Abstract

Sequences of random binary numbers created from polarization measurements of single photons were subjected to a comprehensive runs analysis. Photon pairs from a spontaneous parametric downconversion source were detected in coincidence, with one photon acting as a trigger while the other was analyzed for horizontal or vertical polarization. The resulting sequences of polarization measurements were tested for runs of consecutive vertical or horizontal outcomes against a theory of nonoverlapping runs, without numerical unbiasing. The sequences produced no statistically significant discrepancies with the predicted numbers of runs, even with multiphoton events retained.

© 2011 Optical Society of America

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2011 (1)

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

2010 (2)

2009 (3)

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

S. Scheel, “Single-photon sources—an introduction,” J. Mod. Opt. 56, 141–160 (2009).
[CrossRef]

D. Branning, S. Bhandari, and M. Beck, “Low-cost coincidence-counting electronics for undergraduate quantum optics,” Am. J. Phys. 77, 667–670 (2009).
[CrossRef]

2008 (2)

I. J. Owens, R. J. Hughes, and J. E. Nordholt, “Entangled quantum-key-distribution randomness,” Phys. Rev. A 78, 022307(2008).
[CrossRef]

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[CrossRef]

2007 (2)

M. Stipcevic and B. Medved Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78, 045104 (2007).
[CrossRef] [PubMed]

M. Fiorentino, C. Santori, S. M. Spillane, and R. G. Beausoleil, “Secure self-calibrating quantum random-bit generator,” Phys. Rev. A 75, 032334 (2007).
[CrossRef]

2005 (1)

2004 (1)

H.-Q. Ma, S.-M. Wang, D. Zhang, J.-T. Chang, L.-L. Ji, Y.-X. Hou, and L.-A. Wu, “A random number generator based on quantum entangled photon pairs,” Chin. Phys. Lett. 21, 1961–1964 (2004).
[CrossRef]

2002 (1)

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

2000 (5)

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
[CrossRef]

M. P. Silverman and W. Strange, “Experimental tests for randomness of quantum decay examined as a Markov process,” Phys. Lett. A 272, 1–9 (2000).
[CrossRef]

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests for randomness of spontaneous quantum decay,” Phys. Rev. A 61, 042106 (2000).
[CrossRef]

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact random number generator,” Rev. Sci. Instrum. 71, 1675–1680 (2000).
[CrossRef]

S. Chatterjee, M. R. Yilmaz, M. Habibuliah, and M. Laudato, Commun. Stat. Theory Methods 29, 655–675 (2000).
[CrossRef]

1999 (1)

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests of alpha-, beta-, and electron capture decays for randomness,” Phys. Lett. A 262, 265–273 (1999).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

T. Erber, “Testing the randomness of quantum mechanics: nature’s ultimate cryptogram?,” Ann. N.Y. Acad. Sci. 755, 748–756 (1995).
[CrossRef]

1992 (1)

Y. Peres, “Iterating von Neumann’s procedure for extracting random bits,” Ann. Stat. 20, 590–597 (1992).
[CrossRef]

1986 (1)

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

1982 (1)

G. S. Fishman and L. R. Moore, “A statistical evaluation of multiplicative congruential random number generators with modulus 231−1,” J. Am. Stat. Assoc. 77, 129–136 (1982).
[CrossRef]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

1944 (1)

H. Levene and J. Wolfowitz, “The covariance matrix of runs up and down,” Ann. Math. Stat. 15, 58–69 (1944).
[CrossRef]

1940 (1)

A. M. Mood, “The distribution theory of runs,” Ann. Math. Stat. 11, 367–392 (1940).
[CrossRef]

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact random number generator,” Rev. Sci. Instrum. 71, 1675–1680 (2000).
[CrossRef]

Beausoleil, R. G.

M. Fiorentino, C. Santori, S. M. Spillane, and R. G. Beausoleil, “Secure self-calibrating quantum random-bit generator,” Phys. Rev. A 75, 032334 (2007).
[CrossRef]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Beck, M.

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

D. Branning, S. Bhandari, and M. Beck, “Low-cost coincidence-counting electronics for undergraduate quantum optics,” Am. J. Phys. 77, 667–670 (2009).
[CrossRef]

Bermudez, M. V.

Bevington, P. R.

P. R. Bevington and D. K. Robertson, Data Reduction and Error Analysis (McGraw-Hill, 2003).

Bhandari, S.

D. Branning, S. Bhandari, and M. Beck, “Low-cost coincidence-counting electronics for undergraduate quantum optics,” Am. J. Phys. 77, 667–670 (2009).
[CrossRef]

Bradley, J. V.

J. V. Bradley, Distribution-Free Statistical Tests (Prentice-Hall, 1968); for more information on runs tests, see pp. 250–282.

Branning, D.

