Efficient Independence Testing for Quantum states
Nengkun Yu · arXiv (Cornell University) · 2019
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.