Abstract

Quantum information protocols often rely on tomographic techniques to determine the state of the system. A popular method of encoding information is on the different paths a photon may take, e.g., parallel waveguides in integrated optics. However, reconstruction of states encoded onto a large number of paths is often prohibitively resource intensive and requires complicated experimental setups. Addressing this, we present a simple method for determining the state of a photon in a superposition of d paths using a rotating one-dimensional optical Fourier transform. We establish the theory and experimentally demonstrate the technique by measuring a wide variety of six-dimensional density matrices. The average fidelity of these with the expected state is as high as 0.9852±0.0008. This performance is comparable to or exceeds established tomographic methods for other types of systems.

© 2019 Chinese Laser Press

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References

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2017 (1)

G. B. Silva, S. Glancy, and H. M. Vasconcelos, “Investigating bias in maximum-likelihood quantum-state tomography,” Phys. Rev. A 95, 022107 (2017).
[Crossref]

2016 (1)

2015 (2)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

2014 (2)

C. Bamber and J. S. Lundeen, “Observing Dirac’s classical phase space analog to the quantum state,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

2013 (2)

L. E. Kopilovich, “Construction of non-redundant antenna configurations on square and hexagonal grids of a large size,” Exp. Astron. 36, 425–430 (2013).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

2011 (1)

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

2009 (3)

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

2008 (1)

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

2006 (1)

Z. Hradil, D. Mogilevtsev, and J. Řeháček, “Biased tomography schemes: an objective approach,” Phys. Rev. Lett. 96, 230401 (2006).
[Crossref]

2005 (1)

2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

1996 (1)

G. M. D’Ariano, C. Macchiavello, and M. G. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996).
[Crossref]

1995 (1)

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1989 (2)

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62, 2124–2127 (1989).
[Crossref]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[Crossref]

1979 (1)

J. P. Robinson, “Optimum Golomb rulers,” IEEE Trans. Comput. C-28, 943–944 (1979).
[Crossref]

1941 (1)

P. Erdös and P. Turán, “On a problem of sidon in additive number theory, and on some related problems,” J. London Math. Soc. s1-16, 212–215 (1941).
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Almeida, M. P.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Bamber, C.

C. Bamber and J. S. Lundeen, “Observing Dirac’s classical phase space analog to the quantum state,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

Banaszek, K.

Barbieri, M.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Beck, M.

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

Broome, M. A.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Clarke, L.

Clements, W. R.

Coldenstrodt-Ronge, H.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

D’Ariano, G. M.

G. M. D’Ariano, C. Macchiavello, and M. G. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996).
[Crossref]

Eisert, J.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Erdös, P.

P. Erdös and P. Turán, “On a problem of sidon in additive number theory, and on some related problems,” J. London Math. Soc. s1-16, 212–215 (1941).
[Crossref]

Feito, A.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Gates, J. C.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Gilchrist, A.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Glancy, S.

G. B. Silva, S. Glancy, and H. M. Vasconcelos, “Investigating bias in maximum-likelihood quantum-state tomography,” Phys. Rev. A 95, 022107 (2017).
[Crossref]

Gühne, O.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Horne, M. A.

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[Crossref]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Z. Hradil, D. Mogilevtsev, and J. Řeháček, “Biased tomography schemes: an objective approach,” Phys. Rev. Lett. 96, 230401 (2006).
[Crossref]

Humphreys, P. C.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

James, D. F.

D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005), pp. 509–538.

Jennewein, T.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Jin, X.-M.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Julsgaard, B.

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

Kagalwala, K. H.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Killett, B.

Kleinmann, M.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Knips, L.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Kolthammer, W. S.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Kondakci, H. E.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Kopilovich, L. E.

L. E. Kopilovich, “Construction of non-redundant antenna configurations on square and hexagonal grids of a large size,” Exp. Astron. 36, 425–430 (2013).
[Crossref]

Kröll, S.

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

Kundys, D.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Kwiat, P. G.

D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005), pp. 509–538.

Laing, A.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

Langford, N. K.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Lanyon, B. P.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Lundeen, J.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Lundeen, J. S.

C. Bamber and J. S. Lundeen, “Observing Dirac’s classical phase space analog to the quantum state,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

Macchiavello, C.

G. M. D’Ariano, C. Macchiavello, and M. G. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996).
[Crossref]

Matthews, J. C.

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

Mayer, A.

McAlister, D.

Metcalf, B. J.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
[Crossref]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Milburn, G. J.

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62, 2124–2127 (1989).
[Crossref]

Mogilevtsev, D.

Z. Hradil, D. Mogilevtsev, and J. Řeháček, “Biased tomography schemes: an objective approach,” Phys. Rev. Lett. 96, 230401 (2006).
[Crossref]

Moroder, T.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Munro, W. J.

D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005), pp. 509–538.

O’brien, J. L.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

Paris, M. G.

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M. G. Paris and M. F. Sacchi, “Quantum tomagraphy,” in Advanced Imaging and Electron Physics, 1st ed. (Academic, 2003), Vol. 128, pp. 206–309.

