A Robertson-type uncertainty principle and quantum Fisher information

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 inequalit

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“…In the case N = 1, a reasonable candidate for a lower bound is an expression involving some commutation relation between A and ρ. An inequality of this kind, valid for any N , has been recently proved in [1,11] (see also [5,9,10,12,[15][16][17][18][19][20]30]). To describe this result we need the theory of operator monotone functions.…”

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