Abstract

The uncertainty relation is one of the fundamental principles in quantum mechanics and plays an important role in quantum information science. We experimentally test the error-disturbance uncertainty relation (EDR) with continuous variables for Gaussian states. Two incompatible continuous-variable observables, amplitude and phase quadratures of an optical mode, are measured simultaneously using a heterodyne measurement system. The EDR values with continuous variables for coherent, squeezed, and thermal states are verified experimentally. Our experimental results demonstrate that Heisenberg’s EDR with continuous variables is violated, while Ozawa’s and Branciard’s EDRs with continuous variables are validated.

© 2019 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  5. J. Jin, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
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    [Crossref]
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    [Crossref]
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  14. M. Ozawa, “Uncertainty relations for joint measurements of noncommuting observables,” Phys. Lett. A 320, 367–374 (2004).
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  15. M. Ozawa, “Soundness and completeness of quantum root-mean-square errors,” NPJ Quantum Inf. 5, 1 (2019).
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    [Crossref]
  18. P. Busch, P. Lahti, and R. F. Werner, “Colloquium: quantum root-mean-square error and measurement uncertainty relations,” Rev. Mod. Phys. 86, 1261–1281 (2014).
    [Crossref]
  19. J. Dressel and F. Nori, “Certainty in Heisenberg’s uncertainty principle: revisiting definitions for estimation errors and disturbance,” Phys. Rev. A 89, 022106 (2014).
    [Crossref]
  20. K. Baek, T. Farrow, and W. Son, “Optimized entropic uncertainty relation for successive measurement,” Phys. Rev. A 89, 032108 (2014).
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  23. A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for position and momentum: relative entropy formulation,” Entropy 19, 301 (2017).
    [Crossref]
  24. A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for discrete observables: relative entropy formulation,” Commun. Math. Phys. 357, 1253–1304 (2018).
    [Crossref]
  25. J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
    [Crossref]
  26. G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
    [Crossref]
  27. G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
    [Crossref]
  28. B. Demirel, S. Sponar, G. Sulyok, M. Ozawa, and Y. Hasegawa, “Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states,” Phys. Rev. Lett. 117, 140402 (2016).
    [Crossref]
  29. M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
    [Crossref]
  30. F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
    [Crossref]
  31. L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
    [Crossref]
  32. A. P. Lund and H. M. Wiseman, “Measuring measurement-disturbance relationships with weak values,” New J. Phys. 12, 093011 (2010).
    [Crossref]
  33. S. Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
    [Crossref]
  34. M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
    [Crossref]
  35. W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
    [Crossref]
  36. F. Zhou, L. Yan, S. Gong, Z. Ma, J. He, T. Xiong, L. Chen, W. Yang, M. Feng, and V. Vedral, “Verifying Heisenberg’s error-disturbance relation using a single trapped ion,” Sci. Adv. 2, e1600578 (2016).
    [Crossref]
  37. T. Xiong, L. Yan, Z. Ma, F. Zhou, L. Chen, W. Yang, M. Feng, and P. Busch, “Optimal joint measurements of complementary observables by a single trapped ion,” New J. Phys. 19, 063032 (2017).
    [Crossref]
  38. Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).
  39. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
    [Crossref]
  40. X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
    [Crossref]

2019 (2)

M. Ozawa, “Soundness and completeness of quantum root-mean-square errors,” NPJ Quantum Inf. 5, 1 (2019).
[Crossref]

Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

2018 (1)

A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for discrete observables: relative entropy formulation,” Commun. Math. Phys. 357, 1253–1304 (2018).
[Crossref]

2017 (2)

A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for position and momentum: relative entropy formulation,” Entropy 19, 301 (2017).
[Crossref]

T. Xiong, L. Yan, Z. Ma, F. Zhou, L. Chen, W. Yang, M. Feng, and P. Busch, “Optimal joint measurements of complementary observables by a single trapped ion,” New J. Phys. 19, 063032 (2017).
[Crossref]

2016 (3)

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
[Crossref]

F. Zhou, L. Yan, S. Gong, Z. Ma, J. He, T. Xiong, L. Chen, W. Yang, M. Feng, and V. Vedral, “Verifying Heisenberg’s error-disturbance relation using a single trapped ion,” Sci. Adv. 2, e1600578 (2016).
[Crossref]

