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arXiv:2506.13397 · Physical Review A

Exact quantum capacity of decohering channels in arbitrary dimensions

Shayan Roofeh, Vahid Karimipour

Published Nov 24, 2025• 0 citations
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Abstract

We derive exact analytical expressions for the quantum capacity of a broad subclass of generalized dephasing channels of the form $\mathrm{\ensuremath{\Lambda}}(\ensuremath{\rho})=(1\ensuremath{-}x)\ensuremath{\rho}+xD(\ensuremath{\rho})$, where $D(\ensuremath{\rho})$ represents a structured decoherence process. These channels are degradable for all noise parameters and in arbitrary dimensions, yielding closed-form single-letter capacity formulas. Our analysis includes fully decohering, block-decohering, and weakly decohering channels, the latter involving coherence preservation within overlapping subspaces. Surprisingly, even under maximal decoherence, the channel may retain nonzero capacity due to residual coherence structure. These results reveal a quantitative role for decoherence-free and partially coherent subspaces in preserving quantum information, offering guidance for encoding strategies in quantum memories and fault-tolerant quantum communication systems.

DOI
10.1103/s124-z299

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We derive exact analytical expressions for the quantum capacity of a broad subclass of generalized dephasing channels of the form \ensuremathΛ(\ensuremathρ)=(1\ensuremath−x)\ensuremathρ+xD(\ensuremathρ)\mathrm{\ensuremath{\Lambda}}(\ensuremath{\rho})=(1\ensuremath{-}x)\ensuremath{\rho}+xD(\ensuremath{\rho})\ensuremathΛ(\ensuremathρ)=(1\ensuremath−x)\ensuremathρ+xD(\ensuremathρ), where D(\ensuremathρ)D(\ensuremath{\rho})D(\ensuremathρ) represents a structured decoherence process. These channels are degradable for all noise parameters and in arbitrary dimensions, yielding closed-form single-letter capacity formulas. Our analysis includes fully decohering, block-decohering, and weakly decohering channels, the latter involving coherence preservation within overlapping subspaces. Surprisingly, even under maximal decoherence, the channel may retain nonzero capacity due to residual coherence structure. These results reveal a quantitative role for decoherence-free and partially coherent subspaces in preserving quantum information, offering guidance for encoding strategies in quantum memories and fault-tolerant quantum communication systems.
We show that the amount of coherent quantum information that can be reliably transmitted down a dephasing channel with memory is maximized by separable input states. In particular, we model the channel as a Markov chain or a multimode environment of oscillators. While in the first model, the maximization is achieved for the maximally mixed input state, in the latter it is convenient to exploit the presence of a decoherence-protected subspace generated by memory effects. We explicitly compute the quantum channel capacity for the first model while numerical simulations suggest a lower bound for the latter. In both cases memory effects enhance the coherent information. We present results valid for arbitrary input size.
We introduce Partially Coherent Direct Sum (PCDS) quantum channels, as a generalization of the already known Direct Sum quantum channels. We derive necessary and sufficient conditions to identify the subset of those maps which are degradable, and provide a simplified expression for their quantum capacities. Interestingly, the special structure of PCDS allows us to extend the computation of the quantum capacity formula also for quantum channels which are explicitly not degradable (nor antidegradable). We show instances of applications of the results to dephasing channels, amplitude damping channels and combinations of the two.
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.
Quantum channel capacities give the fundamental performance limits for information flow over a communication channel. However, the prevalence of superadditivity is a major obstacle to understanding capacities, both quantitatively and conceptually. In contrast, examples exhibiting additivity, though relatively rare, offer crucial insights into the origins of nonadditivity and form the basis of our strongest upper bounds on capacity. Degradable channels, whose coherent information is provably additive, stand out as among the few classes of channels for which the quantum capacity is exactly computable. In this paper, we introduce two families of non-degradable channels whose coherent information remains additive, making their quantum capacities tractable. First, we demonstrate that channels capable of “outperforming” their environment, under conditions weaker than degradability, can exhibit either strong or weak additivity of coherent information. Second, we explore a complementary construction that modifies a channel to preserve coherent information additivity while destroying the “outperforming” property. We analyze how structural constraints guarantee strong and weak additivity and investigate how relaxing these constraints leads to the failure of strong additivity, with weak additivity potentially persisting.
Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.
The quantum capacity of a quantum channel captures its capability for noiseless quantum communication. It lies at the heart of quantum information theory. Unfortunately, our poor understanding of nonadditivity of coherent information makes it hard to understand the quantum capacity of all but very special channels. In this Letter, we consider the dephrasure channel, which is the concatenation of a dephasing channel and an erasure channel. This very simple channel displays remarkably rich and exotic properties: we find nonadditivity of coherent information at the two-letter level, a substantial gap between the threshold for zero quantum capacity and zero single-letter coherent information, a big gap between single-letter coherent and private information, and positive quantum capacity for all complementary channels. Its clean form simplifies the evaluation of coherent information substantially and, as such, we hope that the dephrasure channel will provide a much-needed laboratory for the testing of new ideas about nonadditivity.