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Multi-level Amplitude Damping channels: quantum capacity analysis

Stefano Chessa, Vittorio Giovannetti

Published Aug 2, 2020• 7 citations
OpenAlex

Abstract

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 $d$. 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 solvable models known so far. We proceed then to the evaluation of the entanglement assisted, quantum and classical, capacities.

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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.
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.
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.
The primary objective of quantum Shannon theory is to evaluate the capacity of quantum channels. In spite of the existence of rigorous coding theorems that quantify the transmission of information through quantum channels, superadditivity effects limit our understanding of the channel capacities. In this paper, we mainly focus on a family of channels known as multilevel amplitude damping channels. We investigate some of the information capacities of the simplest member of the multilevel amplitude damping channel, a qutrit channel, in the presence of correlations between successive applications of the channel. We find the upper bounds of the single-shot classical capacities and calculate the quantum capacities associated with a specific class of maps after investigating the degradability property of the channels. Additionally, the quantum and classical capacities of the channels have been computed in entanglement-assisted scenarios.
We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.
Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability ppp, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2)h_2((1+\sqrt p)/2)h2​((1+p​)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1p<1p<1, and construct a binary Kraus representation attaining the uniform formation bound.
We introduce a new set of quantum channels: resonant multilevel amplitude damping (ReMAD) channels. Among other instances, they can describe energy dissipation effects in multilevel atomic systems induced by the interaction with a zero-temperature bosonic environment. At variance with the already known class of multilevel amplitude damping (MAD) channels, this new class of maps allows the presence of an environment unable to discriminate transitions with identical energy gaps. After characterizing the algebra of their composition rules, by analyzing the qutrit case, we show that this new set of channels can exhibit degradability and antidegradability in vast regions of the allowed parameter space. There we compute their quantum capacity and private classical capacity. We show that these capacities can be computed exactly also in regions of the parameter space where the channels aren't degradable nor antidegradable.