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arXiv:quant-ph/0402050 · Physical Review Letters

Weak Measurements with Arbitrary Probe States

Lars M. Johansen

Published Sep 14, 2004• 54 citations
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Abstract

The exact conditions on valid probe states for weak measurements are derived. It is demonstrated that weak measurements can be performed with any probe state with vanishing probability current density. This condition is found both for weak measurements of noncommuting observables and for c-number observables. In addition, the interaction between object and probe must be sufficiently weak. Strange weak values can be observed also with mixed probe states, but not for c-number observables.

DOI
10.1103/physrevlett.93.120402

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The exact conditions on valid probe states for weak measurements are derived. It is demonstrated that weak measurements can be performed with any probe state with vanishing probability current density. This condition is found both for weak measurements of noncommuting observables and for c-number observables. In addition, the interaction between object and probe must be sufficiently weak. Strange weak values can be observed also with mixed probe states, but not for c-number observables.
We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation value over an infinite class of different generalized Kirkwood quasiprobability distributions. ``Strange'' weak values for which the real part exceeds the eigenvalue spectrum of the observable can only be found if the Terletsky-Margenau-Hill distribution is negative or, equivalently, if the real part of the weak value of the density operator is negative. We find that a classical model of a weak measurement exists whenever the Terletsky-Margenau-Hill representation of the observable equals the classical representation of the observable and the Terletsky-Margenau-Hill distribution is non-negative. Strange weak values alone are not sufficient to obtain a contradiction with classical models. We propose feasible weak measurements of photon number of the radiation field. Negative weak values of energy contradict all classical stochastic models, whereas negative weak values of photon number contradict all classical stochastic models where the energy is bounded from below by the zero-point energy. We examine coherent states in particular and find negative weak values with probabilities of 16% for kinetic energy (or squared field quadrature), 8% for harmonic oscillator energy, and 50% for photon number. These experiments are robust against detector inefficiency and thermal noise.
Weak measurements with mixed probe states were investigated in the presence of noise in the probe system. We show that the completely mixed state can be used as a probe state for weak measurements of imaginary weak values. The completely mixed probe state has some advantages in terms of noise tolerance. We also experimentally demonstrated weak measurements of polarization rotation via unpolarized light, which corresponds to weak measurements with completely mixed probe states in a two-state system.
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.
Weak measurements allow a probe system to encode the transitionary weak value of a desired observable on a signal system that is both pre- and post-selected. Such interactions alter the state and measurement statistics of the probe system. We establish that by encoding the imaginary part of a weak value onto an initially entangled probe, we can modify its entanglement content, provided the probe observable can distinguish between system states of different von Neumann entropies. Apart from fundamental interest, this result illustrates the utility of the imaginary weak value as a calculational tool in certain entanglement concentration protocols.
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.
Weak values are typically obtained experimentally by performing weak measurements, which involve weak interactions between the measured system and a probe. However, the determination of weak values does not necessarily require weak measurements, and several methods without weak system-probe interactions have been developed previously. In this work, a framework for measuring weak values is proposed to describe the relationship between various weak value measurement techniques in a unified manner. This framework, which uses a probe-controlled system transformation instead of the weak system-probe interaction, improves the understanding of the currently used weak value measurement methods. Furthermore, a diagrammatic representation of the proposed framework is introduced to intuitively identify the complex values obtained in each measurement system. By using this diagram, a new method for measuring weak values with a desired function can be systematically derived. As an example, a scan-free and more efficient direct measurement method of wavefunctions than the conventional techniques using weak measurements is developed.