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Physical Review A

Distillation of the Greenberger-Horne-Zeilinger state from arbitrary tripartite states

Yi-Na Mo, Chuan‐Feng Li, Guang‐Can Guo

Published Jan 4, 2002• 2 citations
Publisher

Abstract

We present a method of distillation of Greenberger-Horne-Zeilinger states from arbitrary tripartite pure states by local operations and classical communication. We go further to discuss the various results we get and calculate the efficiency of the protocol.
DOI
10.1103/physreva.65.024301

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Methods for distilling Greenberger-Horne-Zeilinger (GHZ) states from arbitrary entangled tripartite pure states are described. These techniques work for virtually any input state. Each technique has two stages which we call primary and secondary distillations. Primary distillation produces a GHZ state with some probability, so that when applied to an ensemble of systems a certain percentage is discarded. Secondary distillation produces further GHZs from the discarded systems. These protocols are developed with the help of an approach to quantum information theory based on absolutely selective information, which has other potential applications.
We prove a lower bound on the rate of Greenberger-Horne-Zeilinger states distillable from pure multipartite states by local operations and classical communication (LOCC). Our proof is based on a modification of a combinatorial argument used in the fast matrix multiplication algorithm of Coppersmith and Winograd. Previous use of methods from algebraic complexity in quantum information theory concerned transformations with stochastic LOCC (SLOCC), resulting in an asymptotically vanishing success probability. In contrast, our new protocol works with an asymptotically vanishing error.
We develop an efficient local operation and classical communication (LOCC) scheme for the distillation of Greenberger-Horne-Zeilinger (GHZ) states from tripartite systems subjected to both coherent and incoherent errors. The proposed method employs an iterative process that applies a postselection-based non-linear transformation to increase the entanglement of 3-qubit states. In contrast to traditional distillation protocols that require an exponential number of initial states as a resource, our method achieves subexponential convergence towards a pure GHZ state. The proposed scheme is practical in the sense that it employs a small set of relatively simple unitary operations and projective measurements in the computational basis. We systematically develop a double-iteration protocol by providing a mathematical framework for the transformation processes involved, emphasizing the role of unitary operations in correcting arbitrary small errors in the initial states. Through analytical derivations and numerical simulations, we demonstrate the protocol's ability to progressively eliminate noise and improve fidelity over subsequent iterations. Significantly, our protocol not only corrects for small arbitrary distortions in the GHZ states but also maintains operational simplicity, making it feasible for practical quantum computing applications.