Quantum Bochner theorems and incompatible observables

Robin L. Hudson

Kybernetika (2010)

  • Volume: 46, Issue: 6, page 1061-1068
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Hudson, Robin L.. "Quantum Bochner theorems and incompatible observables." Kybernetika 46.6 (2010): 1061-1068. <http://eudml.org/doc/196490>.

@article{Hudson2010,
abstract = {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.},
author = {Hudson, Robin L.},
journal = {Kybernetika},
keywords = {Bochner's Theorem; multiplier-nonnegative-definiteness; Wigner quasidensities; Pauli matrices; Bochner theorem; Wigner quasi-probability distribution; Pauli matrices},
language = {eng},
number = {6},
pages = {1061-1068},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Quantum Bochner theorems and incompatible observables},
url = {http://eudml.org/doc/196490},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Hudson, Robin L.
TI - Quantum Bochner theorems and incompatible observables
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 1061
EP - 1068
AB - 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.
LA - eng
KW - Bochner's Theorem; multiplier-nonnegative-definiteness; Wigner quasidensities; Pauli matrices; Bochner theorem; Wigner quasi-probability distribution; Pauli matrices
UR - http://eudml.org/doc/196490
ER -

References

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  1. Bochner, S., Lectures on Fourier Integrals, Princeton University Press 1959. (1959) Zbl0085.31802MR0107124
  2. Cushen, C. D., Quasi-characteristic functions of canonical observcables in quantum mechanics, Nottingham PhD Thesis 1970. (1970) 
  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. (1980) MR0681693
  4. Hudson, R. L., 10.1016/0034-4877(74)90007-X, Math. Phys. 6 (1974), 249–252. (1974) MR0384019DOI10.1016/0034-4877(74)90007-X
  5. Gikhman, I. I., Skorohod, A. V., Introduction to the Theory of Random Processes, Philadelphia 1969. (1969) MR0247660
  6. Neumann, J. von, 10.1007/BF01457956, Math. Ann. 104 (1931), 570–578. (1931) MR1512685DOI10.1007/BF01457956
  7. Pool, J. C. T., 10.1063/1.1704817, J. Math. Phys. 7 (1966), 66–76. (1966) Zbl0139.45903MR0204049DOI10.1063/1.1704817
  8. Rudin, W., Fourier Analysis on Groups, Interscience New York 1962. (1962) Zbl0107.09603MR0152834
  9. Wigner, E., 10.1103/PhysRev.40.749, Phys. Rev. 40 (1932), 749–759. (1932) DOI10.1103/PhysRev.40.749

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