  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 the present note, 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, $C \rightarrow PC$, 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 $Tr(C^\dagger P
C)/Tr(C^\dagger C)$. A similar formula yields probabilities for intermediate
measurements, and reproduces the result of the two-vector formalism in the case
of given initial and final states. We apply the history operator formalism to a
few examples: entangler circuit, Mach-Zehnder interferometer, teleportation
circuit, three-box experiment. Not surprisingly, the propagation of coordinate
eigenstates $|q\rangle$ is described by a history operator $C$ containing the
Feynman path-integral.
