Abstract

The ultimate limits of measurement precision are dictated by the laws of quantum mechanics. One of the most fascinating results is that joint or simultaneous measurements of noncommuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for determining what an “optimal” joint measurement is, with corresponding different trade-off relations for the measurements. It is generally a nontrivial task to devise or implement a strategy that minimizes the joint-measurement uncertainty. Here, we implement the simplest possible technique for an optimal four-outcome joint measurement and demonstrate a type of optimal measurement that has not been realized before in a photonic setting. We experimentally investigate a joint-measurement uncertainty relation that is more fundamental in the sense that it refers only to probabilities and is independent of values assigned to measurement outcomes. Using a heralded single-photon source, we demonstrate quantum-limited performance of the scheme on single quanta. Since quantum measurements underpin many concepts in quantum information science, this study is both of fundamental interest and relevant for emerging photonic quantum technologies.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. What is meant by a “sharp” measurement is when an observable is measured by making a projective measurement in its eigenbasis.
  2. W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” Zeitschrift für Physik A Hadrons and Nuclei 43, 172–198 (1927).
  3. H. P. Robertson, “The uncertainty principle,” Phys. Rev. 34, 163–164 (1929).
    [Crossref]
  4. E. Schrödinger, “About Heisenberg uncertainty relation,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 19, 296–303 (1930).
  5. E. Arthurs and J. Kelly, “On the simultaneous measurement of a pair of conjugate observables,” Bell Syst. Tech. J. 44, 725–729 (1965).
    [Crossref]
  6. E. Arthurs and M. Goodman, “Quantum correlations: a generalized Heisenberg uncertainty relation,” Phys. Rev. Lett. 60, 2447–2449 (1988).
    [Crossref]
  7. S. Stenholm, “Simultaneous measurement of conjugate variables,” Ann. Phys. 218, 233–254 (1992).
    [Crossref]
  8. M. J. Hall, “Prior information: how to circumvent the standard joint-measurement uncertainty relation,” Phys. Rev. A 69, 052113 (2004).
    [Crossref]
  9. M. Ozawa, “Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement,” Phys. Rev. A 67, 042105 (2003).
    [Crossref]
  10. P. Busch, P. Lahti, and R. F. Werner, “Proof of Heisenberg’s error-disturbance relation,” Phys. Rev. Lett. 111, 160405 (2013).
    [Crossref]
  11. P. Busch, “Unsharp reality and joint measurements for spin observables,” Phys. Rev. D 33, 2253–2261 (1986).
    [Crossref]
  12. P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, 2001).
  13. P. Busch, P. Lahti, and R. F. Werner, “Heisenberg uncertainty for qubit measurements,” Phys. Rev. A 89, 012129 (2014).
    [Crossref]
  14. 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]
  15. M. M. Weston, M. J. Hall, M. S. Palsson, H. M. Wiseman, and G. J. Pryde, “Experimental test of universal complementarity relations,” Phys. Rev. Lett. 110, 220402 (2013).
    [Crossref]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. 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]
  21. C. Branciard, “Error-tradeoff and error-disturbance relations for incompatible quantum measurements,” Proc. Natl. Acad. Sci. USA 110, 6742–6747 (2013).
    [Crossref]
  22. T. Brougham, E. Andersson, and S. M. Barnett, “Cloning and joint measurements of incompatible components of spin,” Phys. Rev. A 73, 062319 (2006).
    [Crossref]
  23. G. Thekkadath, R. Saaltink, L. Giner, and J. Lundeen, “Determining complementary properties with quantum clones,” Phys. Rev. Lett. 119, 050405 (2017).
    [Crossref]
  24. E. Andersson, S. M. Barnett, and A. Aspect, “Joint measurements of spin, operational locality, and uncertainty,” Phys. Rev. A 72, 042104 (2005).
    [Crossref]
  25. P. Busch, “Some realizable joint measurements of complementary observables,” Found. Phys. 17, 905–937 (1987).
    [Crossref]
  26. S. M. Barnett, “Quantum information via novel measurements,” Philos. Trans. R. Soc. London, Ser. A 355, 2279–2290 (1997).
    [Crossref]
  27. Equivalently, we can assume that the measurement results are ±1/α.
  28. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
    [Crossref]
  29. A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
    [Crossref]
  30. A. A. Abbott and C. Branciard, “Noise and disturbance of qubit measurements: an information-theoretic characterization,” Phys. Rev. A 94, 062110 (2016).
    [Crossref]

