Covariant quantum measurements that maximize the likelihood (original) (raw)

Covariant quantum measurements which maximize the likelihood

We derive the class of covariant measurements which 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 compare collective measurements on identical copies with the scheme of independent measurements on each copy. A "continuous-variables" analogue of the measurement of direction of the angular momentum with two anti-parallel spins by Gisin and Popescu is given.

Covariant quantum measurements may not be optimal

Journal of Modern Optics, 2002

Quantum particles, such as spins, can be used for communicating spatial directions to observers who share no common coordinate frame. We show that if the emitter's signals are the orbit of a group, then the optimal detection method may not be a covariant measurement (contrary to widespread belief). It may be advantageous for the receiver to use a different group and an indirect estimation method: first, an ordinary measurement supplies redundant numerical parameters; the latter are then used for a nonlinear optimal identification of the signal.

Optimal measurements for relative quantum information

Physical Review A, 2004

We provide optimal measurement schemes for estimating relative parameters of the quantum state of a pair of spin systems. We prove that the optimal measurements are joint measurements on the pair of systems, meaning that they cannot be achieved by local operations and classical communication. We also demonstrate that in the limit where one of the spins becomes macroscopic, our results reproduce those that are obtained by treating that spin as a classical reference direction.

Preferred measurements: optimality and stability in quantum parameter estimation

New Journal of Physics, 2010

We explore precision in a measurement process incorporating pure probe states, unitary dynamics and complete measurements via a simple formalism. The concept of 'information complement' is introduced. It undermines measurement precision and its minimization reveals the system properties at an optimal point. Maximally precise measurements can exhibit independence from the true value of the estimated parameter, but demanding this severely restricts the type of viable probe and dynamics, including the requirement that the Hamiltonian be block-diagonal in a basis of preferred measurements. The curvature of the information complement near a globally optimal point provides a new quantification of measurement stability.

Optimal quantum state determination by constrained elementary measurements

The purpose of this short note is to utilize work on isotropic lines in [1], on Wigner distributions for finite-state systems in [2], estimation of the state of a finite level quantum system based on Weyl operators in the L 2 -space over a finite field in [3] to display maximal abelian subsets of certain unitary bases for the matrix algebra M d of complex square matrices of order d > 3; and then, combine these special forms with constrained elementary measurements to obtain optimal ways to determine a d-level quantum state. This enables us to generalise illustrations and strengthen results related to quantum tomography in [4].

Optimal estimation of quantum observables

Journal of Mathematical Physics, 2006

We consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased estimation is achieved by the usual procedure that consists in performing independent measurements of the observable on each system and averaging the measurement outcomes.

Extremal covariant measurements

Journal of Mathematical Physics, 2006

We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally non-transitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.

Optimal generalized quantum measurements for arbitrary spin systems

Physical Review A, 2000

Positive operator valued measurements on a finite number of N identically prepared systems of arbitrary spin J are discussed. Pure states are characterized in terms of Bloch-like vectors restricted by a SU (2J + 1) covariant constraint. This representation allows for a simple description of the equations to be fulfilled by optimal measurements. We explicitly find the minimal POVM for the N = 2 case, a rigorous bound for N = 3 and set up the analysis for arbitrary N .