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ON THE COMPLEMENTARITY OF THE QUADRATURE OBSERVABLES

PEKKA J. LAHTI, Juha-Pekka Pellonpää

Published Aug 14, 2016• 1 citation
OpenAlex

Abstract

In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon transform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.

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In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon transform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.
Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which can be proved in the form of the Fourier-Weyl relation between density matrices and characteristic functions. This relationship constitutes the bridge between phase space and Hilbert space descriptions, which is useful in several applications in the context of quantum information. This chapter shows that the marginal Wigner function along any phase space direction describes the probability distribution of the quantum measurement of the associated quadrature operator. Hence, as far as Gaussian states are concerned, the Wigner function provides one with a local, 'realistic' model to describe quadrature measurements. If one restricts to quadrature measurements, such systems may be mimicked by multivariate classical Gaussian distributions and will never show any signature of quantum non-locality, such as a violation of Bell or CHSH inequalities.
This thesis addresses the use of covariant phase space observables in quantum tomography. Necessary and sufficient conditions for the informational completeness of covariant phase space observables are proved, and some state reconstruction formulae are derived. Different measurement schemes for measuring phase space observables are considered. Special emphasis is given to the quantum optical eight-port homodyne detection scheme and, in particular, on the effect of non-unit detector efficiencies on the measured observable. It is shown that the informational completeness of the observable does not depend on the efficiencies. As a related problem, the possibility of reconstructing the position and momentum distributions from the marginal statistics of a phase space observable is considered. It is shown that informational completeness for the phase space observable is neither necessary nor sufficient for this procedure. Two methods for determining the distributions from the marginal statistics are presented. Finally, two alternative methods for determining the state are considered. Some of their shortcomings when compared to the phase space method are discussed.
We introduce the Wigner operator [Formula: see text] for the rotated quadrature phases and use the technique of integration within an ordered product of operators to derive its explicitly simpler form. Based on this, the mutual relations between [Formula: see text] and the corresponding marginal probability distribution operator can be easily revealed. The Wigner function theory is thus be recasted into a more elegant and concise formalism. The squeezing in rotated quadrature phase is discussed with the same method.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.
We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.