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AIP conference proceedings

Optimal Covariant Measurement of Momentum on a Half Line

Yutaka Shikano, Akio Hosoya, Alexander I. Lvovsky

Published Jan 1, 2009• 1 citation
Publisher

Abstract

We investigate an optimal covariant measurement to measure a momentum of a particle. Given an instantaneous interaction and an infinite measurement time, we explicitly construct a measurement model to satisfy the optimal covariant positive operator valued measure by Holevo and physically consider this measurement model. Furthermore, we apply this model to measure a momentum of a particle on a half line.

DOI
10.1063/1.3131321

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We cannot perform the projective measurement of a momentum on a half line since it is not an observable.Nevertheless, we would like to obtain some physical information of the momentum on a half line.We define an optimality for mea- surement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo.Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self- adjoint and explicitly construct a model Hamiltonian for the measured and probe systems.By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced.We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.

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We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys. 13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.

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We investigate an optimal covariant measurement to measure a momentum of a particle. Given an instantaneous interaction and an infinite measurement time, we explicitly construct a measurement model to satisfy the optimal covariant positive operator valued measure by Holevo and physically consider this measurement model. Furthermore, we apply this model to measure a momentum of a particle on a half line.
We cannot perform the projective measurement of a momentum on a half line since it is not an observable.Nevertheless, we would like to obtain some physical information of the momentum on a half line.We define an optimality for mea- surement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo.Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self- adjoint and explicitly construct a model Hamiltonian for the measured and probe systems.By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced.We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.
We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys. 13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.
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 study the optimization of input states for phase estimation under covariant measurements. Building on Holevo's framework, which provides the optimal covariant measurement for a fixed input state, we further optimize over the input state itself. For a general even 2π2π2π-periodic cost function with non-negative Fourier coefficients, we derive a necessary and sufficient condition for the optimal input state: Its Fock coefficients are determined, up to arbitrary phases, by the eigenvector corresponding to the largest eigenvalue of a Toeplitz matrix defined by the cost function. This characterization yields an explicit expression for the attainable lower bound of the average cost under optimal covariant measurements and shows that this bound asymptotically approaches zero in the infinite-energy limit. For the specific cost function W(θ,θ~)=4sin⁡2[(θ−θ~)/2]W(θ,\tildeθ)=4\sin^2[(θ-\tildeθ)/2]W(θ,θ~)=4sin2[(θ−θ~)/2], we obtain the optimal input state and the corresponding minimum average cost in closed form, demonstrating Heisenberg scaling with respect to the mean photon number.
We derive the class of covariant measurements that are optimal according to the maximum likelihood criterion. The optimization problem is fully resolved in the case of pure input states, under the physically meaningful hypotheses of unimodularity of the covariance group and measurability of the stability subgroup. The general result is applied to the case of covariant state estimation for finite dimension, and to the Weyl-Heisenberg displacement estimation in infinite dimension. We also consider estimation with multiple copies, and analyze the behavior of the likelihood versus the number of copies. A ``continuous-variable'' analog of the measurement of direction of the angular momentum with two antiparallel spins by Gisin and Popescu is given.