Abstract

Uncertainty relations are the hallmarks of quantum physics and have been widely investigated since its original formulation. To understand and quantitatively capture the essence of preparation uncertainty in quantum interference, the uncertainty relations for unitary operators need to be investigated. Here, we report the first experimental investigation of the uncertainty relations for general unitary operators. In particular, we experimentally demonstrate the uncertainty relation for general unitary operators proved by Bagchi and Pati [ Phys. Rev. A 94, 042104 (2016)], which places a non-trivial lower bound on the sum of uncertainties and removes the triviality problem faced by the product of the uncertainties. The experimental findings agree with the predictions of quantum theory and respect the new uncertainty relation.

© 2017 Optical Society of America

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    [Crossref]
  2. W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” Z. Phy. 43, 172 (1927).
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    [Crossref]
  4. J. A. Wheeler and W. H. Zurek, “Quantum Theory and Measurement,” (Princeton University Press, 1983).
    [Crossref]
  5. C. A. Fuchs and A. Peres, “Quantum-state disturbance versus information gain: Uncertainty relations for quantum information,” Phys. Rev. A 53, 2038 (1996).
    [Crossref] [PubMed]
  6. J. M. Renes and J.-C. Boileau, “Conjectured strong complementary information tradeoff,” Phys. Rev. Lett. 103, 020402 (2009).
    [Crossref] [PubMed]
  7. W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
    [Crossref] [PubMed]
  8. O. Gühne, “Characterizing entanglement via uncertainty relations,” Phys. Rev. Lett. 92, 117903 (2004).
    [Crossref] [PubMed]
  9. J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
    [Crossref] [PubMed]
  10. M. J. W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity,” Gen. Relativ. Gravit. 37, 1505 (2005).
    [Crossref]
  11. P. Busch, T. Heinonen, and P. Lahti, “Heisenberg’s uncertainty principle,” Phys. Rep. 452, 155 (2007).
    [Crossref]
  12. 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 (2012).
    [Crossref]
  13. 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]
  14. 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] [PubMed]
  15. C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
    [Crossref]
  16. R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
    [Crossref]
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    [Crossref]
  18. 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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    [Crossref] [PubMed]
  20. 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] [PubMed]
  21. L. Maccone and A. K. Pati, “Stronger uncertainty relations for all incompatible observables,” Phys. Rev. Lett. 113, 260401 (2014).
    [Crossref]
  22. K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
    [Crossref]
  23. S. Bagchi and A. K. Pati, “Uncertainty relations for general unitary operators,” Phys. Rev. A 94, 042104 (2016).
    [Crossref]
  24. D. Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys. Rev. A 54, 1862 (1996).
    [Crossref] [PubMed]
  25. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
    [Crossref]
  26. C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
    [Crossref]
  27. P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
    [Crossref] [PubMed]
  28. Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
    [Crossref] [PubMed]
  29. P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
    [Crossref]
  30. P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
    [Crossref]
  31. Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
    [Crossref]
  32. K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
    [Crossref]
  33. X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
    [Crossref] [PubMed]
  34. X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
    [Crossref]

2017 (2)

Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
[Crossref]

K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[Crossref]

2016 (3)

X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

S. Bagchi and A. K. Pati, “Uncertainty relations for general unitary operators,” Phys. Rev. A 94, 042104 (2016).
[Crossref]

2015 (5)

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[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] [PubMed]

2014 (4)

P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (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] [PubMed]

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] [PubMed]

L. Maccone and A. K. Pati, “Stronger uncertainty relations for all incompatible observables,” Phys. Rev. Lett. 113, 260401 (2014).
[Crossref]

2013 (2)

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] [PubMed]

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]

2012 (3)

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 (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]

C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
[Crossref]

2011 (2)

C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
[Crossref]

R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
[Crossref]

2009 (1)

J. M. Renes and J.-C. Boileau, “Conjectured strong complementary information tradeoff,” Phys. Rev. Lett. 103, 020402 (2009).
[Crossref] [PubMed]

2007 (1)

P. Busch, T. Heinonen, and P. Lahti, “Heisenberg’s uncertainty principle,” Phys. Rep. 452, 155 (2007).
[Crossref]

2005 (1)

M. J. W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity,” Gen. Relativ. Gravit. 37, 1505 (2005).
[Crossref]

2004 (2)

O. Gühne, “Characterizing entanglement via uncertainty relations,” Phys. Rev. Lett. 92, 117903 (2004).
[Crossref] [PubMed]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

2003 (1)

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[Crossref] [PubMed]

1997 (1)

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
[Crossref]

1996 (2)

D. Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys. Rev. A 54, 1862 (1996).
[Crossref] [PubMed]

C. A. Fuchs and A. Peres, “Quantum-state disturbance versus information gain: Uncertainty relations for quantum information,” Phys. Rev. A 53, 2038 (1996).
[Crossref] [PubMed]

1970 (1)

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

1929 (1)

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

1927 (1)

W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” Z. Phy. 43, 172 (1927).
[Crossref]

Adaktylos, T.

