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arXiv:1105.4976 · Journal of Physics A Mathematical and Theoretical

Sequential measurements of conjugate observables

Claudio Carmeli, Teiko Heinosaari, Alessandro Toigo

Published Jun 15, 2011• 19 citations
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Abstract

We present a unified treatment of sequential measurements of two conjugate observables. Our approach is to derive a mathematical structure theorem for all the relevant covariant instruments. As a consequence of this result, we show that every Weyl–Heisenberg covariant observable can be implemented as a sequential measurement of two conjugate observables. This method is applicable both in finite- and infinite-dimensional Hilbert spaces, therefore covering sequential spin component measurements as well as position-momentum sequential measurements.

DOI
10.1088/1751-8113/44/28/285304

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We present a unified treatment of sequential measurements of two conjugate observables. Our approach is to derive a mathematical structure theorem for all the relevant covariant instruments. As a consequence of this result, we show that every Weyl–Heisenberg covariant observable can be implemented as a sequential measurement of two conjugate observables. This method is applicable both in finite- and infinite-dimensional Hilbert spaces, therefore covering sequential spin component measurements as well as position-momentum sequential measurements.
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This article considers quantum systems described by a finite-dimensional complex Hilbert space HHH. We first define the concept of a finite observable on HHH. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.
Abstract We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.
We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find a quantum circuit implementation of a given group-covariant POVM which uses the symmetry of the POVM. Based on representation theory of the symmetry group we develop a general approach for the implementation of group-covariant POVMs which consist of rank-one operators. The construction relies on a method to decompose matrices that intertwine two representations of a finite group. We give several examples for which the resulting quantum circuits are efficient. In particular, we obtain efficient quantum circuits for a class of POVMs generated by Weyl–Heisenberg groups. These circuits allow to implement an approximative simultaneous measurement of the position and crystal momentum of a particle moving on a cyclic chain.
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.
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.