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Physica Scripta

Quantum capacity of Pauli channels with memory

Peng Fei Huang, Guangqiang He, Yuan Lu, Guihua Zeng

Published Dec 8, 2010• 7 citations
Publisher

Abstract

The amount of coherent quantum information that can be reliably transmitted down the memory Pauli channels with Markovian correlated noise is investigated. Two methods for evaluating the quantum capacity of the memory Pauli channels are proposed to try to trace the memory effect on the transmissions of quantum information. We show that the evaluation of quantum capacity can be reduced to the calculation of the initial memory state of each successive transmission. Furthermore, we derive quantum capacities of the memory phase flip channel, bit flip channel and bit-phase flip channel. Also, a lower bound of the quantum capacity of the memory depolarizing channel is obtained. An increase of the degree of memory of the channels has a positive effect on the increase of their quantum capacities.

DOI
10.1088/0031-8949/83/01/015005

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The amount of coherent quantum information that can be reliably transmitted down the memory Pauli channels with Markovian correlated noise is investigated. Two methods for evaluating the quantum capacity of the memory Pauli channels are proposed to try to trace the memory effect on the transmissions of quantum information. We show that the evaluation of quantum capacity can be reduced to the calculation of the initial memory state of each successive transmission. Furthermore, we derive quantum capacities of the memory phase flip channel, bit flip channel and bit-phase flip channel. Also, a lower bound of the quantum capacity of the memory depolarizing channel is obtained. An increase of the degree of memory of the channels has a positive effect on the increase of their quantum capacities.
A general formula for the mutual information of the Pauli channels with memory modelled by correlated noise is derived. It is shown that the mutual information depends on the channel shrinking factor, the input state parameter and the channel memory coefficient. The analyses based on the general formula reveal that the entanglement is always a useful resource to enhance the mutual information of some Pauli channels, such as the bit flip channel and the bit-phase flip channel. Our analyses also show that the entanglement is not advantageous to the reliable transmission of classical information for some Pauli channels at any time, such as the phase flip channel and the phase damping channel.
The amount of classical information that is reliably transmitted over two uses of general Pauli channels with memory, modeled as a correlated noise between a single pair of uses, is investigated. The maximum of the mutual information between the input and the output is proven to be achieved by a class of product states that is explicitly given in terms of the relevant channel parameters below some memory threshold, and by maximally entangled states above this threshold. In particular, this proves a conjecture on the depolarizing channel by Macchiavello and Palma [Phys. Rev. A 65, 050301(R) (2002)]. Furthermore, it also shows that no other scenario can occur for Pauli channels as for example the existence of an intermediate optimal degree of entanglement reported for some Gaussian channels with memory.
The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. The computation of the capacity is reduced to an entropy computation for a function of a Markov process. A reformulation in terms of algebraic measures then enables its calculation. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise correlations is found to increase the capacity.
We show that the amount of coherent quantum information that can be reliably transmitted down a dephasing channel with memory is maximized by separable input states. In particular, we model the channel as a Markov chain or a multimode environment of oscillators. While in the first model, the maximization is achieved for the maximally mixed input state, in the latter it is convenient to exploit the presence of a decoherence-protected subspace generated by memory effects. We explicitly compute the quantum channel capacity for the first model while numerical simulations suggest a lower bound for the latter. In both cases memory effects enhance the coherent information. We present results valid for arbitrary input size.
The quantum capacity captures the value of a quantum channel for transmitting quantum information, establishing the fundamental limits on quantum communication. In spite of its central role in quantum information theory, the quantum capacity of most channels is unknown, with wide gaps between the best upper and lower bounds. Even deciding whether a channel has nonzero capacity -- finding its capacity threshold -- is difficult. In this paper we report significant increases in the capacity thresholds of two prototypical noise models: the depolarizing channel and Pauli channels. In the case of the depolarizing channel, this is the first improvement in 18 years, giving a bigger increase beyond the hashing bound than all previous improvements combined. Our starting point is the representation theoretic framework recently proposed by Bhalerao and Leditzky (2025) to compute coherent information for special permutation invariant states. We generalize their framework to the full symmetric subspace, which allow us to optimize coherent information over rank two states in that space. A representation theoretic calculation shows that exponentially many Kraus operators of the channel annihilate the symmetric space, corresponding to a massive decrease in environment entropy for states on the symmetric space compared to the maximally mixed state. This explains the enhanced coherent information as a manifestation of degeneracy for the resulting codes.