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arXiv:2102.06561 · Physical Review Research

Complex counterpart of variance in quantum measurements for pre- and postselected systems

Kazuhisa Ogawa, Natsuki Abe, H. Kobayashi, Akihisa Tomita

Published Jul 21, 2021• 5 citations
arXivHTMLPDF

Abstract

The variance of an observable in a preselected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and postselected systems, we formulate a complex-valued counterpart of the variance called ``weak variance.'' In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and postselected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and postselected systems.

DOI
10.1103/physrevresearch.3.033077

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The variance of an observable in a preselected quantum system, which is always real and non-negative, appears as an increase in the probe wave packet width in indirect measurements. Extending this framework to pre- and postselected systems, we formulate a complex-valued counterpart of the variance called ``weak variance.'' In our formulation, the real and imaginary parts of the weak variance appear as changes in the probe wave packet width in the vertical-horizontal and diagonal-antidiagonal directions, respectively, on the quadrature phase plane. Using an optical system, we experimentally demonstrate these changes in the probe wave packet width caused by the real negative and purely imaginary weak variances. Furthermore, we show that the weak variance can be expressed as the variance of the weak-valued probability distribution in pre- and postselected systems. These operational and statistical interpretations support the rationality of formulating the weak variance as a complex counterpart of the variance in pre- and postselected systems.
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.
The contribution of the detector dynamics to the weak measurement is analyzed. According to the usual theory [Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)] the outcome of a weak measurement with preselection and postselection can be expressed as the real part of a complex number: the weak value. By accounting for the Hamiltonian evolution of the detector, here we find that there is a contribution proportional to the imaginary part of the weak value to the outcome of the weak measurement. This is due to the coherence of the probe being essential for the concept of complex weak value to be meaningful. As a particular example, we consider the measurement of a spin component and find that the contribution of the imaginary part of the weak value is sizable.
Weak values are the fundamental values for observables in a pre- and postselected system. Weak values are typically measured by weak measurement, in which weak values appear in the change of not the pre- and postselected system but the probe system. This indirect characteristic of weak measurement obscures the meaning of weak values for the pre- and postselected system, in contrast to conventional physical quantities, which have a clear operational meaning. In this study, we operationally formulate weak values as the sensitivity of postselection probability amplitude to small transformation in a pre- and postselected system. This formulation of weak values, which is free from the concept of probe shift assumed in weak measurement, gives a direct interpretation of strange weak values for the pre- and postselected system. Furthermore, this formulation can simplify weak-value measurement experiments because the associated measurements appearing in the modulus change of the postselection probability amplitude do not require external degrees of freedom.
Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.
Weak values are average quantities, therefore investigating their associated variance is crucial in understanding their place in quantum mechanics. We develop the concept of a position-postselected weak variance of momentum as cohesively as possible, building primarily on material from Moyal (Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, Cambridge, 1949 ) and Sonego (Found Phys 21(10):1135, 1991 ) . The weak variance is defined in terms of the Wigner function, using a standard construction from probability theory. We show this corresponds to a measurable quantity, which is not itself a weak value. It also leads naturally to a connection between the imaginary part of the weak value of momentum and the quantum potential. We study how the negativity of the Wigner function causes negative weak variances, and the implications this has on a class of ‘subquantum’ theories. We also discuss the role of weak variances in studying determinism, deriving the classical limit from a variational principle.
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.