| Title:
|
Quantum Bochner theorems and incompatible observables (English) |
| Author:
|
Hudson, Robin L. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
6 |
| Year:
|
2010 |
| Pages:
|
1061-1068 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A quantum version of Bochner's theorem characterising Fourier transforms of probability measures on locally compact Abelian groups gives a characterisation of the Fourier transforms of Wigner quasi-joint distributions of position and momentum. An analogous quantum Bochner theorem characterises quasi-joint distributions of components of spin. In both cases quantum states in which a true distribution exists are characterised by the intersection of two convex sets. This may be described explicitly in the spin case as the intersection of the Bloch sphere with a regular tetrahedron whose edges touch the sphere. (English) |
| Keyword:
|
Bochner's Theorem |
| Keyword:
|
multiplier-nonnegative-definiteness |
| Keyword:
|
Wigner quasidensities |
| Keyword:
|
Pauli matrices |
| MSC:
|
60B15 |
| MSC:
|
81S30 |
| idZBL:
|
Zbl 1219.81175 |
| idMR:
|
MR2797427 |
| . |
| Date available:
|
2011-04-12T12:51:25Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141466 |
| . |
| Reference:
|
[1] Bochner, S.: Lectures on Fourier Integrals.Princeton University Press 1959. Zbl 0085.31802, MR 0107124 |
| Reference:
|
[2] Cushen, C. D.: Quasi-characteristic functions of canonical observcables in quantum mechanics.Nottingham PhD Thesis 1970. |
| Reference:
|
[3] Holevo, A. S.: Veroiatnostnye i statistichneskie aspekty kvantovoi teorii.Nauka, Moscow 1980, English translation Probabilistic and statistical aspects of quantum theory, North Holland 1982. MR 0681693 |
| Reference:
|
[4] Hudson, R. L.: When is the Wigner quasi-probability density nonnegative? Rep.Math. Phys. 6 (1974), 249–252. MR 0384019, 10.1016/0034-4877(74)90007-X |
| Reference:
|
[5] Gikhman, I. I., Skorohod, A. V.: Introduction to the Theory of Random Processes.Philadelphia 1969. MR 0247660 |
| Reference:
|
[6] Neumann, J. von: Die Eindeutigkeit der Schrõdingerschen Operatoren.Math. Ann. 104 (1931), 570–578. MR 1512685, 10.1007/BF01457956 |
| Reference:
|
[7] Pool, J. C. T.: Mathematical aspects of the Weyl correspondence.J. Math. Phys. 7 (1966), 66–76. Zbl 0139.45903, MR 0204049, 10.1063/1.1704817 |
| Reference:
|
[8] Rudin, W.: Fourier Analysis on Groups.Interscience New York 1962. Zbl 0107.09603, MR 0152834 |
| Reference:
|
[9] Wigner, E.: On the quantum correction to thermodynamic equilibrium.Phys. Rev. 40 (1932), 749–759. 10.1103/PhysRev.40.749 |
| . |
