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arXiv:quant-ph/0207113 · IEEE Transactions on Information Theory

Information Rates Achievable With Algebraic Codes on Quantum Discrete Memoryless Channels

M. Hamada

Published Nov 22, 2005• 30 citations
arXivHTMLPDF

Abstract

The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The 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. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably

DOI
10.1109/tit.2005.860824

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The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The 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. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably
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).
Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.
The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1−exp[−nE(R)+o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R0, which implies R0 is a lower bound on the quantum capacity. This work is an extension of the author’s previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002); quant-ph/0109114, LANL (2001); M. Hamada, quant-ph/0112103, LANL (2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this article to those in the previous works is similar to that of Markov chains to sequences of independent identically distributed random variables.
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.
On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function E(R).E(R).E(R). The E(R)E(R)E(R) is positive below some threshold R0,{R}_{0},R0​, which implies R0{R}_{0}R0​ is a lower bound on the quantum capacity.
We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert–Varshamov bound for general stabilizer codes, and establish an analog of the quantum Gilbert–Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straight forward.