Coherence in Property Testing: Quantum-Classical Collapses and Separations

Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain $\{0,1\}^n$. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given $2^{n/16}$ samples, no tester can distinguish distributions of support size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-\Theta(n)}$, even promised they are flat. Quantum states can be in a coherent superposition of states of $\{0,1\}^n$, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, $|\phi_S \rangle=1/\sqrt{|S|}\sum_{i\in S}|i\rangle$. We show that coherence alone is not enough, Coherence limitations: Given $2^{n/16}$ copies, no tester can distinguish subset states of size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-\Theta(n)}$. The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given $2^{O(n)}$ samples from a distribution and $2^{O(n)}$ communication with AM provers, no tester can estimate the support size up to factors $2^{\Omega(n)}$ with probability better than $2^{-\Theta(n)}$. Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.