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Journal of Optics B Quantum and Semiclassical Optics

A method to protect quantum entanglement against certain kinds of phase and exchange errors

Chui‐Ping Yang, Julio R. Gea-Banacloche

Published Jan 25, 2001• 0 citations
Publisher

Abstract

We present a method to protect the entangled states of distant particles against decoherence due to local (but collective) phase errors, and local exchange-type interactions, by pairing up the entangled particles. The method is based on a four-qubit code which forms a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs. We also show how the scheme can be generalized to protect certain entangled states of more than two particles.

DOI
10.1088/1464-4266/3/1/356

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We present a method to protect the entangled states of distant particles against decoherence due to local (but collective) phase errors, and local exchange-type interactions, by pairing up the entangled particles. The method is based on a four-qubit code which forms a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs. We also show how the scheme can be generalized to protect certain entangled states of more than two particles.
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The exchange interaction between identical qubits in a quantum-information processor gives rise to unitary two-qubit errors. It is shown here that decoherence-free subspaces (DFSs) for collective decoherence undergo Pauli errors under exchange, which, however, do not take the decoherence-free states outside of the DFS. In order to protect DFSs against these errors it is sufficient to employ a recently proposed concatenated DFS quantum-error-correcting code scheme [D. A. Lidar, D. Bacon, and K.B. Whaley, Phys. Rev. Lett. 82, 4556 (1999)].
We propose schemes that are efficient when each pair of qubits undergoes some imperfect collective decoherence with different baths. In the proposed scheme, each pair of qubits is first encoded in a decoherence-free subspace composed of two qubits. Leakage out of the encoding space generated by the imperfection is reduced by the quantum Zeno effect. Phase errors in the encoded bits generated by the imperfection are reduced by concatenation of the decoherence-free subspace with either a three-qubit quantum error correcting code that corrects only phase errors or a two-qubit quantum error detecting code that detects only phase errors connected with the quantum Zeno effect again.
We show a way to correct collective error through a three-qubit quantum code. The encoding and decoding (i.e., error recovery) operations for such a code, which protects one qubit of quantum information perfectly (with infinite ``distance'') against collective decoherence, are presented. The code is actually a manifestation of a noiseless subsystem first reported by Knill, Laflamme, and Viola [Phys. Rev. Lett. 84, 2525 (2000)], and provides a good illustration of the relationship between noiseless subsystems and error-correcting codes already noted by these authors. We also show in detail how this code can be used to preserve an arbitrary joint state (including entangled states) of two qubits, by pairing each of them with two other qubits, even in the case in which the two resulting clusters interact in nonequivalent ways with a common environment.
We present a method to protect the entanglement in a maximally entangled n-qubit state of the Greenberger-Horne-Zeilinger or ``Schr\"odinger cat'' form against decoherence, by pairing up each qubit with another one in such a way that each pair of qubits undergoes collective decoherence. Under these conditions, if t pairs experience an error, we show how one can still, in general, extract a maximally entangled state of the remaining $n\ensuremath{-}t$ pairs. Two versions of the method are considered: one of them applies when the locations of the decohered qubits are known, and it might be useful, for instance, in secret-sharing protocols. Another version, which is free from this restriction, provides only partial protection (the resulting state is, generally, not a maximally entangled state), but could nonetheless be useful for the important special case in which the source of error is spontaneous emission.
``Leakage'' errors are particularly serious errors which couple states within a code subspace to states outside of that subspace, thus destroying the error protection benefit afforded by an encoded state. We generalize an earlier method for producing leakage elimination decoupling operations and examine the effects of the leakage eliminating operations on decoherence-free or noiseless subsystems which encode one logical, or protected qubit into three or four qubits. We find that by eliminating a large class of leakage errors, under some circumstances, we can create the conditions for a decoherence-free evolution. In other cases we identify a combined decoherence-free and quantum error correcting code which could eliminate errors in solid-state qubits with anisotropic exchange interaction Hamiltonians and enable universal quantum computing with only these interactions.