Quantum Spectrum Testing

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given $n$ copies of a mixed state $\rho\in\mathbb{C}^{d\times d}$ and the goal is to distinguish whether $\rho$'s spectrum satisfies some property $\mathcal{P}$ or is at least $\epsilon$-far in $\ell_1$-distance from satisfying $\mathcal{P}$. This problem was promoted in the survey of Montanaro and de Wolf under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Here, the hope is for algorithms with subquadratic copy complexity in the dimension $d$. This is because the "empirical Young diagram (EYD) algorithm" can estimate the spectrum of a mixed state up to $\epsilon$-accuracy using only $\widetilde{O}(d^2/\epsilon^2)$ copies. In this work, we show that given a mixed state $\rho\in\mathbb{C}^{d\times d}$: (i) $\Theta(d/\epsilon^2)$ copies are necessary and sufficient to test whether $\rho$ is the maximally mixed state, i.e., has spectrum $(\frac1d, ..., \frac1d)$; (ii) $\Theta(r^2/\epsilon)$ copies are necessary and sufficient to test with one-sided error whether $\rho$ has rank $r$, i.e., has at most $r$ nonzero eigenvalues; (iii) $\widetilde{\Theta}(r^2/\Delta)$ copies are necessary and sufficient to distinguish whether $\rho$ is maximally mixed on an $r$-dimensional or an $(r+\Delta)$-dimensional subspace; and (iv) The EYD algorithm requires $\Omega(d^2/\epsilon^2)$ copies to estimate the spectrum of $\rho$ up to $\epsilon$-accuracy, nearly matching the known upper bound. In addition, we simplify part of the proof of the upper bound. Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov's algebra of polynomial functions on Young diagrams.