B. Demirel, S. Sponar, G. Sulyok, M. Ozawa, and Y. Hasegawa, “Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states,” Phys. Rev. Lett. 117, 140402 (2016).
[Crossref]

2015 (1)

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
[Crossref]

2014 (8)

M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
[Crossref]

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref]

P. Busch, P. Lahti, and R. F. Werner, “Heisenberg uncertainty for qubit measurements,” Phys. Rev. A 89, 012129 (2014).
[Crossref]

P. Busch, P. Lahti, and R. F. Werner, “Colloquium: quantum root-mean-square error and measurement uncertainty relations,” Rev. Mod. Phys. 86, 1261–1281 (2014).
[Crossref]

J. Dressel and F. Nori, “Certainty in Heisenberg’s uncertainty principle: revisiting definitions for estimation errors and disturbance,” Phys. Rev. A 89, 022106 (2014).
[Crossref]

K. Baek, T. Farrow, and W. Son, “Optimized entropic uncertainty relation for successive measurement,” Phys. Rev. A 89, 032108 (2014).
[Crossref]

F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, “Noise and disturbance in quantum measurements: an information-theoretic approach,” Phys. Rev. Lett. 112, 050401 (2014).
[Crossref]

X. M. Lu, S. Yu, K. Fujikawa, and C. H. Oh, “Improved error-tradeoff and error-disturbance relations in terms of measurement error components,” Phys. Rev. A 90, 042113 (2014).
[Crossref]

2013 (5)

C. Branciard, “Error-tradeoff and error-disturbance relations for incompatible quantum measurements,” Proc. Natl. Acad. Sci. USA 110, 6742–6747 (2013).
[Crossref]

G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
[Crossref]

S. Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
[Crossref]

M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
[Crossref]

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

2012 (5)

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
[Crossref]

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

F. Buscemi, “All entangled quantum states are nonlocal,” Phys. Rev. Lett. 108, 200401 (2012).
[Crossref]

F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
[Crossref]

2010 (1)

A. P. Lund and H. M. Wiseman, “Measuring measurement-disturbance relationships with weak values,” New J. Phys. 12, 093011 (2010).
[Crossref]

2004 (2)

M. J. W. Hall, “Prior information: how to circumvent the standard joint-measurement uncertainty relation,” Phys. Rev. A 69, 052113 (2004).
[Crossref]

M. Ozawa, “Uncertainty relations for joint measurements of noncommuting observables,” Phys. Lett. A 320, 367–374 (2004).
[Crossref]

2003 (2)

M. Ozawa, “Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurements,” Phys. Rev. A 67, 042105 (2003).
[Crossref]

J. Jin, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref]

2002 (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref]

1992 (1)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref]

1970 (1)

L. E. Ballentine, “The statistical interpretation of quantum mechanics,” Rev. Mod. Phys. 42, 358–381 (1970).
[Crossref]

1929 (1)

H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
[Crossref]

1927 (2)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[Crossref]

E. H. Kennard, “Zur Quantenmechanik einfacher Bewegungstypen,” Z. Phys. 44, 326–352 (1927).
[Crossref]

Badurek, G.

G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
[Crossref]

J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
[Crossref]

Baek, K.

K. Baek, T. Farrow, and W. Son, “Optimized entropic uncertainty relation for successive measurement,” Phys. Rev. A 89, 032108 (2014).
[Crossref]

Baek, S. Y.

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref]

S. Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
[Crossref]

Ballentine, L. E.

L. E. Ballentine, “The statistical interpretation of quantum mechanics,” Rev. Mod. Phys. 42, 358–381 (1970).
[Crossref]

Barchielli, A.

A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for discrete observables: relative entropy formulation,” Commun. Math. Phys. 357, 1253–1304 (2018).
[Crossref]

A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for position and momentum: relative entropy formulation,” Entropy 19, 301 (2017).
[Crossref]

Bennett, C. H.

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref]

Berta, M.

F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
[Crossref]

Biggerstaff, D. N.

M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
[Crossref]

Branciard, C.

M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
[Crossref]

C. Branciard, “Error-tradeoff and error-disturbance relations for incompatible quantum measurements,” Proc. Natl. Acad. Sci. USA 110, 6742–6747 (2013).
[Crossref]

Broome, M. A.

