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.
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Notes
- 1.
- 2.
For example in [KWW09].
- 3.
\(H_{\min }\) is defined in Sect. 2.
- 4.
Therefore, if we are interested in extracting a key, Theorem 1 only gives better bounds if the sample size is small enough.
- 5.
This definition is equivalent to \(D(\rho ,\phi ) := \frac{1}{2} \Vert \rho - \phi \Vert _1 = \frac{1}{2} {{\mathrm{tr}}}[\sqrt{(\rho - \phi )^\dagger (\rho - \phi )}]\).
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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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Wullschleger, J. (2014). Bitwise Quantum Min-Entropy Sampling and New Lower Bounds for Random Access Codes. In: Bacon, D., Martin-Delgado, M., Roetteler, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2011. Lecture Notes in Computer Science(), vol 6745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54429-3_11
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DOI: https://doi.org/10.1007/978-3-642-54429-3_11
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