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).

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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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

[Crossref]

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

[Crossref]

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]

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]

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

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]

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

[Crossref]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

[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).

[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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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]

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]

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]

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

[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]

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]

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]

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]

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

[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).

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]

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]

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]

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. 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]

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

[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]

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]

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]

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).

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

[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).

[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]

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

[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]

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]

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]

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).

[Crossref]

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]

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]

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

[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]

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]

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).

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]

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]

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]

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]

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]

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

[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]

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).

[Crossref]

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

[Crossref]

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

[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. 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]

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]

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]

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]

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]

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).

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]

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

[Crossref]

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]

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

[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. 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]

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]

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]

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]

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

[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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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]

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]

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]

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

[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]

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]

H. Weyl, Gruppentheorie und Quantenmechanik (University of California, 1928).

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. 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]

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]

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]

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

[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]

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]

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]

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]

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]

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]

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]

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

[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]

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]

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]

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]

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. 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]

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

[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]

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).

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

[Crossref]

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

[Crossref]

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

[Crossref]

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

[Crossref]

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

[Crossref]

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

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]

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]

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]

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]

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]

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, 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]

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]

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. 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]

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]

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]

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]

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

[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]

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

[Crossref]

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

[Crossref]

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]

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]

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]

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]

H. Weyl, Gruppentheorie und Quantenmechanik (University of California, 1928).