A Volume Inequality for Quantum Fisher Information and the Uncertainty Principle |
| |
| Authors: | Paolo Gibilisco Daniele Imparato Tommaso Isola |
| |
| Institution: | (1) Dipartimento SEFEMEQ, Facoltà di Economia, Università di Roma “Tor Vergata”, Via Columbia 2, 00133 Rome, Italy;(2) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy;(3) Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy |
| |
| Abstract: | Let A
1,…,A
N
be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle gives a bound for the quantum generalized covariance 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 〈⋅,⋅〉
ρ,f
be the associated quantum Fisher information. Based on previous results of several authors, we propose here as a conjecture
the inequality whose validity would give a non-trivial bound for any N∈ℕ using the commutators iρ,A
h
]. |
| |
| Keywords: | Generalized variance Uncertainty principle Operator monotone functions Matrix means Quantum Fisher information |
| 本文献已被 SpringerLink 等数据库收录！ |
|
![$$\det\{\mathop{Cov}_{\rho}(A_{h},A_{j})\}\geq \det \biggl\{-\frac{i}{2}\mathop{Tr}(\rho A_{h},A_{j}])\biggr\}$$](/content/p6335p83k2762104/10955_2007_9454_Article_Equa.gif)
![$$\det\{\mathop{Cov}_{\rho}(A_{h},A_{j})\}\geq \det \biggl\{\frac{f(0)}{2}\langle i\rho,A_{h}],i\rho,A_{j}]\rangle_{\rho,f}\biggr\}$$](/content/p6335p83k2762104/10955_2007_9454_Article_Equb.gif)