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

D. Branning and M. V. Bermudez, “Testing quantum randomness of single-photon polarization measurements with the NIST test suite,” J. Opt. Soc. Am. B 27, 1594–1602 (2010).
[CrossRef]

D. Branning, S. Bhandari, and M. Beck, “Low-cost coincidence-counting electronics for undergraduate quantum optics,” Am. J. Phys. 77, 667–670 (2009).
[CrossRef]

D. Branning, A. Katcher, and M. P. Silverman are preparing a manuscript to be called “Concatenation method for approximating distributions of nonoverlapping recurrent events.”

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[CrossRef]

Cerf, N.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Chang, J.-T.

H.-Q. Ma, S.-M. Wang, D. Zhang, J.-T. Chang, L.-L. Ji, Y.-X. Hou, and L.-A. Wu, “A random number generator based on quantum entangled photon pairs,” Chin. Phys. Lett. 21, 1961–1964 (2004).
[CrossRef]

Chatterjee, S.

S. Chatterjee, M. R. Yilmaz, M. Habibuliah, and M. Laudato, Commun. Stat. Theory Methods 29, 655–675 (2000).
[CrossRef]

Clary, B.

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

Cova, S.

Dusek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Dynes, J. F.

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[CrossRef]

Erber, T.

T. Erber, “Testing the randomness of quantum mechanics: nature’s ultimate cryptogram?,” Ann. N.Y. Acad. Sci. 755, 748–756 (1995).
[CrossRef]

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, 1950), Vol.  1, pp. 299–300.

Fiorentino, M.

M. Fiorentino, C. Santori, S. M. Spillane, and R. G. Beausoleil, “Secure self-calibrating quantum random-bit generator,” Phys. Rev. A 75, 032334 (2007).
[CrossRef]

Fishman, G. S.

G. S. Fishman and L. R. Moore, “A statistical evaluation of multiplicative congruential random number generators with modulus 231−1,” J. Am. Stat. Assoc. 77, 129–136 (1982).
[CrossRef]

Furst, M.

Ghioni, M.

Gisin, N.

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

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
[CrossRef]

Guinnard, L.

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
[CrossRef]

Guinnard, O.

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
[CrossRef]

Habibuliah, M.

S. Chatterjee, M. R. Yilmaz, M. Habibuliah, and M. Laudato, Commun. Stat. Theory Methods 29, 655–675 (2000).
[CrossRef]

Hong, C. K.

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

Hou, Y.-X.

H.-Q. Ma, S.-M. Wang, D. Zhang, J.-T. Chang, L.-L. Ji, Y.-X. Hou, and L.-A. Wu, “A random number generator based on quantum entangled photon pairs,” Chin. Phys. Lett. 21, 1961–1964 (2004).
[CrossRef]

Hughes, R. J.

I. J. Owens, R. J. Hughes, and J. E. Nordholt, “Entangled quantum-key-distribution randomness,” Phys. Rev. A 78, 022307(2008).
[CrossRef]

Jennewein, T.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact random number generator,” Rev. Sci. Instrum. 71, 1675–1680 (2000).
[CrossRef]

Ji, L.-L.

H.-Q. Ma, S.-M. Wang, D. Zhang, J.-T. Chang, L.-L. Ji, Y.-X. Hou, and L.-A. Wu, “A random number generator based on quantum entangled photon pairs,” Chin. Phys. Lett. 21, 1961–1964 (2004).
[CrossRef]

Katcher, A.

D. Branning, A. Katcher, and M. P. Silverman are preparing a manuscript to be called “Concatenation method for approximating distributions of nonoverlapping recurrent events.”

Kendall, M. G.

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Griffin1961), Vol.  2, pp. 164–165.

Khanal, S.

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

Kurtsiefer, C.

Lacaita, A.

Laudato, M.

S. Chatterjee, M. R. Yilmaz, M. Habibuliah, and M. Laudato, Commun. Stat. Theory Methods 29, 655–675 (2000).
[CrossRef]

Levene, H.

H. Levene and J. Wolfowitz, “The covariance matrix of runs up and down,” Ann. Math. Stat. 15, 58–69 (1944).
[CrossRef]

Lipscombe, T. C.

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests for randomness of spontaneous quantum decay,” Phys. Rev. A 61, 042106 (2000).
[CrossRef]

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests of alpha-, beta-, and electron capture decays for randomness,” Phys. Lett. A 262, 265–273 (1999).
[CrossRef]

Lütkenhaus, N.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Ma, H.-Q.

H.-Q. Ma, Y. Xie, and L.-A. Wu, “Random number generation based on the time of arrival of single photons,” Appl. Opt. 44, 7760–7763 (2005).
[CrossRef] [PubMed]

H.-Q. Ma, S.-M. Wang, D. Zhang, J.-T. Chang, L.-L. Ji, Y.-X. Hou, and L.-A. Wu, “A random number generator based on quantum entangled photon pairs,” Chin. Phys. Lett. 21, 1961–1964 (2004).
[CrossRef]

Mandel, L.

