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arXiv:1604.01659 · Physical Review A

Decoherent histories and measurement of temporal correlation functions for Leggett-Garg inequalities

J. J. Halliwell

Published Nov 28, 2016• 10 citations
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Abstract

We consider two protocols for the measurement of the temporal correlation functions of a dichotomic variable $Q$ appearing in Leggett-Garg-type inequalities. The protocols measure solely whether $Q$ has the same or a different sign at the end of a given time interval, thereby measuring no more than is required for determination of the correlation function. They are inspired, in part, by a decoherent histories analysis of the two-time histories of $Q$, which yields a number of useful insights, although the protocols are ultimately expressed in macrorealistic form independent of quantum theory. The first type involves an ancilla coupled to the system with two sequential controlled-not (cnot) gates, and the two-time histories of the system (whose probabilities yield the correlation function) are determined in a single final time measurement of the ancilla. It is noninvasive for special choices of initial system states and partially invasive for more general choices. Modified Leggett-Garg-type inequalities which accommodate the partial invasiveness are discussed. The quantum picture of the protocol shows that for certain choices of the primary system initial state, the final state is unaffected by the two cnot gate interactions, hence the protocol is undetectable with respect to final system-state measurements, although it is still invasive at intermediate times. This invasiveness can be reduced with different choices of ancilla states and the protocol is then similar in flavor to a weak measurement. The second type of protocol is based on the fact that the behavior of $Q$ over a time interval can be determined from knowledge of the dynamics together with a measurement of certain initial (or final) data. Its quantum version corresponds to the known fact that when sets of histories are decoherent, their probabilities may be expressed in terms of a record projector, hence the two-time histories in which $Q$ has the same or a different sign can be determined by a single projective measurement. The resulting protocol resembles the decay-type protocol proposed by Huelga and collaborators (which is noninvasive but requires a stationarity assumption).

DOI
10.1103/physreva.94.052131

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We consider two protocols for the measurement of the temporal correlation functions of a dichotomic variable QQQ appearing in Leggett-Garg-type inequalities. The protocols measure solely whether QQQ has the same or a different sign at the end of a given time interval, thereby measuring no more than is required for determination of the correlation function. They are inspired, in part, by a decoherent histories analysis of the two-time histories of QQQ, which yields a number of useful insights, although the protocols are ultimately expressed in macrorealistic form independent of quantum theory. The first type involves an ancilla coupled to the system with two sequential controlled-not (cnot) gates, and the two-time histories of the system (whose probabilities yield the correlation function) are determined in a single final time measurement of the ancilla. It is noninvasive for special choices of initial system states and partially invasive for more general choices. Modified Leggett-Garg-type inequalities which accommodate the partial invasiveness are discussed. The quantum picture of the protocol shows that for certain choices of the primary system initial state, the final state is unaffected by the two cnot gate interactions, hence the protocol is undetectable with respect to final system-state measurements, although it is still invasive at intermediate times. This invasiveness can be reduced with different choices of ancilla states and the protocol is then similar in flavor to a weak measurement. The second type of protocol is based on the fact that the behavior of QQQ over a time interval can be determined from knowledge of the dynamics together with a measurement of certain initial (or final) data. Its quantum version corresponds to the known fact that when sets of histories are decoherent, their probabilities may be expressed in terms of a record projector, hence the two-time histories in which QQQ has the same or a different sign can be determined by a single projective measurement. The resulting protocol resembles the decay-type protocol proposed by Huelga and collaborators (which is noninvasive but requires a stationarity assumption).
In the Leggett-Garg approach to testing macrorealism, the two-time correlation functions, which are normally obtained by sequential measurements of a dichotomic variable QQQ, need to be measured in a noninvasive way in order to exclude certain types of alternative classical explanation. Here, it is shown, for a class of macrorealistic theories, that the correlation functions are readily expressed in terms of a time integral of the velocity corresponding to QQQ and that this expression can be determined from a single final-time measurement of an auxiliary system in continual weak interaction with the primary system. The protocol has the form of a ``waiting detector,'' which clicks only when QQQ changes sign. It shares features with both ideal negative measurements and weak measurements and we argue that it is essentially noninvasive, under certain reasonable assumptions. We show that the noninvasiveness persists to a quantum model of the process.
In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable QQQ must be measured in a noninvasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements in which a null result is argued to be a noninvasive determination of the system's state are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions, which is argued to be noninvasive from both classical and quantum perspectives and hence the noninvasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions, and a number of higher-order conditions for macrorealism are derived.
If the time evolution of a quantum system can be understood classically, then there must exist an underlying probability distribution for the variables describing the system at a sequence of times. It is well known that for systems described by a single time-evolving dichotomic variable QQQ and for which a given set of temporal correlation functions are specified, a necessary set of conditions for the existence of such a probability are provided by the Leggett-Garg (LG) inequalities. Fine's theorem in this context is the nontrivial result that a suitably augmented set of LG inequalities are both necessary and sufficient conditions for the existence of an underlying probability. We present a proof of Fine's theorem for the case of measurements on a dichotomic variable at an arbitrary number of times, thereby generalizing the familiar proofs for three and four times. We demonstrate how the LG framework and Fine's theorem can be extended to the case in which all possible two-time correlation functions are measured (instead of the partial set of two-time correlators normally studied). We examine the limit of a large number of measurements for both of the above cases.
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.
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 (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable QQQ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely, energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three, and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining QQQ is smeared to reflect experimental imprecision. For higher-energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a QQQ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a Gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.