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arXiv:0907.5437

Order Symmetry of Weak Measurements

Lars M. Johansen, Pier A. Mello

Published Jul 30, 2009• 0 citations
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Abstract

Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.

DOI
10.48550/arxiv.0907.5437

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Weak values are usually associated with weak measurements of an observable on a pre- and post-selected ensemble. We show that more generally, weak values are proportional to the correlation between two pointers in a successive measurement. We show that this generalized concept of weak measurements displays a symmetry under reversal of measurement order. We show that the conditions for order symmetry are the same as in classical mechanics. We also find that the imaginary part of the weak value has a counterpart in classical mechanics. This scheme suggests new experimental possibilities.
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Quantum mechanical weak values and their measurement have been a focus of theoretical, experimental, and applied research for more than two decades. The concept of PT symmetry was also introduced into quantum mechanics during this time. This paper defines the notion of a weak value measurement pointer Hamiltonian and establishes equivalences between its Dirac symmetries, its PT symmetries, its eigenvalues, and the associated weak value. The affect of these symmetries upon measurement pointer observables is also identified.
In the weak value formalism of Aharonov et al., the weak value Aw{A}_{w}Aw​ of any observable AAA is generally a complex number. We derive a physical interpretation of its value in terms of the shift in the measurement pointer's mean position and mean momentum. In particular, we show that the mean position shift contains a term jointly proportional to the imaginary part of the weak value and the rate at which the pointer is spreading in space as it enters the measurement interaction.
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the correlations among different pointer degrees of freedom can crucially affect the way the imaginary parts of the weak values are related to the observed shifts of the mean pointer position and momentum. The particular relevance of this analysis is discussed in the case of sequential weak interactions followed by a projective measurement enabling postselection (called sequential WM) which, in the special case, reduces to the usual WM scheme (involving a single weak interaction and postseletion) modified by the effect of pointer-state correlations.
Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.
In this thesis, we develop a probabilistic description of the time-symmetric formulation of quantum mechanics.We derive a generalized form of the probability distribution of a quantum process which is determined by performing pre-and post-selections on a system.It gives the well-known value for an observable associated to a quantum process, called weak value, as an expectation value.We show that the generalized probability satisfies the law of total probability and admits the transitive form by means of conditional probabilities.This motivates us to step forward to a classical ensemble picture in quantum mechanics.In addition, we present the description of quantum measurement in terms of the generalized probability.It is suggested that the difference between the projective measurement and the weak measurement corresponds to the choice of selections of the forward and backward propagating processes. Consistency of the Probability in the Forward and Backward Processes 6 Conclusion and Discussion Bibliography