2017 (2)

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]

G. Thekkadath, R. Saaltink, L. Giner, and J. Lundeen, “Determining complementary properties with quantum clones,” Phys. Rev. Lett. 119, 050405 (2017).
[Crossref]

2016 (1)

A. A. Abbott and C. Branciard, “Noise and disturbance of qubit measurements: an information-theoretic characterization,” Phys. Rev. A 94, 062110 (2016).
[Crossref]

2014 (3)

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

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]

2013 (4)

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

P. Busch, P. Lahti, and R. F. Werner, “Proof of Heisenberg’s error-disturbance relation,” Phys. Rev. Lett. 111, 160405 (2013).
[Crossref]

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

2012 (2)

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]

2011 (1)

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

2009 (1)

2006 (1)

T. Brougham, E. Andersson, and S. M. Barnett, “Cloning and joint measurements of incompatible components of spin,” Phys. Rev. A 73, 062319 (2006).
[Crossref]

2005 (1)

E. Andersson, S. M. Barnett, and A. Aspect, “Joint measurements of spin, operational locality, and uncertainty,” Phys. Rev. A 72, 042104 (2005).
[Crossref]

2004 (1)

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

2003 (1)

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

1997 (1)

S. M. Barnett, “Quantum information via novel measurements,” Philos. Trans. R. Soc. London, Ser. A 355, 2279–2290 (1997).
[Crossref]

1992 (1)

S. Stenholm, “Simultaneous measurement of conjugate variables,” Ann. Phys. 218, 233–254 (1992).
[Crossref]

1988 (1)

E. Arthurs and M. Goodman, “Quantum correlations: a generalized Heisenberg uncertainty relation,” Phys. Rev. Lett. 60, 2447–2449 (1988).
[Crossref]

1987 (1)

P. Busch, “Some realizable joint measurements of complementary observables,” Found. Phys. 17, 905–937 (1987).
[Crossref]

1986 (1)

P. Busch, “Unsharp reality and joint measurements for spin observables,” Phys. Rev. D 33, 2253–2261 (1986).
[Crossref]

1965 (1)

E. Arthurs and J. Kelly, “On the simultaneous measurement of a pair of conjugate observables,” Bell Syst. Tech. J. 44, 725–729 (1965).
[Crossref]

1930 (1)

E. Schrödinger, “About Heisenberg uncertainty relation,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 19, 296–303 (1930).

1929 (1)

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

1927 (1)

W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” Zeitschrift für Physik A Hadrons and Nuclei 43, 172–198 (1927).

Abbott, A. A.

A. A. Abbott and C. Branciard, “Noise and disturbance of qubit measurements: an information-theoretic characterization,” Phys. Rev. A 94, 062110 (2016).
[Crossref]

Alibart, O.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

Andersson, E.

T. Brougham, E. Andersson, and S. M. Barnett, “Cloning and joint measurements of incompatible components of spin,” Phys. Rev. A 73, 062319 (2006).
[Crossref]

E. Andersson, S. M. Barnett, and A. Aspect, “Joint measurements of spin, operational locality, and uncertainty,” Phys. Rev. A 72, 042104 (2005).
[Crossref]

Arthurs, E.

E. Arthurs and M. Goodman, “Quantum correlations: a generalized Heisenberg uncertainty relation,” Phys. Rev. Lett. 60, 2447–2449 (1988).
[Crossref]

E. Arthurs and J. Kelly, “On the simultaneous measurement of a pair of conjugate observables,” Bell Syst. Tech. J. 44, 725–729 (1965).
[Crossref]

Aspect, A.

E. Andersson, S. M. Barnett, and A. Aspect, “Joint measurements of spin, operational locality, and uncertainty,” Phys. Rev. A 72, 042104 (2005).
[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, 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]

Barnett, S. M.

