Quantum Spectrum Testing

Abstract

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