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A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

Plenio, M.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Politi, A.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

Pregnell, K.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Pryde, G. J.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Ralph, T.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

Ralph, T. C.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
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M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
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B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
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Z. Hradil, D. Mogilevtsev, and J. Řeháček, “Biased tomography schemes: an objective approach,” Phys. Rev. Lett. 96, 230401 (2006).
[Crossref]

Resch, K. J.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Richart, D.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
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L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
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J. P. Robinson, “Optimum Golomb rulers,” IEEE Trans. Comput. C-28, 943–944 (1979).
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A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

Sacchi, M. F.

M. G. Paris and M. F. Sacchi, “Quantum tomagraphy,” in Advanced Imaging and Electron Physics, 1st ed. (Academic, 2003), Vol. 128, pp. 206–309.

Saleh, B. E.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Sánchez-Soto, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Schwemmer, C.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
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M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
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J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
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G. B. Silva, S. Glancy, and H. M. Vasconcelos, “Investigating bias in maximum-likelihood quantum-state tomography,” Phys. Rev. A 95, 022107 (2017).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
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B. J. Smith, B. Killett, M. Raymer, I. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
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Smith, P. G. R.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Spring, J. B.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

Stefanov, A.

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Thomas-Peter, N.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
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H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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P. Erdös and P. Turán, “On a problem of sidon in additive number theory, and on some related problems,” J. London Math. Soc. s1-16, 212–215 (1941).
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G. B. Silva, S. Glancy, and H. M. Vasconcelos, “Investigating bias in maximum-likelihood quantum-state tomography,” Phys. Rev. A 95, 022107 (2017).
[Crossref]

Walmsley, I.

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
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B. J. Smith, B. Killett, M. Raymer, I. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
[Crossref]

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[Crossref]

Walther, A.

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

Weinfurter, H.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

White, A. G.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005), pp. 509–538.

Ying, Y.

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[Crossref]

Exp. Astron. (1)

L. E. Kopilovich, “Construction of non-redundant antenna configurations on square and hexagonal grids of a large size,” Exp. Astron. 36, 425–430 (2013).
[Crossref]

IEEE Trans. Comput. (1)

J. P. Robinson, “Optimum Golomb rulers,” IEEE Trans. Comput. C-28, 943–944 (1979).
[Crossref]

J. London Math. Soc. (1)

P. Erdös and P. Turán, “On a problem of sidon in additive number theory, and on some related problems,” J. London Math. Soc. s1-16, 212–215 (1941).
[Crossref]

Nat. Commun. (3)

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
[Crossref]

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5, 3275 (2014).
[Crossref]

Nat. Photonics (1)

J. C. Matthews, A. Politi, A. Stefanov, and J. L. O’brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009).
[Crossref]

Nat. Phys. (2)

J. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. Pregnell, C. Silberhorn, T. Ralph, J. Eisert, M. Plenio, and I. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2009).
[Crossref]

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Opt. Commun. (1)

G. M. D’Ariano, C. Macchiavello, and M. G. Paris, “A fictitious photons method for tomographic imaging,” Opt. Commun. 129, 6–12 (1996).
[Crossref]

Opt. Lett. (2)

Optica (1)

Phys. Rev. A (3)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

L. Rippe, B. Julsgaard, A. Walther, Y. Ying, and S. Kröll, “Experimental quantum-state tomography of a solid-state qubit,” Phys. Rev. A 77, 022307 (2008).
[Crossref]

G. B. Silva, S. Glancy, and H. M. Vasconcelos, “Investigating bias in maximum-likelihood quantum-state tomography,” Phys. Rev. A 95, 022107 (2017).
[Crossref]

Phys. Rev. Lett. (6)

Z. Hradil, D. Mogilevtsev, and J. Řeháček, “Biased tomography schemes: an objective approach,” Phys. Rev. Lett. 96, 230401 (2006).
[Crossref]

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62, 2124–2127 (1989).
[Crossref]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

C. Bamber and J. S. Lundeen, “Observing Dirac’s classical phase space analog to the quantum state,” Phys. Rev. Lett. 112, 070405 (2014).
[Crossref]

M. A. Horne, A. Shimony, and A. Zeilinger, “Two-particle interferometry,” Phys. Rev. Lett. 62, 2209–2212 (1989).
[Crossref]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

Sci. Rep. (1)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Other (2)

M. G. Paris and M. F. Sacchi, “Quantum tomagraphy,” in Advanced Imaging and Electron Physics, 1st ed. (Academic, 2003), Vol. 128, pp. 206–309.

D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” in Asymptotic Theory of Quantum Statistical Inference: Selected Papers (World Scientific, 2005), pp. 509–538.

