Generalized modular-value-based scheme and its generalized modular value

Abstract

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 postselection onto the system, the results of the pointer measurement are so-called the 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 spins 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.

Quantum weak and modular values in enlarged Hilbert spaces

“…Another concept associated with the pre-and postselections is the modular value, which was first proposed by Kedem and Vaidman [5], and was recently studied by us [6,42,43]. For example, we have shown that, by using the spectral decomposition, a modular value can be expressed regarding weak values [42], as…”

Extended validity of weak measurement*

“…In this paper, we analyze the role of the mean value of the meter's input variable in such a process. In order to settle this doubt, we present a modified formulas for the pointer shift, coming from the modified "standard" procedure, and a modified weak value is meanwhile introduced, related to the mean value of meter's input variable and interaction constant, as there are also other theory about the modular value [50][51][52] and general approaches. [53,54] By the modified formulas, the relationship among the pointer shift, the mean value of the meter's input variable and the validity conditions of weak measurement is clarified.…”

Generalized modular values with non-classical pointer states

“…Although their scheme can solve the above problem, this kind of one-qubit pointer state is a limitation for further use of modular values in practical issues, such as state tomography and non-local measurement. In recent works, N. Imoto et al [13,14] overcame that limitation by introducing the generalization modular value scheme; in their model, the pointer is not a qubit but a qudit. In this multilevel pointer system, the resulting value is called a generalized modular value.…”