Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0810.2758 · Journal of Physics A Mathematical and Theoretical

Optimal covariant measurements: the case of a compact symmetry group and phase observables

Claudio Carmeli, Teiko Heinosaari, Juha-Pekka Pellonpää, Alessandro Toigo

Published Mar 17, 2009• 3 citations
arXivHTMLPDF

Abstract

We study various optimality criteria for quantum observables. Observables are represented as covariant positive operator valued measures and we consider the case when the symmetry group is compact. Phase observables are examined as an example.
DOI
10.1088/1751-8113/42/14/145304

Related papers

OPTIMAL COVARIANT MEASUREMENTS: THE CASE OF A COMPACT SYMMETRY GROUP AND PHASE OBSERVABLES

Claudio Carmeli, Teiko Heinosaari, Juha-Pekka Pellonpää +1 more

Abstract. We study various optimality criteria for quantum observables. Observables are represented as covariant positive operator valued measures and we consider the case when the symmetry group is compact. Phase observables are examined as an example. 1.

•Apr 10, 2009•2 citations

Optimal covariant quantum measurements

Erkka Haapasalo, Juha-Pekka Pellonpää

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.

arXiv:2009.14080•Journal of Physics A Mathematical and Theoretical•Feb 17, 2021•5 citations

Extremal covariant positive operator valued measures: The case of a compact symmetry group

Claudio Carmeli, Teiko Heinosaari, Juha-Pekka Pellonpää +1 more

Given a unitary representation U of a compact group G and a transitive G-space Ω, we characterize the extremal elements of the convex set of all U-covariant positive operator valued measures.

arXiv:0801.4721•Journal of Mathematical Physics•Jun 1, 2008•8 citations

Extremal covariant measurements

Giulio Chiribella, Giacomo Mauro D’Ariano

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 nontransitive. 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.

arXiv:quant-ph/0603168•Journal of Mathematical Physics•Sep 1, 2006•14 citations

Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space

Erkka Haapasalo, Juha-Pekka Pellonpää

We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.

arXiv:1105.1084•Journal of Mathematical Physics•Dec 1, 2011•6 citations

Information and quantum measurement

E. B. Davies

Given a finite number of quantum states with {\em a priori} probabilities, the positive operator-valued measure that maximizes the Shannon mutual information is investigated. The group covariant case is examined in detail.

IEEE Transactions on Information Theory•Sep 1, 1978•293 citations
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 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 nontransitive. 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.
We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.