T. Brougham, E. Andersson, and S. M. Barnett, “Cloning and joint measurements of incompatible components of spin,” Phys. Rev. A 73, 062319 (2006).
[Crossref]

E. Andersson, S. M. Barnett, and A. Aspect, “Joint measurements of spin, operational locality, and uncertainty,” Phys. Rev. A 72, 042104 (2005).
[Crossref]

S. M. Barnett, “Quantum information via novel measurements,” Philos. Trans. R. Soc. London, Ser. A 355, 2279–2290 (1997).
[Crossref]

Bell, B.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[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.

A. A. Abbott and C. Branciard, “Noise and disturbance of qubit measurements: an information-theoretic characterization,” Phys. Rev. A 94, 062110 (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]

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]

Brougham, T.

T. Brougham, E. Andersson, and S. M. Barnett, “Cloning and joint measurements of incompatible components of spin,” Phys. Rev. A 73, 062319 (2006).
[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, “Proof of Heisenberg’s error-disturbance relation,” Phys. Rev. Lett. 111, 160405 (2013).
[Crossref]

P. Busch, “Some realizable joint measurements of complementary observables,” Found. Phys. 17, 905–937 (1987).
[Crossref]

P. Busch, “Unsharp reality and joint measurements for spin observables,” Phys. Rev. D 33, 2253–2261 (1986).
[Crossref]

P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, 2001).

Cemlyn, B.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[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]

Clark, A.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[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]

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]

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]

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]

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]

Fulconis, J.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[Crossref]

Giner, L.

G. Thekkadath, R. Saaltink, L. Giner, and J. Lundeen, “Determining complementary properties with quantum clones,” Phys. Rev. Lett. 119, 050405 (2017).
[Crossref]

Goodman, M.

E. Arthurs and M. Goodman, “Quantum correlations: a generalized Heisenberg uncertainty relation,” Phys. Rev. Lett. 60, 2447–2449 (1988).
[Crossref]

Grabowski, M.

P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, 2001).

Halder, M.

Halder, M. M.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

Hall, M. J.

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

Hasegawa, Y.

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]

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).
[Crossref]

Heisenberg, W.

W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” Zeitschrift für Physik A Hadrons and Nuclei 43, 172–198 (1927).

Kaneda, F.

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]

Kelly, J.

E. Arthurs and J. Kelly, “On the simultaneous measurement of a pair of conjugate observables,” Bell Syst. Tech. J. 44, 725–729 (1965).
[Crossref]

Lahti, P.

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, “Proof of Heisenberg’s error-disturbance relation,” Phys. Rev. Lett. 111, 160405 (2013).
[Crossref]

Lahti, P. J.

P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, 2001).

Lundeen, J.

G. Thekkadath, R. Saaltink, L. Giner, and J. Lundeen, “Determining complementary properties with quantum clones,” Phys. Rev. Lett. 119, 050405 (2017).
[Crossref]

Ma, Z.

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]

Mahler, D. H.

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]

Ozawa, M.

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]

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]

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

Palsson, M. S.

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

Pryde, G. J.

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

Rarity, J. G.

A. Clark, B. Bell, J. Fulconis, M. M. Halder, B. Cemlyn, O. Alibart, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Intrinsically narrowband pair photon generation in microstructured fibres,” New J. Phys. 13, 065009 (2011).
[Crossref]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[Crossref]

Ringbauer, M.

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[Crossref]

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Other (3)

What is meant by a “sharp” measurement is when an observable is measured by making a projective measurement in its eigenbasis.

P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, 2001).

Equivalently, we can assume that the measurement results are ±1/α.

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Figures (6)

