Abstract

Photon-number resolving detectors are a fundamental building-block of optical quantum information processing protocols. A loop detector, combined with appropriate statistical processing, can be used to convert a binary on/off photon counter into a photon-number-resolving detector. Here we describe the idea of a signature of photon-counts, which may be used to more robustly reconstruct the photon number distribution of a quantum state. The methodology is applied experimentally in a 9-port loop detector operating at a telecommunications wavelength and compared directly to the approach whereby only the number of photon-counts is used to reconstruct the input distribution. The signature approach is shown to be more robust against calibration errors, exhibit reduced statistical uncertainty, and reduced reliance on a-priori assumptions about the input state.

© 2009 Optical Society of America

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2009 (1)

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

2008 (1)

A. V. Sergienko, “Quantum optics: Beyond single-photon counting”, Nat. Photon. 2, 268–269 (2008).
[Crossref]

2007 (5)

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

M. Fujiwara and M. Sasaki, “Direct measurement of photon number statistics at telecom wavelengths using a charge integration photon detector”, Appl. Opt. 46, 16, 3069–3074 (2007).
[Crossref] [PubMed]

G. A. P. Thé and R. V. Ramos, “Multiple-photon number resolving detector using fibre ring and single-photon detector”, J. Mod. Opt. 54, 8, 1187–1202 (2007).
[Crossref]

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

2006 (4)

Z. Hradil and J. Rehá#x010D;ek, “Likelihood and entropy for statistical inversion”, J. Phys: Conf. Series 36, 55–59 (2006).
[Crossref]

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics”, Phys. Rev. Lett. 97, 043602 (2006).
[Crossref] [PubMed]

G. Zambra and M. G. A. Paris, “Reconstruction of photon-number distribution using low-performance photon counters”, Phys. Rev. A 74, 063830 (2006).
[Crossref]

T. C. Ralph, “Quantum optical systems for the implementation of quantum information processing”, Rep. Prog. Phys. 69, 853–898 (2006).
[Crossref]

2005 (2)

P. P. Rohde, “Non-deterministic approximation of photon number discriminating detectors using non-discriminating detectors”, J. Opt. B: Quant. Semiclass. 7, 82–86 (2005).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

2003 (4)

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector”, Opt. Lett. 28, 1, 52–54 (2003).
[Crossref] [PubMed]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

2001 (1)

P. Kok and S. Braunstein, “Detection devices in entanglement-based optical state preparation”, Phys. Rev. A 63, 033812 (2001).
[Crossref]

1999 (2)

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

1998 (4)

G. M. D’Ariano and C. Macchiavello, “Loss-error compensation in quantum-state measurements”, Phys. Rev. A 57, 4, 3131–3133 (1998).
[Crossref]

D. Mogilevtsev, “Diagonal element inference by direct detection”, Opt. Comm. 156, 307–310 (1998).
[Crossref]

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

G. Ribordy, J-D. Gautier, H. Zbinden, and N. Gisin, “Performance of InGaAs/InP avalanche photodiodes as gated-mode photon counters”, Appl. Opt. 37, 12, 2272–2277 (1998).
[Crossref]

1997 (1)

T. Opatrný and D.-G. Welsch, “Density-matrix reconstruction by unbalanced homodyning”, Phys. Rev. A 55, 1462 (1997).
[Crossref]

1996 (1)

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

1994 (1)

M. Höbel and J. Ricka, “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors”, Rev. Sci. Instrum. 65, 7, 2326–2336 (1994).
[Crossref]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization”, Computer Journal 7, 308–313 (1965).

Achilles, D.

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics”, Phys. Rev. Lett. 97, 043602 (2006).
[Crossref] [PubMed]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Adami, C.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Agliati, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics”, arXiv:quant-ph/0810.4055v1 (2008).

Allevi, A.

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics”, arXiv:quant-ph/0810.4055v1 (2008).

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

Andreoni, A.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics”, arXiv:quant-ph/0810.4055v1 (2008).

Banaszek, K.

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector”, Opt. Lett. 28, 1, 52–54 (2003).
[Crossref] [PubMed]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Barbieri, M.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Bartlett, S. D.

