# Maximal noiseless code rates for collective rotation channels on qudits

Maximal noiseless code rates for collective rotation channels on qudits We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set $$\mathcal {E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm {SU}}(d)\}.$$ E ( d , n ) = { U ⊗ n : U ∈ SU ( d ) } . This is done by analyzing the decomposition of the algebra $$\mathcal {A}(d,n)$$ A ( d , n ) generated by $$\mathcal {E}(d,n)$$ E ( d , n ) . We summarize the results for the channels on qubits ( $$d=2$$ d = 2 ) and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general d. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for $$d = 3$$ d = 3 (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Maximal noiseless code rates for collective rotation channels on qudits

Quantum Information Processing, Volume 14 (11) – Sep 3, 2015
17 pages

/lp/springer-journals/maximal-noiseless-code-rates-for-collective-rotation-channels-on-kyivyWK8TH
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1101-2
Publisher site
See Article on Publisher Site

### Abstract

We study noiseless subsystems on collective rotation channels of qudits, i.e., quantum channels with operators in the set $$\mathcal {E}(d,n) = \{ U^{\otimes n}: U \in {\mathrm {SU}}(d)\}.$$ E ( d , n ) = { U ⊗ n : U ∈ SU ( d ) } . This is done by analyzing the decomposition of the algebra $$\mathcal {A}(d,n)$$ A ( d , n ) generated by $$\mathcal {E}(d,n)$$ E ( d , n ) . We summarize the results for the channels on qubits ( $$d=2$$ d = 2 ) and obtain the maximum dimension of the noiseless subsystem that can be used as the quantum error correction code for the channel. Then we extend our results to general d. In particular, it is shown that the code rate, i.e., the number of protected qudits over the number of physical qudits, always approaches 1 for a suitable noiseless subsystem. Moreover, one can determine the maximum dimension of the noiseless subsystem by solving a non-trivial discrete optimization problem. The maximum dimension of the noiseless subsystem for $$d = 3$$ d = 3 (qutrits) is explicitly determined by a combination of mathematical analysis and the symbolic software Mathematica.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Sep 3, 2015

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