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arXiv:1511.03923 · Physical Review A

Structural features of sequential weak measurements

Lajos Diósi

Published Jul 7, 2016• 24 citations
arXivHTMLPDF

Abstract

We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without postselection yield the corresponding correlations of the Wigner function, offering direct quantum tomography through the moments of the canonical variables. Correlations in spin-$\frac{1}{2}$ sequential weak measurements coincide with those in strong measurements, they are constrained kinematically, and they are equivalent with single measurements. In sequential WMs with postselection, an anomaly occurs, different from the weak value anomaly of single WMs. In particular, the spread of polarization $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ as measured in double WMs of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ will diverge for certain orthogonal pre- and postselected states.

DOI
10.1103/physreva.94.010103

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We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without postselection yield the corresponding correlations of the Wigner function, offering direct quantum tomography through the moments of the canonical variables. Correlations in spin-12\frac{1}{2}21​ sequential weak measurements coincide with those in strong measurements, they are constrained kinematically, and they are equivalent with single measurements. In sequential WMs with postselection, an anomaly occurs, different from the weak value anomaly of single WMs. In particular, the spread of polarization \ensuremathσ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}\ensuremathσ\ifmmode^\elseˆ\fi​ as measured in double WMs of \ensuremathσ\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}\ensuremathσ\ifmmode^\elseˆ\fi​ will diverge for certain orthogonal pre- and postselected states.
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.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
Quantum measurement can be strong or weak, the former being important in readout and initialization of quantum objects and the latter being useful for monitoring and maneuvering quantum evolution. However, the boundary between weak and strong measurement is unclear. Here we show that a phase transition occurs in sequential quantum measurement, which unambiguously separates the weak and strong measurement by a critical value of measurement strength or duration. We formulate the probability distribution of the output of a sequence of quantum measurements as the Boltzmann distribution of an interacting spin model. The measurement results present phase transitions similar to those in the spin model. In particular the sequential commuting positive operator-valued measurement is mapped to a long-range Ising model, and a projective measurement emerges from sequential weak measurements when the strength or the number of measurements becomes above certain critical values, corresponding to the ferromagnetic phase transition of the spin model. This finding sheds insights on sequential quantum measurement, and also provides the theoretical foundation for constructing projective measurements from sequential weak measurements, which have applications in steering the quantum evolution and initializing quantum systems where strong measurement in a single shot is often not possible.
We consider weak measurements (WMs) of a spin observable in quantum mechanics beyond the usual asymptotic regime. This is done by obtaining the exact time-dependent wave functions of the measuring apparatus for general nonideal measurements. Ideal strong measurements and the usual WM regime are obtained as the two extreme limiting cases of our exact treatment. We show that in the intermediate regime nonideal measurements lead to ``semiweak'' eccentric values that differ from the usual weak values. We further show that even in the WM regime the exact treatment leads to a meter behavior that can be markedly distinct from the one predicted by the usual WM formalism. We give several illustrations and discuss an application to the distinction of different realizations of the same density matrix.