S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, “Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics”, arXiv:quant-ph/0204073v1 (2002).

Bondani, M.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics”, arXiv:quant-ph/0810.4055v1 (2008).

Branning, D.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Braunstein, S.

P. Kok and S. Braunstein, “Detection devices in entanglement-based optical state preparation”, Phys. Rev. A 63, 033812 (2001).
[Crossref]

Braunstein, S. L.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Brida, G.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).

D’Ariano, G. M.

G. M. D’Ariano and C. Macchiavello, “Loss-error compensation in quantum-state measurements”, Phys. Rev. A 57, 4, 3131–3133 (1998).
[Crossref]

Diamanti, E.

S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, “Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics”, arXiv:quant-ph/0204073v1 (2002).

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Fitch, M. J.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Franson, J. D.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Fujiwara, M.

Gautier, J-D.

Genovese, M.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Gilchrist, A.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Gisin, N.

Gramegna, M.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Haderka, O.

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

Hamar, M.

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

Höbel, M.

M. Höbel and J. Ricka, “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors”, Rev. Sci. Instrum. 65, 7, 2326–2336 (1994).
[Crossref]

Hockney, G. M.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Hogue, H. H.

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

Hradil, Z.

Z. Hradil and J. Rehá#x010D;ek, “Likelihood and entropy for statistical inversion”, J. Phys: Conf. Series 36, 55–59 (2006).
[Crossref]

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

Huntington, E. H.

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

Jacobs, B. C.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

James, D. F. .V

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Jex, I.

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

Kim, J.

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

Kiss, T.

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

Kok, P.

P. Kok and S. Braunstein, “Detection devices in entanglement-based optical state preparation”, Phys. Rev. A 63, 033812 (2001).
[Crossref]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Lagaris, J. C.

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

Langford, N. K.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Lanyon, B. P.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Lee, H.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Macchiavello, C.

G. M. D’Ariano and C. Macchiavello, “Loss-error compensation in quantum-state measurements”, Phys. Rev. A 57, 4, 3131–3133 (1998).
[Crossref]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization”, Computer Journal 7, 308–313 (1965).

Migdall, A.

A. J. Pearlman, S. V. Polyakov, A. Migdall, and S. W. Nam, “Number-resolving, single photon detection with no deadtime”, Conference on CLEO/QELS, pp 1–2, May 2008.

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1995).

Mogilevtsev, D.

D. Mogilevtsev, “Diagonal element inference by direct detection”, Opt. Comm. 156, 307–310 (1998).
[Crossref]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

Nam, S. W.

A. J. Pearlman, S. V. Polyakov, A. Migdall, and S. W. Nam, “Number-resolving, single photon detection with no deadtime”, Conference on CLEO/QELS, pp 1–2, May 2008.

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization”, Computer Journal 7, 308–313 (1965).

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).

O’Brien, J. L.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Olivares, S.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

Opatrný, T.

T. Opatrný and D.-G. Welsch, “Density-matrix reconstruction by unbalanced homodyning”, Phys. Rev. A 55, 1462 (1997).
[Crossref]

Paris, M. G. A.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

G. Zambra and M. G. A. Paris, “Reconstruction of photon-number distribution using low-performance photon counters”, Phys. Rev. A 74, 063830 (2006).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

Paul, H.

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

Pearlman, A. J.

A. J. Pearlman, S. V. Polyakov, A. Migdall, and S. W. Nam, “Number-resolving, single photon detection with no deadtime”, Conference on CLEO/QELS, pp 1–2, May 2008.

Perina, J.

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

Piacentini, F.

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

Pittman, T. B.

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Polyakov, S. V.

A. J. Pearlman, S. V. Polyakov, A. Migdall, and S. W. Nam, “Number-resolving, single photon detection with no deadtime”, Conference on CLEO/QELS, pp 1–2, May 2008.

Predazzi, E.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

Press, W. H.

W. H. Press et. al, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, England, 2nd edition, 1992).

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Ralph, T. C.

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

T. C. Ralph, “Quantum optical systems for the implementation of quantum information processing”, Rep. Prog. Phys. 69, 853–898 (2006).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

Ramos, R. V.