Fig. 1.
Fig. 1. Joint measurement of incompatible observables A^=a·σ^ and B^=b·σ^. The joint measurement is implemented by doing a projective measurement either of c·σ^ or of d·σ^, with probabilities p and 1p, respectively, where c and d will lie in the plane defined by a and b. The measurements correspond to projective measurements of photon polarization in appropriate bases. Here a,b,c,d are Bloch vectors, denoted in the figure using ket representations of the corresponding polarization states |a, |b, |c, |d; each Bloch vector of unit length corresponds to a pure state.
Fig. 2.
Fig. 2. Schematic of experimental realization of a deterministic scheme for joint quantum measurements: Single signal photons (623 nm) heralded by idler photons (871 nm) were generated from a four-wave mixing source in a PCF pumped at 726 nm. To realize the joint measurement of observables a·σ^ and b·σ^, the signal photons, prepared in a well-defined polarization state, are measured in either a polarization basis corresponding to a measurement of c·σ^ or to that of d·σ^, with probabilities p,1p, respectively. The random selection probability corresponding to the splitting ratio of the beam splitter is p0.7, but could also be implemented with a flip of an unbalanced classical coin. The source generates photon pairs cross-polarized to the pump, which are filtered by both the PBS and additional wideband filters (not shown), resulting in pure horizontally polarized heralded photons entering the measurement stage. Components: pulsed laser (Ti:Sapph), half-wave plate (HWP), quarter-wave plate (QWP), polarizing beam splitter (PBS), non-polarizing beam splitter (BS), dichroic mirror (DM), multimode-fiber-coupled single-photon avalanche diode (APD).
Fig. 3.
Fig. 3. Experimental results. (a) Input state tomography. Tomography has been performed in the “C measurement” arm of the apparatus as well as the “D measurement” arm on the same input state |0, each resulting in a state preparation fidelity of Fp=99.5%. The fidelity between the two resulting state reconstructions is Fc=99.9993(2)%, indicating that the measurement setups are well calibrated to each other. (b) Sharpness of the joint measurement. The quantity on the left-hand side of Eq. (4) as a function of θ=arccos(|a·b|)/2, i.e., Δα2Δβ2, which represents the contribution to measurement uncertainty purely due to performing the measurements jointly. The error bars are determined only by Poisson statistics of raw count rates. The plot shows three sets of experiments, each corresponding to a, b defining a distinct plane on the Bloch sphere (see Fig. 5). Each data point is an average of 100 runs, each of which involved the detection of 1.5×104 heralded single photons. The solid black line represents the quantum limit.
Fig. 4.
Fig. 4. Experimental values of (a) α and (b) β as functions of θ, plotted with the corresponding theoretical values; (c) product of experimental variances for the sharp measurements; (d) product of experimental variances for the joint measurements and their comparison with theory for the ideal case and infinite number of runs of the experiment.
Fig. 5.
Fig. 5. Examples of pairs of incompatible observables used in the experiments. Spin directions a, b (states |a, |b) defining the incompatible observables, along with c, d (states |c, |d) for implementing their joint measurements with (a) θ=13° in experiment 1, (b) θ=25° in experiment 1, (c) θ=13° in experiment 2, (d) θ=25° in experiment 2, (e) θ=13° in experiment 3, (f) θ=25° in experiment 3. For the experiments, a is kept constant, and b is varied such that θarccos(|a·b|)/2=1,4,7,,25°, for each of the three sets of experiments, corresponding to azimuthal angles ϕ1=160.7°, ϕ2=51.6°, ϕ3=83.7°, for experiments 1, 2, and 3, respectively.
Fig. 6.
Fig. 6. Expectation values for the individual “sharp” and joint measurements. For the three experiments, the plots show expectation values for sharp measurements of (a) a·σ^ and (b) b·σ^, (c) c·σ^ and (d) d·σ^, and expectation values for the joint measurements (e) Aj¯ and (f) Bj¯. Also plotted for comparison are the ideal theoretical predictions, which do not include the effects of experimental imperfections.

Equations (13)

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P±a=12(1^±a·σ^),
Π±a=12(1^±αa·σ^),Π±b=12(1^±βb·σ^),
|αa+βb|+|αaβb|2,
Δα2Δβ2(1α2)(1β2)α2β2sin2(2θ),
Π±=γ±21^±γk2k·σ^,
Δ2Aj/α2=(1α2A^2)/α2=(1α2)/α2+1A^2,
Aj¯=pc·σ^+(1p)d·σ^,Bj¯=pc·σ^(1p)d·σ^.
c=(αa+βb)2pandd=(αaβb)2(1p).
p=|(αa+βb)|/21p=|(αaβb)|/2.
αopt=(2p1)βoptcos(2θ),whereβopt=±{±[2(p1)p+1]2(12p)2sec2(2θ)+2(p1)p+1}12.
C^=C+CC++C,D^=D+DD++D.
a·σ^=A+AA++A,b·σ^=B+BB++B.
α=Aj¯/a·σ^,β=Bj¯/b·σ^,