arXiv:2112.07520 · Journal of Physics A Mathematical and Theoretical

Uncertainties in quantum measurements: a quantum tomography

A. P. Balachandran, F Calderón, V. P. Nair, Aleksandr Pinzul, Andrés Fernando Reyes-Lega, Sachindeo Vaidya

Published Apr 25, 2022• 3 citations

Abstract

Abstract The observables associated with a quantum system S form a non-commutative algebra A S . It is assumed that a density matrix ρ can be determined from the expectation values of observables. But A S admits inner automorphisms a ↦ u a u − 1 , a , u ∈ A S , u * u = u u * = 1 , so that its individual elements can be identified only up to unitary transformations. So since Tr ρ ( uau *) = Tr( u * ρu ) a , only the spectrum of ρ , or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, ρ cannot be determined at all, as we shall explain. However, abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables in abelian algebras A M ⊂ A S ( M for measurement, S for system). We study the uncertainties in extending ρ | A M to ρ | A S (the determination of which means measurement of A S ) and devise a protocol to determine ρ

DOI: 10.1088/1751-8121/ac6a2c

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