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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.

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Hi John, This looks like an improved version of mathoverflow.net/questions/88133, but I haven't read the question closely so I'm not sure if they are asking the same thing or are two questions on the same topic (certainly it's great to ask a series of related questions!). If the former, you can probably leave a link to this question at the top of the old one, and close the old one yourself; if you can't close it yourself, I'm sure you can leave a comment on meta asking it be closed. If the latter, then I hope you get good answers to both questions! –  Theo Johnson-Freyd Feb 22 '12 at 17:02
    
A mathematical clarification question: You define the matrices in $\mathcal M_r^{(kk)}$ as "having unit trace norm", which is not a phrase I often come across --- I could imagine it meaning "unit trace", or "unit (matrix) norm", or.... But then you say that $\mathcal M_k^{(kk)}\cong\mathbb C^{k^2}$, so it seems that this condition is vacuous. In any case, am I right to think that the distance is asking "how far can a full-rank matrix be from any lower-rank matrix?"? If so, then you certainly do want some condition, or to normalize the distance somehow, else you get the answer $D_k=\infty$.... –  Theo Johnson-Freyd Feb 22 '12 at 17:08
    
... I will have to read the linked paper to understand why absolute-value-of-trace is the natural distance (I assume this is what you mean --- I don't know how to take the trace of "the absolute value of a matrix"), and not some other matrix norm that seems more natural to me (e.g. Hilbert-Schmidt norm). But probably for the maximum asymptotics you don't care, because there are probably AM-GM style inequalities. Anyway, once $\mathcal M_r^{(kk)}$ is made somehow compact or the norm is otherwise normalized, I like this question a lot. –  Theo Johnson-Freyd Feb 22 '12 at 17:14

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