Efficient Independence Testing for Quantum states

Abstract

We provide simple and efficient testers of problems about quantum independence by collective measurements and independent measurements, respectively. We show that given mixed states $\rho,\sigma\in \mathcal{D}(\mathbb{C}^{d})$, $O(\frac{d^2}{\epsilon^2})$ copies are sufficient to test whether $\rho=\sigma$ using independent measurements. Together with the identity tester in collective measurements setting \cite{BOW17}, we prove the following. Given multipartite mixed state $\rho\in \mathcal{D}(\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}\otimes\cdots\otimes\mathbb{C}^{d_m})$, $\tilde{O}(\frac{\Pi_{i=1}^m d_i}{\epsilon^2})$ copies are sufficient and necessary to test whether $\rho$ is independent, i.e., in the tensor product form, by collective measurements, and $O(\frac{\Pi_{i=1}^m d_i^2}{\epsilon^2})$ copies are sufficient by independent measurements. Given $\rho_1,\rho_2, \cdots,\rho_n\in \mathcal{D}(\mathbb{C}^{d})$ in the query model, and $O(\frac{d}{\epsilon^2})$ copies are sufficient and necessary to test whether $\rho_i$s are all identical by collective measurements, $O(\frac{d^2}{\epsilon^2})$ copies are sufficient by independent measurements. Given $\rho_1,\rho_2, \cdots,\rho_n\in \mathcal{D}(\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}\otimes\cdots\otimes\mathbb{C}^{d_m})$, $\tilde{O}(\frac{\Pi_{i=1}^m d_i}{\epsilon^2})$ copies are sufficient and necessary to test whether $\rho_i$s are independent by collective measurements, and $O(\frac{\Pi_{i=1}^m d_i^2}{\epsilon^2})$ copies are sufficient by independent measurements. The identity testing and independence testing in $\mathcal{D}(\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2})$ can be accomplished with $O(\frac{d_1^2d_2^2}{\epsilon^2})$ copies using just local measurements, respectively. This technique is used to provide efficient testers of conditional independence for classical-quantum-quantum states.