The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration shows that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration shows that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.

arXiv:quant-ph/0207081Physical Review ASep 19, 200311 citations

Uncertainty relations in simultaneous measurements for arbitrary observables

Shiro Ishikawa

Reports on Mathematical PhysicsJun 1, 199165 citations

Joint measurements of spin, operational locality, and uncertainty

Erika Andersson, Stephen Mark Barnett, Alain Aspect

Joint, or simultaneous, measurements of noncommuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin- $1∕2$ particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.Joint, or simultaneous, measurements of noncommuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin-$1∕2$ particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.

arXiv:quant-ph/0509152Physical Review AOct 13, 200555 citations

Uncertainty Relations Between Measurement Errors of Two Observables

Yu Watanabe

Springer thesesDec 17, 20130 citations

Joint measurement of multiple noncommuting parameters

Jiamin Li, Yuhong Liu, Liang Cui +4 more

Although quantum metrology allows us to make precision measurements beyond the standard quantum limit, it mostly works on the measurement of only one observable due to the Heisenberg uncertainty relation on the measurement precision of noncommuting observables for one system. In this paper, we study the schemes of joint measurement of multiple observables which do not commute with each other using the quantum entanglement between two systems. We focus on analyzing the performance of a SU(1,1) nonlinear interferometer on fulfilling the task of joint measurement. The results show that the information encoded in multiple noncommuting observables on an optical field can be simultaneously measured with a signal-to-noise ratio higher than the standard quantum limit, and the ultimate limit of each observable is still the Heisenberg limit. Moreover, we find a resource conservation rule for the joint measurement.Although quantum metrology allows us to make precision measurements beyond the standard quantum limit, it mostly works on the measurement of only one observable due to the Heisenberg uncertainty relation on the measurement precision of noncommuting observables for one system. In this paper, we study the schemes of joint measurement of multiple observables which do not commute with each other using the quantum entanglement between two systems. We focus on analyzing the performance of a SU(1,1) nonlinear interferometer on fulfilling the task of joint measurement. The results show that the information encoded in multiple noncommuting observables on an optical field can be simultaneously measured with a signal-to-noise ratio higher than the standard quantum limit, and the ultimate limit of each observable is still the Heisenberg limit. Moreover, we find a resource conservation rule for the joint measurement.

arXiv:1711.10857Physical Review AMay 29, 201826 citations

The Remaining Uncertainty in Quantum Measurement of Noncommuting Discrete Observables

Vladimír Majerník, M. Vlček

Acta Physica Hungarica A) Heavy Ion PhysicsFeb 1, 20020 citations

1∕2

particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.