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arXiv:2008.00494 · Quantum

Partially Coherent Direct Sum Channels

Stefano Chessa, Vittorio Giovannetti

Published Jul 15, 2021• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2021-07-15-504

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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.
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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.
Abstract Evaluating capacities of quantum channels is the first purpose of quantum Shannon theory, but in most cases the task proves to be very hard. Here, we introduce the set of Multi-level Amplitude Damping quantum channels as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension d . In the special case of d = 3, by exploiting degradability, data-processing inequalities, and channel isomorphism, we compute the associated quantum and private classical capacities for a rather wide class of maps, extending the set of models whose capacity can be computed known so far. We proceed then to the evaluation of the entanglement assisted quantum and classical capacities.
The set of Multi-level Amplitude Damping (MAD) quantum channels is introduced as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension ddd. In the special case of d=3d=3d=3, by exploiting degradability, data-processing inequalities, and channel isomorphism, we compute the associated quantum and private classical capacities for a rather wide class of maps, extending the set of solvable models known so far. We proceed then to the evaluation of the entanglement assisted, quantum and classical, capacities.
Traditionally, Quantum Information, and Quantum Communication specifically, have been focused on qubit-based architectures. Recent results, however, highlighted that higher dimensional architectures (qudit-based) may present advantages both in terms of communication and computation; a family of channels called Multi-level Amplitude Damping (MAD) channels, which are a possible qudit generalization of the well known Amplitude Damping Channels, is able to model energy decay processes that may happen during signal transmission. In this work, the Quantum Capacity of 4-dimensional MAD's is studied, relying on a technique for computing it even outside of degradable and antidegradable conditions. We also characterized the complete region of antidegradability and degradability in the parameter space for a generic d-dimensional MAD using both analytical and semi-numerical methods.
This thesis investigates a characterization for the set of antidegradable qubit channels. The characterization comes from a correspondence between the antidegradability of a channel, and the symmetric extendibility of the Choi operator. Using an inequality intended to describe the set of bipartite qubit states which admit symmetric extension, we demonstrate that it additionally provides a condition to characterize the set of antidegradable qubit channels. Using the characterization we investigate the antidegradability of qubit channels with respect to the rank of the Choi matrix. We also apply the antidegradability criterion to the qubit amplitude damping and qubit depolarizing channel. We end by discussing degradable and antidegradable channels in terms of their channel capacities and the applications of these results to investigating phenomena such as super-additive coherent information and superactivation.