Accuracy matrix in generalized simultaneous measurement of a qubit system

Abstract

We formulate the accuracy of quantum measurement for a qubit system in terms of a 3 by 3 matrix. This matrix, which we refer to as the accuracy matrix, can be calculated from a positive operator-valued measure (POVM) corresponding to the quantum measurement. Based on the accuracy matrix, we derive new trade-off relations between the measurement accuracy of two or three noncommuting observables of a qubit system. These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity. They can be interpreted as the uncertainty relations between measurement errors in simultaneous measurements, and also as the trade-off relations between the measurement error and back-action of measurement. A no-cloning inequality is derived from the trade-off relations. Furthermore, our formulation and the obtained results can be applied to analyze quantum state tomography. We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by quantum measurement.

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Cited by in corpus (6)

• Quantum Cloning Machines and the Applications
• Uncertainty Relation Revisited from Quantum Estimation Theory
• Duality relations in a two-path interferometer with an asymmetric beam splitter
• The information about the state of a qubit gained by a weakly coupled detector
• Approximating incompatible von Neumann measurements simultaneously
• Virtual Cloning of Quantum States