  Here we present the time-bidirectional state formalism (TBSF) unifying in a
general manner the standard quantum mechanical formalism with no postselection
and the time-symmetrized two-state (density) vector formalism, which deals with
postselected states. In the proposed approach, a quantum particle's state,
called a time-bidirectional state, is equivalent to a joined state of two
particles propagating in opposite time directions. For a general
time-bidirectional state, we derive outcome probabilities of generalized
measurements, as well as mean and weak values of Hermitian observables. We also
show how the obtained expressions reduce to known ones in the special cases of
no postselection and generalized two-state (density) vectors. Then we develop
tomography protocols based on mutually unbiased bases and a symmetric
informationally complete positive operator-valued measure, allowing
experimental reconstruction of an unknown single qubit time-bidirectional
state. Finally, we employ the developed techniques for tracking of a qubit's
time-reversal journey in a quantum teleportation protocol realized with a
cloud-accessible noisy superconducting quantum processor. The obtained results
justify an existence of a postselection-induced qubit's proper time-arrow,
which is different from the time-arrow of a classical observer, and demonstrate
capabilities of the TBSF for exploring quantum phenomena brought forth by a
postselection in the presence of noise.
