Optimal Second-Order Rates for Quantum Soft Covering and Privacy
Amplification
- URL: http://arxiv.org/abs/2202.11590v1
- Date: Wed, 23 Feb 2022 16:02:31 GMT
- Title: Optimal Second-Order Rates for Quantum Soft Covering and Privacy
Amplification
- Authors: Yu-Chen Shen, Li Gao, Hao-Chung Cheng
- Abstract summary: We study quantum soft covering and privacy amplification against quantum side information.
For both tasks, we use trace distance to measure the closeness between the processed state and the ideal target state.
Our results extend to the moderate deviation regime, which are the optimal rates when the trace distances vanish at sub-exponential speed.
- Score: 19.624719072006936
- License: http://creativecommons.org/publicdomain/zero/1.0/
- Abstract: We study quantum soft covering and privacy amplification against quantum side
information. The former task aims to approximate a quantum state by sampling
from a prior distribution and querying a quantum channel. The latter task aims
to extract uniform and independent randomness against quantum adversaries. For
both tasks, we use trace distance to measure the closeness between the
processed state and the ideal target state. We show that the minimal amount of
samples for achieving an $\varepsilon$-covering is given by the
$(1-\varepsilon)$-hypothesis testing information (with additional logarithmic
additive terms), while the maximal extractable randomness for an
$\varepsilon$-secret extractor is characterized by the conditional
$(1-\varepsilon)$-hypothesis testing entropy.
When performing independent and identical repetitions of the tasks, our
one-shot characterizations lead to tight asymptotic expansions of the
above-mentioned operational quantities. We establish their second-order rates
given by the quantum mutual information variance and the quantum conditional
information variance, respectively. Moreover, our results extend to the
moderate deviation regime, which are the optimal asymptotic rates when the
trace distances vanish at sub-exponential speed. Our proof technique is direct
analysis of trace distance without smoothing.
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