G. A. P. Thé and R. V. Ramos, “Multiple-photon number resolving detector using fibre ring and single-photon detector”, J. Mod. Opt. 54, 8, 1187–1202 (2007).
[Crossref]

Reeds, J. A.

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

Rehá#x010D;ek, J.

Z. Hradil and J. Rehá#x010D;ek, “Likelihood and entropy for statistical inversion”, J. Phys: Conf. Series 36, 55–59 (2006).
[Crossref]

Rehácek, J.

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

Ribordy, G.

Ricka, J.

M. Höbel and J. Ricka, “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors”, Rev. Sci. Instrum. 65, 7, 2326–2336 (1994).
[Crossref]

Rohde, P. P.

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

P. P. Rohde, “Non-deterministic approximation of photon number discriminating detectors using non-discriminating detectors”, J. Opt. B: Quant. Semiclass. 7, 82–86 (2005).
[Crossref]

Rossi, A.

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

Sanders, B. C.

S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, “Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics”, arXiv:quant-ph/0204073v1 (2002).

Sasaki, M.

Sergienko, A. V.

A. V. Sergienko, “Quantum optics: Beyond single-photon counting”, Nat. Photon. 2, 268–269 (2008).
[Crossref]

Silberhorn, C.

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics”, Phys. Rev. Lett. 97, 043602 (2006).
[Crossref] [PubMed]

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Sliwa, C.

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Takeuchi, S.

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

Thé, G. A. P.

G. A. P. Thé and R. V. Ramos, “Multiple-photon number resolving detector using fibre ring and single-photon detector”, J. Mod. Opt. 54, 8, 1187–1202 (2007).
[Crossref]

Törmä, P.

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

Traina, P.

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

Vallauri, E.

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1995).

Walmsley, I.

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

Walmsley, I. A.

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics”, Phys. Rev. Lett. 97, 043602 (2006).
[Crossref] [PubMed]

K. Banaszek and I. A. Walmsley, “Photon counting with a loop detector”, Opt. Lett. 28, 1, 52–54 (2003).
[Crossref] [PubMed]

Webb, J. G.

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

Weinhold, T. J.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

Welsch, D.-G.

T. Opatrný and D.-G. Welsch, “Density-matrix reconstruction by unbalanced homodyning”, Phys. Rev. A 55, 1462 (1997).
[Crossref]

White, A. G.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Wright, M. H.

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

Yamamoto, Y.

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, “Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics”, arXiv:quant-ph/0204073v1 (2002).

Yurtsever, U.

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

Zambra, G.

G. Zambra and M. G. A. Paris, “Reconstruction of photon-number distribution using low-performance photon counters”, Phys. Rev. A 74, 063830 (2006).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

Zbinden, H.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multiphoton detection using visible light photon counter”, Appl. Phys. Lett. 74, 7, 902 (1999).
[Crossref]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system”, Appl. Phys. Lett. 74, 8, 1063 (1999).
[Crossref]

Computer Journal (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization”, Computer Journal 7, 308–313 (1965).

Int. J. Quant. Inf. (1)

G. Brida, M. Genovese, M. Gramegna, P. Traina, E. Predazzi, S. Olivares, and M. G. A. Paris, “Toward a full reconstruction of density matrix by on/off measurements”, Int. J. Quant. Inf. 7, 27–32 (2009).
[Crossref]

J. Mod. Opt. (1)

G. A. P. Thé and R. V. Ramos, “Multiple-photon number resolving detector using fibre ring and single-photon detector”, J. Mod. Opt. 54, 8, 1187–1202 (2007).
[Crossref]

J. Opt. B: Quant. Semiclass. (1)

P. P. Rohde, “Non-deterministic approximation of photon number discriminating detectors using non-discriminating detectors”, J. Opt. B: Quant. Semiclass. 7, 82–86 (2005).
[Crossref]

J. Phys: Conf. Series (1)

Z. Hradil and J. Rehá#x010D;ek, “Likelihood and entropy for statistical inversion”, J. Phys: Conf. Series 36, 55–59 (2006).
[Crossref]

