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.