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Figures (5)

Fig. 1.
Fig. 1. The working principle of path-encoded quantum state reconstruction. (a) Geometry: the spatial arrangement of paths. (b), (c) Paths passing through a cylindrical lens to an image sensor. Along one direction, the paths are interfered by the lens. Along the other direction, the paths are unaltered. The off-diagonal elements of the density matrix ρ are found by performing a discrete Fourier transform of the recorded interference pattern. The cylindrical lens allows for only chosen sets of paths to be interfered at a time. This allows the method to accommodate duplicate path spacings in the geometry.
Fig. 2.
Fig. 2. State reconstruction method. (a) Six paths are shown in the figure and encode the state ρ. The optical Fourier transform (OFT) axis (blue solid line) rotates to particular angles θij, of which a few are shown, to interfere each pair of paths at a time (angles are with respect to a horizontal axis along the bottom-most paths). We assign to each pair of points a line segment Lij. (b) Corresponding OFT for the eight angles required to reconstruct this particular density matrix. As only paths with angle θij between them interfere, the diagonal elements can be recovered from the remaining paths. The ky axis is always in the direction of the interference, and the x axis is perpendicular to it (example shown for θ15). (c) Each pattern is recorded and analyzed one at a time via discrete Fourier transform (DFT) by taking a one pixel wide “slice” through the interference pattern. This process is repeated for every interference pattern present in a given image. (d) Fourier transform of the interference pattern (for illustrative purposes, we plot the magnitude). The magnitude ρij is recovered from the height of the DFT at the position y¯=Lij. The normalization is obtained by summing the zero frequency peaks of each interference pattern present in the θ65 subpanel (in this example) in panel (b). All panels contain real data.
Fig. 3.
Fig. 3. Experiment demonstrating the state reconstruction method. (a) State preparation in blue box: the Rayleigh length of an 808 nm diode laser is set by a beam expander. A series of displacement crystals (xtal) and half- and quarter-wave plates (labeled by the angles ϕ, ζ, and Ω) generate the state ρ. The resulting eight-path geometry is not compatible with the tomography method, and so two paths are blocked to produce a compatible six-dimensional state. A set of HWP and QWP may be inserted to form a mixed state by rapidly spinning HWPs. The purity is a function of the wave-plate angle τ. We can also produce photon pairs via SPDC at 808 nm using a diode at λ=404  nm to pump a 15 mm ppKTP crystal. The measured g(2)(0) of the source is 0.1979±0.0005. (b) Analysis is presented in the purple box: lenses f1=1000  mm and f2=400  mm image the six paths onto a camera (an electron multiplying CCD in the case of down-converted photons). A rotating cylindrical lens (f=250  mm) performs the optical Fourier transform (OFT) along the OFT axis. A one pixel wide slice of each interference pattern is analyzed with DFT on a computer. Wider slices can be used and averaged over; however, this may reduce the visibility if there are imperfections in the interference pattern. This would include tilting of the dark fringes, or, as can be seen in the figure, if the intensity in each bright fringe is not evenly distributed. The coherences are obtained by the heights of the Fourier transform peaks, normalized by the total intensity. No filters are applied to the raw data.
Fig. 4.
Fig. 4. Experimental results. (a) Experimental (dots) and theoretical (curves) coherences ρij of the density matrix ρ. These are produced by varying the QWP angle ζ in Fig. 3. As the coherences are constrained by the experimental setup, only a few unique values appear in any given matrix. As such, data points for multiple coherences overlap. Note that error bars, obtained by averaging over multiple pictures, were omitted for clarity but range from 103 to 102. (b) Experimentally reconstructed six-dimensional state ρ(ζ=30°). Each diagram represents a 6×6 matrix, with theoretical elements to the right of each experimental element. The fidelity with the nominal input state is 0.9911 (fidelity is one if the states are identical).
Fig. 5.
Fig. 5. Experimental results. (a) Fidelity as a function of the wave-plate angles ϕ, ζ, and Ω, shown in Fig. 3. The fidelity is close to unity, meaning ρ and ρth are nearly equal. Averaged over all points, the fidelity is 0.9852±0.0008 (dashed line). (b) Purity Tr {ρ2} as a function of the HWP angle τ for a classical source and single-photon source. The single-photon source deviates due to the much shorter coherence length. (c) Reconstructed density matrix of single photons in four paths. The experimental values are labeled. The corresponding theoretical values are either 0.25 or 0. The calculated fidelity is 1.00±0.03.

Equations (7)

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P(kx)=|ψ˜(kx)|2[ρ11+ρ22+2|ρ12|cos(L12xkx+ϕ12)],
Fkx{P(kx)}(x¯)=(ρ00+ρ11)δ(x¯)+ρ12δ(x¯L12x)+ρ21δ(x¯+L12x).
ρx(ky)=ky|ρ|ky=ijdρijky|ψiyψj|ky|ψixψj|.
ρx(ky)=|ψ˜(ky)|2ijdρijeiLijyky|ψixψj|.
ρx(ky)=idρii|ψixψi|+ijdρijeiLijyky|ψixψj|.
P(xm,ky)=xm|ρx(ky)|xm=|ψ(0)|2(idρiiδxmxi+ijdρijeiLijykyδxmxiδxmxj).
Fky{P(xm,ky)}(y¯)=dkyP(xm,ky)eikyy¯=idρiiδxmxiδ(y¯)+ijdρijδ(y¯Lijy)δxmxiδxmxj.

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