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- Quantum mechanics as an operationally time symmetric probabilistic theory

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13110057

The standard formulation of quantum mechanics is

operationally asymmetric with respect to time reversal---in the language of

compositions of tests, tests in the past can influence the outcomes of test in

the future but not the other way around. The question of whether this represents

a fundamental asymmetry or it is an artifact of the formulation is not a new

one, but even though various arguments in favor of an inherent symmetry have

been made, no complete time-symmetric formulation expressed in rigorous

operational terms has been proposed. Here, we discuss such a possible

formulation based on a generalization of the usual notion of test. We propose

to regard as a test any set of events between an input and an output system

which can be obtained by an autonomously defined laboratory procedure. This

includes standard tests, as well as proper subsets of the complete set of

outcomes of standard tests, whose realization may require post-selection in

addition to pre-selection. In this approach, tests are not expected to be

operations that are up to the choices of agents---the theory simply says what

circuits of tests may occur and what the probabilities for their outcomes would

be, given that they occur. By virtue of the definition of test, the

probabilities for the outcomes of past tests can depend on tests that take

place in the future.

Such theories have been previously called non-causal, but

here we revisit that notion of causality. Using the Choi-Jamiolkowski

isomorphism, every test in that formulation, commonly regarded as inducing

transformations from an input to an output system, becomes equivalent to a

passive detection measurement applied jointly on two input systems---one from

the past and one from the future. This is closely related to the two-state

vector formalism, but it comes with a conceptual revision: every measurement is

a joint measurement on two separate systems and not on one system described by

states in the usual Hilbert space and its dual. We thus obtain a static picture

of quantum mechanics in space-time or more general structures, in which every

experiment is a local measurement on a global quantum state that generalizes

the recently proposed quantum process matrix. The existence of two types of

systems in the proposed formalism allows us to define causation in terms of

correlations without invoking the idea of intervention, offering a possible

answer to the problem of the meaning of causation. The framework is naturally

compatible with closed time-like curves and other exotic causal structures.

©2012 Institut Périmètre de Physique Théorique