  We investigate the statistical arrow of time for a quantum system being
monitored by a sequence of measurements. For a continuous qubit measurement
example, we demonstrate that time-reversed evolution is always physically
possible, provided that the measurement record is also negated. Despite this
restoration of dynamical reversibility, a statistical arrow of time emerges,
and may be quantified by the log-likelihood difference between forward and
backward propagation hypotheses. We then show that such reversibility is a
universal feature of non-projective measurements, with forward or backward
Janus measurement sequences that are time-reversed inverses of each other.
