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arXiv:2202.10995

Error Exponent and Strong Converse for Quantum Soft Covering

Hao–Chung Cheng, Li Gao

Published Feb 22, 2022• 0 citations
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Abstract

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.

DOI
10.48550/arxiv.2202.10995

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How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.
How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem, which uses a pairwise-independent random codebook to approximate the target output state of a quantum channel. We establish a one-shot error exponent bound and a one-shot strong converse bound on the approximation error measured in terms of the expected trace distance between the codebook-induced state and the true channel output state. When using independent and identically-distributed random codebook with a rate above the quantum mutual informationI(X : B)ρ, we prove that the error decays exponentially with its error exponent expressed by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the error converges to one exponentially fast. Similar results are obtained using a random constant composition codebook, whereas the sandwiched Augustin information gives the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime.
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders α∈[12,∞)α\in[\frac{1}{2},\infty)α∈[21​,∞). For α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for α∈[1,∞)α\in[1,\infty)α∈[1,∞), it is characterized by the order-ααα sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for α∈(2,∞)α\in(2,\infty)α∈(2,∞), expressed in terms of the corresponding order-ααα sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the KKK-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1).
How well can we approximate a classical-quantum channel output state by using a random codebook with a certain size? In this work, we study the quantum soft covering problem and establish exponential achievability and strong converse bounds on the expected trace distance between the codebook-induced state and the true state. When using independent and identically distributed random codebook or constant composition random codebook with a rate above the quantum mutual information I(X : B)ρ, we prove that the trace distances decay exponentially with error exponents expressed by the sandwiched Rényi and Augustin information. For a rate below I(X : B)ρ, we show that both the trace distances converge to 1 exponentially fast. The full manuscript can be found at [1].
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 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.
In this work, we propose a soft covering problem for fully quantum channels using relative entropy as a criterion for operator closeness. We establish covering lemmas by deriving one-shot bounds on the achievable rates in terms of smooth min-entropies. In the asymptotic regime, we show that the infimum of the rate, defined as the logarithm of the minimum rank of the encoded input state, is given by the minimal coherent information between the reference and output systems that yields the target output state. Furthermore, we present a one-shot quantum decoupling theorem that also employs a relative-entropy criterion. Due to the Pinsker inequality, our one-shot results based on the relative-entropy criterion are tighter than the corresponding results based on the trace norm considered in the literature. In addition, we establish achievable error exponents and second-order rates for quantum soft covering under both trace-distance and relative-entropy criteria.
We show two properties of i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states. The first property is that the probability distribution of the error exponent across the ensemble accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the works by Dalai, Renes, Li and Yang and Cheng and Liu, implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate region the threshold we derive coincides with the ensemble-average of the exponent, that is, the CQ typical random coding (TRC) exponent. The second property we derive is that the probability that a randomly selected code from an ensemble contains a nested code of the same rate such that each of its codewords achieves the CQ expurgated exponent goes to one asymptotically in the codeword length. Such nested code can be obtained by expurgating a vanishingly small fraction of codewords as the codeword length increases. This result refines Holevo's work [5], which proved that the expurgated exponent can be achieved by discarding the worst half of the codewords from at least one code in the ensemble.