Abstract

It is a fundamental consequence of the superposition principle for quantum states that there must exist nonorthogonal states, that is, states that, although different, have a nonzero overlap. This finite overlap means that there is no way of determining with certainty in which of two such states a given physical system has been prepared. We review the various strategies that have been devised to discriminate optimally between nonorthogonal states and some of the optical experiments that have been performed to realize these.

© 2009 Optical Society of America

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2009 (1)

S. M. Barnett, S. Croke, “On the conditions for discrimination between quantum states with minimum error,” J. Phys. A 42, 062001 (2009).
[CrossRef]

2008 (2)

D. Qiu, “Minimum-error discrimination between mixed quantum states,” Phys. Rev. A 77, 012328 (2008).
[CrossRef]

S. Croke, E. Andersson, S. M. Barnett, “No-signaling bound on quantum state discrimination,” Phys. Rev. A 77, 012113 (2008).
[CrossRef]

2007 (7)

S. Croke, P. J. Mosley, S. M. Barnett, I. A. Walmsley, “Maximum confidence measurements and their optical implementation,” Eur. Phys. J. D 41, 589–598 (2007).
[CrossRef]

A. Montanaro, “On the distinguishability of random quantum states,” Commun. Math. Phys. 273, 619–636 (2007).
[CrossRef]

C. Zhang, G. Wang, M. Ying, “Discrimination between pure states and mixed states,” Phys. Rev. A 75, 062306 (2007).
[CrossRef]

P. Raynal, N. Lütkenhaus, “Optimal unambiguous state discrimination of two density matrices: a second class of exact solutions,” Phys. Rev. A 76, 052322 (2007).
[CrossRef]

U. Herzog, “Optimum unambiguous discrimination of two mixed states and application to a class of similar states,” Phys. Rev. A 75, 052309 (2007).
[CrossRef]

M. A. P. Touzel, R. B. A. Adamson, A. M. Steinberg, “Optimal bounded-error strategies for projective measurements in non-orthogonal state discrimination,” Phys. Rev. A 76, 062314 (2007).
[CrossRef]

J. A. Bergou, “Quantum state discrimination and selected applications,” J. Phys.: Conf. Ser. 84, 012001 (2007).
[CrossRef]

2006 (4)

C. Mochon, “Family of generalized ‘pretty good’ measurements and the minimal-error pure-state discrimination problems for which they are optimal,” Phys. Rev. A 73, 032328 (2006).
[CrossRef]

P. J. Mosley, S. Croke, I. A. Walmsley, S. M. Barnett, “Experimental realization of maximum confidence quantum state discrimination for the extraction of quantum information,” Phys. Rev. Lett. 97, 193601 (2006).
[CrossRef] [PubMed]

L. Masanes, A. Acin, N. Gisin, “General properties of nonsignaling theories,” Phys. Rev. A 73, 012112 (2006).
[CrossRef]

S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, J. Jeffers, “Maximum confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
[CrossRef] [PubMed]

2005 (3)

S. E. Ahnert, M. C. Payne, “General implementation of all possible positive-operator-value measurements of single-photon polarization states,” Phys. Rev. A 71, 012330 (2005).
[CrossRef]

U. Herzog, J. A. Bergou, “Optimum unambiguous discrimination of two mixed quantum states,” Phys. Rev. A 71, 050301 (2005).
[CrossRef]

P. Raynal, N. Lütkenhaus, “Optimal unambiguous state discrimination of two density matrices: lower bound and class of exact solutions,” Phys. Rev. A 72, 022342 (2005).
[CrossRef]

2004 (8)

Y. C. Eldar, M. Stojnic, B. Hassibi, “Optimal quantum detectors for unambiguous detection of mixed states,” Phys. Rev. A 69, 062318 (2004).
[CrossRef]

Y. Feng, R. Duan, M. Ying, “Unambiguous discrimination between mixed quantum states,” Phys. Rev. A 70, 012308 (2004).
[CrossRef]

C.-L. Chou, “Minimum-error discrimination among mirror-symmetric mixed quantum states,” Phys. Rev. A 70, 062316 (2004).
[CrossRef]

Y. C. Eldar, A. Megretski, G. C. Verghese, “Optimal detection of symmetric mixed quantum states,” IEEE Trans. Inf. Theory 50, 1198–1207 (2004).
[CrossRef]

S. M. Barnett, “Optical demonstrations of statistical decision theory for quantum systems,” Quantum Inf. Comput. 4, 450–459 (2004).

J. A. Bergou, U. Herzog, M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
[CrossRef]

A. Chefles, “Quantum states: discrimination and classical information transmission. A review of experimental progress,” Lect. Notes Phys. 649, 467–511 (Springer, 2004).
[CrossRef]

M. Mohseni, A. M. Steinberg, J. A. Bergou, “Optical realization of optimal unambiguous discrimination for pure and mixed quantum states,” Phys. Rev. Lett. 93, 200403 (2004).
[CrossRef] [PubMed]

2003 (9)

K. Hunter, “Measurement does not always aid state discrimination,” Phys. Rev. A 68, 012306 (2003).
[CrossRef]

J. Fiurášek, M. Ježek, “Optimal discrimination of mixed quantum states involving inconclusive results,” Phys. Rev. A 67, 012321 (2003).
[CrossRef]

P. Raynal, N. Lütkenhaus, S. J. van Enk, “Reduction theorems for optimal unambiguous state discrimination of density matrices,” Phys. Rev. A 68, 022308 (2003).
[CrossRef]

J. A. Bergou, U. Herzog, M. Hillery, “Quantum filtering and discrimination between sets of Boolean functions,” Phys. Rev. A 90, 257901 (2003).

