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Quantum mechanics as an operationally time symmetric probabilistic theory

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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.