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arXiv:quant-ph/0606208 · Journal of Physics A Mathematical and Theoretical

Backward evolving quantum states

Lev Vaidman A

Published Mar 7, 2007• 10 citations
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Abstract

The basic concept of the two-state vector formalism, which is the time symmetric approach to quantum mechanics, is the backward evolving quantum state. However, due to the time asymmetry of the memory's arrow of time, the possible ways to manipulate a backward evolving quantum state differ from those for a standard, forward evolving quantum state. The similarities and the differences between forward and backward evolving quantum states regarding the no-cloning theorem, nonlocal measurements and teleportation are discussed. The results are relevant not only in the framework of the two-state vector formalism, but also in the framework of retrodictive quantum theory.

DOI
10.1088/1751-8113/40/12/s23

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The basic concept of the two-state vector formalism, which is the time symmetric approach to quantum mechanics, is the backward evolving quantum state. However, due to the time asymmetry of the memory's arrow of time, the possible ways to manipulate a backward evolving quantum state differ from those for a standard, forward evolving quantum state. The similarities and the differences between forward and backward evolving quantum states regarding the no-cloning theorem, nonlocal measurements and teleportation are discussed. The results are relevant not only in the framework of the two-state vector formalism, but also in the framework of retrodictive quantum theory.
Although nondemolition, reliable, and instantaneous quantum measurements of some nonlocal variables are impossible, demolition reliable instantaneous measurements are possible for all variables. It is shown that this is correct also in the framework of the time-symmetric quantum formalism, i.e. nonlocal variables of composite quantum systems with quantum states evolving both forward and backward in time are measurable in a demolition way. The result follows from the possibility to reverse with certainty the time direction of a backward evolving quantum state. Demolition measurements of nonlocal backward evolving quantum states require remarkably small resources. This is so because the combined operation of time reversal and teleportation of a local backward evolving quantum state requires only a single quantum channel and no transmission of classical information.
The two-state-vector formalism and the entangled histories formalism are attempts to better understand quantum correlations in time. Both formalisms share some similarities, but they are not identical, having subtle differences in their interpretation and manipulation of quantum temporal structures. However, the main objective of this paper is to prove that, with appropriately defined scalar products, both formalisms can be made isomorphic. We show, for instance, that they treat operators and states on equal footing, leading to the same statistics for all measurements. In particular, we discuss the topic of quantum correlations in time and show how they can be generated and analyzed in a consistent way using these formalisms. Furthermore, we elaborate on an unconventional behavior of quantum histories of evolving multipartite systems which do not exhibit global nonlocal correlations in time but nevertheless can lead to entangled reduced histories characterizing evolution of an arbitrarily chosen subsystem.
In this paper we present the two-state vector formalism of quantum mechanics. It is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre- and post-selected ensembles. Several peculiar effects which naturally arise in this approach are considered. In particular, the concept of ``weak measurements'' (standard measurements with weakening of the interaction) is discussed in depth revealing a very unusual but consistent picture. Also, a design of a gedanken experiment which implements a kind of quantum ``time machine'' is described. The issue of time-symmetry in the context of the two-state vector formalism is clarified.
In this paper we present the two-state vector formalism of quantum mechanics. It is a time-symmetrized approach to standard quantum theory particularly helpful for the analysis of experiments performed on pre- and post-selected ensembles. Several peculiar effects which naturally arise in this approach are considered. In particular, the concept of ``weak measurements'' (standard measurements with weakening of the interaction) is discussed in depth revealing a very unusual but consistent picture. Also, a design of a gedanken experiment which implements a kind of quantum ``time machine'' is described. The issue of time-symmetry in the context of the two-state vector formalism is clarified.
It is brought forward that viable theories of the physical world that have no variable at all that can play the role of time, do not exist; some notion of time is one of the very first ingredients a candidate theory should possess. Almost by definition, time has an arrow. In contrast, time reversibility, or even the possibility to run the equations of motion backwards in time, is not at all a primary requirement. This means that the direction of the arrow of time may well be uniquely defined in the theory, even locally. It is explained that a rigorous definition of time, as well as a formulation of the causality and locality concepts, can only be given when one has a model for the physical phenomena described. The only viable causality condition is one that is symmetric under time reversal. We explain these statements in terms of the author’s favoured deterministic cellular automaton interpretation of quantum mechanics, also to be referred to as 'vector space analysis', and expand on these ideas. It is also summarised how our more rigorous causality condition affects Bell’s theorems. What distinguishes quantum systems from classical ones is our fundamental inability to control the microscopic details of the initial state when phenomena are studied in the light of some theoretical model.