard deviations, allowing us to exclude the property of macroscopic state definiteness from our system. Our experiment is then equivalent to the test of quantum noncontextuality [Kochen S, Specker E (1967) J Math Mech 17(1):59-87] that successfully addresses the measurement detectability loophole.
Quantum coherence is one of the primary non-classical features of quantum systems. While protocols such as the Leggett-Garg inequality (LGI) and quantum tomography can be used to test for the existence of quantum coherence and dynamics in a given system, unambiguously detecting inherent "quantumness" still faces serious obstacles in terms of experimental feasibility and efficiency, particularly in complex systems. Here we introduce two "quantum witnesses" to efficiently verify quantum coherence and dynamics in the time domain, without the expense and burden of non-invasive measurements or full tomographic processes. Using several physical examples, including quantum transport in solid-state nanostructures and in biological organisms, we show that these quantum witnesses are robust and have a much finer resolution in their detection window than the LGI has. These robust quantum indicators may assist in reducing the experimental overhead in unambiguously verifying quantum coherence in complex systems.
The possibility of observing violations of temporal Bell inequalities,
originally proposed by Leggett and Garg as a mean of testing the quantum-mechanical
delocalization of suitably chosen macroscopic bodies, is
discussed by taking into account the effect of the measurement process.
A general criterion
quantifying this possibility is defined and shown not to be fulfilled by
the various experimental configurations proposed so far to test
inequalities of different forms.
We consider the violation of the Leggett-Garg inequality in electronic
Mach-Zehnder inteferometers. This set-up has two distinct advantages over
earlier quantum-transport proposals: firstly, the required correlation
functions can be obtained without time-resolved measurements. Secondly, the
geometry of an interferometer allows one to construct the correlation functions
from ideal negative measurements, which addresses the non-invasiveness
requirement of the Leggett-Garg inequality. We discuss two concrete
realisations of these ideas: the first in quantum Hall edge-channels, the
second in a double quantum dot interferometer.
Quantum mechanics violates Leggett-Garg inequalities because the operator
formalism predicts correlations between different spin components that would
correspond to negative joint probabilities for the outcomes of joint
measurements. However, the uncertainty principle ensures that such joint
measurements cannot be implemented without errors. In a sequential measurement
of the spin components, the resolution and back-action errors of the
intermediate measurement can be described by random spin flips acting on an
intrinsic joint probability. If the error rates are known, the intrinsic joint
probability can be reconstructed from the noisy statistics of the actual
measurement outcomes. In this paper, we use the spin-flip model of measurement
errors to analyze experimental data on photon polarization obtained with an
interferometric setup that allows us to vary the measurement strength and hence
the balance between resolution and back-action errors. We confirm that the
intrinsic joint probability obtained from the experimental data is independent
of measurement strength and show that the same violation of the Leggett-Garg
inequality can be obtained for any combination of measurement resolution and
back-action.
The quantum superposition principle states that an entity can exist in two different states simultaneously, counter to our 'classical' intuition. Is it possible to understand a given system's behaviour without such a concept? A test designed by Leggett and Garg can rule out this possibility. The test, originally intended for macroscopic objects, has been implemented in various systems. However to date no experiment has employed the 'ideal negative result' measurements that are required for the most robust test. Here we introduce a general protocol for these special measurements using an ancillary system, which acts as a local measuring device but which need not be perfectly prepared. We report an experimental realization using spin-bearing phosphorus impurities in silicon. The results demonstrate the necessity of a non-classical picture for this class of microscopic system. Our procedure can be applied to systems of any size, whether individually controlled or in a spatial ensemble.
We present a detailed motivation for and definition of the contextual values
of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104
240401 (2010)]. The theory of contextual values extends the well-established
theory of generalized state measurements by bridging the gap between partial
state collapse and the observables that represent physically relevant
information about the system. To emphasize the general utility of the concept,
we first construct the full theory of contextual values within an operational
formulation of classical probability theory, paying special attention to
observable construction, detector coupling, generalized measurement, and
measurement disturbance. We then extend the results to quantum probability
theory built as a superstructure on the classical theory, pointing out both the
classical correspondences to and the full quantum generalizations of both
L\"uder's rule and the Aharonov-Bergmann-Lebowitz rule in the process. We find
in both cases that the contextual values of a system observable form a
generalized spectrum that is associated with the independent outcomes of a
partially correlated and generally ambiguous detector; the eigenvalues are a
special case when the detector is perfectly correlated and unambiguous. To
illustrate the approach, we apply the technique to both a classical example of
marble color detection and a quantum example of polarization detection. For the
quantum example we detail two devices: Fresnel reflection from a glass
coverslip, and continuous beam displacement from a calcite crystal. We also
analyze the three-box paradox to demonstrate that no negative probabilities are
necessary in its analysis. Finally, we provide a derivation of the quantum weak
value as a limit point of a pre- and postselected conditioned average and
provide sufficient conditions for the derivation to hold.
