Quantum Spectrum Testing

In this work, we study the problem of testing properties spectrum a mixed quantum state. Here one is given n copies state $$\rho \in \mathbb {C}^{d\times d}$$ ρ ∈ C d × and goal to distinguish (with high probability) whether $$ ’s satisfies some property $${\mathcal {P}}$$ P or it at least $$\epsilon ϵ -far in $$\ell _1$$ ℓ 1 -distance from satisfying . This was promoted survey Montanaro de Wolf (A testing. Technical report, arXiv:1310.2035 , 2013) under name unitarily invariant states. It natural analogue classical symmetric probability distributions. Unlike distributions—where generally hopes for algorithms with sample complexity that sublinear domain size—here hope subquadratic copy dimension d. because (frequently rediscovered) “empirical Young diagram (EYD) algorithm” (Alicki et al. J Math Phys 29(5):1158–1162, 1988; Keyl Werner Rev A 64(5):052311, 2011; Hayashi Matsumoto 66(2):022311, 2002; Christandl Mitchison Commun. Math. Phys. 261(3):789–797, 2006) can estimate any up -accuracy using only $${\widetilde{O}}(d^2/\epsilon ^2)$$ O ~ ( 2 / ) copies. show {C}^{d \times : $$\Theta (d/\epsilon Θ are necessary sufficient test maximally state, i.e., has $$(\frac{1}{d}, \dots \frac{1}{d})$$ ⋯ be viewed as Paninski (IEEE Trans Inf Theory 54(10):4750–4755, 2008) sharp bounds uniformity-testing. (r^2/\epsilon )$$ r one-sided error rank r, most nonzero eigenvalues. For two-sided error, lower bound $$\Omega (r/\epsilon Ω holds. $${\widetilde{\Theta }}(r^2)$$ on an r-dimensional $$(r+1)$$ + -dimensional subspace. More generally, vs. $$r+\Delta Δ $$1 \le \Delta r$$ ≤ ), }}(r^2/\Delta sufficient. The EYD algorithm requires (d^2/\epsilon -accuracy, nearly matching known upper bound. addition, simplify part proof Our techniques involve asymptotic representation theory group; particular Kerov’s algebra polynomial functions diagrams.