C.-L. Chou, L. Y. Hsu, “Minimum-error discrimination between symmetric mixed quantum states,” Phys. Rev. A 68, 042305 (2003).
[CrossRef]

Y. C. Eldar, A. Mergretski, G. C. Verghese, “Designing optimal quantum detectors via semidefinite programming,” IEEE Trans. Inf. Theory 49, 1007–1012 (2003).
[CrossRef]

Y. C. Eldar, “A semidefinite programming approach to optimal unambiguous discrimination of quantum states,” IEEE Trans. Inf. Theory 49, 446–456 (2003).
[CrossRef]

T. Rudolph, R. W. Spekkens, P. S. Turner, “Unambiguous discrimination of mixed states,” Phys. Rev. A 68, 010301 (2003).
[CrossRef]

S. M. Barnett, A. Chefles, I. Jex, “Comparison of two unknown pure quantum states,” Phys. Lett. A 307, 189–195 (2003).
[CrossRef]

2002 (10)

S. Zhang, M. Ying, “Set discrimination of quantum states,” Phys. Rev. A 65, 062322 (2002).
[CrossRef]

M. Ježek, J. Řeháček, J. Fiurášek, “Finding optimal strategies for minimum-error quantum-state discrimination,” Phys. Rev. A 65, 060301 (2002).
[CrossRef]

X. Sun, S. Zhang, Y. Feng, M. Ying, “Mathematical nature of and a family of lower bounds for the success probability of unambiguous discrimination,” Phys. Rev. A 65, 044306 (2002).
[CrossRef]

Y. Sun, J. A. Bergou, M. Hillery, “Optimum unambiguous discrimination between subsets of nonorthogonal quantum states,” Phys. Rev. A 66, 032315 (2002).
[CrossRef]

U. Herzog, J. A. Bergou, “Minimum-error discrimination between subsets of linearly dependent quantum states,” Phys. Rev. A 65, 050305 (2002).
[CrossRef]

E. Andersson, S. M. Barnett, C. R. Gilson, K. Hunter, “Minimum-error discrimination between three mirror-symmetric states,” Phys. Rev. A 65, 052308 (2002).
[CrossRef]

H. Barnum, E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity,” J. Math. Phys. 43, 2097–2106 (2002).
[CrossRef]

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

S. M. Barnett, E. Andersson, “Bound on measurement based on the no-signaling condition,” Phys. Rev. A 65, 044307 (2002).
[CrossRef]

Y. Feng, S. Zhang, R. Duan, M. Ying, “Lower bound on inconclusive probability of unambiguous discrimination,” Phys. Rev. A 66, 062313 (2002).
[CrossRef]

2001 (7)

S. Franke-Arnold, E. Andersson, S. M. Barnett, S. Stenholm, “Generalized measurements of atomic qubits,” Phys. Rev. A 63, 052301 (2001).
[CrossRef]

R. B. M. Clarke, V. M. Kendon, A. Chefles, S. M. Barnett, E. Riis, “Experimental realization of optimal detection strategies for overcomplete states,” Phys. Rev. A 64, 012303 (2001).
[CrossRef]

J. Mizuno, M. Fujiwara, M. Akiba, T. Kawanishi, S. M. Barnett, M. Sasaki, “Optimum detection for extracting maximum information from symmetric qubit sets,” Phys. Rev. A 65, 012315 (2001).
[CrossRef]

S. M. Barnett, “Quantum limited state discrimination,” Fortschr. Phys. 49, 909–913 (2001).
[CrossRef]

Y. C. Eldar, G. D. Forney, “On quantum detection and the square-root measurement,” IEEE Trans. Inf. Theory 47, 858–872 (2001).
[CrossRef]

S. M. Barnett, “Minimum-error discrimination between multiply symmetric states,” Phys. Rev. A 64, 030303 (2001).
[CrossRef]

S. Zhang, Y. Feng, X. Sun, M. Ying, “Upper bound for the success probability of unambiguous discrimination among quantum states,” Phys. Rev. A 64, 062103 (2001).
[CrossRef]

2000 (1)

A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401–424 (2000).
[CrossRef]

1999 (3)

S. Ghosh, G. Kar, A. Roy, “Optimal cloning and no signaling,” Phys. Lett. A 261, 17–19 (1999).
[CrossRef]

M. Sasaki, S. M. Barnett, R. Jozsa, M. Osaki, O. Hirota, “Accessible information and optimal strategies for real symmetrical quantum sources,” Phys. Rev. A 59, 3325–3335 (1999).
[CrossRef]

C.-W. Zhang, C.-F. Li, G.-C. Guo, “General strategies for discrimination of quantum states,” Phys. Lett. A 261, 25–29 (1999).
[CrossRef]

1998 (7)

A. Chefles, S. M. Barnett, “Strategies for discriminating between non-orthogonal quantum states,” J. Mod. Opt. 45, 1295–1302 (1998).
[CrossRef]

N. Gisin, “Quantum cloning without signaling,” Phys. Lett. A 242, 1–2 (1998).
[CrossRef]

M. Sasaki, K. Kato, M. Izutsu, O. Hirota, “Quantum channels showing superadditivity in classical capacity,” Phys. Rev. A 58, 146–158 (1998).
[CrossRef]

L.-M. Duan, G.-C. Guo, “Probabilistic cloning and identification of linearly independent quantum states,” Phys. Rev. Lett. 80, 4999–5002 (1998).
[CrossRef]

A. Chefles, “Unambiguous discrimination between linearly independent quantum states,” Phys. Lett. A 239, 339–347 (1998).
[CrossRef]

A. Peres, D. R. Terno, “Optimal distinction between non-orthogonal quantum states,” J. Phys. A 31, 7105–7111 (1998).
[CrossRef]

A. Chefles, S. M. Barnett, “Optimum unambiguous discrimination between linearly independent symmetric states,” Phys. Lett. A 250, 223–229 (1998).
[CrossRef]

1997 (3)

M. Ban, K. Kurokawa, R. Momose, O. Hirota, “Optimum measurements for discrimination among symmetric states and parameter estimation,” IEEE Transl. J. Magn. Jpn. 36, 1269–1288 (1997).