We generalize the derivation of Leggett-Garg inequalities to systematically
treat a larger class of experimental situations by allowing multi-particle
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 semi-weak polarization measurement based on Fresnel reflection. We also
point out a non- trivial connection between specific two-party Leggett-Garg
inequality violations and convex sums of strange weak values.
The violation of J. Bell's inequality with two entangled and spatially separated quantum two- level systems (TLS) is often considered as the most prominent demonstration that nature does not obey ?local realism?. Under di?erent but related assumptions of ?macrorealism?, plausible for macroscopic systems, Leggett and Garg derived a similar inequality for a single degree of freedom undergoing coherent oscillations and being measured at successive times. Such a ?Bell's inequality in time?, which should be violated by a quantum TLS, is tested here. In this work, the TLS is a superconducting quantum circuit whose Rabi oscillations are continuously driven while it is continuously and weakly measured. The time correlations present at the detector output agree with quantum-mechanical predictions and violate the inequality by 5 standard deviations. Comment: 26 pages including 10 figures, preprint format
By means of the quantum potential interpretation we show that there is no need for a break or ``cut'' in the way we regard reality between quantum and classical levels.
The seminal paper of Aharonov, Albert, and Vaidman introduces the
concept of a weak value as a statistical average over realizations of a
weak measurement, where the system is both pre- and post-selected. By
taking restricted averages, weak values can exceed the range of
eigenvalues associated with the observable in question. We discuss how
to implement a weak values measurement with solid-state qubits. In
parallel activity, Leggett and Garg have devised a test of quantum
mechanics for a single system using different ensembles of (projective)
measurements at different times and correlation functions of those
outcomes. The original motivation was to test if there was a size scale
where quantum mechanics would break down. Introduced as a
``Bell-inequality in time'', the assumptions of macrorealism that could
be verified by a non-invasive detector imply that their correlation
function obeys a Leggett-Garg inequality that quantum mechanics would
violate, formally similar to the inequality of Bell. We demonstrate that
the proper notion of a classical weak value also demands these
assumptions, and that furthermore a weak value can be non-classical if
and only if a Leggett-Garg inequality can also be violated. We will
discuss generalized weak values, where post-selection occurs on a range
of weak measurement results. Our analysis is presented in terms of
kicked quantum nondemolition measurements on a quantum double-dot charge
qubit.
Ambiguous measurements do not reveal complete information about the system under test. Their quantum-mechanical counterparts are semi-weak (or in the limit, weak-) measurements and here we discuss their role in tests of the Leggett-Garg inequalities. We show that, whilst ambiguous measurements allow one to forgo the usual non-invasive measureability assumption, to derive an LGI that may be violated, we are forced to introduce another assumption that equates the invasive influence of ambiguous and unambiguous detectors. We then derive signalling conditions that should be fulfilled for the plausibility of the Leggett-Garg test and propose an experiment on a three-level system with a direct quantum-optics realisation that satisfies all signalling constraints and violates a Leggett-garg inequality.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating on that sum of variances are valid in a state-dependent manner and the lower bound is guaranteed to be nontrivial for two observables being incompatible on the state of the system being measured. The behaviour we find agrees with the predictions of quantum theory and obeys the new uncertainty relations even for the special states which trivialize Heisenberg-Robertson relation. We realize a direct measurement model and give the first experimental investigation of the strengthened relations.
We investigate how discrete internal degrees of freedom in a quasi-macroscopic system affect the violation of the Leggett--Garg inequality, a test of macroscopic-realism based on temporal correlation functions. As a specific example, we focus on an ensemble of qubits subject to collective and individual noise. This generic model can describe a range of physical systems, including atoms in cavities, electron or nuclear spins in NV centers in diamond, erbium in YSiO, bismuth impurities in silicon, or arrays of superconducting circuits, to indicate but a few. Such large ensembles are potentially more macroscopic than other systems that have been used so far for testing the Leggett--Garg inequality, and open a route toward probing the boundaries of quantum mechanics at macroscopic scales. We find that, because of the non-trivial internal structure of such an ensemble, the behavior of different measurement schemes, under the influence of noise, can be surprising. We discuss which measurement schemes are optimal for flux qubits and NV centers, and some of the technological constraints and difficulties for observing such violations with present-day experiments.
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 the quantum witness test of macroscopic realism and derive an
upper bound for possible violations of this equality due to quantum mechanics.
The bound depends only on the number of possible outcomes for the blind
measurement at the heart of the witness protocol. Mirroring recent results for
the related Leggett-Garg inequality, we show that quantum mechanics can
saturate the algebraic bound for large systems. We also establish a connection
between the quantum witness and the trace distance between density matrices.
