Entropy of a Quantum Error Correction Code

Kribs, David W.; Pasieka, Aron; Życzkowski, Karol

doi:10.1142/s1230161208000225

Cited by 7 publications

(11 citation statements)

References 45 publications

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“…For instance, in [24] it is suggested that the entanglement fidelity [25] is the important quantity to maximize in schemes for quantum error correction. In [26], it is proposed that the code entropy is an important quantitative measure of the quantum correction schemes by determining a hierarchical structure in the set of protected spaces [27].…”

Section: Introductionmentioning

confidence: 99%

Quantifying the Performance of Quantum Codes

Cafaro

L’Innocente

Lupo

et al. 2011

Open Syst. Inf. Dyn.

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We study the properties of error correcting codes for noise models in the presence of asymmetries and/or correlations by means of the entanglement fidelity and the code entropy. First, we consider a dephasing Markovian memory channel and characterize the performance of both a repetition code and an error avoiding code in terms of the entanglement fidelity. We also consider the concatenation of such codes and show that it is especially advantageous in the regime of partial correlations. Finally, we characterize the effectiveness of the codes and their concatenation by means of the code entropy and find, in particular, that the effort required for recovering such codes decreases when the error probability decreases and the memory parameter increases. Second, we consider both symmetric and asymmetric depolarizing noisy quantum memory channels and perform quantum error correction via the five qubit stabilizer code. We characterize this code by means of the entanglement fidelity and the code entropy as function of the asymmetric error probabilities and the degree of memory. Specifically, we uncover that while the asymmetry in the depolarizing errors does not affect the entanglement fidelity of the five qubit code, it becomes a relevant feature when the code entropy is used as a performance quantifier.Comment: 21 pages, 10 figure

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Section: Introductionmentioning

confidence: 99%

Quantifying the Performance of Quantum Codes

Cafaro

L’Innocente

Lupo

et al. 2011

Open Syst. Inf. Dyn.

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“…In some cases communication with a given code is free of decoherence, which means that either the output of a channel is already the same as the input or the output needs only a unitary operation to be recovered to the initial state. In the former case we talk about decoherence-free subspaces (DFSs), in the latter about unitarily correctable codes (UCCs) [25]. Both correspond to L being a pure state, i.e., α ij = α * i α j holds and for this reason they are also called the zero entropy codes [25].…”

Section: Quantum Error Correctionmentioning

confidence: 99%

Decoherence-Free Communication over Multiaccess Quantum Channels

Demianowicz

2013

Open Syst. Inf. Dyn.

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In this paper we consider decoherence-free communication over multiple access and k-user quantum channels. First, we concentrate on a hermitian unitary noise model U for a two-access bi-unitary channel and show that in this case a decoherence-free code exists if the space of Schmidt matrices of an eigensubspace of U exhibits certain properties of decomposability. Then, we show that our technique is also applicable for generic random unitary two-access channels. Finally, we consider the applicability of the result to the case of a larger number of senders and general Kraus operators.

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“…As a quantitative measure, which characterizes to what extent a given protected space is close to a decoherence free space, one can use the von Neumann entropy of this state, S = −Trα ln α. This code entropy [22] is equal to zero if the protected space is decoherence free or if the information lost can be recovered by a reversible unitary operation. Observe that the code entropy S characterizes the map Ψ and the code space C, but does not depend on the particular Kraus form used to represent Ψ.…”

Section: Protection and Darkness: The Knill-laflamme Conditionmentioning

confidence: 99%

Protected subspaces in quantum information

Majgier

Maassen

Życzkowski

2009

Quantum Inf Process

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Please be advised that this information was generated on 2018-05-13 and may be subject to change.arXiv:0902.4126v2 [quant-ph] In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by "coding" the state space of the system before transmission into a "protected" part of a larger state space, and by applying a proper "decoding" map afterwards. By a version of the Heisenberg Principle, which we prove, such a protected space must be "dark" in the sense that no information leaks out during the transmission. We explain the role of the Knill-Laflamme condition in relation to protection and darkness, and we analyze several degrees of protection, whether related to error correction, or to state restauration after a measurement. Recent results on higher rank numerical ranges of operators are used to construct examples. In particular, dark spaces are constructed for any map of rank 2, for a biased permutations channel and for certain separable maps acting on multipartite systems. Furthermore, error correction subspaces are provided for a class of tri-unitary noise models. P r o te c te d S u b sp a ces in Q u an tu m In form ation I. IN T R O D U C T IO NWe consider a quantum channel of finite dimension through which a quantum system in some state is sent. The output consists of another quantum state, and possibly some classical information. We are interested in the question to what extent the original quantum state can be recovered from that state and that information. In particular, we investigate if there are subspaces of the Hilbert space of the original system , on which the state can be perfectly restored.In the literature a hierarchy of such spaces, which we shall call protected subspaces here, has been described. The strongest protection possible is provided in the case of a "decoherence free subspace" [1-4]. In this case the channel acts on the subspace as a isometric transformation. All we have to do in order to recover the state, is to rotate it back.The next strongest form of protection occurs when the channel acts on the subspace as a random choice between isometries, whose image spaces are m utually orthogonal. Then by measuring along a suitable partition of the output Hilbert space, it can be inferred from the output state which isometry has occurred, so that it can be rotated back. This situation is characterized by the well-known Knill-Laflamme criterion, [5, 6] and the protected subspace in this case is usually called an error correction subspace.The weakest form of protection is provided in yet a third situation, which was encountered in the context of quantum trajectories and the purification tendency of states along these paths [7]. In this case the deformation of the state is not caused by some given external device, but by the experimenter himself, who is performing a Kraus measurement [8]. Also in this case the "channel" acts as a random isometry, but the image spaces need not be orthogonal. It is now the m eas...

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Entropy of a Quantum Error Correction Code

Cited by 7 publications

References 45 publications

Quantifying the Performance of Quantum Codes

Quantifying the Performance of Quantum Codes

Decoherence-Free Communication over Multiaccess Quantum Channels

Protected subspaces in quantum information

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