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

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).