Optimal joint measurements of complementary observables by a single trapped ion

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 \textsl{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 $^{40}Ca^{+}$ 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.

[1]  E. H. Kennard.  Zur Quantenmechanik einfacher Bewegungstypen , 1927 .

[2]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[3]  H. Weyl.  Gruppentheorie und Quantenmechanik , 1928 .

[4]  H. P. Robertson.  The Uncertainty Principle , 1929 .

[5]  Leslie E Ballentine,et al.  The statistical interpretation of quantum mechanics , 1970 .

[6]  M. Ozawa.  Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement , 2002, quant-ph/0207121.

[7]  Masanao Ozawa.  Uncertainty relations for noise and disturbance in generalized quantum measurements , 2003 .

[8]  Masanao Ozawa.  Uncertainty relations for joint measurements of noncommuting observables , 2004 .

[9]  M. Hall,et al.  Prior information: How to circumvent the standard joint-measurement uncertainty relation , 2003, quant-ph/0309091.

[10]  S. Lloyd,et al.  Quantum-Enhanced Measurements: Beating the Standard Quantum Limit , 2004, Science.

[11]  M. Koashi.  Unconditional security of quantum key distribution and the uncertainty principle , 2006 .

[12]  P. Busch,et al.  Heisenberg's uncertainty principle , 2006, quant-ph/0609185.

[13]  G. Tóth,et al.  Entanglement detection , 2008, 0811.2803.

[14]  Paul Busch,et al.  Approximate joint measurements of qubit observables , 2007, Quantum Inf. Comput..

[15]  M. Wolf,et al.  Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. , 2009, Physical review letters.

[16]  S. Wehner,et al.  The Uncertainty Principle Determines the Nonlocality of Quantum Mechanics , 2010, Science.

[17]  R. Renner,et al.  The uncertainty principle in the presence of quantum memory , 2009, 0909.0950.

[18]  Aephraim M. Steinberg,et al.  Violation of Heisenberg's measurement-disturbance relationship by weak measurements. , 2012, Physical review letters.

[19]  K. Fujikawa.  Universally valid Heisenberg uncertainty relation , 2012, 1205.1360.

[20]  M. Ozawa,et al.  Experimental demonstration of a universally valid error–disturbance uncertainty relation in spin measurements , 2012, Nature Physics.

[21]  Geoff J Pryde,et al.  Experimental test of universal complementarity relations. , 2012, Physical review letters.

[22]  Marco Tomamichel,et al.  The link between entropic uncertainty and nonlocality , 2011, 1108.5349.

[23]  M. Ozawa,et al.  Violation of Heisenberg's error-disturbance uncertainty relation in neutron-spin measurements , 2013, 1305.7251.

[24]  Keiichi Edamatsu,et al.  Experimental violation and reformulation of the Heisenberg's error-disturbance uncertainty relation , 2013, Scientific Reports.

[25]  R. Werner,et al.  Proof of Heisenberg's error-disturbance relation. , 2013, Physical review letters.

[26]  C. Branciard.  Error-tradeoff and error-disturbance relations for incompatible quantum measurements , 2013, Proceedings of the National Academy of Sciences.

[27]  C. H. Oh,et al.  Optimal joint measurement of two observables of a qubit , 2014, 1402.3785.

[28]  S. Baek,et al.  Experimental test of error-disturbance uncertainty relations by weak measurement. , 2013, Physical review letters.

[29]  R. Werner,et al.  Heisenberg uncertainty for qubit measurements , 2013, 1311.0837.

[30]  R. Werner,et al.  Colloquium: Quantum root-mean-square error and measurement uncertainty relations , 2013, 1312.4393.

[31]  Andrew G. White,et al.  Experimental joint quantum measurements with minimum uncertainty. , 2013, Physical review letters.

[32]  Jingning Zhang,et al.  Experimental test of the quantum Jarzynski equality with a trapped-ion system , 2014, Nature Physics.

[33]  Otfried Gühne,et al.  One-to-One Mapping between Steering and Joint Measurability Problems. , 2015, Physical review letters.

[34]  P. Busch,et al.  Direct tests of measurement uncertainty relations: what it takes. , 2014, Physical review letters.

[35]  P. Busch,et al.  Measurement uncertainty: the problem of characterising optimal error bounds , 2015 .

[36]  M. Feng,et al.  Verifying Heisenberg’s error-disturbance relation using a single trapped ion , 2016, Science Advances.

[37]  Y. Liu,et al.  Experimental Test of Heisenberg's Measurement Uncertainty Relation Based on Statistical Distances. , 2015, Physical review letters.

[38]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.