C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
[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 (2012).
[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] [PubMed]

Bagchi, S.

S. Bagchi and A. K. Pati, “Uncertainty relations for general unitary operators,” Phys. Rev. A 94, 042104 (2016).
[Crossref]

Ballentine, L. E.

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

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Bian, Z.

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

Bian, Z. H.

Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
[Crossref]

K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Bian, Z.-h.

X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[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] [PubMed]

Boileau, J.-C.

J. M. Renes and J.-C. Boileau, “Conjectured strong complementary information tradeoff,” Phys. Rev. Lett. 103, 020402 (2009).
[Crossref] [PubMed]

Bowen, W. P.

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
[Crossref] [PubMed]

Boyd, R. W.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

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] [PubMed]

Brierley, S.

C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
[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] [PubMed]

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] [PubMed]

Busch, P.

P. Busch, T. Heinonen, and P. Lahti, “Heisenberg’s uncertainty principle,” Phys. Rep. 452, 155 (2007).
[Crossref]

Calderbank, A. R.

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
[Crossref]

Colbeck, R.

R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
[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.

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] [PubMed]

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] [PubMed]

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 (2012).
[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] [PubMed]

Fisher, K.

R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
[Crossref]

Fuchs, C. A.

C. A. Fuchs and A. Peres, “Quantum-state disturbance versus information gain: Uncertainty relations for quantum information,” Phys. Rev. A 53, 2038 (1996).
[Crossref] [PubMed]

Gottesman, D.

D. Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” Phys. Rev. A 54, 1862 (1996).
[Crossref] [PubMed]

Gühne, O.

O. Gühne, “Characterizing entanglement via uncertainty relations,” Phys. Rev. Lett. 92, 117903 (2004).
[Crossref] [PubMed]

Guo, G. C.

C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
[Crossref]

Hall, M. J. W.

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] [PubMed]

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] [PubMed]

M. J. W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity,” Gen. Relativ. Gravit. 37, 1505 (2005).
[Crossref]

Hamel, D. R.

R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
[Crossref]

Hasegawa, Y.

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] [PubMed]

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 (2012).
[Crossref]

Hayat, 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).
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K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
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C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
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K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
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Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
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X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
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Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
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P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
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C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
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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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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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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 (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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L. Maccone and A. K. Pati, “Stronger uncertainty relations for all incompatible observables,” Phys. Rev. Lett. 113, 260401 (2014).
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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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Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
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P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
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X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
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P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
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A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
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W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
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X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
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P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
[Crossref]

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W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90, 043601 (2003).
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A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
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A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
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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).
[Crossref]

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C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
[Crossref]

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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] [PubMed]

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 (2012).
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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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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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[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 (2012).
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P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
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Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
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K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

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K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[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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J. A. Wheeler and W. H. Zurek, “Quantum Theory and Measurement,” (Princeton University Press, 1983).
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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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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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C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
[Crossref]

Xu, X. Y.

C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
[Crossref]

Xue, P.

Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
[Crossref]

K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[Crossref]

X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[Crossref]

P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
[Crossref]

Zhan, X.

Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
[Crossref]

K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[Crossref]

X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
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K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[Crossref]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

Zhang, R.

Z. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, B. C. Sanders, and P. Xue, “Realization of single-qubit positive-operator-valued measurement via a one-dimensional photonic quantum walk,” Phys. Rev. Lett. 114, 203602 (2015).
[Crossref] [PubMed]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, “Experimental quantum-walk revival with a time-dependent coin,” Phys. Rev. Lett. 114, 140502 (2015).
[Crossref] [PubMed]

Zhang, X.

X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

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X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

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J. A. Wheeler and W. H. Zurek, “Quantum Theory and Measurement,” (Princeton University Press, 1983).
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M. J. W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity,” Gen. Relativ. Gravit. 37, 1505 (2005).
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Nat. Phys. (3)

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 (2012).
[Crossref]

C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, “Experimental investigation of the entanglement-assisted entropic uncertainty principle,” Nat. Phys. 7, 752 (2011).
[Crossref]

R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, “Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement,” Nat. Phys. 7, 757 (2011).
[Crossref]

New J. Phys. (1)

P. Xue, H. Qin, B. Tang, and B. C. Sanders, “Observation of quasiperiodic dynamics in a one-dimensional quantum walk of single photons in space,” New J. Phys. 16, 053009 (2014).
[Crossref]

Opt. Exp. (1)

X. Zhan, J. Li, H. Qin, Z.-h. Bian, and P. Xue, “Linear optical demonstration of quantum speed-up with a single qudit,” Opt. Exp. 23, 18422 (2015).
[Crossref]

Phys. Rep. (1)

