Estimating Quantum Entropy
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Abstract
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of estimating the von Neumann entropy, <inline-formula> <tex-math notation="LaTeX">$S(\rho)$ </tex-math></inline-formula>, and Rényi entropy, <inline-formula> <tex-math notation="LaTeX">$S_{\alpha }(\rho)$ </tex-math></inline-formula> of an unknown mixed quantum state <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> dimensions, given access to independent copies of <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>. We provide algorithms with copy complexity <inline-formula> <tex-math notation="LaTeX">$O(d^{2/\alpha })$ </tex-math></inline-formula> for estimating <inline-formula> <tex-math notation="LaTeX">$S_{\alpha }(\rho)$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$\alpha < 1$ </tex-math></inline-formula>, and copy complexity <inline-formula> <tex-math notation="LaTeX">$O(d^{2})$ </tex-math></inline-formula> for estimating <inline-formula> <tex-math notation="LaTeX">$S(\rho)$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$S_{\alpha }(\rho)$ </tex-math></inline-formula> for non-integral <inline-formula> <tex-math notation="LaTeX">$\alpha >1$ </tex-math></inline-formula>. These bounds are at least quadratic in <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>, which is the order dependence on the number of copies required for estimating the entire state <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>. For integral <inline-formula> <tex-math notation="LaTeX">$\alpha >1$ </tex-math></inline-formula>, on the other hand, we provide an algorithm for estimating <inline-formula> <tex-math notation="LaTeX">$S_{\alpha }(\rho)$ </tex-math></inline-formula> with a sub-quadratic copy complexity of <inline-formula> <tex-math notation="LaTeX">$O(d^{2-2/\alpha })$ </tex-math></inline-formula>, and we show the optimality of the algorithms by providing a matching lower bound.