The Leggett-Garg Inequalities and Linear Positivity: an Alternative to Non-Invasive Measurability
J. J. Halliwell
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.
•Aug 10, 2015•0 citations