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

S. M. Barnett, E. Riis, “Experimental demonstration of polarization discrimination at the Helstrom bound,” J. Mod. Opt. 44, 1061–1064 (1997).
[CrossRef]

1996 (2)

B. Huttner, A. Muller, G. Gautier, H. Zbinden, N. Gisin, “Unambiguous quantum measurement of nonorthogonal states,” Phys. Rev. A 54, 3783–3789 (1996).
[CrossRef] [PubMed]

P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, W. K. Wootters, “Classical information capacity of a quantum channel,” Phys. Rev. A 54, 1869–1876 (1996).
[CrossRef] [PubMed]

1995 (3)

S. J. D. Phoenix, P. D. Townsend, “How to beat the code breakers using quantum mechanics,” Contemp. Phys. 36, 165–195 (1995).
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G. Jaeger, A. Shimony, “Optimal distinction between two non-orthogonal quantum states,” Phys. Lett. A 197, 83–87 (1995).
[CrossRef]

B. Huttner, N. Imoto, N. Gisin, T. Mor, “Quantum cryptography with coherent states,” Phys. Rev. A 51, 1863–1869 (1995).
[CrossRef] [PubMed]

1994 (2)

P. Hausladen, W. K. Wootters, “A ‘pretty good’ measurement for distinguishing quantum states,” J. Mod. Opt. 41, 2385–2390 (1994).
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S. Popescu, D. Rohrlich, “Quantum nonlocality as an axiom,” Found. Phys. 24, 379–385 (1994).
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1993 (1)

L. P. Hughston, R. Jozsa, W. K. Wootters, “A complete classification of quantum ensembles having a given density matrix,” Phys. Lett. A 183, 14–18 (1993).
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1992 (1)

S. Stenholm, “Simultaneous measurement of conjugate variables,” Ann. Phys. (N.Y.) 218, 233–254 (1992).
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1990 (1)

A. Peres, “Neumark’s theorem and quantum inseparability,” Found. Phys. 20, 1441–1453 (1990).
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1988 (2)

D. Dieks, “Overlap and distinguishability of quantum states,” Phys. Lett. A 126, 303–306 (1988).
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A. Peres, “How to differentiate between non-orthogonal states,” Phys. Lett. A 128, 19–19 (1988).
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1987 (3)

I. D. Ivanovic, “How to differentiate between non-orthogonal states,” Phys. Lett. A 123, 257–259 (1987).
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N. G. Walker, “Quantum theory of multiport optical homodyning,” J. Mod. Opt. 34, 15–60 (1987).
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P. J. Bussey, “Communication and non-communication in Einstein–Rosen experiments,” Phys. Lett. A 123, 1–3 (1987).
[CrossRef]

1984 (1)

N. G. Walker, J. E. Carroll, “Simultaneous phase and amplitude measurements on optical signals using a multiport junction,” Electron. Lett. 20, 981–983 (1984).
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1983 (1)

T. F. Jordan, “Quantum correlations do not transmit signals,” Phys. Lett. A 94, 264–264 (1983).
[CrossRef]

1982 (1)

P. J. Bussey, ““Super-luminal communication in Einstein–Podolsky–Rosen experiments,” Phys. Lett. A 90, 9–12 (1982).
[CrossRef]

1980 (1)

G. C. Ghirardi, A. Rimini, T. Weber, “A general argument against superluminal transmission through the quantum mechanical measurement process,” Lett. Nuovo Cimento 27, 293–298 (1980).
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1979 (1)

A. S. Kholevo, “On asymptotically optimal hypothesis testing in quantum statistics,” Theor. Probab. Appl. 23, 411–415 (1979).
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1978 (2)

E. B. Davies, “Information and quantum measurement,” IEEE Trans. Inf. Theory IT-24, 596–599 (1978).
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E. B. Davies, “Information and quantum measurement,” IEEE Trans. Inf. Theory IT-24, 596–599 (1978).
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1975 (1)

H. P. Yuen, R. S. Kennedy, M. Lax, “Optimum testing of multiple hypotheses in quantum detection theory,” IEEE Trans. Inf. Theory IT-21, 125–134 (1975).
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1973 (1)

A. S. Holevo, “Statistical decision theory for quantum systems,” J. Multivariate Anal. 3, 337–394 (1973).
[CrossRef]

1968 (1)

C. W. Helstrom, “Detection theory and quantum mechanics II,” Inf. Control. 13, 156–171 (1968).
[CrossRef]

1967 (1)

C. W. Helstrom, “Detection theory and quantum mechanics,” Inf. Control. 10, 254–291 (1967).
[CrossRef]

1966 (1)

C. Y. She, H. Heffner, “Simultaneous measurement of noncommuting observables,” Phys. Rev. 152, 1103–1110 (1966).
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Acin, A.

L. Masanes, A. Acin, N. Gisin, “General properties of nonsignaling theories,” Phys. Rev. A 73, 012112 (2006).
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Adamson, R. B. A.

M. A. P. Touzel, R. B. A. Adamson, A. M. Steinberg, “Optimal bounded-error strategies for projective measurements in non-orthogonal state discrimination,” Phys. Rev. A 76, 062314 (2007).
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Ahnert, S. E.

S. E. Ahnert, M. C. Payne, “General implementation of all possible positive-operator-value measurements of single-photon polarization states,” Phys. Rev. A 71, 012330 (2005).
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Akiba, M.

J. Mizuno, M. Fujiwara, M. Akiba, T. Kawanishi, S. M. Barnett, M. Sasaki, “Optimum detection for extracting maximum information from symmetric qubit sets,” Phys. Rev. A 65, 012315 (2001).
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Andersson, E.

S. Croke, E. Andersson, S. M. Barnett, “No-signaling bound on quantum state discrimination,” Phys. Rev. A 77, 012113 (2008).
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S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, J. Jeffers, “Maximum confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
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S. M. Barnett, E. Andersson, “Bound on measurement based on the no-signaling condition,” Phys. Rev. A 65, 044307 (2002).
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E. Andersson, S. M. Barnett, C. R. Gilson, K. Hunter, “Minimum-error discrimination between three mirror-symmetric states,” Phys. Rev. A 65, 052308 (2002).
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S. Franke-Arnold, E. Andersson, S. M. Barnett, S. Stenholm, “Generalized measurements of atomic qubits,” Phys. Rev. A 63, 052301 (2001).
[CrossRef]

Ban, M.

M. Ban, K. Kurokawa, R. Momose, O. Hirota, “Optimum measurements for discrimination among symmetric states and parameter estimation,” IEEE Transl. J. Magn. Jpn. 36, 1269–1288 (1997).

Barnett, S. M.

