Two-state vector formalism
The two-state vector formalism (TSVF) is a description of quantum mechanics in terms of a causal relation in which the present is caused by quantum states of the past and of the future taken in combination.
The two-state vector formalism is one example of a time-symmetric interpretation of quantum mechanics (see Minority 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:
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