Properties of Noncommutative Rényi and Augustin Information

Cheng, Hao-Chung; Gao, Li; Hsieh, Min-Hsiu

doi:10.1007/s00220-022-04319-8

Cited by 12 publications

(2 citation statements)

References 98 publications

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“…Proof of a concavity property.Proof of Lemma 14. As stated in the proof of Proposition 4-(b) in[46], for any classical-quantum state σ XE = x∈X p X (x)|x x| ⊗ σ x E and an arbitrary state τ E ∈ D(E), we haveD * α (σ XE σ X ⊗ τ E ) α (σ XE σ X ⊗τ E ) = x p X (x)e (α−1)D * α (ρ x E τ E ) 1 α…”

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confidence: 82%

Strong Converse for Privacy Amplification against Quantum Side Information

Shen¹,

Li²,

Cheng³

2022

Preprint

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We establish a one-shot strong converse bound for privacy amplification against quantum side information using trace distance as a security criterion. This strong converse bound implies that in the independent and identical scenario, the trace distance exponentially converges to one in every finite blocklength when the rate of the extracted randomness exceeds the quantum conditional entropy. The established one-shot bound has an application to bounding the information leakage of classical-quantum wiretap channel coding and private communication over quantum channels. That is, the trace distance between Alice and Eavesdropper's joint state and its decoupled state vanishes as the rate of randomness used in hashing exceeds the quantum mutual information. On the other hand, the trace distance converges to one when the rate is below the quantum mutual information, resulting in an exponential strong converse. Our result also leads to an exponential strong converse for entropy accumulation, which complements a recent result by Dupuis [arXiv:2105.05342]. Lastly, our result and its applications apply to the moderate deviation regime. Namely, we characterize the asymptotic behaviors of the trace distances when the associated rates approach the fundamental thresholds with speeds slower than O( 1 / √ n).

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confidence: 82%

Strong Converse for Privacy Amplification against Quantum Side Information

Shen¹,

Li²,

Cheng³

2022

Preprint

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“…The large deviation analysis [16,[61][62][63][64][65][66][67][68][69][70][71] of privacy amplification against quantum side information and quantum soft covering has been investigated in previous literature [21,26,27,38,72,73], wherein one fixes the rate or the size of |Z| and |C| and studies the optimal errors in terms of the trace distance. Also, some moderate deviation analysis [44,45] were studied for characterizing the minimal trace distance while the rates approach the first-order limits with certain speed [27,38].…”

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confidence: 99%

Optimal Second-Order Rates for Quantum Soft Covering and Privacy Amplification

Shen¹,

Li²,

Cheng³

2022

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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 ε-covering is given by the (1 − ε)-hypothesis testing information (with additional logarithmic additive terms), while the maximal extractable randomness for an ε-secret extractor is characterized by the conditional (1 − ε)-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 secondorder 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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