arXiv:quant-ph/0305062

Renyi extrapolation of Shannon entropy

Karol Życzkowski

Published May 12, 2003• 13 citations

Abstract

Relations between Shannon entropy and Renyi entropies of integer order are discussed. For any N-point discrete probability distribution for which the Renyi entropies of order two and three are known, we provide an lower and an upper bound for the Shannon entropy. The average of both bounds provide an explicit extrapolation for this quantity. These results imply relations between the von Neumann entropy of a mixed quantum state, its linear entropy and traces.

DOI: 10.48550/arxiv.quant-ph/0305062

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arXiv:quant-ph/0305062

Renyi extrapolation of Shannon entropy

Karol Życzkowski

Published May 12, 2003• 13 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/0305062

Related papers

Rényi Extrapolation of Shannon Entropy

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Open Systems & Information DynamicsSep 1, 2003181 citations

R enyi Extrapolation of Shannon Entropy

Centrum Fizyki Teoretycznej, Polska Akademia Nauk

Jan 1, 20033 citations

The structure of Rényi entropic inequalities

Noah Linden, Milán Mosonyi, Andreas Winter

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H_α(\mathbf{p})

for any distribution

\mathbf{p} = (p_i)_{i=1}^n

given its purified quantum query-access. For non-integer

α> 1

, our algorithm estimates

H_α(\mathbf{p})

to within additive error

ε

using only

\widetilde{\mathcal{O}}\big(n^{\max\{1/2,1-1/α\}}/ε\big)

queries, which improves the best known upper bound for dependence on both

n

and

1/ε

. For

0<α<1

, our algorithm has query complexity

\widetilde{\mathcal{O}}\big(n^{1/(2α)}/ε^{\max\{1,1/(2α)\}}\big)

, which improves the dependence on

1/ε

of the current best upper bound without incresing its

n

dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to

Ω\big(1/ε^{\max\{1,1/(2α)\}}\big)

for

0<α<1

, which identifies the optimal precision dependence of our algorithm.