History operators in quantum mechanics

It is convenient to describe a quantum system at all times by means of a "history operator" C, encoding measurements and unitary time evolution between measurements. These operators naturally arise when computing the probability of measurement sequences, and generalize the "sum over position histories" of the Feynman path-integral. As we argue in this work, this description has some computational advantages over the usual state vector description, and may help to clarify some issues regarding nonlocality of quantum correlations and collapse. A measurement on a system described by C modifies the history operator, C→PC, where P is the projector corresponding to the measurement. We refer to this modification as "history operator collapse". Thus, C keeps track of the succession of measurements on a system, and contains all histories compatible with the results of these measurements. The collapse modifies the history content of C, and therefore modifies also the past (relative to the measurement), but never in a way to violate causality. Probabilities of outcomes are obtained as Tr(C†PC)/Tr(C†C). A similar formula yields probabilities for intermediate measurements, and reproduces the result of the two-vector formalism in the case of the given initial and final states. We apply the history operator formalism to a few examples: entangler circuit, Mach-Zehnder interferometer, teleportation circuit and three-box experiment. Not surprisingly, the propagation of coordinate eigenstates |q) is described by a history operator C containing the Feynman path-integral.