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arXiv:quant-ph/0109114 · Physical Review A

Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

Mitsuru Hamada

Published Apr 15, 2002• 19 citations
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Abstract

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 $E(R).$ The $E(R)$ is positive below some threshold ${R}_{0},$ which implies ${R}_{0}$ is a lower bound on the quantum capacity.

DOI
10.1103/physreva.65.052305

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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.
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.
We consider a generalization of a classical ARQ protocol for the case of quantum error correcting codes and the quantum depolarizing channel. We define the fidelity of the ARQ protocol as the probability that the outcome of a measurement of the received quantum state is collinear to the transmitted quantum state, under the condition that the measurement outcome belongs to the code space. Further, we derive tight upper and lower bounds on the exponent of the fidelity of the ARQ protocol. The obtained bounds show a threshold behavior of the fidelity. Namely, in asymptotics, as the code length tends to infinity, the fidelity tends to either 1 or 0 depending on the code rate and the probability of error of the quantum depolarizing channel.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
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).