Uncertainties in Quantum Measurements: A Quantum Tomography

• URL: http://arxiv.org/abs/2112.07520v1
• Date: Tue, 14 Dec 2021 16:29:53 GMT
• Title: Uncertainties in Quantum Measurements: A Quantum Tomography
• Authors: A.P. Balachandran, F. Calder\'on, V.P. Nair, Aleksandr Pinzul, A.F. Reyes-Lega and S. Vaidya
• Abstract summary: The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
• Score: 52.77024349608834
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The observables associated with a quantum system $S$ form a non-commutative algebra ${\mathcal A}_S$. It is assumed that a density matrix $\rho$ can be determined from the expectation values of observables. But $\mathcal A_S$ admits inner automorphisms $a\mapsto uau^{-1},\; a,u\in {\mathcal A}_S$, $u^*u=u^*u=1$, so that its individual elements can be identified only up to unitary transformations. So since $\mathrm{Tr} \rho (uau^*)= \mathrm{Tr} (u^*\rho u)a$, only the spectrum of $\rho$, or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, $\rho$ 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 ${\mathcal A}_M\subset {\mathcal A}_S$ ($M$ for measurement, $S$ for system). We study the uncertainties in extending $\rho|_{{\mathcal A}_M}$ to $\rho|_{{\mathcal A}_S}$ (the determination of which means measurement of ${\mathcal A}_S$) and devise a protocol to determine $\rho|_{{\mathcal A}_S}\equiv \rho$ by determining $\rho|_{{\mathcal A}_M}$ for different choices of ${\mathcal A}_M$. 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 $\rho$ of the system $S$ due to operator mixing. Associated uncertainty principles for von Neumann entropy are discussed in the appendix, adapting the earlier work of Bia{\l}ynicki-Birula and Mycielski to the present case.

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