Balachandran, AP and Calderón, F and Nair, VP and Pinzul, A and Reyes-Lega, AF and Vaidya, S (2022) Uncertainties in quantum measurements: A quantum tomography. In: Journal of Physics A: Mathematical and Theoretical, 55 (22).
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Abstract
The observables associated with a quantum system S form a non-commutative algebra AS . It is assumed that a density matrix ρ can be determined from the expectation values of observables. But AS admits inner automorphisms a▪uau-1,a,ûAS, u∗u=uu∗=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 AM⊂AS (M for measurement, S for system). We study the uncertainties in extending ρ|AM to ρ|AS (the determination of which means measurement of AS ) and devise a protocol to determine ρ|ASρ by determining ρ|AM for different choices of AM . 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.
| Item Type: | Journal Article |
|---|---|
| Publication: | Journal of Physics A: Mathematical and Theoretical |
| Publisher: | Institute of Physics |
| Additional Information: | The copyright for this article belongs to the Author. |
| Keywords: | abelian algebras; Kadison-Singer; quantum measurement; quantum tomography |
| Department/Centre: | Division of Physical & Mathematical Sciences > Centre for High Energy Physics |
| Date Deposited: | 21 Sep 2022 08:49 |
| Last Modified: | 21 Sep 2022 08:49 |
| URI: | https://eprints.iisc.ac.in/id/eprint/76623 |
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