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arXiv:0909.4115

The coherent information of Pauli channels with coded inputs

Xiaoyu Chen, Li-zhen Jiang

Published Sep 23, 2009• 0 citations
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Abstract

The calculating of the coherent information is a fundamental step in obtaining the quantum capacity of a quantum channel. We introduce orthogonal and complete code basis to evaluate the coherent information per channel use when the input is the maximal mixture of stabilizer codewords. In the code basis, the output density matrix is diagonal, the joint output of the system and the auxiliary is block diagonal. The coherent information is worked out by counting the weights of error operators.

DOI
10.48550/arxiv.0909.4115

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The calculating of the coherent information is a fundamental step in obtaining the quantum capacity of a quantum channel. We introduce orthogonal and complete code basis to evaluate the coherent information per channel use when the input is the maximal mixture of stabilizer codewords. In the code basis, the output density matrix is diagonal, the joint output of the system and the auxiliary is block diagonal. The coherent information is worked out by counting the weights of error operators.
Quantum transmission rate may be improved by using input of entangled state instead of product state for Pauli channel. Using GHZ state as input for the depolarizing channel (a special Pauli channel), the channel fidelity of non-zero quantum capacity was improved from> 0.81071 to> 0.80944. We will introduce orthogonal and complete code basis to evaluate the coherent information per channel use when the input is the maximal mixture of stabilizer codewords. In the code basis, the output density matrix is diagonal, the joint output is block diagonal. The coherent information is worked out by counting the weights of error operators. Quantum capacity is analytically proved to be non-zero for all values of channel fidelity except
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Quantum capacities of Pauli channels are not additive, a degenerate quantum code may improve the hashing bound of the capacity. The difficulty in approaching the capacity is how to calculate the coherent information of a generic degenerate quantum code. Using graph state basis, we greatly reduce the problem for the input of quantum error-correcting code. We show that for a graph diagonal state passing through a Pauli channel the output state is diagonalizable and the joint output state of the system and ancilla is block diagonalizable. When the input state is an equal probable mixture of codewords of a stabilizer code, the coherent information can be analytically obtained.
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.
We prove a strong converse for quantum communication over Pauli channels within the class of stabilizer codes. If a code whose code space is a full joint eigenspace of a stabilizer group transmits above the coherent information of its own input state, its entanglement fidelity decays exponentially in the block length; the encoder may be any isometry onto that space and the decoder any channel. For memoryless channels this determines the ε\varepsilonε-quantum capacity of the class for every ε<1\varepsilon < 1ε<1, so that tolerating a constant error buys no rate; for antidegradable channels, such as the depolarizing channel with error probability p∈[1/4,3/4]p \in [1/4, 3/4]p∈[1/4,3/4], that capacity is zero, while for p∈[1/4,1/2)p \in [1/4,1/2)p∈[1/4,1/2) partial-transposition bounds provably cannot certify a strong converse. The proof uses neither additivity assumptions nor semidefinite relaxations: optimal decoding succeeds precisely on an event in a product probability space, so the blowing-up lemma of Ahlswede, Gács and Körner applies, and the side information it produces is charged against the coherent information. The argument also constrains near-deterministic decoding for codes of any kind, and we isolate the encoder-side statement that would extend it to all of them.
The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic (stabilizer) codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have the desired performance are symplectic codes. On the depolarizing channel, this work's bound is actually the highest possible rate at which symplectic codes work reliably. The details of this work can be found in (M. Hamada, 2002).