The Monge metric on the sphere and geometry of quantum states (Q2774553)

In quantum mechanics, states are unit vectors in a complex-valued Hilbert space. Thus, traditionally, a distance between two quantum states is defined in terms of the corresponding Hilbert-space metric; in particular, the distance between any two orthogonal pure states -- e.g., between (almost) delta-functions located at two different points \(x_1\neq x_2\) -- is the same: \(\sqrt{2}\). From the physical viewpoint, it is desirable to supplement this Hilbert metric by another metric that would lead to Euclidean distance for classical points. As a starting point, the authors use the Monge metric that describes the distance between two probability distributions on a metric space \(X\) (e.g., on \(\mathbb{R}^n\) or on a sphere), with densities \(\rho_1(x)\) and \(\rho_2(x)\). The idea behind Monge metric is as follows: we imagine that \(\rho_1(x)\) represents the initial density of some substance at each point \(x\), and \(d(\rho_1,\rho_2)\) is the smallest amount of effort (in mass \(\times\) distance) that is needed to move this substance so as the resulting distribution is \(\rho_2(x)\). In precise terms, \(d(\rho_1,\rho_2)\) is the minimum of \(\int d(x,Tx)\cdot\rho_1(x) dx\) over all 1-1 volume-preserving mappings \(T:X\to X\) that transform \(\rho_1(x)\) into \(\rho_2(x)\) (i.e., for which \(\rho_1(x)=\rho_2(Tx)\cdot\text{det}(T'(x))\)). NEWLINENEWLINENEWLINEThe authors propose a natural quantum analogue of this definition, analyze it for \(X=\) sphere, and show that the resulting metric is physically appropriate for describing decoherence -- a process extremely important for quantum computing.