###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][Nielsen]* $D_k$ 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 $k\gg1$?

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###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][SHLT] 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) $D_8 \simeq 0.08$. A reasonable conjecture (for example) may be $D_k = \mathcal{O}\,(1/k)$.

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###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.

[Nielsen]: http://arxiv.org/abs/quant-ph/0408063
[SHLT]: http://mathoverflow.net/questions/88133/harris-algebraic-geometry-quantum-dynamics-on-varieties-and-a-more-than-mone