S. M. Barnett, S. Croke, “On the conditions for discrimination between quantum states with minimum error,” J. Phys. A 42, 062001 (2009).
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S. Croke, E. Andersson, S. M. Barnett, “No-signaling bound on quantum state discrimination,” Phys. Rev. A 77, 012113 (2008).
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S. Croke, P. J. Mosley, S. M. Barnett, I. A. Walmsley, “Maximum confidence measurements and their optical implementation,” Eur. Phys. J. D 41, 589–598 (2007).
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S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, J. Jeffers, “Maximum confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
[CrossRef] [PubMed]

P. J. Mosley, S. Croke, I. A. Walmsley, S. M. Barnett, “Experimental realization of maximum confidence quantum state discrimination for the extraction of quantum information,” Phys. Rev. Lett. 97, 193601 (2006).
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S. M. Barnett, “Optical demonstrations of statistical decision theory for quantum systems,” Quantum Inf. Comput. 4, 450–459 (2004).

S. M. Barnett, A. Chefles, I. Jex, “Comparison of two unknown pure quantum states,” Phys. Lett. A 307, 189–195 (2003).
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S. M. Barnett, E. Andersson, “Bound on measurement based on the no-signaling condition,” Phys. Rev. A 65, 044307 (2002).
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E. Andersson, S. M. Barnett, C. R. Gilson, K. Hunter, “Minimum-error discrimination between three mirror-symmetric states,” Phys. Rev. A 65, 052308 (2002).
[CrossRef]

J. Mizuno, M. Fujiwara, M. Akiba, T. Kawanishi, S. M. Barnett, M. Sasaki, “Optimum detection for extracting maximum information from symmetric qubit sets,” Phys. Rev. A 65, 012315 (2001).
[CrossRef]

S. Franke-Arnold, E. Andersson, S. M. Barnett, S. Stenholm, “Generalized measurements of atomic qubits,” Phys. Rev. A 63, 052301 (2001).
[CrossRef]

R. B. M. Clarke, V. M. Kendon, A. Chefles, S. M. Barnett, E. Riis, “Experimental realization of optimal detection strategies for overcomplete states,” Phys. Rev. A 64, 012303 (2001).
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S. M. Barnett, “Quantum limited state discrimination,” Fortschr. Phys. 49, 909–913 (2001).
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S. M. Barnett, “Minimum-error discrimination between multiply symmetric states,” Phys. Rev. A 64, 030303 (2001).
[CrossRef]

M. Sasaki, S. M. Barnett, R. Jozsa, M. Osaki, O. Hirota, “Accessible information and optimal strategies for real symmetrical quantum sources,” Phys. Rev. A 59, 3325–3335 (1999).
[CrossRef]

A. Chefles, S. M. Barnett, “Strategies for discriminating between non-orthogonal quantum states,” J. Mod. Opt. 45, 1295–1302 (1998).
[CrossRef]

A. Chefles, S. M. Barnett, “Optimum unambiguous discrimination between linearly independent symmetric states,” Phys. Lett. A 250, 223–229 (1998).
[CrossRef]

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

S. M. Barnett, E. Riis, “Experimental demonstration of polarization discrimination at the Helstrom bound,” J. Mod. Opt. 44, 1061–1064 (1997).
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R. B. M. Clarke, A. Chefles, S. M. Barnett, E. Riis, “Experimental demonstration of optimal unambiguous state discrimination,” Phys. Rev. A 63, 040305.
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U. Herzog, J. A. Bergou, “Optimum unambiguous discrimination of two mixed quantum states,” Phys. Rev. A 71, 050301 (2005).
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J. A. Bergou, U. Herzog, M. Hillery, “Discrimination of quantum states,” Lect. Notes Phys. 649, 417–465 (2004).
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J. A. Bergou, U. Herzog, M. Hillery, “Quantum filtering and discrimination between sets of Boolean functions,” Phys. Rev. A 90, 257901 (2003).

U. Herzog, J. A. Bergou, “Minimum-error discrimination between subsets of linearly dependent quantum states,” Phys. Rev. A 65, 050305 (2002).
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Y. Sun, J. A. Bergou, M. Hillery, “Optimum unambiguous discrimination between subsets of nonorthogonal quantum states,” Phys. Rev. A 66, 032315 (2002).
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M. Kleinmann, H. Kampermann, D. Bruss, “Structural approach to unambiguous discrimination of two mixed states,” arXiv.org, arXiv:0803.1083v1 (March 7, 2008).

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P. Busch, M. Grabowski, P. Lahti, Operational Quantum Physics (Springer, 1995).

Bussey, P. J.

P. J. Bussey, “Communication and non-communication in Einstein–Rosen experiments,” Phys. Lett. A 123, 1–3 (1987).
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P. J. Bussey, ““Super-luminal communication in Einstein–Podolsky–Rosen experiments,” Phys. Lett. A 90, 9–12 (1982).
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Carroll, J. E.

N. G. Walker, J. E. Carroll, “Simultaneous phase and amplitude measurements on optical signals using a multiport junction,” Electron. Lett. 20, 981–983 (1984).
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Chefles, A.

A. Chefles, “Quantum states: discrimination and classical information transmission. A review of experimental progress,” Lect. Notes Phys. 649, 467–511 (Springer, 2004).
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S. M. Barnett, A. Chefles, I. Jex, “Comparison of two unknown pure quantum states,” Phys. Lett. A 307, 189–195 (2003).
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R. B. M. Clarke, V. M. Kendon, A. Chefles, S. M. Barnett, E. Riis, “Experimental realization of optimal detection strategies for overcomplete states,” Phys. Rev. A 64, 012303 (2001).
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A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401–424 (2000).
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A. Chefles, “Unambiguous discrimination between linearly independent quantum states,” Phys. Lett. A 239, 339–347 (1998).
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A. Chefles, S. M. Barnett, “Strategies for discriminating between non-orthogonal quantum states,” J. Mod. Opt. 45, 1295–1302 (1998).
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A. Chefles, S. M. Barnett, “Optimum unambiguous discrimination between linearly independent symmetric states,” Phys. Lett. A 250, 223–229 (1998).
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R. B. M. Clarke, A. Chefles, S. M. Barnett, E. Riis, “Experimental demonstration of optimal unambiguous state discrimination,” Phys. Rev. A 63, 040305.
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C.-L. Chou, “Minimum-error discrimination among mirror-symmetric mixed quantum states,” Phys. Rev. A 70, 062316 (2004).
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M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

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R. B. M. Clarke, V. M. Kendon, A. Chefles, S. M. Barnett, E. Riis, “Experimental realization of optimal detection strategies for overcomplete states,” Phys. Rev. A 64, 012303 (2001).
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R. B. M. Clarke, A. Chefles, S. M. Barnett, E. Riis, “Experimental demonstration of optimal unambiguous state discrimination,” Phys. Rev. A 63, 040305.
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T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, 1991).

