The Two-State Vector Formalism of Quantum Mechanics

Abstract

We present the two-state vector formalism of quantum mechanics. It is a timesymmetrized 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. Descriptions of Quantum Systems The Quantum StateIn 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.

Measurement and collapse within the two-state vector formalism

“…However, we can also arrive at Eq. 2 from another direction [13,14]. Notice that if the final state is one of the eigenstates, Ψ f = a k , then Eq.…”

“…However, adding another constraint, namely, the final (backwardevolving) state-vector, results in a more complete description of the quantum system in between these two boundary conditions, that has bearings on the determination of measurement outcomes. The usefulness of the backward-evolving state-vector was demonstrated in the works of Aharonov et al [8,9,10,11,12,13,14,15,16,17,18,19,20,21].…”

Quantum-mechanical realization of a Popescu-Rohrlich box

“…Now let us examine the maximal bound on the CHSH inequality for the third causality satisfying class -the swapped class (3). Here the CHSH inequality may only saturate the maximal bound of 4.…”

Weak Values and Quantum Nonlocality

“…Such a final state may arise due to a post-selection, that is, performing an additional measurement on the system and considering only the cases with the desired outcome. Since then, the time-symmetric formalism was further generalized (see for instance [5,6]) and was shown to be very helpful for understanding conceptual ideas in quantum mechanics, such as the past of the quantum particle [7], the measurement problem [8] and more.…”