Kari Ylinen

Are number and phase complementary observables

We study various ways of characterizing the quantum optical number and phase as complementary observables.

The moment operators of phase space observables and their number margins

Abstract We show that the moment operators of any of the phase space observables B( C ) ) Z ↦ A ¦s〉 (Z)≔ 1 π ∫ Z/LL> D¦s〉〈s¦D z ∗ dλ(z) ¬E L(H) associated with the number states ¦s〉 are the powers of the lowering operator. We also determine all the moment operators of the number margins of these observables. They are integer valued polynomials of the number operator, the degree of the polynomial being the degree of the moment and the integer coefficients depending on ¦s〉 . In the Appendix we develop the necessary tools for integrating unbounded functions with respect to operator measures.

Phase space quantization and the operator moment problem

We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

The norm-1-property of a quantum observable

A normalized positive operator measure

Covariant fuzzy observables and coarse-graining

X\mapsto E(X)

Operator integrals and phase space observables

has the norm-1-property if

Positive sesquilinear form measures and generalized eigenvalue expansions

\no{E(X)}=1

Coexistent observables and effects in a convexity approach

whenever

The uniqueness question in the multidimensional moment problem with applications to phase space observables

E(X)\ne O

Positive operator measures determined by their moment sequences

. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrary close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.A normalized positive operator measure X⟼E(X) has the norm-1-property if ‖E(X)‖=1 whenever E(X)≠O. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrarily close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.