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arXiv:2608.16452

Strong Converse Exponents of Quantum Soft Covering and Privacy Amplification

Shi-Bing Li, Hongsen Qiu, Xinyu Zhang

quant-phmath.FAmath.OASubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
v1 · Aug 17, 2026v2 · Aug 18, 2026
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Abstract

We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders $α\in[\frac{1}{2},\infty)$. For $α\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $α\in[1,\infty)$, it is characterized by the order-$α$ 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 $α\in(2,\infty)$, expressed in terms of the corresponding order-$α$ 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 $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $α\in[\frac{1}{2},1)$.

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In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order α∈[2,∞)α\in[2,\infty)α∈[2,∞). The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.
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We determine the strong converse exponent for the entanglement cost of quantum state merging, showing that it is characterized by the optimized ααα-zzz conditional Rényi entropies with z=α/2∈[1/2,1]z=α/2\in[1/2,1]z=α/2∈[1/2,1]. This contrasts with the sandwiched conditional Rényi entropies that typically govern strong converse exponents in quantum information theory. As a consequence, we derive the strong converse exponent of the partially smoothed conditional min-entropy in purified distance. This exponent is governed by club-sandwiched conditional entropies, whereas global smoothing leads to a sandwiched expression.
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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].