M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
[Crossref]

Buscemi, F.

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
[Crossref]

F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, “Noise and disturbance in quantum measurements: an information-theoretic approach,” Phys. Rev. Lett. 112, 050401 (2014).
[Crossref]

F. Buscemi, “All entangled quantum states are nonlocal,” Phys. Rev. Lett. 108, 200401 (2012).
[Crossref]

Busch, P.

T. Xiong, L. Yan, Z. Ma, F. Zhou, L. Chen, W. Yang, M. Feng, and P. Busch, “Optimal joint measurements of complementary observables by a single trapped ion,” New J. Phys. 19, 063032 (2017).
[Crossref]

P. Busch, P. Lahti, and R. F. Werner, “Heisenberg uncertainty for qubit measurements,” Phys. Rev. A 89, 012129 (2014).
[Crossref]

P. Busch, P. Lahti, and R. F. Werner, “Colloquium: quantum root-mean-square error and measurement uncertainty relations,” Rev. Mod. Phys. 86, 1261–1281 (2014).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Chen, L.

T. Xiong, L. Yan, Z. Ma, F. Zhou, L. Chen, W. Yang, M. Feng, and P. Busch, “Optimal joint measurements of complementary observables by a single trapped ion,” New J. Phys. 19, 063032 (2017).
[Crossref]

F. Zhou, L. Yan, S. Gong, Z. Ma, J. He, T. Xiong, L. Chen, W. Yang, M. Feng, and V. Vedral, “Verifying Heisenberg’s error-disturbance relation using a single trapped ion,” Sci. Adv. 2, e1600578 (2016).
[Crossref]

Chen, Z.

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
[Crossref]

Darabi, A.

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

Demirel, B.

B. Demirel, S. Sponar, G. Sulyok, M. Ozawa, and Y. Hasegawa, “Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states,” Phys. Rev. Lett. 117, 140402 (2016).
[Crossref]

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
[Crossref]

Deng, X.

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

Dressel, J.

J. Dressel and F. Nori, “Certainty in Heisenberg’s uncertainty principle: revisiting definitions for estimation errors and disturbance,” Phys. Rev. A 89, 022106 (2014).
[Crossref]

Du, J.

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
[Crossref]

Edamatsu, K.

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref]

S. Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
[Crossref]

Erhart, J.

G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
[Crossref]

J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
[Crossref]

Farrow, T.

K. Baek, T. Farrow, and W. Son, “Optimized entropic uncertainty relation for successive measurement,” Phys. Rev. A 89, 032108 (2014).
[Crossref]

Fedrizzi, A.

M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
[Crossref]

Fei, S.

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
[Crossref]

Feng, M.

T. Xiong, L. Yan, Z. Ma, F. Zhou, L. Chen, W. Yang, M. Feng, and P. Busch, “Optimal joint measurements of complementary observables by a single trapped ion,” New J. Phys. 19, 063032 (2017).
[Crossref]

F. Zhou, L. Yan, S. Gong, Z. Ma, J. He, T. Xiong, L. Chen, W. Yang, M. Feng, and V. Vedral, “Verifying Heisenberg’s error-disturbance relation using a single trapped ion,” Sci. Adv. 2, e1600578 (2016).
[Crossref]

Franz, T.

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A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for discrete observables: relative entropy formulation,” Commun. Math. Phys. 357, 1253–1304 (2018).
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X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
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X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
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X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
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W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

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[Crossref]

W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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[Crossref]

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
[Crossref]

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref]

F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, “Noise and disturbance in quantum measurements: an information-theoretic approach,” Phys. Rev. Lett. 112, 050401 (2014).
[Crossref]

G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
[Crossref]

S. Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
[Crossref]

J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
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M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
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X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

J. Jin, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref]

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W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
[Crossref]

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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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B. Demirel, S. Sponar, G. Sulyok, M. Ozawa, and Y. Hasegawa, “Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states,” Phys. Rev. Lett. 117, 140402 (2016).
[Crossref]

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
[Crossref]

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[Crossref]

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L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

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B. Demirel, S. Sponar, G. Sulyok, M. Ozawa, and Y. Hasegawa, “Experimental test of residual error-disturbance uncertainty relations for mixed spin-1/2 states,” Phys. Rev. Lett. 117, 140402 (2016).
[Crossref]

