The positive operator valued measure (POVM), also known as the probability operator valued measure, is useful in quantum information processing. The POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and history of POVMs in quantum theory, and an expository discussion of the quantum mechanics of photonic qubits, a particular implementation of a POVM for use in the measurement of photonic qubits is reviewed.
REFERENCES
1.
A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993).
2.
J. M.
Jauch
and C.
Piron
, “Generalized Localizability
,” Helv. Phys. Acta
40
, 559
–570
(1967
).3.
S. K. Berberian, Notes on Spectral Theory (Van Nostrand, Princeton, 1966).
4.
M. A.
Neumark
, “On a Representation of Additive Operator Set Functions
,” Dokl. Acad. Sci. URSS
41
, 359
–361
(1943
).5.
B. Sz. Nagy, “Extensions of Linear Transformations in Hilbert Space which Extend Beyond this Space,” Appendix in F. Riesz and B. Sz. Nagy, Functional Analysis (Dover, New York, 1990).
6.
E. B.
Davies
and J. T.
Lewis
, “An Operational Approach to Quantum Probability
,” Commun. Math. Phys.
17
, 239
–260
(1970
).7.
E. B. Davies, Quantum Theory of Open Systems (Academic, New York, 1976).
8.
A. S.
Holevo
, “An Analogue of the Theory of Statistical Decisions in Noncommutative Probability Theory
,” Trans. Moscow Math. Soc.
26
, 133
–149
(1972
).9.
A. S.
Holevo
, “Statistical Decision Theory for Quantum Systems
,” J. Multivariate Anal.
3
, 337
–394
(1973
).10.
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
11.
P. A.
Benioff
, “Operator Valued Measures in Quantum Mechanics: Finite and Infinite Processes
,” J. Math. Phys.
13
, 231
–242
(1972
).12.
P. A.
Benioff
, “Decision Procedures in Quantum Mechanics
,” J. Math. Phys.
13
, 908
–915
(1972
).13.
P. A.
Benioff
, “Procedures in Quantum Mechanics without von Neumann’s Projection Axiom
,” J. Math. Phys.
13
, 1347
–1355
(1972
).14.
C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
15.
E. B.
Davies
, “Information and Quantum Measurement
,” IEEE Trans. Inf. Theory
IT-24
, 596
–599
(1978
).16.
P. Busch, P. J. Lathi, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, Berlin, 1996), 2nd ed.
17.
P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics (Springer, Berlin, 1995).
18.
A. K.
Ekert
, B.
Huttner
, G. M.
Palma
, and A.
Peres
, “Eavesdropping on Quantum-Cryptographical Systems
,” Phys. Rev. A
50
, 1047
–1056
(1994
).19.
H. E.
Brandt
, J. M.
Myers
, and S. J.
Lomonaco
,Jr. , “Aspects of Entangled Translucent Eavesdropping in Quantum Cryptography
,” Phys. Rev. A
56
, 4456
–4465
(1997
);Erratum, 58, 2617 (1998).
20.
C. A.
Fuchs
and A.
Peres
, “Quantum-State Disturbance Versus Information Gain: Uncertainty Relations for Quantum Information
,” Phys. Rev. A
53
, 2038
–2045
(1996
).21.
C. A.
Fuchs
, “Information Gain vs State Disturbance in Quantum Theory,” lanl e-print
quant-ph/9611010 (1996
).22.
N.
Lutkenhaus
, “Security Against Eavesdropping in Quantum Cryptography
,” Phys. Rev. A
54
, 97
–111
(1996
).23.
C. A.
Fuchs
, “Nonorthogonal Quantum States Maximize Classical Information Capacity
,” Phys. Rev. Lett.
79
, 1162
–1165
(1997
).24.
E.
Biham
, M.
Boyer
, G.
Brassard
, J.
van de Graaf
, and T.
Mor
, “Security of Quantum Key Distribution Against All Collective Attacks,” lanl e-print
quant-ph/9801022 (1998
).25.
H. E. Brandt and J. M. Myers, Invention Disclosure: POVM Receiver for Quantum Cryptography (U.S. Army Research Laboratory, Adelphi, MD, 1996).
26.
J. M.
Myers
and H. E.
Brandt
, “Converting a Positive Operator-Valued Measure to a Design for a Measuring Instrument on the Laboratory Bench
,” Meas. Sci. Technol.
8
, 1222
–1227
(1997
).27.
B.
Huttner
, A.
Muller
, J. D.
Gautier
, H.
Zbinden
, and N.
Gisin
, “Unambiguous Quantum Measurement of Nonorthogonal States
,” Phys. Rev. A
54
, 3783
–3789
(1996
).28.
C. H.
Bennett
, “Quantum Cryptography Using Any Two Nonorthogonal States
,” Phys. Rev. Lett.
68
, 3121
–3124
(1992
).29.
W. K. Wootters (private communication).
30.
J. Grossman, “Realizing Generalized Quantum Measurements on the Polarization of Photons,” Williams College senior thesis, 1996.
31.
N.
Gisin
, “Hidden Quantum Nonlocality Revealed by Local Filters
,” Phys. Lett. A
210
, 151
–156
(1996
).32.
A.
Barenco
, C. H.
Bennett
, R.
Cleve
, D. P.
DiVincenzo
, N.
Margolus
, P.
Shor
, T.
Sleator
, J. A.
Smolin
, and H.
Weinfurter
, “Elementary Gates for Quantum Computation
,” Phys. Rev. A
52
, 3457
–3467
(1995
).33.
J. Schwinger, Quantum Kinematics and Dynamics (Addison–Wesley, Redwood City, CA, 1991).
34.
J. Schwinger, “Hermann Weyl and Quantum Kinematics,” in Exact Sciences and Their Philosophical Foundations (Peter Lang, Frankfurt am Main, 1988), pp. 107–129.
35.
K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966).
36.
W. K.
Wootters
, and W. H.
Zurek
, “A Single Quantum Cannot be Cloned
,” Nature (London)
299
, 802
–803
(1982
).37.
38.
V.
Degiorgio
, “Phase Shift Between the Transmitted and the Reflected Optical Fields of a Semireflecting Lossless Mirror is π/2
,” Am. J. Phys.
48
, 81
–82
(1980
).39.
A.
Zeilinger
, “General Properties of Lossless Beam Splitters in Interferometry
,” Am. J. Phys.
49
, 882
–883
(1981
).40.
Z. Y.
Ou
and L.
Mandel
, “Derivation of Reciprocity Relations for a Beam Splitter from Energy Balance
,” Am. J. Phys.
57
, 66
–67
(1989
).
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