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986).
[CrossRef] [PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Marangon, D. G.

Migdall, A.

Mood, A. M.

A. M. Mood, “The distribution theory of runs,” Ann. Math. Stat. 11, 367–392 (1940).
[CrossRef]

Moore, L. R.

G. S. Fishman and L. R. Moore, “A statistical evaluation of multiplicative congruential random number generators with modulus 231−1,” J. Am. Stat. Assoc. 77, 129–136 (1982).
[CrossRef]

Nauerth, S.

Nordholt, J. E.

I. J. Owens, R. J. Hughes, and J. E. Nordholt, “Entangled quantum-key-distribution randomness,” Phys. Rev. A 78, 022307(2008).
[CrossRef]

Owens, I. J.

I. J. Owens, R. J. Hughes, and J. E. Nordholt, “Entangled quantum-key-distribution randomness,” Phys. Rev. A 78, 022307(2008).
[CrossRef]

Peev, M.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Peres, Y.

Y. Peres, “Iterating von Neumann’s procedure for extracting random bits,” Ann. Stat. 20, 590–597 (1992).
[CrossRef]

Ribordy, G.

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

Robertson, D. K.

P. R. Bevington and D. K. Robertson, Data Reduction and Error Analysis (McGraw-Hill, 2003).

Rogina, B. Medved

M. Stipcevic and B. Medved Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78, 045104 (2007).
[CrossRef] [PubMed]

Samori, C.

Santori, C.

M. Fiorentino, C. Santori, S. M. Spillane, and R. G. Beausoleil, “Secure self-calibrating quantum random-bit generator,” Phys. Rev. A 75, 032334 (2007).
[CrossRef]

Scarani, V.

V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[CrossRef]

Scheel, S.

S. Scheel, “Single-photon sources—an introduction,” J. Mod. Opt. 56, 141–160 (2009).
[CrossRef]

Sharpe, A. W.

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[CrossRef]

Shields, A. J.

J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008).
[CrossRef]

Shin, Y. H.

D. Branning, S. Khanal, Y. H. Shin, B. Clary, and M. Beck, “Scalable multi-photon coincidence-counting electronics,” Rev. Sci. Instrum. 82, 016102 (2011).
[CrossRef] [PubMed]

Silverman, C. R.

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests for randomness of spontaneous quantum decay,” Phys. Rev. A 61, 042106 (2000).
[CrossRef]

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests of alpha-, beta-, and electron capture decays for randomness,” Phys. Lett. A 262, 265–273 (1999).
[CrossRef]

Silverman, M. P.

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests for randomness of spontaneous quantum decay,” Phys. Rev. A 61, 042106 (2000).
[CrossRef]

M. P. Silverman and W. Strange, “Experimental tests for randomness of quantum decay examined as a Markov process,” Phys. Lett. A 272, 1–9 (2000).
[CrossRef]

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests of alpha-, beta-, and electron capture decays for randomness,” Phys. Lett. A 262, 265–273 (1999).
[CrossRef]

M. P. Silverman is preparing a manuscript to be called “A certain uncertainty—physical insights from random events.”

D. Branning, A. Katcher, and M. P. Silverman are preparing a manuscript to be called “Concatenation method for approximating distributions of nonoverlapping recurrent events.”

Soto, J.

J. Soto, “Statistical testing of random number generators,” Proceedings of the 22nd National Information Systems Security Conference (National Institute of Standards and Technology, 1999), csrc.nist.gov/groups/ST/toolkit/rng/documents/nissc-paper.pdf.

Spillane, S. M.

M. Fiorentino, C. Santori, S. M. Spillane, and R. G. Beausoleil, “Secure self-calibrating quantum random-bit generator,” Phys. Rev. A 75, 032334 (2007).
[CrossRef]

Stefanov, A.

A. Stefanov, N. Gisin, O. Guinnard, L. Guinnard, and H. Zbinden, “Optical quantum random number generator,” J. Mod. Opt. 47, 595–598 (2000).
[CrossRef]

Stipcevic, M.

M. Stipcevic and B. Medved Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78, 045104 (2007).
[CrossRef] [PubMed]

Strange, W.

M. P. Silverman, W. Strange, C. R. Silverman, and T. C. Lipscombe, “Tests for randomness of spontaneous quantum decay,” Phys. Rev. A 61, 042106 (2000).
[CrossRef]

M. P. Silverman and W. Strange, “Experimental tests for randomness of quantum decay examined as a Markov process,” Phys. Lett. A 272, 1–9 (2000).
[CrossRef]

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Other (9)

M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Griffin1961), Vol.  2, pp. 164–165.

