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Physical Review A

Quantum weak and modular values in enlarged Hilbert spaces

Le Bin Ho, Nobuyuki Imoto

Published Jan 16, 2018• 9 citations
Publisher

Abstract

We introduce an enlarged state, which combines both pre- and postselection states at a given time $t$ 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.

DOI
10.1103/physreva.97.012112

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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.
We present a general formalism for characterizing 2-time quantum states, describing pre- and postselected quantum systems. The most general 2-time state is characterized by a ``density vector'' that is independent of measurements performed between the preparation and postselection. We provide a method for performing tomography of an unknown 2-time density vector. This procedure, which cannot be implemented by weak or projective measurements, brings new insight to the fundamental role played by Kraus operators in quantum measurements. Finally, after showing that general states and measurements are isomorphic, we show that any measurement on a 2-time state can be mapped to a measurement on a preselected bipartite state.
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.
It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.
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.
Weak values have been shown to be helpful especially when considering them as the outcomes of weak measurements. In this paper we show that, in principle, the real and imaginary parts of the weak value of any operator may be elucidated from expectation values of suitably defined density, flux, and Hermitian commutator operators. Expectation values are the outcomes of strong (projective) measurements, implying that weak values are general properties of operators in association with pre- and postselection and they need not be preferentially associated with weak measurements. They should be considered as an important measurable property which provides added information compared with the ``standard'' diagonal expectation value of an operator. As the first specific example we consider the determination of the real and imaginary parts of the weak value of the momentum operator employing projective time-of-flight experiments. Then the results are analyzed from the point of view of Bohmian mechanics. Finally, we consider recent neutron interferometry experiments used to determine the weak values of the neutron spin.