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Degradable and Antidegradable Qubit Channels

Connor Paddock

Published Sep 7, 2016• 1 citation
OpenAlex

Abstract

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.

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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.
This paper provides a characterization for the set of antidegradable qubit channels. The characterization arises from the correspondence between the antidegradability of a channel and the symmetric extendibility of its Choi operator. Using an inequality derived to describe the set of bipartite qubit states which admit symmetric extension, we are able to characterize the set of all antidegradable qubit channels. Using the characterization we investigate the antidegradability of unital qubit channels and arbitrary qubit channels with respect to the dimension of the environment. We additionally provide a condition which describes qubit channels which are simultaneously degradable and antidegradable along with a classification of self-complementary qubit channels.
To decide whether a quantum channel is degradable is relatively easy: one has to find at least one example of a degrading quantum channel. But in general, no conclusive criterion exists to show the opposite. Using elementary methods we derive a necessary and sufficient condition to decide under what circumstances the conclusion is unambiguous. The findings lead to an extension of the antidegradability region for qubit and qutrit transpose depolarizing channels. In the qubit case we reproduce the known results for the class of qubit depolarizing channels (due to their equivalence). One of the consequences is that the optimal qubit and qutrit asymmetric cloners possess a single-letter quantum capacity formula. We also investigate the ramifications of the criterion for the search of exclusively conjugate degradable channels.
We analyze bipartite quantum states that admit a symmetric extension. Any such state can be decomposed into a convex combination of states that allow a pure symmetric extension. A necessary condition for a state to admit a pure symmetric extension is that the spectra of the local and global density matrices are equal. This condition is also sufficient for two qubits but not for any larger systems. Using this condition, we present a conjectured necessary and sufficient condition for a two-qubit state to admit symmetric extension, which we prove in some special cases. The results from symmetric extension carry over to degradable and antidegradable channels and we use this to prove that all degradable channels with qubit output have a qubit environment.
The quantum capacity of degradable quantum channels has been proven to be additive. On the other hand, there is no general rule for the behavior of quantum capacity for antidegradable quantum channels. We introduce the set of partially degradable (PD) quantum channels to answer the question of additivity of quantum capacity for a well-separable subset of antidegradable channels. A quantum channel is PD if the channel output can be used to simulate the degraded environment state. The PD channels could exist both in the degradable, antidegradable and conjugate degradable family. We define the term partial simulation, which is a clear benefit that arises from the structure of the complementary channel of a PD channel. We prove that the quantum capacity of an arbitrary dimensional PD channel is additive. We also demonstrate that better quantum data rates can be achieved over a PD channel in comparison with standard (non-PD) channels. Our results indicate that the partial degradability property can be exploited and yet still hold many benefits for quantum communications.
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.
A bipartite state ${\ensuremath{\rho}}_{AB}$ is symmetric extendible if there exists a tripartite state ${\ensuremath{\rho}}_{AB{B}^{\ensuremath{'}}}$ whose ABABAB and $A{B}^{\ensuremath{'}}$ marginal states are both identical to ${\ensuremath{\rho}}_{AB}$. Symmetric extendibility of bipartite states is of vital importance in quantum information because of its central role in separability tests, one-way distillation of Einstein-Podolsky-Rosen pairs, one-way distillation of secure keys, quantum marginal problems, and antidegradable quantum channels. We establish a simple analytic characterization for symmetric extendibility of any two-qubit quantum state ${\ensuremath{\rho}}_{AB}$; specifically, $\mathrm{tr}({\ensuremath{\rho}}_{B}^{2})\ensuremath{\ge}\mathrm{tr}({\ensuremath{\rho}}_{AB}^{2})\ensuremath{-}4\sqrt{det{\ensuremath{\rho}}_{AB}}$. As a special case we solve the bosonic three-representability problem for the two-body reduced density matrix.