J. Soto, “Statistical testing of random number generators,” Proceedings of the 22nd National Information Systems Security Conference (National Institute of Standards and Technology, 1999), csrc.nist.gov/groups/ST/toolkit/rng/documents/nissc-paper.pdf.

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, 1950), Vol.  1, pp. 299–300.

D. Branning, A. Katcher, and M. P. Silverman are preparing a manuscript to be called “Concatenation method for approximating distributions of nonoverlapping recurrent events.”

P. R. Bevington and D. K. Robertson, Data Reduction and Error Analysis (McGraw-Hill, 2003).

M. P. Silverman is preparing a manuscript to be called “A certain uncertainty—physical insights from random events.”

J. Von Neumann, “Various techniques used in connection with random digits,” in Monte Carlo Method, Vol. 12 of National Bureau of Standards Applied Mathematics Series (1951), pp. 36–38.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

J. V. Bradley, Distribution-Free Statistical Tests (Prentice-Hall, 1968); for more information on runs tests, see pp. 250–282.

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Figures (7)

Fig. 1
Fig. 1

Experimental arrangement for measuring single-photon polarizations. Signal and idler photon pairs are created in the PDC and counted in coincidence either at detectors A B or A B , depending on the measurement outcome for the diagonally polarized idler photon in the HV basis. A binary sequence is created by assigning “0” to the coincidence events A B and “1” to the events A B .

Fig. 2
Fig. 2

Observed frequencies of runs of 1s of length t = 6 occurring in M subsequences of Sequence Lo - μ of length 8192 bits , with interruptors (a) removed and (b) retained. The solid curves are the theoretical distributions, approximated by a concatenation method. The interruptors do not change the distribution appreciably because they occur in only 0.6% of the occupied time bins.

Fig. 3
Fig. 3

Observed frequencies of runs of length t = 6 , for subsequences of Sequence Hi - μ with interruptors (a) removed and (b) retained. The solid curves are the theoretical distributions, approximated by a concatenation method. The interruptors, which constitute 17% of the occupied time bins, shift the distribution substantially by decreasing the frequency of occurrence of runs.

Fig. 4
Fig. 4

Observed frequencies of runs of length t = 10 , for subsequences of Sequence Lo - μ with interruptors (a) removed and (b) retained. As in Fig. 2, the distributions are nearly identical because the interruptors are rare. Because the mean number of occurrences is low, the distributions for t = 10 are not well represented by the normal distribution, but can be closely approximated with the concatenation method (solid curves).

Fig. 5
Fig. 5

Observed frequencies of runs of length t = 10 , for subsequences of Sequence Hi - μ with interruptors (a) removed and (b) retained. As in Fig. 3, the interruptors shift the distribution substantially in favor of lower numbers, and the concatenation method allows the theoretical distribution to be well approximated (solid curves) even where the Gaussian approximation fails.

Fig. 6
Fig. 6

P values from χ 2 tests of Sequence Lo - μ with interruptors retained (solid diamonds) and removed (open circles) for run lengths 2–13.

Fig. 7
Fig. 7

P values from χ 2 tests of Sequence Hi - μ , with interruptors retained (solid diamonds) and removed (open circles) for run lengths 2–13.

Tables (3)

Tables Icon

Table 1 Characteristics of the Polarization Measurement Sequences Lo - μ and Hi - μ

Tables Icon

Table 2 Predicted and Observed Numbers of Runs of 1s for Sequence Lo - μ , With and Without Interruptors

Tables Icon

Table 3 Predicted and Observed Numbers of Runs of 1s for Sequence Hi - μ , With and Without Interruptors

Equations (12)

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| ψ = M | vac + η | H s | H i + O ( η 2 ) | 2 H s | 2 H i ,
( n a + n b n a )
μ t = 1 p t q p t ,
σ t 2 = 1 ( q p t ) 2 2 t + 1 q p t p q 2 .
E ( N n , t ) = n + 1 μ t + σ t 2 μ t ( μ t + 1 ) 2 μ t 2 ,
var ( N n , t ) = n σ t 2 μ t 3 + 7 μ t 2 + 2 μ t 3 μ t 4 + 2 μ t σ t 2 ( μ t 1 ) σ t 4 4 μ t 4 .
E ( N n , t ) n μ t ,
var ( N n , t ) n μ t 3 σ t 2 .
P n , t ( z ) = k = 0 p n , t , k z k .
H t ( z , s ) = n = 1 P n , t ( z ) s n = 1 P t ( s ) ( 1 s ) ( 1 z P t ( s ) ) ,
P t ( s ) = p t s t ( 1 p s ) 1 s + q p t s t + 1 .
{ Z i = A Z i 1 ( mod 2 31 1 ) ; i = 1 , 2 , } ,

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