Geodesics for mixed quantum states via their geometric mean operator

Abstract

We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators $ρ_1$ and $ρ_2$ in the base space $M$ is given by the shortest distance in the (Hilbert Schmidt) bundle space $E$ of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in $E$ of the geodesic between the $ρ_1$ and $ρ_2$ in $M$. The horizontal lift is that unique curve in $E$ that orthogonally traverses the fibers $F\subset E$ above the curve in $M$, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states $ρ(s)$ along the base space geodesic parameterized by affine parameter $s$ between the initial $ρ_1$ and final $ρ_2$ states. We emphasize the role played by geometric mean operator $M(s) = ρ_1^{-1/2}\, \sqrt{ρ_1^{1/2}ρ(s)ρ_1^{1/2}}\,ρ_1^{-1/2}$, where the Uhlmann root fidelity between $ρ_1$ and $ρ(s)$ is given by $\sqrt{F}(ρ_1,ρ(s)) = Tr[M(s)\,ρ_1] = Tr[\sqrt{ρ_1^{1/2}ρ(s)ρ_1^{1/2}}]$, and $ρ(s) = M(s)\,ρ_1\,M(s)$. We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states $ρ(p) = (1-p) I/N + p\,|Ψ\rangle\langle Ψ|$ with $|Ψ\rangle = \{|GHZ\rangle, |W\rangle\}$ in dimension $N=2^3$. For the latter, we compare expressions in the limit $p\to1$ to the infinite number of possible geodesics between the orthogonal pure states $|GHZ\rangle$ and $|W\rangle$. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension $N=2$.