G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
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G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements,” Phys. Rev. A 88, 022110 (2013).
[Crossref]

J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185–189 (2012).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

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A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for discrete observables: relative entropy formulation,” Commun. Math. Phys. 357, 1253–1304 (2018).
[Crossref]

A. Barchielli, M. Gregoratti, and A. Toigo, “Measurement uncertainty relations for position and momentum: relative entropy formulation,” Entropy 19, 301 (2017).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
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[Crossref]

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W. Ma, Z. Ma, H. Wang, Z. Chen, Y. Liu, F. Kong, Z. Li, X. Peng, M. Shi, F. Shi, S. Fei, and J. Du, “Experimental test of Heisenberg’s measurement uncertainty relation based on statistical distances,” Phys. Rev. Lett. 116, 160405 (2016).
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Y. Liu, Z. Ma, H. Kang, D. Han, M. Wang, Z. Qin, X. Su, and K. Peng, “Experimental test of error-tradeoff uncertainty relation using a continuous-variable entangled state,” NPJ Quantum Inf. 5, 68 (2019).

X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
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P. Busch, P. Lahti, and R. F. Werner, “Colloquium: quantum root-mean-square error and measurement uncertainty relations,” Rev. Mod. Phys. 86, 1261–1281 (2014).
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F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109, 100502 (2012).
[Crossref]

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M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
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M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
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F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, “Noise and disturbance in quantum measurements: an information-theoretic approach,” Phys. Rev. Lett. 112, 050401 (2014).
[Crossref]

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M. M. Weston, M. J. W. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
[Crossref]

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X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, “Gate sequence for continuous variable one-way quantum computation,” Nat. Commun. 4, 2828 (2013).
[Crossref]

J. Jin, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
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Figures (3)

Fig. 1.
Fig. 1. (a) Principle of the test of EDR with continuous variables. A joint measurement apparatus implements the approximations of incompatible observables A and B with the compatible observables C and D by coupling the signal and meter modes via a beam-splitter. Coherent state (CS), squeezed state (SS), and thermal state (TS) serve as signal modes, and a vacuum state serves as meter mode. (b) Schematic of the experimental setup. Signal state is prepared by a NOPA. The measurement apparatus is composed by a BS, which is a combination of PBS–HWP–PBS, and two HDs. Two output modes of the BS are detected by HD1 and HD2, respectively. NOPA, nondegenerate optical parametric amplifier; BS, beam-splitter; HWP, half-waveplate; PBS, polarization beam-splitter; HD, homodyne detector; LO, local oscillator.
Fig. 2.
Fig. 2. Experimental results. (a), (b) and (c) Dependence of error (black curve) and disturbance (red curve) on the transmission efficiency of BS (T) for coherent, squeezed, and thermal states, respectively. (d), (e) and (f) Lefthand sides of the EDRs with continuous variables for coherent, squeezed, and thermal states, respectively. Green curve, Heisenberg’s EDR; red curve, Ozawa’s EDR; blue curve, Branciard’s EDR. Black line, righthand side of the EDR. All experimental data agree well with the theoretical predictions. The error bars are obtained by RMS of measurements repeated ten times.
Fig. 3.
Fig. 3. Comparison of the lower bounds of EDRs for three Gaussian states. (a) Coherent state as signal mode. (b) Squeezed state as signal mode. (c) Thermal state as signal mode. Blue curve, Heisenberg bound; orange curve, Ozawa bound; green curve, Branciard bound. Black circles show experimental data. Black dotted curve shows the theoretical prediction for the experimental parameters.

Equations (5)

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ε(A)η(B)CAB,
ε(A)η(B)+ε(A)σ(B)+σ(A)η(B)CAB.
[ε(A)2σ(B)2+σ(A)2η(B)2+2ε(A)η(B)σ(A)2σ(B)2CAB2]1/2CAB,
ε(A)=(CA)2=(T1)2σ(x^ρ)2+Rσ(x^ν)2=[(1T)x^cRx^d]2,
η(B)=(DB)2=(R1)2σ(p^ρ)2+Tσ(p^ν)2=[(1R)p^cTp^d]2,