Approximating quantum states by states of low rank
Nathaniel Johnston, Chi-Kwong Li
TL;DR
This work characterizes the closest density matrix of rank at most $k$ to a given density matrix under any unitary similarity invariant norm, showing the optimizer has the spectral form $Y = P_k X P_k + \gamma P_k$ with $\gamma = \tfrac{1}{k}\sum_{j=k+1}^n x_j$, derived from the eigen-decomposition $X = \sum_{j=1}^n x_j v_j v_j^*$. The authors prove this minimizer is unique whenever the norm is strictly convex and extend the Eckart–Mirsky–Young truncation to all USI norms. They then analyze the opposite problem—which state is farthest from the set of low-rank density matrices—showing the maximally mixed state is not always extremal under general USI norms, though it is for the trace norm; for Schatten norms, maximality holds for $p \in \{1\} \cup [2,4]$ and fails for $p>4$. For Ky Fan norms they prove that it suffices to examine at most five candidate ranks, yielding explicit maximizers in many regimes. Overall, the paper provides a unified framework for low-rank density-matrix approximation across a broad class of unitary-invariant norms, with concrete formulas and algorithmic implications based on spectral data.
Abstract
Given a positive integer k, it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k. This problem has already been solved when "distance" is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally mixed state, but this is not true in many other unitary similarity invariant norms.