On some properties of orthogonal Weingarten functions
We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar m...
We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar m...
Asymptotic behavior for logarithmic diffusion
In this paper we prove, via the entropy dissipation method, that the solutions of the d-dimensional logarithmic diffusion equation, with nonhomogeneous Dirichlet boundary data, decay exponentially in ...
In this paper we prove, via the entropy dissipation method, that the solutions of the d-dimensional logarithmic diffusion equation, with nonhomogeneous Dirichlet boundary data, decay exponentially in ...
Markov property of Gaussian states of canonical commutation relation algebras
J. Math. Phys. 50, 113517 (2009); doi:10.1063/1.3253974
Published 19 November 2009
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The Markov property of Gaussian states of canonical commutation relation algebras is studied. The detailed description is given by the representing block matrix. The proof is short and allows infinite dimension. The relation to classical Gaussian Markov triplets is also described. The minimizer of relative entropy with respect to a Gaussian Markov state has the Markov property. The appendix contains formulas for the relative entropy.
©2009 American Institute of Physics
| History: | Received 30 September 2009; accepted 30 September 2009; published 19 November 2009 |
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http://link.aip.org/link/?JMAPAQ/50/113517/1 |
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- Accardi, L., “On the noncommutative Markov property,” Funkc. Anal. Priloz. 9, 1 (1975).
- Accardi, L. and Frigerio, A., “Markovian cocycles,”
Proc. Roy Irish Acad., Sect. A 83, 251 (1983) . - Ando, T. and Petz, D., “Gaussian Markov triplets approached by block matrices,”
Acta Sci. Math. 75, 265 (2009) . - Beghi, A., “On the relative entropy of discrete-time Markov processes with given end-point densities,”
IEEE Trans. Inf. Theory 42, 1529 (1996) . - Bhatia, R., Matrix Analysis (Springer, New York, 1996).
- Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics (Springer, New York, 1979), Vol. I
- Operator Algebras and Quantum Statistical Mechanics (Springer, New York, 1981), Vol. II.
- van Daele, A. and Verbeure, A., “Unitary equivalence of Fock representations on the Weyl algebra,”
Commun. Math. Phys. 20, 268 (1971) . - Evans, D. E. and Lewis, J. T., “Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A 24 (1977).
- Fannes, M., Nachtergaele, B., and Werner, R. F., “Finitely correlated states on quantum spin chains,”
Commun. Math. Phys. 144, 443 (1992) . - Hayden, P., Jozsa, R., Petz, D., and Winter, A., “Structure of states which satisfy strong subadditivity of quantum entropy with equality,”
Commun. Math. Phys. 246, 359 (2004) . - Holevo, A. S., Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982).
- Jenčová, A., Petz, D., and Pitrik, J., “Markov triplets on CCR-algebras,” Acta Sci. Math. 76, 625 (2009).
- Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras (Academic, New York, 1986), Vol. II.
- Kosaki, H., “Relative entropy for states: a variational expression,”
J. Oper. Theory 16, 335 (1986) . - Lieb, E. H. and Ruskai, M. B., “Proof of the strong subadditivity of quantum-mechanical entropy,” J. Math. Phys. 14, 1938 (1973).
- Manuceau, F. and Verbeure, A., “Quasi-free states of the C.C.R.-algebra and Bogoliubov transformations,”
Commun. Math. Phys. 9, 293 (1968) . - Ohya, M. and Petz, D., Quantum Entropy and Its Use (Springer-Verlag, Heidelberg, 1993).
- Petz, D., An Invitation to the Algebra of the Canonical Commutation Relation (Leuven University Press, Leuven, 1990).
- Petz, D., “Entropy, the central limit theorem and the algebra of the canonical commutation relation,”
Lett. Math. Phys. 24, 211 (1992) . - Petz, D., “Entropy of Markov states,”
Riv. di Math. Pura ed Appl. 14, 33 (1994) . - Petz, D., Quantum Information Theory and Quantum Statistics (Springer-Verlag, Heidelberg, 2008).
- Pitrik, J., “Markovian quasifree states on canonical anticommutation relation algebras,” J. Math. Phys. 48, 112110 (2007).
- Slawny, F., “On factor representations and the C*-algebra of canonical commutation relation,”
Commun. Math. Phys. 24, 151 (1972) .
