= Copy-number trace-distance obstruction
{title2=$\frac{\epsilon_u+\epsilon_v}{2}\geq\frac14\left(\sqrt{1-c^{2m}}-\sqrt{1-c^{2n}}\right)^2,\quad m>n$}
Let distinct nonorthogonal <pure states> have overlap magnitude $0<c<1$, and let one <quantum channel> take $n$ identical inputs to $m$ outputs. Let $\epsilon_u,\epsilon_v$ be the respective <probabilities> that a test for the ideal $m$-copy <pure state> fails. <Trace distance> contraction, its triangle inequality and the <pure-target upper bound on trace distance> give $\sqrt{1-c^{2m}}\leq\sqrt{\epsilon_u}+\sqrt{1-c^{2n}}+\sqrt{\epsilon_v}$. Squaring with the <Cauchy-Schwarz inequality> gives the displayed strategy-independent positive bound. It establishes both a worst-case bound and an equal-prior average bound. Averaging rotated pairs establishes a positive Haar-average bound for unknown qubit inputs, without assuming independent clone errors.
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