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arXiv:0712.0320 · Physical Review A

Multiple-time states and multiple-time measurements in quantum mechanics

Yakir Aharonov, Sandu Popescu, Jeff Tollaksen, Lev Vaidman A

Published May 12, 2009• 104 citations
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Abstract

We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to an alternative approach to quantum mechanics. In particular, we introduce the idea of multitime quantum states which are the appropriate tools for describing these experimental situations. We also describe multitime measurements and discuss their relation to multitime states. A consequence of our formalism is to put states and operators on an equal footing. Finally we discuss the implications of our approach to quantum mechanics for the problem of the flow of time.

DOI
10.1103/physreva.79.052110

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We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to an alternative approach to quantum mechanics. In particular, we introduce the idea of multitime quantum states which are the appropriate tools for describing these experimental situations. We also describe multitime measurements and discuss their relation to multitime states. A consequence of our formalism is to put states and operators on an equal footing. Finally we discuss the implications of our approach to quantum mechanics for the problem of the flow of time.
In quantum mechanics time usually appears as classical parameter which means that it is treated as being essentially different from spatial coordinates that are represented by operators. On the other hand, relativity theory demands to treat space and time on an equal footing and there are many approaches to the problem how to introduce time operator. In order to support these approaches we propose the experiment that will clearly expose the equivalence of space and time regarding the relation towards quantum interference. The proposed experiment can show that it is possible to have superposition of being created in two different moments in time, and by measuring standard interference pattern, it will be shown that there is interference in time. It is similar to the case of verification of the possibility of single quantum system to be in the state that is superposition of two different positions in space. If so, this would mean that there should be exactly the same formal approach to the space coordinates and time in quantum mechanics since they are all quantum in their nature.
Our formalism for describing quantum measurements is generalized to the multitime case and used to derive rigorous time-energy uncertainty relations. For a free particle, we find that $T\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}$, where $\ensuremath{\delta}H$ is the dispersion in energy and TTT is the measurement duration as given by an external clock of arbitrarily high accuracy. Moreover, any system used as a clock obeys $\ensuremath{\delta}t\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}$, where $\ensuremath{\delta}t$ is the dispersion in the values of time as measured by the system.
The one-time and two-time measurements are studied by employing the idea of Aharonov et al. that the strength of a measurement depends on the state of the measuring device in the context of von Neumann measurement theory. A measurement is classified as either a strong or a weak measurement depending on whether the interaction between the system and the measuring device is infinitely large or negligibly small. A single strong measurement yields one of the eigenvalues of the operator while a single weak measurement, in addition to the eigenvalues, has infinite other possible outcomes. Nevertheless, the expectation values of the strong and the weak one-time measurements are the same. However, the expectation values of the two-time measurements depend on the strength of the first measurement. For the strong case the result is consistent with the conventional result without having to introduce the state collapsing. For the weak case the result is the real part of the expectation value of the time-ordered two-time operator. The fact that this expectation value cannot be factorized indicates that the entanglement between the system and the measuring device introduced by a weak measurement cannot be neglected.
Circumstances are described wherein no state at a given time (nor any definite evolution from one time to another) can be ascribed to a given physical system, but wherein the system can nonetheless be associated with definite dispersion-free values of a new sort of observable, which we call a "multiple-time" observable. The description of physical systems in terms of these new observables is discussed. It emerges as a by-product of our work that no experiment whatever (albeit that its result is certain) can be carried out on a system without disturbing the values of other measurable quantities.
Time – what it is and its nature – has from the beginning held mysteries for layman, philosopher, and physicist alike. Periodic phenomena from motions of heavenly bodies and pendulums to quantal atomic clocks provide the means to measure time intervals, and Newton’s absolute time is a background flow against which classical and non-relativistic quantum physics is played out. While internally consistent, these two fields of physics are incompatible with the requirement of treating space and time on an equal footing. Quantum field theories provide the answer, as both treated as parameters, physics residing instead in the states and operators. But even in classical physics and quantum mechanics, time-independent descriptions provide an alternative to an explicitly time-dependent one. These, including a general method of approach called invariant imbedding, are discussed as alternative representations of an underlying reality that is the concern of physics.
Entangled states provide the necessary tools for conceptual tests of quantum mechanics and other alternative theories. These tests include local hidden variables theories, pre- and post-selective quantum mechanics, QND measurements, complementarity, and tests of mechanics itself against, e.g. the so-called causal communication constraint. We show how to produce nonlocal entangled states of multiple cavity fields. We first discuss two-cavity entanglement, namely the generation of the Bell basis states for this system, when there is at most one photon in either of the cavities. We then discuss a straightforward generalization to similar N-cavity states. These states can be produced by sending appropriately prepared atoms through the cavities. As applications we briefly review a proposal to test quantum mechanics against the causal communication constraint using the two-cavity state and a test of pre- and post-selective quantum mechanics using a three-cavity state and the close connection of the latter to the consistent histories interpretation. Finally we point out the relation between these schemes and the schemes suggested for quantum computing, teleportation and quantum copying.