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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Exact pointer theories for von Neumann projector measurements of pre- and postselected and preselected-only quantum systems: statistical mixtures and weak value persistence

Allen Parks, Scott E. Spence, John Edward Gray

Published Dec 18, 2013• 10 citations
Publisher

Abstract

Abstract Exact expressions for the mean pointer position, the mean pointer momentum and their variances are obtained for projection operator measurements performed upon ensembles of pre- and postselected (PPS) and preselected-only (PSO) quantum systems. These expressions are valid for any interaction strength which couples a measurement pointer to a quantum system, and consequently should be of general interest to both experimentalists and theoreticians. To account for the ‘collapse’ of PPS states to PSO states that occurs as interaction strength increases and to introduce the concept of ‘weak value persistence’, the exact PPS and PSO pointer theories are combined to provide a pointer theory for statistical mixtures. For the purpose of illustrating ‘weak value persistence’, mixture weights defined in terms of the Bhattacharyya coefficient are used and the statistical mixture theory is applied to mean pointer position data associated with weak value projector measurements obtained from a recent dynamical quantum non-locality detection experiment.

DOI
10.1098/rspa.2013.0651

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Physical Review A•Mar 31, 2017•13 citations
Abstract Exact expressions for the mean pointer position, the mean pointer momentum and their variances are obtained for projection operator measurements performed upon ensembles of pre- and postselected (PPS) and preselected-only (PSO) quantum systems. These expressions are valid for any interaction strength which couples a measurement pointer to a quantum system, and consequently should be of general interest to both experimentalists and theoreticians. To account for the ‘collapse’ of PPS states to PSO states that occurs as interaction strength increases and to introduce the concept of ‘weak value persistence’, the exact PPS and PSO pointer theories are combined to provide a pointer theory for statistical mixtures. For the purpose of illustrating ‘weak value persistence’, mixture weights defined in terms of the Bhattacharyya coefficient are used and the statistical mixture theory is applied to mean pointer position data associated with weak value projector measurements obtained from a recent dynamical quantum non-locality detection experiment.
Exact pointer states are obtained for projection operator measurements performed upon pre-selected (PS) and upon pre- and post-selected (PPS) quantum systems. These states are used to provide simple exact expressions for both the pointer spatial probability distribution profiles and the mean values of arbitrary pointer observables associated with PS and PPS projection operator measurements that are valid for any strength of the interaction which couples a measurement pointer to the quantum system. These profiles and mean values are compared in order to identify the effects of post-selection upon projector measurement pointers. As a special case, these mean value results are applied to the weak measurement regime - yielding PS and PPS mean value expressions which are valid for any operator (projector or non-projector). Measurement sensitivities which are useful for estimating weak measurement accuracies for PS and PPS systems are also obtained and discussed.
It is well known that the pointer state |Φ> resulting from the von Neumann measurement of a projection operator A performed upon an ensemble of quantum systems in the preselected state |ψ> depends upon A|ψ>. Here it is shown that the pointer state |Ψ> obtained from such a measurement performed upon an ensemble that is also postselected depends upon the weak value of A - regardless of the measurement interaction strength. It is also found that, while the spatial distribution of |Ψ> exhibits interference, the idempotency of A prohibits interference in that of |Φ>. This is explained in terms of welcher Weg information.
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.
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.
We consider a generalized modular-value-based scheme based on the standard von Neumann measurement. We model the scheme as an interaction between a quantum system and a discrete quantum pointer where the pointer operator is a projection operator onto one of the states of the basis of the pointer Hilbert space. The interaction strength is made arbitrarily large. After post-selection onto the system, the results of the pointer measurement are the so-called conditional probabilities. We first explicitly derive the analytical expressions of the conditional probabilities, the expectation value, and the average displacement in the measured value of a pointer observable that we name as the pointer quantities. We also provide an expression for a generalized modular value and discuss the relationship between the generalized modular value and generalized weak values. The study then shows that the generalized modular value can characterize these pointer quantities. Then we give applications of our proposal to the cases of a spin-sss particle pointer and a semiclassical pointer state. One of the key results is that the amplification effect, similar to the weak-value case, is also observed in the case of the generalized modular value. Our study can also apply to the cases of nonclassical pointer states.