A Volume Inequality for Quantum Fisher Information and the Uncertainty Principle

Abstract

Let A1, ..., AN be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principleff gives a bound for the quantum generalized covariance in terms of the commutators [A h , Aj]. The right side matrix is antisymmetric and therefore the bound is trivial (equal to zero) in the odd caseLet f be an arbitrary normalized symmetric operator monotone function and let •, • ρ,f be the associated quantum Fisher information. In this paper we conjecture the inequalityff that gives a non-trivial bound for any N ∈ N using the commutators i[ρ, A h ]. The inequality has been proved in the cases N = 1, 2 by the joint efforts of many authors (see the Introduction). In this paper we prove the (real) case N = 3.

A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

“…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.…”

From Quasi-Entropy to Skew Information

“…The skew information is nothing else but the Fisher information restricted to M c D , but it is parametrized by the commutator. Skew information appears, for example, in uncertainty relations [1,5,6,7,13,17,18], see also Theorem 4. In that application, the skew information is regarded as a bilinear form.…”

“…Since A, A ≤ A, A , the determinant inequality holds (see Lemma 2 in [9]). This theorem is interpreted as quantum uncertainty principle [1,6,8,13]. In the earlier works the function g from the left-hand-side was (x + 1)/2 and the proofs were more complicated.…”

A Robertson-type uncertainty principle and quantum Fisher information

“…Let us denote the associated volume by Vol f ρ . We shall prove that for any N ∈ N + (this is one of the main differences from (1.2)) and for arbitrary self-adjoint matrices A 1 , ..., A N one has [7]).…”