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Experimental investigation of joint measurement uncertainty relations for three incompatible observables at a single-spin level

K. Rehan, Tian‐Jian Xiong, L.-L. Yan, Fei Zhou, Jiawei Zhang, J. C. Li, Liang Chen, W. L. Yang, Mang Feng

Published Aug 8, 2020• 3 citations
Publisher

Abstract

In the light of the Busch, Lathi and Werner proposal, we explore, for the first time, the joint measurements and confirmation of uncertainty relations for three incompatible observables that reflect the original spirit proposed by Heisenberg in 1927. We first develop the error trade-off relations theoretically and then demonstrate the first experimental witness of joint measurements using a single ultracold 40Ca+ ion trapped in a harmonic potential. In addition, we report, that in contrast to the case of two observables, scarifying accuracy of any one of the three observables the rest of two can be measured with ultimate accuracy.

DOI
10.1364/oe.401337

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In the light of the Busch, Lathi and Werner proposal, we explore, for the first time, the joint measurements and confirmation of uncertainty relations for three incompatible observables that reflect the original spirit proposed by Heisenberg in 1927. We first develop the error trade-off relations theoretically and then demonstrate the first experimental witness of joint measurements using a single ultracold 40Ca+ ion trapped in a harmonic potential. In addition, we report, that in contrast to the case of two observables, scarifying accuracy of any one of the three observables the rest of two can be measured with ultimate accuracy.
The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first direct test and confirmation of the tight bounds prescribed by such an error trade-off relation, based on an experimental realisation of optimal joint measurements of complementary observables using a single ultracold ion trapped in a harmonic potential. Our work provides a prototypical determination of ultimate joint measurement error bounds with potential applications in quantum information science for high-precision measurement and information security.
Being one of the centroidal concepts in quantum theory, the fundamental constraint imposed by Heisenberg uncertainty relations has always been a subject of immense attention and challenging in the context of joint measurements of general quantum mechanical observables. In particular, the recent extension of the original uncertainty relations has grabbed a distinct research focus and set a new ascendent target in quantum mechanics and quantum information processing. In the present work we explore the joint measurements of three incompatible observables, following the basic idea of a newly proposed error trade-off relation. In comparison to the counterpart of two incompatible observables, the joint measurements of three incompatible observables are more complex and of more primal interest in understanding quantum mechanical measurements. Attributed to the pristine idea proposed by Heisenberg in 1927, we develop the error trade-off relations for compatible observables to categorically approximate the three incompatible observables. Implementing these relations we demonstrate the first experimental witness of the joint measurements for three incompatible observables using a single ultracold 40Ca+^{40}Ca^{+}40Ca+ ion in a harmonic potential. We anticipate that our inquisition would be of vital importance for quantum precision measurement and other allied quantum information technologies.
ion trapped in a harmonic potential. By performing unitary operations under carrier transitions, we verify the uncertainty relation proposed by Busch, Lahti, and Werner (BLW) based on a general error-trade-off relation for joint measurements on two compatible observables. The positive operator-valued measure, required by the compatible observables, is constructed by single-qubit operations, and the lower bound of the uncertainty, as observed, is satisfied in a state-independent manner. Our results provide the first evidence confirming the BLW-formulated uncertainty at a single-spin level and will stimulate broad interests in various fields associated with quantum mechanics.
Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [P. Busch et al., Phys. Rev. Lett. 111, 160405 (2013); P. Busch et al., Phys. Rev. A 89, 012129 (2014)]. Here we reformulate their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible observables designed to approximate two incompatible observables on one qubit are measured simultaneously.
Heisenberg-type measurement uncertainty relations (MURs) of two quantum observables are essential for contemporary research in quantum foundations and quantum information science. Going beyond, here we report the first experimental study of MUR of three quantum observables. We establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower bounded by an incompatibility measure, inspired by the proposal of Busch et al. [Phys. Rev. A 89, 012129 (2014)PLRAAN1050-294710.1103/PhysRevA.89.012129]. We develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the optimal measurements. We propose a novel implementation of the optimal joint measurements and present several experimental demonstrations with a single-photon qubit. We stress that our method is universally applicable to the study of many qubit observables. Besides, we theoretically show that MURs for joint measurement can be attained by sequential measurements in two of our explored cases. We anticipate that this work may stimulate broad interests associated with Heisenberg's uncertainty principle in the case of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.
Heisenberg's measurement uncertainty relations (MUR) of two quantum observables are essential for contemporary researches in quantum foundations and quantum information science. Going beyond, here we report the first experimental test of MURs for three quantum observables. Following the proposal of Bush, Lahti, and Werner [Phys. Rev. A 89, 012129 (2014)], we first establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower-bounded by an incompatibility measure. We then develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the corresponding optimal measurements. Furthermore, we propose a novel implementation of optimal joint measurements and experimentally test our MURs using a single-photon qubit. Lastly, we discuss to analytically calculate the exact value of incompatibility measure for some symmetric triplets. We anticipate that this work may stimulate broad interests associated with the Heisenberg's uncertainty relation of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.