- Sublinear quantum algorithms for estimating von Neumann entropy(arXiv)
Author : Tom Gur, Min-Hsiu Hsieh, Sathyawageeswar Subramanian
Abstract : Entropy is a fundamental property of both classical and quantum systems, spanning myriad theoretical and practical applications in physics and computer science. We study the problem of obtaining estimates to within a multiplicative factor γ>1 of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. Our main results are: ∙ an O˜(n1+η2γ2)-query quantum algorithm that outputs a γ-multiplicative approximation of the Shannon entropy H(p) of a classical probability distribution p=(p1,…,pn); ∙ an O˜(n12+1+η2γ2)-query quantum algorithm that outputs a γ-multiplicative approximation of the von Neumann entropy S(ρ) of a density matrix ρ∈Cn×n. In both cases, the input is assumed to have entropy bounded away from zero by a quantity determined by the parameter η>0, since, as we prove, no polynomial query algorithm can multiplicatively approximate the entropy of distributions with arbitrarily low entropy. In addition, we provide Ω(n13γ2) lower bounds on the query complexity of γ-multiplicative estimation of Shannon and von Neumann entropies. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature
2.Device-independent lower bounds on the conditional von Neumann entropy (arXiv)
Author : Peter Brown, Hamza Fawzi, Omar Fawzi
Abstract : The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascués-Pironio-Acín hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a new minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering instances of known tight analytical bounds. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.