The average result of a weak measurement of some observable A can, under
post-selection of the measured quantum system, exceed the largest eigenvalue of
A. The nature of weak measurements, as well as the presence of post-selection
and hence possible contribution of measurement-disturbance, has led to a
long-running debate about whether or not this is surprising. Here, it is shown
that such "anomalous weak values" are non-classical in a precise sense: a
sufficiently weak measurement of one constitutes a proof of contextuality. This
clarifies, for example, which features must be present (and in an experiment,
verified) to demonstrate an effect with no satisfying classical explanation.
We define a quantum witness for high-mass matter-wave interferometers that
allows us to test fundamental assumptions of macroscopic realism. We propose an
experimental realisation using absorptive laser gratings and show that such
systems can strongly violate a macrorealistic quantum-witness equality. The
measurement of the witness can therefore provide clear evidence of physics
beyond macrorealism for macromolecules and nanoparticles.
The Kibble-Zurek mechanism (KZM) captures the key physics of nonequilibrium dynamics in second order phase transitions, and accurately predicts the density of topological defects formed in such processes. However, the central prediction of KZM-i.e., the scaling of the density of defects with the quench rate-still needs further experimental confirmation, particularly for quantum transitions. Here, we perform a quantum simulation of the nonequilibrium dynamics of the Landau-Zener model based on a nine-stage optical interferometer with an overall visibility of 0.975±0.008. The results support the adiabatic-impulse approximation, which is the core of Kibble-Zurek theory. Moreover, the developed high-fidelity multistage optical interferometer can support more complex linear optical quantum simulations.
We show that the quantum bound for temporal correlations in a Leggett-Garg
test, analogous to the Tsirelson bound for spatial correlations in a Bell test,
strongly depends on the number of levels N that can be accessed by the
measurement apparatus via projective measurements. We provide exact bounds for
small N, that exceed the known bound for the Leggett-Garg inequality, and
show that in the limit the Leggett-Garg inequality can be
violated up to its algebraic maximum.
I consider the extent to which the applicability of the concept of classical realism is constrained, irrespective of the validity or not of the quantum formalism, by existing experiments both in the EPR–Bell setup, including recent experiments testing 'nonlocal realistic' theories, and in the area of 'macroscopic quantum coherence'. Unless we are willing to sacrifice one or more other intuitively plausible notions such as that of the conventional 'arrow of time', it appears impossible, in either context, to maintain the classical notion of realism.
Quantum Mechanics (QM) is one of the pillars of modern physics: an impressive amount of experiments have confirmed this theory and many technological applications are based on it. Nevertheless, at one century since its development, various aspects concerning its very foundations still remain to be clarified. Among them, the transition from a microscopic probabilistic world into a macroscopic deterministic one and quantum non-locality. A possible way out from these problems would be if QM represents a statistical approximation of an unknown deterministic theory.This review is addressed to present the most recent progresses on the studies related to hidden variable theories (HVT), both from an experimental and a theoretical point of view, giving a larger emphasis to results with a direct experimental application.More in details, the first part of the review is a historical introduction to this problem. The Einstein–Podolsky–Rosen argument and the first discussions about HVT are introduced, describing the fundamental Bell's proposal for a general experimental test of every local HVT and the first attempts to realise it.The second part of the review is devoted to elucidate the recent progresses toward a conclusive Bell inequalities experiment obtained with entangled photons and other physical systems.Finally, the last sections are targeted to shortly discuss non-local HVT.
By weakly measuring the polarization of a photon between two strong polarization measurements, we experimentally investigate the correlation between the appearance of anomalous values in quantum weak measurements and the violation of realism and nonintrusiveness of measurements. A quantitative formulation of the latter concept is expressed in terms of a Leggett-Garg inequality for the outcomes of subsequent measurements of an individual quantum system. We experimentally violate the Leggett-Garg inequality for several measurement strengths. Furthermore, we experimentally demonstrate that there is a one-to-one correlation between achieving strange weak values and violating the Leggett-Garg inequality.
An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given. Our recursive algorithm factorizes any N×N unitary matrix into a sequence of two-dimensional beam splitter transformations. The experiment is built from the corresponding devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix. Thus optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems become feasible.
Quantum-vs. macro-realism: What does the Leggett-Garg inequality actually test
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Beyond the temporal Tsirelson bound: an experimental test of a Leggett-Garg inequality in a three-level system
- K Wang
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- H Katiyar
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H. Katiyar, A. Brodutch, D. Lu, and R. Laflamme, "Experimental violation of the leggett-garg inequality in a
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Condition for macroscopic realism beyond the Leggett-Garg inequalities
- J Kofler
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J. Kofler and C. Budroni, "Condition for macroscopic realism beyond the Leggett-Garg inequalities,"
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Necessary and sufficient conditions for macroscopic realism from quantum mechanics
- L Clemente
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for macrorealism: Limitations of the Leggett-Garg inequality," Phys. Rev. Lett. 116, 150401 (2016).
Comparing conditions for macrorealism: Leggett-Garg inequalities versus no-signaling in time
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J. J. Halliwell, "Comparing conditions for macrorealism:
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