Measuring the quantum state of a single system with minimum state disturbance

Abstract

Conventionally, unknown quantum states are characterized using quantum-state tomography based on strong or weak measurements carried out on an ensemble of identically prepared systems. By contrast, the use of protective measurements offers the possibility of determining quantum states from a series of weak, long measurements performed on a single system. Because the fidelity of a protectively measured quantum state is determined by the amount of state disturbance incurred during each protective measurement, it is crucial that the initial quantum state of the system is disturbed as little as possible. Here we show how to systematically minimize the state disturbance in the course of a protective measurement, thus enabling the maximization of the fidelity of the quantum-state measurement. Our approach is based on a careful tuning of the time dependence of the measurement interaction and is shown to be dramatically more effective in reducing the state disturbance than the previously considered strategy of weakening the measurement strength and increasing the measurement time. We describe a method for designing the measurement interaction such that the state disturbance exhibits polynomial decay to arbitrary order in the inverse measurement time 1/T . We also show how one can achieve even faster, subexponential decay, and we find that it represents the smallest possible state disturbance in a protective measurement. In this way, our results show how to optimally measure the state of a single quantum system using protective measurements.Journal reference: Phys. Rev. A 93, 012115 (2016),

Determining the quantum expectation value by measuring a single photon

“…(f)) that all the photons accumulate around a specific position, corresponding to x = a P : as stated in the main text, this means that each single photon carries information about the expectation value of its polarisation, granting the possibility of extracting such value even in a one-shot experiment with just a single photon. The expectation values obtained in the protected case are respectively P π −0.012(14), P 17π 60 = −0.19(2) and P π 8 = 0.72(2), in good agreement with the theory, as well as with the ones obtained without protecting the single photons.…”

“…The conceptual strength of the PM argument for the reality of the quantum state is a highly controversial issue [6][7][8][9][10][11][12], namely, does the procedure allow observing the state or only the protection mechanism? Nevertheless, the idea of PM triggered and helped studying various foundational topics beyond exploring the meaning of the wavefunction, such as the analysis of Bohmian trajectories [13], determination of the stationary basis [12] and analysis of measurement optimisation for minimising the state disturbance [14]. The concept of protection was also extended to measurement of a two-state vector [15].…”

Protective measurement of open quantum systems

“…where θ n (x) and φ n (x) are the polar and azimuthal angles of the net field directionr n (x) given by (16). Then the state (11) at t = T can be evaluated to…”

A few reflections on protective measurements and more.

“…An application to quantum state measurements has also been proposed [22]: here the optimization of protective measurement is used for minimising the state disturbance when performing a protective measurement of a single system. As a further interesting study, we can mention the extension of the concept of protection to include protection in pre-and post-selected systems [23,20].…”