Let A1,...,A(N) be complex self-adjoint matrices and let rho be a density matrix. The Robertson uncertainty principle det{Cov rho(A(h), A(j))} >= det{ -1/2Tr(rho[A(h), A(j)])} gives a bound for the quantum generalized variance in terms of the commutators [A(h), A(j)]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case N = 2m + 1. Let f be an arbitrary normalized symmetric operator monotone function and let (.,.)(rho,f) be the associated quantum Fisher information. We have conjectured the inequality det{Cov rho(A(h), A(j))} >= det {f(0)/2 < i[rho, A(h)], i[rho, Aj]>rho,f} that gives a non-trivial bound for any N is an element of N using the commutators [rho, A(h)]. In the present paper the conjecture is proved by mean of the Kubo-Ando mean inequality

Gibilisco, P., Imparato, D., Isola, T. (2008). A Robertson-type uncertainty principle and quantum Fisher information. LINEAR ALGEBRA AND ITS APPLICATIONS, 428(7), 1706-1724 [10.1016/j.laa.2007.10.013].

A Robertson-type uncertainty principle and quantum Fisher information

GIBILISCO, PAOLO;ISOLA, TOMMASO
2008-01-01

Abstract

Let A1,...,A(N) be complex self-adjoint matrices and let rho be a density matrix. The Robertson uncertainty principle det{Cov rho(A(h), A(j))} >= det{ -1/2Tr(rho[A(h), A(j)])} gives a bound for the quantum generalized variance in terms of the commutators [A(h), A(j)]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd case N = 2m + 1. Let f be an arbitrary normalized symmetric operator monotone function and let (.,.)(rho,f) be the associated quantum Fisher information. We have conjectured the inequality det{Cov rho(A(h), A(j))} >= det {f(0)/2 < i[rho, A(h)], i[rho, Aj]>rho,f} that gives a non-trivial bound for any N is an element of N using the commutators [rho, A(h)]. In the present paper the conjecture is proved by mean of the Kubo-Ando mean inequality
2008
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/06 - PROBABILITA' E STATISTICA MATEMATICA
English
Con Impact Factor ISI
generalized variance; matrix means; operator monotone functions; quantum Fisher information; uncertainty principle
Gibilisco, P., Imparato, D., Isola, T. (2008). A Robertson-type uncertainty principle and quantum Fisher information. LINEAR ALGEBRA AND ITS APPLICATIONS, 428(7), 1706-1724 [10.1016/j.laa.2007.10.013].
Gibilisco, P; Imparato, D; Isola, T
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/41287
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