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Different approach

Condition (1) requires that the coefficients $\{ \alpha_i^{(n)} \}$ depend upon the Hamiltonian eigenvalues $\{ \varepsilon_i \}$. Resulting condition number $\kappa$ of the Hessian of $Q[\rho]$ is[*]

\begin{displaymath}\kappa \sim \frac{\varepsilon_{\rm max} - \varepsilon_{\rm min}}
{\varepsilon_{\rm gap}} \end{displaymath}

Instead start by imposing condition (2), making the $\{ \alpha_i^{(n)} \}$ independent of state $i$ (so that $\kappa \approx 1$) e.g.

\begin{displaymath}Q[\rho] = E[\rho] + \alpha {\rm Tr}\left[ \rho^2 \left( 1 - \rho \right)^2
\right] \end{displaymath}

For $\alpha \gg \mathop{\rm max}\limits _i{\left\vert \tilde{\varepsilon}_i - \lambda
\right\vert}$, the minimum occurs for

i.e. for approximately idempotent density-matrices.

Peter D. Haynes 2001-11-19