The Two-state Vector Formalism

The two-state vector formalism (TSVF) [1] is a time-symmetric description of the standard quantum mechanics originated in Aharonov, Bergmann and Lebowitz [2]. The TSVF describes a quantum system at a particular time by two quantum states: the usual one, evolving forward in time, defined by the results of a complete measurement at the earlier time, and by the quantum state evolving backward in time, defined by the results of a complete measurement at a later time. According to the standard quantum formalism, an ideal (von Neumann) measurement at time t of a nondegenerate variable A tests for existence at this time of the forward evolving state |A = a (it yields the outcome A = a with certainty if this was the state) and creates the state evolving towards the future: |Ψ(t ′) = e − i ¯ h t ′ t Hdt |A = a, t ′ > t. (1) (In general, the Hamiltonians H(t) at different times do not commute and a time ordering has to be performed.) In the TSVF this ideal measurement also tests for backward evolving state arriving from the future A = a| and creates the state evolving towards the past: Φ(t ′′)| = A = a|e i ¯ h t ′′ t Hdt , t ′′ < t. (2) Apart from some differences (discussed below) following from the asymmetry of the memory arrow of time, one can perform similar manipulations of the forward and backward evolving states. In particular, neither can be cloned and both can be teleported.