Fixed points of quantum operations. (English) Zbl 1060.81009

Arias, A.; Gheondea, A.; Gudder, S.

J. Math. Phys. 43, No. 12, 5872-5881 (2002).

Summary: Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator \(A\) is invariant under a quantum operation \(\phi\), we call \(A\) a \(\phi\)-fixed point. Physically, the \(\phi\)-fixed points are the operators that are not disturbed by the action of \(\phi\). Our main purpose is to answer the following question. If \(A\) is a \(\phi\)-fixed point, is \(A\) compatible with the operation elements of \(\phi\)? We show in general that the answer is no and we give some sufficient conditions under which the answer is yes. Our results follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

MSC:

81R15	Operator algebra methods applied to problems in quantum theory
81P15	Quantum measurement theory, state operations, state preparations
47H10	Fixed-point theorems
47N50	Applications of operator theory in the physical sciences

References:

[1]	Gudder S., J. Math. Phys. 42 pp 5212– (2001) · Zbl 1018.81005 · doi:10.1063/1.1407837
[2]	Busch P., Phys. Lett. A 249 pp 10– (1998) · doi:10.1016/S0375-9601(98)00704-X
[3]	Bratteli O., J. Operator Theory 43 pp 97– (2000)
[4]	Tomiyama J., Proc. Jpn. Acad. 33 pp 608– (1957) · Zbl 0081.11201 · doi:10.3792/pja/1195524885
[5]	Hakeda J., Tohoku Math. J. 19 pp 315– (1967) · Zbl 0175.14201 · doi:10.2748/tmj/1178243281
[6]	Choi M.-D., Ill. J. Math. 18 pp 565– (1974)
[7]	Schwartz J., Commun. Pure Appl. Math. 16 pp 19– (1963) · Zbl 0131.33201 · doi:10.1002/cpa.3160160104