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arXiv:1602.01594

An interpretation and understanding of complex modular values

Le Bin Ho, Nobuyuki Imoto

Published Feb 4, 2016• 1 citation
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Abstract

In contrast to that a weak value of an observable is usually divided into real and imaginary parts, here we show that separation into modulus and argument is important for modular values. We first show that modular values are expressed by the average of dynamic phase factors with complex conditional probabilities. We then relate, using the polar decomposition, the modulus of the modular value to the relative change in the qubit pointer post-selection probabilities, and relate the argument of the modular value to the summation of a geometric phase and an intrinsic phase.

DOI
10.48550/arxiv.1602.01594

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In contrast to that a weak value of an observable is usually divided into real and imaginary parts, here we show that separation into modulus and argument is important for modular values. We first show that modular values are expressed by the average of dynamic phase factors with complex conditional probabilities. We then relate, using the polar decomposition, the modulus of the modular value to the relative change in the qubit pointer post-selection probabilities, and relate the argument of the modular value to the summation of a geometric phase and an intrinsic phase.
We theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches. We consider both the finite-dimensional discrete pointer state and continuous pointer state cases. In contrast to that, a weak value of an observable is usually divided into its real and imaginary parts; here, we show that separation from the modulus and phase is necessary to a modular value. We show that the modulus of the modular value is related to the pointer post-selection conditional probability, and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase. We also discuss a relationship between the modulus and phase, and therein, the derivative of the phase is related to the derivative of the logarithm of the modulus via a Berry-Simon-like connection which is in the form of a weak value. As a consequence, the modulus-phase relation allows us to obtain these polar components whenever the connection is specified. One of the possible applications of our results is to evaluate the weak value (the Berry-Simon-like connection) when the polar modular value is experimentally obtained.
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.
Abstract We express modular and weak values of observables of three- and higher-level quantum systems in their polar form. The Majorana representation of N -level systems in terms of symmetric states of N − 1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N -level quantum systems can be factored in N − 1 contributions. Their modulus is determined by the product of N − 1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N − 1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three-box paradox (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen . 24 2315–28) from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre- and postselected.
Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.
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.