A commutative positive operator valued (POV) measure with real spectrum is characterized by the existence of a projection valued measure (the sharp reconstruction of ) with real spectrum such that can be interpreted as a randomization of . This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of and the sharp reconstruction of . The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.
REFERENCES
1.
E. B.
Davies
and J. T.
Lewis
, Commun. Math. Phys.
17
, 239
(1970
).2.
G.
Ludwig
, Foundations of Quantum Mechanics
(Springer-Verlag
, New York
, 1983
), Vol. I
.3.
A. S.
Holevo
, Probabilistics and Statistical Aspects of Quantum Theory
(North Holland
, Amsterdam
, 1982
).4.
A. S.
Holevo
, Statistical Structure of Quantum Theory
, Lecture Notes in Physics
, Vol. 67
(Springer
, New York
, 2001
).5.
A. S.
Holevo
, Rep. Math. Phys.
22
, 385
(1985
).6.
Y.
Yamamoto
and H. A.
Haus
, Rev. Mod. Phys.
58
, 1001
(1986
).7.
M.
Hillery
, R. F.
O‘Connell
, M. O.
Scully
, and E. P.
Wigner
, Phys. Rep.
106
, 121
(1984
).8.
E.
Prugovecki
, Stochastic Quantum Mechanics and Quantum Spacetime
(Reidel
, Dordrecht
, 1984
).9.
10.
P.
Busch
, M.
Grabowski
, and P.
Lahti
, Operational Quantum Physics
, Lecture Notes in Physics
Vol. 31
(Springer-Verlag
, Berlin
, 1995
).11.
M. A.
Neumark
, Izv. Akad. Nauk SSSR, Ser. Mat.
4
, 277
(1940
).12.
A. S.
Holevo
, Trans. Mosc. Math. Soc.
26
, 133
(1972
).13.
14.
S. T.
Ali
, Lect. Notes Math.
905
, 207
(1982
).15.
G.
Cattaneo
and G.
Nisticó
, J. Math. Phys.
41
, 4365
(2000
).16.
R.
Beneduci
and G.
Nisticó
, J. Math. Phys.
44
, 5461
(2003
).17.
18.
19.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics
(Academic
, New York
, 1980
).20.
H.
Martens
and W. M.
de Muynck
, Found. Phys.
20
, 357
(1990
).21.
H.
Martens
and W. M.
de Muynck
, Found. Phys.
20
, 255
(1989
).22.
F. E.
Schroeck
, Jr., Int. J. Theor. Phys.
28
, 247
(1989
).23.
24.
S. T.
Alí
and H. D.
Doebner
, J. Math. Phys.
17
, 1105
(1976
).25.
J.
Uffink
, Int. J. Theor. Phys.
33
, 199
(1994
).26.
27.
S. K.
Berberian
, Notes on Spectral Theory
, Van Nostrand Mathematical Studies
Vol. 5
(Van Nostrand
, New Jersey
, 1966
).28.
N.
Dunford
and J. T.
Schwartz
, Linear Operators
(Interscience
, New York
, 1963
), Pt. II.29.
N. I.
Akhiezer
and I. M.
Glazman
, Theory of Linear Operators in Hilbert Space
(Ungar
, New York
, 1963
).30.
J.
von Neumann
, Mathematical Foundations of Quantum Mechanics
(Princeton University Press
, Princeton
, 1955
).31.
32.
33.
J. F. C.
Kingman
and S. J.
Taylor
, Introduction to Measure and Probability
(Cambridge University Press
, Cambridge
, 1966
).34.
G.
Cattaneo
, T.
Marsico
, G.
Nisticó
, and G.
Bacciagaluppi
, Found. Phys.
27
, 1323
(1997
).35.
P.
Bush
and F. E.
Schroeck
, Found. Phys.
19
, 807
(1989
).36.
© 2007 American Institute of Physics.
2007
American Institute of Physics
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