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arXiv:2003.00583 · IEEE Transactions on Information Theory

Positivity and Nonadditivity of Quantum Capacities Using Generalized Erasure Channels

Vikesh Siddhu, Robert B. Griffiths

Published May 17, 2021• 26 citations
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Abstract

We consider various forms of a process, which we call gluing, for combining two or more complementary quantum channel pairs (B,C) to form a composite. One type of gluing combines a perfect channel with a second channel to produce a generalized erasure channel pair (Bg,Cg). We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the dephrasure channel of Leditzky et al. For both (i) and (ii), (Bg,Cg) depends on the damping parameter 0 ≤ p ≤ 1 and a parameter 0 ≤ λ ≤ 1 that characterizes the gluing process. In both cases we study Q(1)(Bg) and Q(1)(Cg), where Q(1)is the channel coherent information, and determine the regions in the (p, λ) plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that Q(1)(Cg) is zero for λ ≤ 1/2 when p=0, but is strictly positive (though perhaps extremely small) for all values of λ > 0 when p is positive by even the smallest amount. In addition we study the nonadditivity of Q(1)(Bg) for two identical channels in parallel. It occurs in a well-defined region of the (p, λ) plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of (p, λ) values where nonadditivity occurs. Again, an intuitive explanation is lacking.

DOI
10.1109/tit.2021.3080819

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We consider various forms of a process, which we call gluing, for combining two or more complementary quantum channel pairs (B,C) to form a composite. One type of gluing combines a perfect channel with a second channel to produce a generalized erasure channel pair (Bg,Cg). We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the dephrasure channel of Leditzky et al. For both (i) and (ii), (Bg,Cg) depends on the damping parameter 0 ≤ p ≤ 1 and a parameter 0 ≤ λ ≤ 1 that characterizes the gluing process. In both cases we study Q(1)(Bg) and Q(1)(Cg), where Q(1)is the channel coherent information, and determine the regions in the (p, λ) plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that Q(1)(Cg) is zero for λ ≤ 1/2 when p=0, but is strictly positive (though perhaps extremely small) for all values of λ > 0 when p is positive by even the smallest amount. In addition we study the nonadditivity of Q(1)(Bg) for two identical channels in parallel. It occurs in a well-defined region of the (p, λ) plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of (p, λ) values where nonadditivity occurs. Again, an intuitive explanation is lacking.
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.
The quantum capacity of a channel is often used as a single measure to characterize the ability of a channel to transmit quantum information coherently. The capacity determines the maximal asymptotic rate at which we can code reliably over a channel. Here we argue that this asymptotic treatment is insufficient to the point of being irrelevant in the quantum setting when decoherence limits our ability to manipulate large quantum systems. For all practical purposes we should instead focus on the tradeoff between three parameters: the rate of the code, the number of coherent uses of the channel, and the fidelity of the transmission. The aim is then to specify the region determined by allowed combinations of these parameters. Towards this goal, we find an approximate characterization of the region of allowed triplets for the qubit dephasing channel and for the erasure channel with classical post-processing. In each case the region is parametrized by a second channel parameter, the dispersion of the quantum capacity. In the process we also develop general bounds on the achievable region that are valid for all quantum channels.
Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to superadditivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of superadditive channels are scarce. Here we study a family of channels called platypus channels. Its simplest member, a qutrit channel, is shown to display superadditivity of coherent information when used jointly with a variety of qubit channels. Higher-dimensional family members display superadditivity of quantum capacity together with an erasure channel. Subject to the "spin-alignment conjecture" introduced in our companion paper [F. Leditzky, D. Leung, V. Siddhu, G. Smith, and J. A. Smolin, The platypus of the quantum channel zoo, IEEE Transactions on Information Theory (IEEE, 2023), 10.1109/TIT.2023.3245985], our results on superadditivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, superadditivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves superadditivity in all examples. Our results show that superadditivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity.
We present an experimental approach to construct a dephrasure channel that contains both dephasing and erasure noises and can be used as an efficient tool to study the superadditivity of coherent information. Using a three-fold dephrasure channel, the superadditivity of coherent information is observed, and a substantial gap is found between the zero single-letter coherent information and zero quantum capacity. Particularly, we find that, when the coherent information of n channel uses is zero, with a larger number of channel uses the quantum capacity becomes positive. These phenomena exhibit a more obvious superadditivity of coherent information than previous works and demonstrate a higher threshold for nonzero quantum capacity. Such novel channels built in our experiment also can provide a useful platform to study the nonadditive properties of coherent information and quantum channel capacity.
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.
Recently there has been considerable activity on the subject of the additivity of various quantum channel capacities. Here, we construct a family of channels with a sharply bounded classical and, hence, private capacity. On the other hand, their quantum capacity when combined with a zero private (and zero quantum) capacity erasure channel becomes larger than the previous classical capacity. As a consequence, we can conclude for the first time that the classical private capacity is nonadditive. In fact, in our construction even the quantum capacity of the tensor product of two channels can be greater than the sum of their individual classical private capacities. We show that this violation occurs quite generically: every channel can be embedded into our construction, and a violation occurs whenever the given channel has a larger entanglement-assisted quantum capacity than (unassisted) classical capacity.