Croke, S.

S. M. Barnett, S. Croke, “On the conditions for discrimination between quantum states with minimum error,” J. Phys. A 42, 062001 (2009).
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S. Croke, E. Andersson, S. M. Barnett, “No-signaling bound on quantum state discrimination,” Phys. Rev. A 77, 012113 (2008).
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S. Croke, P. J. Mosley, S. M. Barnett, I. A. Walmsley, “Maximum confidence measurements and their optical implementation,” Eur. Phys. J. D 41, 589–598 (2007).
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S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, J. Jeffers, “Maximum confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
[CrossRef] [PubMed]

P. J. Mosley, S. Croke, I. A. Walmsley, S. M. Barnett, “Experimental realization of maximum confidence quantum state discrimination for the extraction of quantum information,” Phys. Rev. Lett. 97, 193601 (2006).
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Davies, E. B.

E. B. Davies, “Information and quantum measurement,” IEEE Trans. Inf. Theory IT-24, 596–599 (1978).
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E. B. Davies, “Information and quantum measurement,” IEEE Trans. Inf. Theory IT-24, 596–599 (1978).
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Dieks, D.

D. Dieks, “Overlap and distinguishability of quantum states,” Phys. Lett. A 126, 303–306 (1988).
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Duan, L.-M.

L.-M. Duan, G.-C. Guo, “Probabilistic cloning and identification of linearly independent quantum states,” Phys. Rev. Lett. 80, 4999–5002 (1998).
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Duan, R.

Y. Feng, R. Duan, M. Ying, “Unambiguous discrimination between mixed quantum states,” Phys. Rev. A 70, 012308 (2004).
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Y. Feng, S. Zhang, R. Duan, M. Ying, “Lower bound on inconclusive probability of unambiguous discrimination,” Phys. Rev. A 66, 062313 (2002).
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Eldar, Y.

R. L. Kosut, I. Walmsley, Y. Eldar, H. Rabitz, “Quantum state detector design: optimal worst-case a posteriori performance,” arXiv.org, arXiv:quant-ph/0403150v1 (March 21, 2004).

Eldar, Y. C.

Y. C. Eldar, M. Stojnic, B. Hassibi, “Optimal quantum detectors for unambiguous detection of mixed states,” Phys. Rev. A 69, 062318 (2004).
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Y. C. Eldar, A. Megretski, G. C. Verghese, “Optimal detection of symmetric mixed quantum states,” IEEE Trans. Inf. Theory 50, 1198–1207 (2004).
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Y. C. Eldar, A. Mergretski, G. C. Verghese, “Designing optimal quantum detectors via semidefinite programming,” IEEE Trans. Inf. Theory 49, 1007–1012 (2003).
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Y. C. Eldar, “A semidefinite programming approach to optimal unambiguous discrimination of quantum states,” IEEE Trans. Inf. Theory 49, 446–456 (2003).
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Y. C. Eldar, G. D. Forney, “On quantum detection and the square-root measurement,” IEEE Trans. Inf. Theory 47, 858–872 (2001).
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Y. Feng, R. Duan, M. Ying, “Unambiguous discrimination between mixed quantum states,” Phys. Rev. A 70, 012308 (2004).
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X. Sun, S. Zhang, Y. Feng, M. Ying, “Mathematical nature of and a family of lower bounds for the success probability of unambiguous discrimination,” Phys. Rev. A 65, 044306 (2002).
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Y. Feng, S. Zhang, R. Duan, M. Ying, “Lower bound on inconclusive probability of unambiguous discrimination,” Phys. Rev. A 66, 062313 (2002).
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S. Zhang, Y. Feng, X. Sun, M. Ying, “Upper bound for the success probability of unambiguous discrimination among quantum states,” Phys. Rev. A 64, 062103 (2001).
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J. Fiurášek, M. Ježek, “Optimal discrimination of mixed quantum states involving inconclusive results,” Phys. Rev. A 67, 012321 (2003).
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Y. C. Eldar, G. D. Forney, “On quantum detection and the square-root measurement,” IEEE Trans. Inf. Theory 47, 858–872 (2001).
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S. Franke-Arnold, E. Andersson, S. M. Barnett, S. Stenholm, “Generalized measurements of atomic qubits,” Phys. Rev. A 63, 052301 (2001).
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Fujiwara, M.

J. Mizuno, M. Fujiwara, M. Akiba, T. Kawanishi, S. M. Barnett, M. Sasaki, “Optimum detection for extracting maximum information from symmetric qubit sets,” Phys. Rev. A 65, 012315 (2001).
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Gautier, G.

B. Huttner, A. Muller, G. Gautier, H. Zbinden, N. Gisin, “Unambiguous quantum measurement of nonorthogonal states,” Phys. Rev. A 54, 3783–3789 (1996).
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G. C. Ghirardi, A. Rimini, T. Weber, “A general argument against superluminal transmission through the quantum mechanical measurement process,” Lett. Nuovo Cimento 27, 293–298 (1980).
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S. Ghosh, G. Kar, A. Roy, “Optimal cloning and no signaling,” Phys. Lett. A 261, 17–19 (1999).
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S. Croke, E. Andersson, S. M. Barnett, C. R. Gilson, J. Jeffers, “Maximum confidence quantum measurements,” Phys. Rev. Lett. 96, 070401 (2006).
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E. Andersson, S. M. Barnett, C. R. Gilson, K. Hunter, “Minimum-error discrimination between three mirror-symmetric states,” Phys. Rev. A 65, 052308 (2002).
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L. Masanes, A. Acin, N. Gisin, “General properties of nonsignaling theories,” Phys. Rev. A 73, 012112 (2006).
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C.-W. Zhang, C.-F. Li, G.-C. Guo, “General strategies for discrimination of quantum states,” Phys. Lett. A 261, 25–29 (1999).
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L.-M. Duan, G.-C. Guo, “Probabilistic cloning and identification of linearly independent quantum states,” Phys. Rev. Lett. 80, 4999–5002 (1998).
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Y. C. Eldar, M. Stojnic, B. Hassibi, “Optimal quantum detectors for unambiguous detection of mixed states,” Phys. Rev. A 69, 062318 (2004).
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P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, W. K. Wootters, “Classical information capacity of a quantum channel,” Phys. Rev. A 54, 1869–1876 (1996).
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P. Hausladen, W. K. Wootters, “A ‘pretty good’ measurement for distinguishing quantum states,” J. Mod. Opt. 41, 2385–2390 (1994).
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C. Y. She, H. Heffner, “Simultaneous measurement of noncommuting observables,” Phys. Rev. 152, 1103–1110 (1966).
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C. W. Helstrom, “Detection theory and quantum mechanics II,” Inf. Control. 13, 156–171 (1968).
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U. Herzog, J. A. Bergou, “Optimum unambiguous discrimination of two mixed quantum states,” Phys. Rev. A 71, 050301 (2005).
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J. A. Bergou, U. Herzog, M. Hillery, “Quantum filtering and discrimination between sets of Boolean functions,” Phys. Rev. A 90, 257901 (2003).

