  Classical measurements are passive, in the sense that they do not affect the
physical properties of the measured system. Normally, quantum measurements are
not passive in that sense. In the infinite dimensional Hilbert space, however,
we find that quantum projective measurement can be passive in a way which is
impossible in finite dimensional Hilbert spaces. Specifically, we find that
expectation value of a hermitian Hamiltonian can have an imaginary part in the
infinite dimensional Hilbert space and that such an imaginary part implies a
possibility to avoid quantum Zeno effect, which can physically be realized in
quantum arrival experiments. The avoidance of quantum Zeno effect can also be
understood as avoidance of a quantum version of gambler's fallacy, leading to
the notion of passive quantum measurement that updates information about the
physical system without affecting its physical properties. The arrival time
probability distribution of a particle is found to be given by the flux of the
probability current. Possible negative fluxes correspond to regimes at which
there is no arrival at all, physically understood as regimes at which the
particle departs rather than arrives.
