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arXiv:0711.4149 · Physical Review A

Test of weak measurement on a two- or three-qubit computer

Todd A. Brun, Lajos Diósi, Walter T. Strunz

Published Mar 10, 2008• 22 citations
arXivHTMLPDF

Abstract

Current quantum computer technology is sufficient to realize weak measurements and the corresponding concept of ``weak values.'' We demonstrate how the weak value anomaly can be tested, along with consistency and simultaneity of weak values, using only discrete degrees of freedom. All that is needed is a quantum computer with two---or better, three---qubits. We also give an interpretation of the weak value as an effective field strength in a post-selected spin measurement.

DOI
10.1103/physreva.77.032101

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Current quantum computer technology is sufficient to realize weak measurements and the corresponding concept of ``weak values.'' We demonstrate how the weak value anomaly can be tested, along with consistency and simultaneity of weak values, using only discrete degrees of freedom. All that is needed is a quantum computer with two---or better, three---qubits. We also give an interpretation of the weak value as an effective field strength in a post-selected spin measurement.
Historically, weak values have been associated with weak measurements performed on quantum systems. Over the past two decades, a series of works have shown that weak values can be determined via measurements of arbitrary strength. One such proposal by Denkmayr et al. [Phys. Rev. Lett. 118, 010402 (2017)], carried out in neutron interferometry experiments, yielded better outcomes for strong than for weak measurements. We extend this scheme and explain how to implement it in an optical setting as well as in a quantum computational context. Our implementation in a quantum computing system provided by IBM confirms that weak values can be measured, with varying degrees of performance, over a range of measurement strengths. However, at least for this model, strong measurements do not always perform better than weak ones.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
Weak values are conventionally extracted in the limit of vanishing measurement strength. At finite strength, the corresponding post-selected conditional averages generally acquire nonlinear corrections and need not retain their weak value interpretation. Here we identify a class of measurement protocols for which this expectation fails: the conditional value reconstructed from the measuring probe is exactly independent of the probe strength and therefore coincides with the real part of the weak value. We first analyze this phenomenon in the measurement--disturbance protocol introduced by Lund and Wiseman. We then establish conditions under which this invariance persists, showing that it is determined by the interplay between the probe interaction, a subsequent measurement interaction with an ancillary apparatus, and the final post-selection. These results demonstrate that weak coupling, although generally sufficient for accessing weak values, is not always necessary.
Many quantum paradoxes based on a realistic view of weak values were discussed in the last decades. They lead to astonishing conclusions such as the measurement of a spin component of a spin-1/2 particle resulting in $100\ensuremath{\hbar}$, the separation of a photon from its polarization, and the possibility of having three particles in two boxes without any two particles being in the same box, among others. Here, we show that the realistic view of the weak values present in these (and other) works is equivalent to a realistic (and highly controversial) view of quantum measurements, where a measurement reveals the underlying reality of the measured quantity. We discuss that all quantum paradoxes based on weak values simply disappear if we deny these realistic views of quantum measurements and weak values. Our work thus aims to demonstrate the strong assumptions and the corresponding problems present in the interpretation of these quantum paradoxes.
The concept of a \emph{weak value} of a quantum observable was developed in the late 1980s by Aharonov and colleagues to characterize the value of an observable for a quantum system in the time interval between two projective measurements. Curiously, these values often lie outside the eigenspectrum of the observable, and can even be complex-valued. Nevertheless, the weak value of a quantum observable has been shown to be a valuable resource in quantum metrology, and has received recent attention in foundational aspects of quantum mechanics. This paper is driven by a desire to more fully understand the underlying mathematical structure of weak values. In order to do this, we allow an observable to be \emph{any} Hermitian operator, and use the pre- and post-selected states to develop well-defined linear maps between the Hermitian operators and their corresponding weak values. We may then use the inherent Euclidean structure on Hermitian space to geometrically decompose a weak value of an observable. In the case in which the quantum systems are qubits, we provide a full geometric characterization of weak values.