TQC 2011: Theory of Quantum Computation, Communication, and Cryptography pp 164-173 | Cite as
Bitwise Quantum Min-Entropy Sampling and New Lower Bounds for Random Access Codes
Abstract
Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. König and Renner showed a min-entropy sampling theorem that holds relative to quantum knowledge. Their result achieves the optimal rate, but it can only be applied if the bits are sampled in blocks, and only gives weak bounds for the non-smooth min-entropy.
We give two new quantum min-entropy sampling theorems that do not have the above weaknesses. The first theorem shows that the result by König and Renner also applies to bitwise sampling, and the second theorem gives a strong bound for the non-smooth min-entropy. Our results imply a new lower bound for \(k\)-out-of-\(n\) random access codes: while previous results by Ben-Aroya, Regev, and de Wolf showed that the decoding probability is exponentially small in \(k\) if the storage rate is smaller than \(0.7\), our results imply that this holds for any storage rate strictly smaller than \(1\), which is optimal.
Keywords
Optimal Rate Statistical Distance Sampling Theorem Storage Rate Strong ExtractorNotes
Acknowledgements
I thank Robert König, Thomas Vidick and Stephanie Wehner for helpful discussions and the anonymous reviewers for useful comments. This work was funded by the U.K. EPSRC grant EP/E04297X/1 and the Canada-France NSERC-ANR project FREQUENCY. Most of this work was done while I was at the University of Bristol.
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