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arXiv:2605.15953 · IEEE International Symposium on Information Theory (ISIT)

Fast convergence of Dynamic Capacities of GNS-Symmetric Quantum Channels

Omar Fawzi, Li Gao, Mizanur Rahaman, Mostafa Taheri

quant-phmath-phmath.OASubmitted May 15, 2026• Announced May 18, 2026• 0 citations
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Abstract

We consider a quantum system described by a quantum channel $Φ$ that is applied at every time step and study the time evolution of its information capacities. When $Φ$ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of $Φ$. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.

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We consider a quantum system described by a quantum channel ΦΦΦ that is applied at every time step and study the time evolution of its information capacities. When ΦΦΦ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of ΦΦΦ. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.
This thesis studies the capacities of quantum channels under repeated noisy evolution, with a focus on the long-time behavior of classical and quantum information transmission. The main goal is to understand how much information can still be preserved or recovered when a quantum channel is applied many times, and to develop structural, analytic, and algorithmic tools for characterizing these capacities.The core results are presented in Chapter 2. First, the thesis investigates the infinite-time regime and shows that the asymptotic capacities of quantum channels are determined by the structure of their peripheral subspace. This leads to explicit characterizations of the infinite-time classical capacity and bounds on the infinite-time quantum capacity, together with results on additivity and extensions to more general settings. The thesis then studies the finite-time behavior for GNS-symmetric channels and proves explicit exponential convergence bounds, showing that the capacities approach their infinite-time values rapidly. In addition, algorithmic methods are developed for computing the peripheral structure and the associated capacity quantities efficiently.Chapter 3 explores a related optimization-based direction motivated by Sum-of-Squares methods and semidefinite programming. Using ideas inspired by separability and SDP hierarchies, this part investigates whether machine-learning-assisted ansatz discovery can help construct more efficient SOS certificates. Although this direction is not yet complete in the specific capacity setting, it provides promising intermediate results and suggests new computational approaches for related problems in quantum information theory. Overall, this thesis provides a structural and quantitative understanding of the long-time information-carrying capabilities of noisy quantum dynamical systems, while also opening new perspectives through optimization and machine-learning-based methods.
This dissertation deals with entropic inequalities for bosonic channels in quantum information theory. Such inequalities have applications in foundational questions of quantum communication and can be used to make statements about the convergence of quantum dynamical semigroups. We prove, among other results, an inequality for the output entropy of a quantum channel which models additive classical noise and apply this inequality to derive bounds on the classical capacity of a general family of non-Gaussian quantum channels.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
We study encodings that give the best-known thresholds for the nonzero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a nonzero capacity for a fidelity (probability of no error) of f=0.80870f=0.808\phantom{\rule{0.2em}{0ex}}70f=0.80870.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.