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Entanglementassisted capacity of a quantum channel and the reverse shannon theorem
 IEEE Trans. Inf. Theory
, 2002
"... Abstract—The entanglementassisted classical capacity of a noisy quantum channel ( ) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entangleme ..."
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Cited by 114 (6 self)
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Abstract—The entanglementassisted classical capacity of a noisy quantum channel ( ) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of after half of it has passed through the channel. We calculate entanglementassisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglementassisted communication—communication over a discrete memoryless channel (DMC) between parties who share prior random information—we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency. Index Terms—Channel capacity, entanglement, quantum information, Shannon theory. I.
Nonbinary stabilizer codes over finite fields
 IEEE Trans. Inform. Theory
, 2006
"... One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant quantum comput ..."
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Cited by 51 (11 self)
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One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the wellknown notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper. 1
Graphs, Quadratic Forms, and Quantum Codes
, 2002
"... We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms. We prov ..."
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Cited by 29 (4 self)
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We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms. We provide some simple examples to illustrate our results.
An introduction to quantum error correction and faulttolerant quantum computation
, 2009
"... Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. ..."
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Cited by 25 (2 self)
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Abstract. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers.
Beyond Stabilizer Codes I: Nice Error Bases
, 2001
"... Nice error bases have been introduced by Knill as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with ..."
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Cited by 21 (7 self)
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Nice error bases have been introduced by Knill as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with
Information rates achievable with algebraic codes on quantum discrete memoryless channels
 IEEE Trans. Information Theory
, 2005
"... The highest information rate at which quantum errorcorrection 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 ..."
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Cited by 16 (7 self)
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The highest information rate at which quantum errorcorrection 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 stabilizer codes. On the depolarizing channel, this work’s bound is actually the highest possible rate at which symplectic stabilizer codes work reliably.
Beyond Stabilizer Codes II: Clifford Codes
, 2001
"... Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the ..."
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Cited by 13 (5 self)
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Knill introduced a generalization of stabilizer codes, in this note called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the
Asymptotic bounds on quantum codes from algebraic geometry codes
 IEEE Trans. Inf. Theory
, 2006
"... © 2006 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to s ..."
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Cited by 10 (6 self)
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© 2006 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at:
Graphbased classification of selfdual additive codes over finite fields
 Adv. Math. Commun
, 2009
"... Abstract. Quantum stabilizer states over Fm can be represented as selfdual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used thi ..."
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Cited by 9 (5 self)
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Abstract. Quantum stabilizer states over Fm can be represented as selfdual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify selfdual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDS conjecture holds, we are able to classify all selfdual additive MDS codes over F9 by using an extension technique. We prove that the minimum distance of a selfdual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations. 1.