The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators on a complex Hilbert space , and the quantum effects for a physical system can be described by the set of positive contractive operators on a complex Hilbert space . In this note, by the techniques of operator block and spectral, we give the simpler representation of and obtained the new necessary and sufficient conditions for , for and , where is the set of all orthogonal projection operators on . In particular, we get that if exists, then for and . In addition, we consider the relations between the existence of , , and , where , , , and are the positive and negative parts of .
REFERENCES
1.
2.
H.
Du
, C. Y.
Deng
, and Q. H.
Li
, “On the infimum problem of Hilbert space effects
,” Sci. China, Ser. A: Math.
51
, 320
(2006
).3.
A.
Gheondea
, S.
Gudder
, and P.
Jonas
“On the infimum of quantum effects
,” J. Math. Phys.
46
, 062102
(2005
).4.
5.
S.
Gudder
, “Examples, problems, and results in effect algebras
,” Int. J. Theor. Phys.
35
, 2365
(1996
).6.
S.
Gudder
, “Lattice properties of quantum effects
,” J. Math. Phys.
37
, 2637
(1996
).7.
R.
Kadison
, “Order properties of bounded self-adjoint operators
,” Proc. Am. Math. Soc.
34
, 505
(1951
).8.
P.
Lahti
and M.
Maczynski
, “Partial order of quantum effects
,” J. Math. Phys.
36
, 1673
(1995
).9.
Y.
Li
and H.
Du
, “A note on the infimum problem of Hilbert space effects
,” J. Math. Phys.
47
, 102103
(2006
).10.
W.
Liu
and J.
Wu
, “A representation theorem of infimum of bounded quantum observables
,” J. Math. Phys.
49
, 073521
(2008
).11.
T.
Moreland
and S.
Gudder
, “Infima of Hilbert space effects
,” Linear Algebr. Appl.
286
, 1
(1999
).12.
S.
Pulmannová
and E.
Vinceková
, “Remarks on the order for quantum observables
,” Math. Slovaca.
57
, 589
(2007
).13.
M.
Xu
, H.
Du
, and C.
Fang
, “An explicit expression of supremum of bounded quantum observables
,” J. Math. Phys.
50
, 033502
(2009
).© 2009 American Institute of Physics.
2009
American Institute of Physics
You do not currently have access to this content.