Two-state vector formalism

Theory

The two-state vector formalism is one example of a time-symmetric interpretation of quantum mechanics (see Interpretations of quantum mechanics). Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921, and later by several other scientists. The two-state vector formalism was first developed by Satosi Watanabe in 1955, who named it the Double Inferential state-Vector Formalism (DIVF). Watanabe proposed that information given by forwards evolving quantum states is not complete; rather, both forwards and backwards evolving quantum states are required to describe a quantum state: a first state vector that evolves from the initial conditions towards the future, and a second state vector that evolves backwards in time from future boundary conditions. Past and future measurements, taken together, provide complete information about a quantum system. Watanabe's work was later rediscovered by Yakir Aharonov, Peter Bergmann and Joel Lebowitz in 1964, who later renamed it the Two-State Vector Formalism (TSVF). Conventional prediction, as well as retrodiction, can be obtained formally by separating out the initial conditions (or, conversely, the final conditions) by performing sequences of coherence-destroying operations, thereby cancelling out the influence of the two state vectors.

The two-state vector is represented by:

${\displaystyle ⟨\mathrm{\Phi }||\mathrm{\Psi }⟩}$

where the state ${\displaystyle ⟨\mathrm{\Phi }|}$ evolves backwards from the future and the state ${\displaystyle |\mathrm{\Psi }⟩}$ evolves forwards from the past.

In the example of the double-slit experiment, the first state vector evolves from the electron leaving its source, the second state vector evolves backwards from the final location of the electron on the detection screen, and the combination of forwards and backwards evolving state vectors determines what occurs when the electron passes the slits.

The two-state vector formalism provides a time-symmetric description of quantum mechanics, and is constructed such as to be time-reversal invariant. It can be employed in particular for analyzing pre- and post-selected quantum systems. Building on the notion of two-state, Reznik and Aharonov constructed a time-symmetric formulation of quantum mechanics that encompasses probabilistic observables as well as nonprobabilistic weak observables.