Uncertainties in Quantum Measurements: A Quantum Tomography

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.