P. Busch, T. Heinonen, and P. Lahti, “Heisenberg’s uncertainty principle,” Phys. Rep. 452, 155 (2007).
[Crossref]

Phys. Rev. (1)

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

Phys. Rev. A (9)

C. A. Fuchs and A. Peres, “Quantum-state disturbance versus information gain: Uncertainty relations for quantum information,” Phys. Rev. A 53, 2038 (1996).
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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]

P. Xue, R. Zhang, Z. Bian, X. Zhan, H. Qin, and B. C. Sanders, “Localized state in a two-dimensional quantum walk on a disordered lattice,” Phys. Rev. A 92, 042316 (2015).
[Crossref]

C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B. C. Hiesmayr, “Entanglement detection via mutually unbiased bases,” Phys. Rev. A 86, 022311 (2012).
[Crossref]

Z. H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95, 052338 (2017).
[Crossref]

K. K. Wang, G. C. Knee, X. Zhan, Z. H. Bian, J. Li, and P. Xue, “Optimal experimental demonstration of error-tolerant quantum witnesses,” Phys. Rev. A 95, 032122 (2017).
[Crossref]

K. Wang, X. Zhan, Z. Bian, J. Li, Y. Zhang, and P. Xue, “Experimental investigation of the stronger uncertainty relations for all incompatible observables,” Phys. Rev. A 93, 052108 (2016).
[Crossref]

S. Bagchi and A. K. Pati, “Uncertainty relations for general unitary operators,” Phys. Rev. A 94, 042104 (2016).
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Phys. Rev. Lett. (14)

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction and orthogonal geometry,” Phys. Rev. Lett. 78, 405 (1997).
[Crossref]

X. Zhan, X. Zhang, J. Li, Y. Zhang, B. C. Sanders, and P. Xue, “Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair,” Phys. Rev. Lett. 116, 090401 (2016).
[Crossref] [PubMed]

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Figures (2)
Fig. 1
Fig. 1 Experimental setup. The herald single photons are produced via type-I spontaneous parametric down-conversion in a β-barium-borate nonlinear crystal and are injected into the optical network. The first polarizing beam splitter, half-wave plate (H1) and beam displacer (BD1) are used to prepare a qutrit state |ψθ〉. Two sandwich-type sets of wave plates (Q1-H2-Q2 and Q1-H3-Q2) are used to realize the evolution operator U. The halfwave plates (H4–H7) and two beam displacers (BD2 and BD3) are used to realize the evolution operator V (or V†). The projection measurement {P1, P2} can be realized by four half-wave plates (H8–H11) and two beam displacers (BD4–BD5). Tomography of the qutrit state can be realized by half-wave plates (H12–H14), quarter-wave plate (Q3), BD6 and a polarizing beam splitter.

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Fig. 2
Fig. 2 Experimental results. The solid black line corresponds to the theoretical prediction of the LHS of the inequality, i.e., ΔU2 + ΔV2. The black triangles represent the sum of the measured uncertainties of ΔU2 and ΔV2 with the eleven states |ψθ〉. The red dashed line corresponds to the theoretical prediction of the RHS of the inequality. The red squares represent the experimental results of the RHS of inequality with the eleven states |ψθ〉. Error bar indicates the statistical uncertainty which is obtained based on assuming Poissonian statistics.

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Tables (2)

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Table 1 The setting angles of the wave plates for the unitary evolutions and projection measurements. Here “−” denotes the corresponding wave plate is removed from the optical circuit.

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Table 2 The setting angles of the wave plates for state tomography.

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Equations (5)

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Δ U 2 = 1 − | 〈 ψ | U | ψ 〉 | 2 = 1 − Tr ( | ψ U 〉 〈 ψ U | ψ 〉 〈 ψ | ) Δ V 2 = 1 − | 〈 ψ | V | ψ 〉 | 2 = 1 − Tr ( | ψ V 〉 〈 ψ V | ψ 〉 〈 ψ | )
Δ U 2 + Δ V 2 ≥ 1 + | 〈 ψ U | ψ V 〉 | 2 − 2 cos ϕ | Δ ( 3 ) |
| 〈 ψ U | ψ V 〉 | 2 = 〈 ψ | V † U | ψ 〉 〈 ψ | U † V | ψ 〉 = Tr ( V † U | ψ 〉 〈 ψ | U † V | ψ 〉 〈 ψ | )
Δ ( 3 ) = 〈 ψ | ψ U 〉 〈 ψ U | ψ V 〉 〈 ψ V | ψ 〉 = Tr ( U | ψ 〉 〈 ψ | U † V | ψ 〉 〈 ψ | V † | ψ 〉 〈 ψ | )
U = ( 1 0 0 0 e − i 2 3 π 0 0 0 e − i 4 3 π ) ,     V = ( 0 1 0 0 0 1 1 0 0 )

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