U. Herzog, J. A. Bergou, “Minimum-error discrimination between subsets of linearly dependent quantum states,” Phys. Rev. A 65, 050305 (2002).
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Figures (13)

Fig. 1
Fig. 1

The optimal minimum error measurement for discriminating between the pure states ψ 0 , ψ 1 is a von Neumann measurement. For p 0 = p 1 = 1 2 this is a projective measurement onto the states φ 0 , φ 1 , symmetrically located on either side of the signal states and shown in blue here. For p 0 > p 1 the optimal measurement performs better when the state ψ 0 sent, shown here in light blue (labeled φ 0 , φ 1 ), is the case p 0 = 3 4 .

Fig. 2
Fig. 2

Bloch sphere representation of states. The states used in the example, along with the states onto which the optimal maximum confidence and minimum error POM elements project, are shown [81]. © 2006 by the American Physical Society.

Fig. 3
Fig. 3

Graphs showing the maximum confidence (left) and minimum error (right) figures of merit, for various values of the parameter θ for the example discussed in the text. In each case the values achieved by the optimal maximum confidence measurement are indicated by a dashed curve, and those corresponding to the optimal minimum error measurement are indicated by a solid curve.

Fig. 4
Fig. 4

Polarization of light as a two-level system, or qubit.

Fig. 5
Fig. 5

A beam splitter can be used to superpose or separate field modes. The input and output modes are labeled with the associated annihilation operators.

Fig. 6
Fig. 6

Schematic of the Barnett–Riis experiment achieving the Helstrom bound for state discrimination between two pure states. GTP, Glan–Thompson polarizer; PBS, polarizing beam splitter; PD0, PD1, photodetectors.

Fig. 7
Fig. 7

Results from the Barnett–Riis experiment demonstrating minimum error state discrimination at the Helstrom bound. Reproduced with permission from [109], http://www.informaworld.com.

Fig. 8
Fig. 8

Representation of the trine (left) and tetrad (right) states on the Poincaré sphere.

Fig. 9
Fig. 9

Schematic of the Clarke et al. experimental realization of minimum error discrimination between the trine states. PBS1–PBS4, polarizing beam splitters; HWP1–HWP3, half-wave plates; PD1–PD3, photodetectors. For details see [110].

Fig. 10
Fig. 10

Schematic of the Clarke et al. experimental realization of unambiguous discrimination between two nonorthogonal polarization states.

Fig. 11
Fig. 11

Results of the Clarke et al. experimental realization of unambiguous discrimination between two nonorthogonal polarization states. The rate of inconclusive results is shown on the left, and the error rate for each initial state is given on the right. A model taking into account the nonideal characteristics of the beam splitters was used to generate the nonideal theory plots in each case. For full details see [112]. © 2001 by the Americal Physical Society.

Fig. 12
Fig. 12

Schematic of the experimental apparatus used to demonstrate maximum confidence discrimination between three elliptical polarization states. PBS1–PBS4, polarizing beam splitters; QWP1–QWP4, quarter-wave plates; HWP1–HWP4, half-wave plates; PD0–PD2, PD?, photodetectors.

Fig. 13
Fig. 13

Results of the maximum confidence discrimination experiment, showing the confidence figure of merit for measurement outcomes 0 (red), 1 (green), and 2 (blue). Lines indicate the theoretical value of the figure of merit for the maximum confidence (dotted) and minimum error (dashed) measurement strategies. Shaded areas indicate the range of values consistent with a nonideal model, taking into account errors introduced at the polarizing beam splitters; for details see [116]. Figure reproduced from [115], © American Physical Society.

Equations (126)

