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Physical Review Letters

Reversible Quantum Measurements on a Spin 1/2 and Measuring the State of a Single System

Antoine Royer

Published Aug 15, 1994• 58 citations
Publisher

Abstract

A measurement procedure on a spin 1/2 system is described, whose effect on the (possibly unknown) state of the system can be reversed, by means of other similar measurements, with a sizable probability of success ${P}^{\mathrm{rev}}$ (e.g., $\frac{1}{2}<{P}^{\mathrm{rev}}<1$, depending on the state). Repeated such measurements followed by reversal, on a single system, allow us to determine its (a priori unknown) state operator with a probability of success which is nonzero, though extremely small (because ${P}^{\mathrm{rev}}<1$).

DOI
10.1103/physrevlett.73.913

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A measurement procedure on a spin 1/2 system is described, whose effect on the (possibly unknown) state of the system can be reversed, by means of other similar measurements, with a sizable probability of success Prev{P}^{\mathrm{rev}}Prev (e.g., 12<Prev<1\frac{1}{2}<{P}^{\mathrm{rev}}<121​<Prev<1, depending on the state). Repeated such measurements followed by reversal, on a single system, allow us to determine its (a priori unknown) state operator with a probability of success which is nonzero, though extremely small (because Prev<1{P}^{\mathrm{rev}}<1Prev<1).
A measurement process is logically reversible if the premeasurement density operator of the measured system is uniquely determined from the postmeasurement density operator and the outcome of the measurement. This paper analyzes the necessary and sufficient condition for a measurement process to be logically reversible and discusses specific examples on quantum-nondemolition measurements, quantum counting, and measurement of spin systems. It is shown that for any sharp measurement we can construct a logically reversible measurement that continuously approaches the sharp measurement with a decrease in the measurement error. A general condition for a measurement process to be reversed by another with a nonzero probability of success is given, and the implications of such physical reversibility are discussed. \textcopyright{} 1996 The American Physical Society.
Experimental determination of an unknown quantum state usually requires several incompatible measurements. However, it is also possible to determine the full quantum state from a single, repeated measurement. For this purpose, the quantum system whose state is to be determined is first coupled to a second quantum system (the ``assistant'') in such a way that part of the information in the quantum state is transferred to the assistant. The actual measurement is then performed on the enlarged system including the original system and the assistant. We discuss in detail the requirements of this procedure and experimentally implement it on a simple quantum system consisting of nuclear spins.
The unknown state rho of a quantum system S is determined by letting it interact with an auxiliary system A, the initial state of which is known. A one-to-one mapping can thus be realized between the density matrix rho and the probabilities of the occurrence of the eigenvalues of a single and factorized observable of S+A, so that rho can be determined by repeated measurements using a single apparatus. If S and A are spins, it suffices to measure simultaneously their z components after a controlled interaction. The most robust setups are determined in this case for an initially pure or a completely disordered state of A. They involve an Ising or anisotropic Heisenberg coupling and an external field.
We propose a class of probabilistic reversing operations on the state of a system that was disturbed by a weak measurement. It can approximately recover the original state from the disturbed state especially with an additional information gain using the Hermitian conjugate of the measurement operator. We illustrate the general scheme by considering a quantum measurement consisting of spin systems with an experimentally feasible interaction and show that the reversing operation simultaneously increases both the fidelity to the original state and the information gain with such a high probability of success that their average values increase simultaneously.
Here we show that, in principle it is possible to clone (measure) a single arbitrary unknown quantum state of a spin-12\frac{1}{2}21​ particle (an electron) with arbitrary precision and with success probability tending to one, using protective measurement. We first transfer the information from spin to spatial degree of freedom (d.o.f) of system electron, then trap it in a double well potential, and finally measure it protectively using a probe electron (which donot get entangled with system electron, but still extracts expectation value of an observable from a single quantum system (system electron)) to obtain information about the unknown spin polarization. Nonorthogonal state discrimination being a subclass of cloning, part of the paper (till finding out θmθ_{m}θm​, polar angle corresponding to the unknown spin polarization) is sufficient for discrimination.