The Two-State Vector Formalism of Quantum Mechanics

Abstract

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. 13.1 Descriptions of Quantum Systems 13.1.1 The Quantum State In the standard quantum mechanics, a system at a given time t is described completely by a quantum state |Ψ , (13.1) defined by the results of measurements performed on the system in the past relative to the time t. (It might be that the system at time t is not described by a pure quantum state, but by a mixed state (density matrix). However, we can always assume that there is a composite system including this system which is in a pure state.) The status of a quantum state is controversial: there are many papers on reality of a quantum state and numerous interpretations of this "reality". However, it is non-controversial to say that the quantum state yields maximal information about how this system can affect other systems (in particular, measuring devices) interacting with it at time t. Of course, the results of all measurements in the past, or just the results of the last complete measurement, also have this information, but these results include other facts too, so the quantum state is the most concise information about how the quantum system can affect other systems at time t.