Intrinsic volumes of the quantum state space and mutually unbiased bases
Zsombor Szilágyi, Mihály Weiner
TL;DR
This work advances the convex-geometric analysis of quantum state spaces by deriving explicit closed-form expressions for the second and third intrinsic volumes of the $d$-dimensional state space $\mathcal S_d$ (with dimension $D=d^2-1$) and by extending the computation of intrinsic volumes for the complementarity polytope $\mathcal P_d$. The authors express $V_{D-2}(\mathcal S_d)$ and $V_{D-3}(\mathcal S_d)$ in terms of derivatives $p_d^{(2)}(0)$ and $p_d^{(3)}(0)$ of the volume polynomial $p_d(\varepsilon)$ of the $\varepsilon$-neighborhood, providing explicit gamma-function formulas. They also compute the next two unnormalized intrinsic volumes of $\mathcal P_d$ using facet-cone decompositions and verify with Monte Carlo methods, showing that $V_N(\mathcal P_d) \le V_N(\mathcal S_d)$ for $N=D,D-1,D-2,D-3$, hence four intrinsic volumes do not exclude inscription of $\mathcal P_d$ into $\mathcal S_d$. Finally, four example vector configurations in $\mathbb C^6$ demonstrate how higher intrinsic volumes can rule out certain candidate configurations, highlighting the practical utility of intrinsic-volume comparisons beyond volume and surface area. The results enrich the geometric understanding of quantum state spaces and offer concrete obstructions for specific inscribability conjectures in quantum information theory.
Abstract
Previous studies on the geometrical properties of the state space of a finite-level quantum system have determined its volume and surface area. Building on this foundation, we derive explicit formulas for two additional intrinsic volume quantities. The question of whether a complete set of mutually unbiased bases exists in dimension $d$ can be equivalently framed as whether a specific convex polytope can be inscribed within the state space of a $d$-level quantum system. One motivation for our work was the hypothesis that a smaller intrinsic volume of the state space compared to the corresponding intrinsic volume of the mentioned polytope could rule out such an inscription. While our computations of these two intrinsic volumes do not lead to this conclusion, they nonetheless provide fundamental insights into the geometric structure of quantum state spaces. In particular, we show that these quantities can be used to rule out the existence of some unit-vector ``configurations'' (though not the one formed by the bases vectors of a complete set of mutually unbiased bases).