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Frontiers in Optics

Quantum Communication over Partially Degradable Quantum Channels

László Gyöngyösi, Sándor Imre

Published Jan 1, 2013• 1 citation
Publisher

Abstract

A quantum channel is partially degradable if the output can be used to simulate the degraded environment. We demonstrate that better quantum data rates can be achieved over PD channels in comparison to standard channels.
DOI
10.1364/fio.2013.fm3c.7

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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.
We investigate a quantum coding for quantum communication over a partially degradable (PD) quantum channel. For a PD channel, the degraded environment state can be expressed from the channel output state up to a degrading map. PD channels can be restricted to the set of optical channels, which allows for the parties to exploit the benefits in experimental quantum communications. We show that for a PD channel, the partial degradability property leads to higher quantum data rates in comparison with those of a degradable channel. The PD property is particularly convenient for quantum communications and allows one to implement the experimental quantum protocols with higher performance. We define a coding scheme for PD channels and give the achievable rates of quantum communication.
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 a environment that is “small ” in the sense dE ≤ dB. Such channels include
A quantum channel is conjugate degradable if the channel’s environment can be simulated up to complex conjugation using the channel’s output. For all such channels, the quantum capacity can be evaluated using a single-letter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to calculate the quantum capacity of N to N+1 and 1 to M universal quantum cloning machines as well as the quantum capacity of a channel that arises naturally when data are being transmitted to an accelerating receiver. All the channels considered turn out to have strictly positive quantum capacity, meaning they could be used as part of a communication system to send quantum states reliably.
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.