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L = 1 2 ( H + i V ) H L = 1 2 0 .
O ̂ = m o m o m o m .
P ( o m ) = o m ψ 2 .
Pol ̂ = H H H + V V V .
P ( H ) = H L 2 = 1 2 .
P ( H ) = H H = P ̂ H ,
P ( o m ) = o m o m = P ̂ m .
P ( m ) = π ̂ m .
P err = P ( ψ 0 ) P ( 1 ψ 0 ) + P ( ψ 1 ) P ( 0 ψ 1 ) = p 0 ψ 0 π ̂ 1 ψ 0 + p 1 ψ 1 π ̂ 0 ψ 1 = p 0 Tr ( ( p 0 ψ 0 ψ 0 p 1 ψ 1 ψ 1 ) π ̂ 0 ) ,
ψ 0 = cos θ 0 + sin θ 1 ,
ψ 1 = cos θ 0 sin θ 1 ,
λ ± = 1 2 ( p 0 p 1 ± 1 4 p 0 p 1 cos 2 2 θ ) .
P err = 1 2 ( 1 1 4 p 0 p 1 ψ 0 ψ 1 2 ) ,
φ 0 = 1 2 ( 0 + 1 ) ,
φ 1 = 1 2 ( 0 1 ) .
P err = i = 0 N 1 p i j i Tr ( ρ ̂ i π ̂ j )
P corr = 1 P err = i = 0 N 1 p i Tr ( ρ ̂ i π ̂ i ) .
i p i ρ ̂ i π ̂ i p j ρ ̂ j 0 , j ,
π ̂ i ( p i ρ ̂ i p j ρ ̂ j ) π ̂ j = 0 , i , j .
i p i Tr ( ρ ̂ i π ̂ i ) j p j Tr ( ρ ̂ j π ̂ j ) .
j Tr ( ( i p i ρ ̂ i π ̂ i p j ρ ̂ j ) π ̂ j ) 0 .
j ( i p i π ̂ i ρ ̂ i p j ρ ̂ j ) π ̂ j = i π ̂ i ( p i ρ ̂ i j p j ρ ̂ j π ̂ j ) = 0 ,
( Γ ̂ p j ρ ̂ j ) π ̂ j = π ̂ i ( p i ρ ̂ i Γ ̂ ) = 0 , i , j .
π ̂ i = p i ρ ̂ 1 2 ρ ̂ i ρ ̂ 1 2 ,
ψ i = V ̂ ψ i 1 = V ̂ i ψ 0 , i = 0 , , N 1 ,
ρ ̂ = 1 N i = 0 N 1 ψ i ψ i = 1 N i = 0 N 1 V ̂ i ψ 0 ψ 0 V ̂ i ,
V ̂ ρ ̂ V ̂ = 1 N i = 0 N 1 V ̂ ψ i ψ i V ̂ = 1 N i = 0 N 1 V ̂ i + 1 ψ 0 ψ 0 V ̂ i + 1 = 1 N ( i = 1 N 1 V ̂ i ψ 0 ψ 0 V ̂ i + V ̂ N ψ 0 ψ 0 V ̂ N ) = ρ ̂ ,
V ̂ ρ ̂ = V ̂ ρ ̂ V ̂ V ̂ = ρ ̂ V ̂ ,
π ̂ i = 1 N ρ ̂ 1 2 ψ i ψ i ρ ̂ 1 2 = 1 N ρ ̂ 1 2 V ̂ i ψ 0 ψ 0 V ̂ i ρ ̂ 1 2 ,
ψ i ρ ̂ 1 2 ψ i ψ i ρ ̂ 1 2 ψ j ψ i ρ ̂ 1 2 ψ j ψ j ρ ̂ 1 2 ψ j = 0 .
ψ i ρ ̂ 1 2 ψ i = ψ 0 V ̂ i ρ ̂ 1 2 V ̂ i ψ 0 = ψ 0 ρ ̂ 1 2 ψ 0 , i ,
Γ ̂ = 1 N i = 0 N 1 ψ i ψ i 1 N ρ ̂ 1 2 ψ i ψ i ρ ̂ 1 2 = 1 N ψ 0 ρ ̂ 1 2 ψ 0 i = 0 N 1 1 N ψ i ψ i ρ ̂ 1 2 = 1 N ψ 0 ρ ̂ 1 2 ψ 0 ρ ̂ 1 2 .
φ ( Γ ̂ 1 N ψ i ψ i ) φ 0 , i , φ .
φ Γ ̂ φ = 1 N ψ i ρ ̂ 1 2 ψ i φ ρ ̂ 1 2 φ = 1 N ψ i ρ ̂ 1 4 ρ ̂ 1 4 ψ i φ ρ ̂ 1 4 ρ ̂ 1 4 φ 1 N ψ i ρ ̂ 1 4 ρ ̂ 1 4 φ 2 = 1 N ψ i φ 2 ,
ψ 0 = 0 ,
ψ 1 = 1 2 0 + 3 2 1 ,
ψ 2 = 1 2 0 3 2 1
π ̂ i = σ ̂ 1 2 p i ρ ̂ i σ ̂ 1 2 ,
p i ρ ̂ i p j ρ ̂ j 0 , j .
π ̂ ? = ψ 1 ψ 1 ,
π ̂ 0 = ( sin θ 0 + cos θ 1 ) ( sin θ 0 + cos θ 1 ) .
π ̂ 0 = a 0 ( sin θ 0 + cos θ 1 ) ( sin θ 0 + cos θ 1 ) ,
π ̂ 1 = a 1 ( sin θ 0 cos θ 1 ) ( sin θ 0 cos θ 1 ) ,
π ̂ ? = 1 ̂ π ̂ 0 π ̂ 1 .
P ( ? ) = p 0 ψ 0 π ̂ ? ψ 0 + p 1 ψ 1 π ̂ ? ψ 1 = 1 sin 2 2 θ ( p 0 a 0 + p 1 a 1 ) ,
π ̂ 0 = 1 2 cos 2 θ ( sin θ 0 + cos θ 1 ) ( sin θ 0 + cos θ 1 ) ,
π ̂ 1 = 1 2 cos 2 θ ( sin θ 0 cos θ 1 ) ( sin θ 0 cos θ 1 ) ,
π ̂ ? = ( 1 tan 2 θ ) 0 0 .
a 0 = 1 p 1 / p 0 cos 2 θ sin 2 2 θ ,
a 1 = 1 p 0 / p 1 cos 2 θ sin 2 2 θ ,
α i α 0 i 2 α ,
α i α 2 α 0 .
P ? = i 2 α 0 2 = 2 α 0 2 = α α ,
ψ i π ̂ j ψ i = P i δ i j ,
ψ i ψ j = ψ j ψ j δ i j ,
π ̂ j = P j ψ j ψ j 2 ψ j ψ j
π ̂ ? = 1 ̂ j π ̂ j .
ρ ̂ 0 = i λ i ( 0 ) λ i ( 0 ) λ i ( 0 ) , ρ ̂ 1 = i λ i ( 1 ) λ i ( 1 ) λ i ( 1 ) .
Λ ̂ ker ( 0 ) = 1 ̂ i λ i ( 0 ) λ i ( 0 ) ,
