Jukka Kiukas

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.

Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains

We consider the problem of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. The starting point of the analysis is the fact that the knowledge of the output state completely fixes the dynamics up to an equivalence class of ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators.Assuming that the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard’ rate n−1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e., it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check, we prove that a parameter related to the ‘coordinate transformation’ unitaries has zero quantum Fisher information.

A proof for the informational completeness of the rotated quadrature observables

We give a new mathematically rigorous proof for the fact that, when S is a dense subset of [0, 2π), the rotated quadrature operators Qθ, θ S, of a single-mode electromagnetic field constitute an informationally complete set of observables.

Informationally complete sets of Gaussian measurements

We prove the necessary and sufficient conditions for the informational completeness of an arbitrary set of Gaussian observables on continuous variable systems with a finite number of degrees of freedom. In particular, we show that an informationally complete set either contains a single informationally complete observable, or includes infinitely many observables. We show that for a single informationally complete observable, the minimal outcome space is the phase space, and the corresponding probability distribution can always be obtained from the quantum optical Q-function by linear postprocessing and Gaussian convolution, in a suitable symplectic coordinatization of the phase space. In the case of projection valued Gaussian observables, e.g., generalized field quadratures, we show that an informationally complete set of observables is necessarily infinite. Finally, we generalize the treatment to the case where the measurement coupling is given by a general linear bosonic channel, and characterize informational completeness for an arbitrary set of the associated observables.

Moment operators of the Cartesian margins of the phase space observables

The theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the x-margin are polynomials of the position operator and those of the y-margin are polynomials of the momentum operator.

Measuring position and momentum together

We describe an operational scheme for determining both the position and momentum distributions in a large class of quantum states, together with an experimental implementation.

Coexistence of effects from an algebra of two projections

The coexistence relation of quantum effects is a fundamental structure, describing those pairs of experimental events that can be implemented in a single setup. Only in the simplest case of qubit effects is an analytic characterization of coexistent pairs known. We generalize the qubit coexistence characterization to all pairs of effects in arbitrary dimensions that belong to the von Neumann algebra generated by two projections. We demonstrate the presented mathematical machinery by several examples, and show that it covers physically relevant classes of effect pairs.

Semispectral measures as convolutions and their moment operators

The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables.

Quantization and noiseless measurements

In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable is associated with a unique positive operator measure (POM) Ef, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly, we note that the noiseless measurements are those which are determined by a selfadjoint operator. The POM Ef in our quantization is defined through its moment operators, which are required to be of the form , with Γ being a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical questions, that is, functions taking only values 0 and 1. We compare two concrete realizations of the map Γ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.

Moment Operators of Observables in the Balanced Homodyne Detection Scheme

We review the results of article [1], where we gave a mathematically rigorous proof that the rotated quadrature observables 12(e−iθa+eiθa* of a single mode electromagnetic field can be determined in the so called “high amplitude limit” of the balanced homodyne detection. In the proof, we used the moment operators of observables actually measured by the detector; the high amplitude limit was described by certain limits of a sequence (En)n∈N of such observables.