Nat. Photon. (1)

A. V. Sergienko, “Quantum optics: Beyond single-photon counting”, Nat. Photon. 2, 268–269 (2008).
[Crossref]

Nature (1)

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-not gate”, Nature 426, 264–267 (2003).
[Crossref] [PubMed]

New J. Phys. (1)

P. P. Rohde, J. G. Webb, E. H. Huntington, and T. C. Ralph, “Photon number projection using non-numberre-solving detectors”, New J. Phys. 9, 233 (2007).
[Crossref]

Opt. Comm. (1)

D. Mogilevtsev, “Diagonal element inference by direct detection”, Opt. Comm. 156, 307–310 (1998).
[Crossref]

Opt. Lett. (1)

Opt. Spect. (1)

G. Brida, M. Genovese, M. G. A. Paris, F. Piacentini, E. Predazzi, and E. Vallauri, “On Reconstructing Photon Statistics by on/off Detectors: Toward the Multi-Partite Case”, Opt. Spect. 103, 1, 90–97 (2007).
[Crossref]

Phys. Rev. A (6)

G. M. D’Ariano and C. Macchiavello, “Loss-error compensation in quantum-state measurements”, Phys. Rev. A 57, 4, 3131–3133 (1998).
[Crossref]

J. Reháček, Z. Hradil, O. Haderka, J. Perina, and M. Hamar, “Multiple-photon resolving fiber-loop detector”, Phys. Rev. A 67, 061801(R) (2003).

G. Zambra and M. G. A. Paris, “Reconstruction of photon-number distribution using low-performance photon counters”, Phys. Rev. A 74, 063830 (2006).
[Crossref]

T. Opatrný and D.-G. Welsch, “Density-matrix reconstruction by unbalanced homodyning”, Phys. Rev. A 55, 1462 (1997).
[Crossref]

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors”, Phys. Rev. A 68, 043814 (2003).
[Crossref]

P. Kok and S. Braunstein, “Detection devices in entanglement-based optical state preparation”, Phys. Rev. A 63, 033812 (2001).
[Crossref]

Phys. Rev. Lett. (4)

H. Paul, P. Törmä, T. Kiss, and I. Jex, “Photon Chopping: New Way to Measure the Quantum State of Light”, Phys. Rev. Lett. 76, 2464 (1996).
[Crossref] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. .V James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement”, Phys. Rev. Lett. 99, 250505 (2007).
[Crossref]

G. Zambra, A. Andreoni, M. Bondani, M. Gramegna, M. Genovese, G. Brida, A. Rossi, and M. G. A. Paris, “Experimental reconstruction of photon statistics without photon counting”, Phys. Rev. Lett. 95, 063602 (2005).
[Crossref] [PubMed]

D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics”, Phys. Rev. Lett. 97, 043602 (2006).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

T. C. Ralph, “Quantum optical systems for the implementation of quantum information processing”, Rep. Prog. Phys. 69, 853–898 (2006).
[Crossref]

Rev. Sci. Instrum. (1)

M. Höbel and J. Ricka, “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors”, Rev. Sci. Instrum. 65, 7, 2326–2336 (1994).
[Crossref]

SIAM J. Optim. (1)

J. C. Lagaris, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions”, SIAM J. Optim. 9, 1, 112–147 (1998).
[Crossref]

Other (10)

W. H. Press et. al, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, England, 2nd edition, 1992).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1995).

G. Zambra, A. Allevi, A. Andreoni, M. Bondani, and M. G. A. Paris, “Nontrivial photon statistics with low resolution-threshold photon counters”, Int. J. Quant. Inf.5, 1–2, 305–309 (2007).

H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, “Towards photo-statistics from photon-number discriminating detectors”, J. Mod. Opt.51, 9–10, 1517–1528 (2004).

M. Bondani, A. Allevi, A. Agliati, and A. Andreoni, “Self-consistent characterization of light statistics”, arXiv:quant-ph/0810.4055v1 (2008).

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Review article: Linear optical quantum computing”, arXiv:quant-ph/0512071 (2005).

