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

Journal of Mathematical Physics

Exact and Simultaneous Measurements

Platon C. Deliyannis

Published Apr 1, 1972• 1 citation
Publisher

Abstract

The observables which can be determined through their states of exact measurement are characterized, and a criterion for the set of states in which two observables are both measured exactly is established so that the two observables will be compatible.
DOI
10.1063/1.1666003

Related papers

Uncertainty relations in simultaneous measurements for arbitrary observables

Shiro Ishikawa

Reports on Mathematical Physics•Jun 1, 1991•65 citations

Preparation and Measurement Uncertaintyin Quantum Mechanics

Oliver Reardon-Smith

This thesis addresses two forms of quantum uncertainty. In part I, we focus on preparation uncertainty, an expression of the fact that there are sets of observables for which the induced probability distributions are not simultaneously sharp in any state. We exactly characterise the preparation uncertainty regions for several finite dimensional case studies, including a new derivation of the preparation uncertainty region for the Pauli observables of qubits, and two qutrit case studies which have not previously been addressed in the literature. We also consider the variance based preparation uncertainty for position and momentum observables for the well known “particle in a box” system. We see that the appropriate momentum observable is not given by the spectral measure of a self-adjoint operator, although the position observable is. The box system lacks the phase-space symmetry used to determine the free particle and particle on a ring systems so determining the box uncertainty region is rather more difficult than in these cases. We give upper and lower bounds on the boundary of the uncertainty region, and show that our upper bound is exact in an interval. In part II we turn our attention to measurement uncertainty, exploring the space of compatible joint approximations to incompatible target observables. We prove a general theorem, which shows that, for a broad class of figures of merit, the optimal compatible approximations to covariant targets are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for covariant observables, since the space of covariant compatible approximations is smaller than the space of all compatible approximations. We employ this theorem to derive measurement uncertainty regions for three mutually orthogonal Pauli observables, and for the quantum Fourier pair acting in any finite dimension.

•Jul 31, 2019•0 citations

Observables measured simultaneously with the potential

A. H. Nasr

The general form of the observables measured simultaneously with some class of potentials is given.

Journal of Mathematical Physics•Dec 1, 1982•1 citation

Simultaneous measurement of two quantum observables: Compatibility, broadcasting, and in-between

Teiko Heinosaari

One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.

arXiv:1604.00775•Physical Review A•Apr 25, 2016•24 citations

On simultaneous measurements of incompatible observables

Heinz Dieter Dombrowski

Archive for Rational Mechanics and Analysis•Jan 1, 1969•1 citation

Compatible-Marginal Spin Alignment for Three Qubits

Haoran Wang

The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.

•Oct 9, 2026•0 citations
This thesis addresses two forms of quantum uncertainty. In part I, we focus on preparation uncertainty, an expression of the fact that there are sets of observables for which the induced probability distributions are not simultaneously sharp in any state. We exactly characterise the preparation uncertainty regions for several finite dimensional case studies, including a new derivation of the preparation uncertainty region for the Pauli observables of qubits, and two qutrit case studies which have not previously been addressed in the literature. We also consider the variance based preparation uncertainty for position and momentum observables for the well known “particle in a box” system. We see that the appropriate momentum observable is not given by the spectral measure of a self-adjoint operator, although the position observable is. The box system lacks the phase-space symmetry used to determine the free particle and particle on a ring systems so determining the box uncertainty region is rather more difficult than in these cases. We give upper and lower bounds on the boundary of the uncertainty region, and show that our upper bound is exact in an interval. In part II we turn our attention to measurement uncertainty, exploring the space of compatible joint approximations to incompatible target observables. We prove a general theorem, which shows that, for a broad class of figures of merit, the optimal compatible approximations to covariant targets are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for covariant observables, since the space of covariant compatible approximations is smaller than the space of all compatible approximations. We employ this theorem to derive measurement uncertainty regions for three mutually orthogonal Pauli observables, and for the quantum Fourier pair acting in any finite dimension.
One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.
The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.