Uncertainties in quantum measurements: a quantum tomography. (7th June 2022)

• Record Type: Journal Article
• Authors: Balachandran, A P; Calderón, F; Nair, V P; Pinzul, Aleksandr; Reyes-Lega, A F; Vaidya, S
• Is Part Of: Journal of physics. Volume 55:Number 22(2022)
• Publication Date: 2022-06-07
• Subjects: quantum tomography -- Kadison–Singer -- quantum measurement -- abelian algebras; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ac6a2c
• Languages: English
• ISSNs: 1751-8113

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 | A S ≡ ρ by determining ρ | A M for different choices of A M . The problem we formulate and study is a generalization of the Kadison–Singer theorem. We give an example where the system S is a particle on a circle and the experiment measures the abelian algebra of a magnetic field B coupled to S . The measurement of B gives information about the state ρ of the system S due to operator mixing. Associated uncertainty principles for von Neumann entropy are discussed in the appendix, adapting the earlier work by Białynicki-Birula and Mycielski (1975 Commun. Math. Phys. 44 129) to the present case.