A discrete entropic uncertainty relation. (English) Zbl 0725.46039

Maassen, H.

Quantum probability and applications V, Proc. 4th Workshop, Heidelberg/FRG 1988, Lect. Notes Math. 1442, 263-266 (1990).

[For the entire collection see Zbl 0702.00025.]
Let \(M_n\) be the algebra of all \(n\times n\) complex matrices. Given a maximal Abelian subalgebra \(\mathcal A\) and a state \(\omega\) put
\[ H(\mathcal A,\omega)=-\sum_{i=1}^n \omega (P_i)\log \omega (P_i), \] where \(P_i\), \(i=1,2,\ldots,n\), are minimal projections in \(\mathcal A\). Consider the following “degree of incompatibility” of maximal abelian subalgebras \(\mathcal A\) and \(\mathcal B\):
\[ d(\mathcal A,\mathcal B)=\inf_\omega (H(\mathcal A,\omega)+H(\mathcal B,\omega)). \] Consider also the following function on pairs of abelian von Neumann algebras in \(M_n\), \[ m(\mathcal A,\mathcal B)=\max \{\text{tr}\, PQ,\;P\in\mathcal A,\;Q\in\mathcal B\;\text{minimal\;projections}\}. \] The main result of the present paper is the following “entropic” uncertainty relation, which has been conjectured by K. Kraus [Phys. Rev. D (3) 35, No. 10, 3070–3075 (1987; doi:10.1103/PhysRevD.35.3070)].
Theorem. For all maximal abelian von Neumann subalgebras \(\mathcal A\) and \(\mathcal B\) in \(M_n\) one has \[ d(\mathcal A,\mathcal B)\geq -\log m(\mathcal A,\mathcal B). \]

Reviewer: Sh. A. Ayupov (Tashkent)

MSC:

46L60	Applications of selfadjoint operator algebras to physics
46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras
81P05	General and philosophical questions in quantum theory
94A17	Measures of information, entropy

Keywords:

entropy; maximal abelian subalgebra; state; minimal projections; degree of incompatibility; function on pairs of abelian von Neumann algebras; uncertainty relation