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arXiv:1007.5281 · Physical Review Letters

Modular Values and Weak Values of Quantum Observables

Yaron Kedem, Lev Vaidman A

Published Nov 30, 2010• 72 citations
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Abstract

The concept of a modular value of an observable of a pre- and postselected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multiqubit meter provides an explanation of some current experiments.

DOI
10.1103/physrevlett.105.230401

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The concept of a modular value of an observable of a pre- and postselected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multiqubit meter provides an explanation of some current experiments.
Abstract Modular value of an observable of a pre- and postselected quantum system is a concept similar to weak value, but in contrast to describing only weak coupling, it can describe the coupling of any strength but only for qubit meters, so it may be used more widely in some scenarios. Besides, modular value has been proved to have advantages on measuring weak values of nonlocal joint observables and nonlocal entanglement states. In this paper, we extend the work of J. S. Lundeen and K. J. Reschb on connection between the measured values of multiparticle observables of a pre- and postselected quantum system and annihilation operator by modular value and give their relation. Based on the above, we propose a new method different from tomography to measure modular values, and we find that our annihilation operator measurement (AOM) method can significantly reduce the number of measurements from O (4 n ) to O (2 n ) compared with the original tomography method.
We introduce a novel concept which we call as potent value of system observable for pre- and post-selected quantum states. This describes, in general, how a quantum system affects the state of the apparatus during the time between two strong measurements corresponding to pre- and post-selections. The potent value can be realized for any interaction strength and for arbitrary coupling between the system and the apparatus observables. Most importantly, potent values generalize and unify the notion of the weak values and modular values of observables in quantum theory. Furthermore, we define a potent operator which describes the action of one system on the another and show that superposition of time-evolutions and time-translation machines are potent operators. These concepts may find useful applications in quantum information processing and can lead to technological benefits.
We introduce an enlarged state, which combines both pre- and postselection states at a given time ttt in between the pre- and postselection. Based on this form, quantum weak and modular values can be completely interpreted as expectation values of a linear combination of given operators in the enlarged Hilbert space. This formalism thus enables us to describe and measure the weak and modular values at any time dynamically. A protocol for implementing an enlarged Hamiltonian has also been proposed and applied to a simple example of a single spin under an external magnetic field. In addition, the time-dependent weak and modular values for pre- and postselection density matrices mapping onto an enlarged density matrix are also discussed.
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.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.
We present a procedure to completely determine the complex modular values of arbitrary observables of pre- and postselected ensembles, which works experimentally for all measurement strengths and all postselected states. This procedure allows us to discuss the physics of modular and weak values in interferometric experiments involving a qubit meter. We determine both the modulus and the argument of the modular value for any measurement strength in a single step, by simultaneously controlling the visibility and the phase in a quantum eraser interference experiment. Modular and weak values are closely related. Using entangled qubits for the probed and meter systems, we show that the phase of the modular and weak values has a topological origin. This phase is completely defined by the intrinsic physical properties of the probed system and its time evolution. The physical significance of this phase can thus be used to evaluate the quantumness of weak values.