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Deccoherent 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 different sign at the ends of a given time interval. They are inspired, in part, by a decoherent histories analysis of the two-time histories of Q 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 CNOT gates, and the two-time histories of the system are determined in a single final time measurement of the ancilla. It is non-invasive 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 primary system initial state 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 flavour to a weak measurement. The second type of protocol is based on the fact that the behaviour 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 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 non-invasive but requires a stationarity assumption).