S. D. Bartlett, E. Diamanti, B. C. Sanders, and Y. Yamamoto, “Photon counting schemes and performance of non-deterministic nonlinear gates in linear optics”, arXiv:quant-ph/0204073v1 (2002).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000).

D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, I. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing”, J. Mod. Opt.51, 9–10, 1499–1515 (2004).

A. J. Pearlman, S. V. Polyakov, A. Migdall, and S. W. Nam, “Number-resolving, single photon detection with no deadtime”, Conference on CLEO/QELS, pp 1–2, May 2008.

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Figures (12)

Fig. 1.
Fig. 1.

Experimental configuration of the loop detector and diode laser source of pulsed coherent states. The fibre loop may be broken (disconnected) at point ’X’ for characterisation purposes as described in the text.

Fig. 2.
Fig. 2.

Determination of a) optimum SPDM trigger delay and b) clock frequency for the loop detector. The SPDM trigger delay is chosen to maximise the detection probability of the first time-bin. The clock frequency is chosen to maximise ∑5 i=2 pi.

Fig. 3.
Fig. 3.

Raw time bin detection (click) probabilities for ηd =10%, n̄=8.1 photons/pulse.

Fig. 4.
Fig. 4.

Characteristic conditional probabilities P(m|n) for the 1550 nm loop detector. Note the logarithmic vertical axis used to accommodate the rapidly diminishing higher order terms.

Fig. 5.
Fig. 5.

Measured time-bin detection probabilities for a) ηd =10% and b) ηd =25%, with the loop open-circuit.

Fig. 6.
Fig. 6.

Measurement of afterpulsing from time series data of Fig. 5 (ηd =25%).

Fig. 7.
Fig. 7.

Raw and corrected time bin detection probabilities for a)ηd =10% and b)ηd =25% with n̄=3.25 photons/pulse.

Fig. 8.
Fig. 8.

Predicted 9-bit signature probabilities for ηd =10%, n̄=8 photons/pulse.

Fig. 9.
Fig. 9.

Mean-square-error for coherent-state fits to raw, corrected and signature probability data for n̄=6.5±0.8 photons/pulse.

Fig. 10.
Fig. 10.

Photon-number distributions corresponding to the observed ε (MSE) minima of Fig. 9. The best fit to the a) corrected binomial data occurs with n̄=4.54 photons/pulse, and the b) signature data at n̄=5.95 photons/pulse. The photostatistics (and derived error bounds) of the n̄=6.5±0.8 photons/pulse. input state is shown for comparison.

Fig. 11.
Fig. 11.

Estimated mean photon numbers for binomial probability and signature based reconstruction techniques.

Fig. 12.
Fig. 12.

Reconstructed photon-number distributions for an input coherent state with a) n̄ i =4.6±0.8 and b) 6.5±0.8 photons/pulse. Poissonian distributions representing the same mean photon number as the reconstructed states (r =4.3 and 6.0 photons/pulse. respectively) are shown for comparison.

Tables (2)

Tables Icon

Table 1. Detection efficiencies corresponding to loop TDM bins.

Tables Icon

Table 2. P(m|n) for loop detector.

Equations (15)

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Psig (d) = n1+...+nN=nn!n1!...nN! i=1NPc (i)ni
×jd [pdc+(1pdc)[1ploss(j)nj]]
×ld[(1pdc)ploss(l)nj] .
p (m|n) =d=mpsig(d) .
pout(m) =i=mim ηm (1η)im ρii .
pclick ηρ = pdc +(1pdc) n=1k jnk jn ηn (1η)jn ρnn
p mρ =d=m psig dρ
psig(d,ρ)=n=0kρnn[n1++nN=nn!n1!nN!i=1Npc(i)ni
×jd[pdc+(1pdc)[1ploss(j)nj]]
×ld[(1pdc)ploss(l)nl]]
ε=iS(pip̂i)2
psig dρN=9 psig ([d0],ρ) N=10+psig([d1],ρ)N=10
dpsig dρ N=9=1 n , ρ
ρnn =n̅nn!Σi=1Kn̅ii!
ε =(Tr[ρ]1)2 +d(p̂sigdρN=9psigdρN=9)2

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