The Leggett-Garg Inequalities and No-Signalling in Time: A Quasi-Probability Approach

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 quasi-probability linear in the projection operators to describe the properties of the system at two times. We show that this quasi-probability 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-signalling in time condition of Kofler and Brukner. We argue that this quasi-probability, appropriately measured, provides a non-invasive 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 quasi-probability is non-negative, the LG system has a natural parallel with the EPRB system and Fine's theorem. A simple spin model illustrating key features of the approach is exhibited.Comment: 23 pages. Significant revisions. Change of titl

Leggett-Garg tests of macrorealism: checks for non-invasiveness and generalizations to higher-order correlators

In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable Q must be measured in a non-invasive 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 non-invasive 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 non-invasive from both classical and quantum perspectives and hence the non-invasiveness 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.Comment: 21 pages, Late

Two Proofs of Fine's Theorem

Fine's theorem concerns the question of determining the conditions under which a certain set of probabilities for pairs of four bivalent quantities may be taken to be the marginals of an underlying probability distribution. The eight CHSH inequalities are well-known to be necessary conditions, but Fine's theorem is the striking result that they are also a sufficient condition. It has application to the question of finding a local hidden variables theory for measurements of pairs of spins for a system in an EPRB state. Here we present two simple and self-contained proofs of Fine's theorem in which the origins of this non-obvious result can be easily seen. The first is a physically motivated proof which simply notes that this matching problem is solved using a local hidden variables model given by Peres. The second is a straightforward algebraic proof which uses a representation of the probabilities in terms of correlation functions and takes advantage of certain simplifications naturally arising in that representation. A third, unsuccessful attempt at a proof, involving the maximum entropy technique is also briefly describedComment: 17 pages, latex. Revised argument for setting average spins to zero. References added. Corrected figur

Arrival Times in Quantum Theory from an Irreversible Detector Model

We investigate a detector scheme designed to measure the arrival of a particle at $x=0$ during a finite time interval. The detector consists of a two state system which undergoes a transition from one state to the other when the particle crosses $x=0$, and possesses the realistic feature that it is effectively irreversible as a result of being coupled to a large environment. The probabilities for crossing or not crossing $x=0$ thereby derived coincide with earlier phenomenologically proposed expressions involving a complex potential. The probabilities are compared with similar previously proposed expressions involving sums over paths, and a connection with time operator approaches is also indicated.Comment: 19 pages, plain Tex (Fourth revision). To appear in Prog.Th.Phys. Vol. 102, No.

How the Quantum Universe Became Classical

This is an informal introduction to the ideas of decoherence and emergent classicality, including a simple account of the decoherent histories approach to quantum theory. It is aimed at undergraduates with a basic appreciation of quantum theory. The emphasis is on simple physical ideas and pictures.Comment: 24 pages, 11 figure

Classical Limit of the Quantum Zeno Effect by Environmental Decoherence

We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region -- this is the Zeno effect. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which kills interference between histories. We show how this works explicitly by coupling to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally-induced fluctuations in momentum are sufficiently large and we establish the associated timescale. We link our results to earlier work on the hbar -->0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper demonstrating the suppression of quantum-mechanical reflection by decoherenceComment: 45 pages, 8 figure