The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the Second-Hand Lion trace distance $D_k$ is by definition $$D_k = \max_{M_1\in \mathcal{M}^{(kk)}_k}\ \, \min_{M_2\in \mathcal{M}^{(kk)}_{k{-}1}}\ \,\tfrac{1}{2} \text{tr}\,|M_1-M_2|$$
The question asked is, what is the asymptotic behavior of $D_k$ for $k\gg1$?
Physics and Engineering Motivation
The set $\mathcal{M}^{(kk)}_k$ is isomorphic to the quantum Hilbert space $\mathbb{C}^{k^2}$, and the set $\mathcal{M}^{(kk)}_{k{-}1}$ is an determinantal variety that we shall call the SHL variety of order $k-1$; and the SHL variety has a natural embedding in the larger Hilbert space.
Moreover the Second-Hand Lion Theorem assures us:
The SHL variety of order $k-1$ has dimension $k^2-1$, that is, the SHL variety lacks precisely one dimension with respect to the embedding Hilbert space $\mathbb{C}^{k^2}$.
The SHL variety of order $k-1$ is naturally equipped with algebraic coordinates that are well-suited to the efficient numerical integration of trajectories (both metric and symplectic) on the SHL variety.
Physically the question asked therefore amounts to this: For quantum states adversarially chosen within a $k\times k$ Hilbert space, what is the worst-case quantum fidelity with which that Hilbert state can be approximated as an SHL varietal state of order $k-1$?
This question is motivated partly by numerical experiments that indicate (for example) $D_8 \simeq 0.0568$. A reasonable conjecture (for example) may be $D_k = \mathcal{O}\,(1/k)$.
Broader Motivations
More broadly, the question asked was conceived with a view toward illuminating the general problems of approximating Hilbert-space dynamics with varietal dynamics, in regard to the mathematically natural, physically fundamental, computationally practicable (and perennially surprising) features of these varietal dynamical systems.
