A Robertson-type Uncertainty Principle by Quantum Fisher

A Robertson-type Uncertainty Principle by Quantum Fisher
Information.
Daniele Imparato
The Heisenberg uncertainty principle for selfadjoint matrices can be stated as
1
|Tr(ρ[A, B])|2 .
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Schrödinger and Robertson improved that result adding the squared covariance
Varρ (A) · Varρ (B) ≥
1
|Tr(ρ[A, B])|2 .
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Robertson himself realized that for N observables A1 , ..., AN one can prove the general result
i
det {Covρ (Ah , Aj )} ≥ det − Tr(ρ[Ah , Aj ]) .
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Varρ (A) · Varρ (B) − Covρ (A, B)2 ≥
Let f be an arbitrary normalized symmetric operator monotone function and let h·, ·iρ,f be the
associated quantum Fisher information. In this talk I will present the recently proved inequality
f (0)
hi[ρ, Ah ], i[ρ, Aj ]iρ,f
det {Covρ (Ah , Aj )} ≥ det
2
that, contrary to the Robertson uncertainty principle, gives a non-trivial bound also in the odd case
N = 2m + 1. The above inequality is a consequence of the fundamental Kubo-Ando inequality
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(A + B),
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saying that any operator mean is larger than the harmonic mean and smaller than the arithmetic
mean. This is a joint work with P. Gibilisco and T. Isola.
2(A−1 + B −1 )−1 ≤ mf (A, B) ≤
References
[1] Gibilisco, P. and Isola, T., Uncertainty principle and quantum Fisher information. Proceedings
of IGAIA, Univ. of Tokio, pag. 154-161, 2005 and Ann. Inst. Stat. Math, 59: 147–159, 2007.
[2] Gibilisco, P., Imparato, D. and Isola, T., Uncertainty principle and quantum Fisher information
II. J. Math. Phys., 48: 072109, 2007. 2007.
[3] Gibilisco, P., Imparato, D. and Isola, T., A volume inequality for quantum Fisher information
and the uncertainty principle. arXiv:math-ph/0706.0791v1, to appear on J. Stat. Phys. 2007.
[4] Gibilisco, P., Imparato, D. and Isola, T., A Robertson-type Uncertainty Principle and Quantum Fisher Information. arXiv:math-ph/0707.1231v1, to appear on Linear Algebra and its
Applications. 2007.
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