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arXiv:quant-ph/0007062 · Journal of Optics B Quantum and Semiclassical Optics

Joint measurements via quantum cloning

Giacomo Mauro D’Ariano, Chiara Macchiavello, Massimiliano Federico Sacchi

Published Mar 16, 2001• 29 citations
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Abstract

We explore the possibility of achieving the optimal joint measurement of noncommuting observables on a single quantum system by performing conventional measurements of commuting self-adjoint operators on the clones of the original quantum system. We consider the case of both finite- and infinite-dimensional Hilbert spaces. In the former we study the joint measurement of three orthogonal components of a spin ½; in the latter we consider the case of the joint measurement of any pair of noncommuting quadratures of one mode of the electromagnetic field. We show that universal covariant cloning is not ideal for joint measurements, and a suitable nonuniversal covariant cloning is needed.

DOI
10.1088/1464-4266/3/2/305

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We explore the possibility of achieving the optimal joint measurement of noncommuting observables on a single quantum system by performing conventional measurements of commuting self-adjoint operators on the clones of the original quantum system. We consider the case of both finite- and infinite-dimensional Hilbert spaces. In the former we study the joint measurement of three orthogonal components of a spin ½; in the latter we consider the case of the joint measurement of any pair of noncommuting quadratures of one mode of the electromagnetic field. We show that universal covariant cloning is not ideal for joint measurements, and a suitable nonuniversal covariant cloning is needed.
Cloning of observables, unlike standard cloning of states, aims at copying the information encoded in the statistics of a class of observables rather then on quantum states themselves. In such a process the emphasis is on the quantum operation (evolution plus measurement) necessary to retrieve the original information. We analyze, for qubit systems, the cloning of a class generated by two noncommuting observables, elucidating the relationship between such a process and joint measurements. This helps in establishing an optimality criterion for cloning of observables. We see that, even if the cloning machine is designed to act on the whole class generated by two noncommuting observables, the same optimal performances of a joint measurement can be attained. Finally, the connection with state dependent cloning is enlightened.
A statistical analysis of optimal universal cloning shows that it is possible to identify an ideal (but nonpositive) copying process that faithfully maps all properties of the original Hilbert space onto two separate quantum systems, resulting in perfect correlations for all observables. The joint probabilities for noncommuting measurements on separate clones then correspond to the real parts of the complex joint probabilities observed in weak measurements on a single system, where the measurements on the two clones replace the corresponding sequence of weak measurement and postselection. The imaginary parts of weak measurement statics can be obtained by replacing the cloning process with a partial swap operation. A controlled-swap operation combines both processes, making the complete weak measurement statistics accessible as a well-defined contribution to the joint probabilities of fully resolved projective measurements on the two output systems.
A joint measurement of two observables is a simultaneous measurement of both quantities upon the same quantum system. When two quantum-mechanical observables do not commute, then a joint measurement of these observables cannot be accomplished directly by projective measurements alone. In this paper we shall discuss the use of quantum cloning to perform a joint measurement of two components of spin associated with a qubit system. We introduce cloning schemes which are optimal with respect to this task. The cloning schemes may be thought to work by cloning two components of spin onto their outputs. We compare the proposed cloning machines to existing cloners.
Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.