Partially smoothed information measures are fundamental tools in one-shot quantum information theory. In this work, we determine the exact strong converse exponents of these measures for both pure quantum states and classical states. Notably, we find that the strong converse exponents based on trace distance takes different forms between pure and classical states, indicating that they are not uniform across all quantum states. Leveraging these findings, we derive the strong converse exponents for quantum data compression, intrinsic randomness extraction, and classical state splitting. A key technical step in our analysis is the determination of the strong converse exponent for classical privacy amplification, which is of independent interest.

We introduce a minimax approach for characterizing the capacities of fully quantum arbitrarily varying channels (FQAVCs) under different shared resource models. In contrast to previous methods, our technique avoids de Finetti-type reductions, providing a more streamlined proof without dependency on the dimension of the jamming system. Consequently, we show that the entanglement-assisted and shared-randomness-assisted capacities of FQAVCs match those of the corresponding compound channels, even in the presence of general quantum adversaries.

The quantum dichotomies problem asks at what rate one pair of quantum states can be approximately mapped into another pair of quantum states. In the many copy limit and for vanishing error, the optimal rate is known to be given by the ratio of the respective quantum relative distances. Here, we study the large-deviation behavior of quantum dichotomies and determine the exact strong converse exponent based on the purified distance. This is the first time to establish the exact high-error large-deviation analysis for this task in fully quantum setting.

We determine the exact error and strong converse exponent for entanglement-assisted classical-quantum channel simulation in worst case input purified distance. The error exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [1, \infty)$, notably without the need for a critical rate--a sharp contrast to the error exponent for classical-quantum channel coding. The strong converse exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [\frac{1}{2},1]$. As in the classical work [Oufkir et al., arXiv:2410.07051], we start with the goal of asymptotically expanding the meta-converse for channel simulation in the relevant regimes. However, to deal with non-commutativity issues arising from classical-quantum channels and entanglement-assistance, we critically use various properties of the quantum fidelity, additional auxiliary channel techniques, approximations via Chebyshev inequalities, and entropic continuity bounds.

The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel’s entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.

We determine the exact strong converse exponent for entanglement-assisted classical communication of a quantum channel. Our main contribution is the derivation of an upper bound for the strong converse exponent which is characterized by the sandwiched Rényi divergence. It turns out that this upper bound coincides with the lower bound of Gupta and Wilde (Commun. Math. Phys. 334:867-887, 2015). Thus, the strong converse exponent follows from the combination of these two bounds. Our result has two implications. Firstly, it implies that the exponential bound for the strong converse property of quantum-feedback-assisted classical communication, derived by Cooney, Mosonyi and Wilde (Commun. Math. Phys. 344:797-829, 2016), is optimal. This answers their open question in the affirmative. Hence, we have determined the exact strong converse exponent for this problem as well. Secondly, due to an observation of Leung and Matthews, it can be easily extended to deal with the transmission of quantum information under the assistance of entanglement or quantum feedback, yielding similar results. The above findings provide, for the first time, a complete operational interpretation to the channel’s sandwiched Rényi information of order α > 1.

The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order$s\in (1,2]$, which has not been addressed previously in the quantum setting.

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award.The smooth max-relative entropy is a basic tool in quantum information theory and cryptography. In this paper, we derive the exact exponent for the decay of the small modification of the quantum state in smoothing the max-relative entropy. We then apply this result to the problem of privacy amplification against quantum side information and obtain an upper bound for the exponent of the decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general.Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched Rényi divergence of order between 1 and 2, which has not been addressed previously in the quantum setting.