Λ ̂ ker ( 1 ) = 1 ̂ i λ i ( 1 ) λ i ( 1 ) ,
Tr ( ρ ̂ 0 π ̂ 1 ) = Tr ( ρ ̂ 0 Λ ̂ ker ( 0 ) π ̂ 1 Λ ̂ ker ( 0 ) ) = 0 .
P ( ρ ̂ i i ) = P ( ρ ̂ i ) P ( i ρ ̂ i ) P ( i ) .
P ( ρ ̂ i i ) = p i Tr ( ρ ̂ i π ̂ i ) Tr ( ρ ̂ π ̂ i ) ,
i π ̂ i 1 ̂ ,
π ̂ i = ρ ̂ 1 2 Q ̂ i ρ ̂ 1 2 ,
P ( ρ ̂ i i ) = Tr ( ρ ̂ 1 2 p i ρ ̂ i ρ ̂ 1 2 Q ̂ i Tr ( Q ̂ i ) ) ,
π ̂ i ρ ̂ 1 ρ ̂ i ρ ̂ 1
π ̂ i ρ ̂ 1 2 σ ̂ i ρ ̂ 1 2 ,
[ P ( ρ ̂ i i ) ] max = γ max ( ρ ̂ 1 2 p i ρ ̂ i ρ ̂ 1 2 ) ,
ψ 0 = cos θ 0 + sin θ 1 ,
ψ 1 = cos θ 0 + e 2 π i 3 sin θ 1 ,
ψ 2 = cos θ 0 + e 2 π i 3 sin θ 1 ,
ρ ̂ = cos 2 θ 0 0 + sin 2 θ 1 1 ,
φ 0 = sin θ 0 + cos θ 1 ,
φ 1 = sin θ 0 + e 2 π i 3 cos θ 1 ,
φ 2 = sin θ 0 + e 2 π i 3 cos θ 1 .
P ( ? ) = Tr ( ρ ̂ π ̂ ? ) = 1 2 ( α 0 + α 1 + α 2 ) cos 2 θ sin 2 θ .
π ̂ ? = ( 1 tan 2 θ ) 0 0 .
π ̂ i ME = 1 3 ρ ̂ 1 2 ψ i ψ i ρ ̂ 1 2 = 2 3 φ i ME φ i ME ,
φ 0 ME = 1 2 ( 0 + 1 ) ,
φ 1 ME = 1 2 ( 0 + e 2 π i 3 1 ) ,
φ 2 ME = 1 2 ( 0 + e 2 π i 3 1 ) .
γ max ( ρ ̂ 1 2 p i ρ ̂ i ρ ̂ 1 2 ) γ max ( ρ ̂ 1 2 p j ρ ̂ j ρ ̂ 1 2 ) ρ ̂ j ,
P ( ρ ̂ i i ) = p i Tr ( ρ ̂ i π ̂ i ) p i Tr ( ρ ̂ i π ̂ i ) + j i p j Tr ( ρ ̂ j π ̂ i ) .
ρ 0 = ψ 0 ψ 0 ,
ρ 1 = p 1 p 1 + p 2 ψ 1 ψ 1 + p 2 p 1 + p 2 ψ 2 ψ 2 = q 0 0 + ( 1 q ) 1 1 ,
π ̂ 0 = ψ 0 ψ 0 ,
π ̂ 1 = 1 ̂ ψ 0 ψ 0
P ( ρ ̂ i i ) avg = i P ( i ) P ( ρ ̂ i i ) = i P ( ρ ̂ i ) P ( i ρ ̂ i ) ,
H ( A : B ) = i j P ( a i , b j ) log ( P ( a i , b j ) P ( a i ) P ( b j ) ) ,
H ( A : B ) = i j p i Tr ( ρ ̂ i π ̂ j ) log ( Tr ( ρ ̂ i π ̂ j ) Tr ( ρ ̂ π ̂ j ) ) ,
π ̂ 0 = 2 3 1 1 ,
π ̂ 1 = 2 3 ( 1 2 1 + 3 2 0 ) ( 1 2 1 + 3 2 0 ) ,
π ̂ 2 = 2 3 ( 1 2 1 3 2 0 ) ( 1 2 1 3 2 0 ) .
ψ i π ̂ j ψ i = 1 2 ( 1 δ i j ) .
Ψ = p 0 ψ 0 L 0 R + 1 p 0 ψ 1 L 1 R ,
ρ ̂ R = Tr L ( Ψ Ψ ) = ( p 0 p 0 ( 1 p 0 ) cos 2 θ p 0 ( 1 p 0 ) cos 2 θ 1 p 0 ) .
ρ ̂ ? = ( ρ ? 00 ρ ? 01 ρ ? 10 ρ ? 11 )
ρ ̂ R = ( q 0 0 0 q 1 ) + q ? ( ρ ? 00 ρ ? 01 ρ ? 10 ρ ? 11 ) .
Ψ = i = 0 N 1 p i ψ i L i R ,
P ( ψ j j ) = R j ρ ̂ R j j R .
P ( ψ j j ) = R j ρ ̂ R j j R R j P ̂ D j R .
0 = H , 1 = V .
+ 45 ° = 1 2 ( 0 + 1 ) , 45 ° = 1 2 ( 0 1 ) ,
L = 1 2 ( 0 + i 1 ) , R = 1 2 ( 0 i 1 ) .
a ̂ 3 H , V = r a ̂ 1 H , V + t a ̂ 2 H , V ,
a ̂ 4 H , V = t a ̂ 1 H , V + r a ̂ 2 H , V ,
t 2 + r 2 = 1 , r t * + t r * = 0 .
a ̂ 3 H = a ̂ 2 H , a ̂ 3 V = a ̂ 1 V ,
a ̂ 4 H = a ̂ 1 H , a ̂ 4 V = a ̂ 2 V .
P err = 1 2 ( 1 sin 2 θ ) .
ψ 1 3 = 1 2 H 3 2 V ,
ψ 2 3 = 1 2 H + 3 2 V ,
ψ 3 3 = H
ψ 1 4 = 1 3 ( H + 2 e 2 π i 3 V ) ,
ψ 2 4 = 1 3 ( H + 2 e 2 π i 3 V ) ,
ψ 3 4 = 1 3 ( H + 2 V ) ,
ψ 4 4 = H
ψ 1 3 1 6 P 3 2 3 P 1 1 6 P 2 ,
ψ 2 3 1 6 P 3 + 1 6 P 1 + 2 3 P 2 ,
ψ 3 3 2 3 P 3 + 1 6 P 1 1 6 P 2 ,
ψ 0 = cos θ R + sin θ L ,
ψ 1 = cos θ R + e 2 π i 3 sin θ L ,
ψ 2 = cos θ R + e 2 π i 3 sin θ L .
H 2 states ( A : B ) = 0.196 ± 0.007 bits ,
H trine ( A : B ) = 0.491 0.027 + 0.011 bits, H tetrad ( A : B ) = 0.363 0.024 + 0.09 bits ,

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