Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures

Gustavo Martin Bosyk (1, 2) , Steeve Zozor (2, 1) , Mariela Portesi (2, 1) , Tristan Martín Osán (3) , Pedro Walter Lamberti (3)

1 GIPSA-CICS - GIPSA - Communication Information and Complex Systems
2 IFLP - Instituto de Física La Plata [La Plata]
3 FaMAF - Facultad de Matemática, Astronomía y Física [Cordoba]

We provide a twofold extension of Landau-Pollak uncertainty relations for mixed quantum states and for positive operator valued measures, by recourse to geometric considerations. The generalization is based on metrics between pure states, having the form of a function of the square of the inner product between the states. The triangle inequality satisfied by such metrics plays a crucial role in our derivation. The usual Landau-Pollak inequality is thus a particular case (derived from Wootters metric) of the family of inequalities obtained, and, moreover, we show that it is the most restrictive relation within the family.

Domaines: Physique Quantique [quant-ph]; Théorie de l'information [cs.IT]; Physique mathématique [math-ph]

Gustavo Martin Bosyk, Steeve Zozor, Mariela Portesi, Tristan Martín Osán, Pedro Walter Lamberti. Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures. Physical Review A : Atomic, molecular, and optical physics [1990-2015], 2014, 90 (5), pp.052114. ⟨10.1103/PhysRevA.90.052114⟩. ⟨hal-01085032⟩