Quantum Closeness Testing: A Streaming Algorithm and Applications

One of the main subjects of this paper is to study quantum property testing with local measurement. In particular, we establish a novel $\ell_2$ norm connection between quantum property testing problems and the corresponding distribution testing problems. This connection opens up the potential to derive efficient testing algorithms using techniques developed for classical property testing. As the first demonstration of these possibilities, we designed two streaming algorithms: one for quantum state tomography, the other for quantum closeness testing. By using the idea of our tomography algorithm, we obtain a streaming algorithm which provide good estimations for each $k$-qubit reduced density matrice of $m$-qubit state using only $\log m$ copies for constant $k$. This is tight and exponential speedup compare with optimal tomography for each $k$-qubit reduced density matrice. To the best of our knowledge, no streaming algorithm has yet been used for quantum property testing. So, to illustrate their usefulness, we achieve the following: independence testing for quantum states; identity and independence testing for quantum state collections; and conditional independence for classical-quantum-quantum states. Additionally, with a dimension splitting technique, we derive matching lower bound up to log factor for independence testing with joint measurement.