We present a method for multipartite entanglement purification of any stabilizer state shared by several parties. In our protocol each party measures the stabilizer operators of a quantum error-correcting code on his or her qubits. The parties exchange their measurement results, detect or correct errors, and decode the desired purified state. We give sufficient conditions on the stabilizer codes that may be used in this procedure and find that Steane's seven-qubit code is the smallest error-correcting code sufficient to purify any stabilizer state. An error-detecting code that encodes two qubits in six can also be used to purify any stabilizer state. We further specify which classes of stabilizer codes can purify which classes of stabilizer states.
We present a method for multipartite entanglement purification of any stabilizer state shared by several parties. In our protocol each party measures the stabilizer operators of a quantum error-correcting code on his or her qubits. The parties exchange their measurement results, detect or correct errors, and decode the desired purified state. We give sufficient conditions on the stabilizer codes that may be used in this procedure and find that Steane's seven-qubit code is the smallest error-correcting code sufficient to purify any stabilizer state. An error-detecting code that encodes two qubits in six can also be used to purify any stabilizer state. We further specify which classes of stabilizer codes can purify which classes of stabilizer states.
We introduce a class of multiparticle entanglement purification protocols that allow us to distill a large class of entangled states. These include cluster states, Greenberger-Horne-Zeilinger states, and various error correction codes all of which belong to the class of two-colorable graph states. We analyze these schemes under realistic conditions and observe that they are scalable; i.e., the threshold value for imperfect local operations does not depend on the number of parties for many of these states. When compared to schemes based on bipartite entanglement purification, the protocol is more efficient and the achievable quality of the purified states is larger. As an application we discuss an experimental realization of the protocol in optical lattices which allows one to purify cluster states.
We investigate multiparticle entanglement purification schemes which allow one to purify all two colorable graph states, a class of states which includes, e.g., cluster states, Greenberger-Horne-Zeilinger states, and code words of various error correction codes. The schemes include both recurrence protocols and hashing protocols. We analyze these schemes under realistic conditions and observe for a generic error model that the threshold value for imperfect local operations depends on the structure of the corresponding interaction graph, but is otherwise independent of the number of parties. The qualitative behavior can be understood from an analytically solvable model which deals only with a restricted class of errors. We compare direct multiparticle entanglement purification protocols with schemes based on bipartite entanglement purification and show that the direct multiparticle entanglement purification is more efficient and the achievable fidelity of the purified states is larger. We also show that the purification protocol allows one to produce private entanglement, an important aspect when using the produced entangled states for secure applications. Finally we discuss an experimental realization of a multiparty purification protocol in optical lattices which is issued to improve the fidelity of cluster states created in such systems.
In this paper, we propose a novel bipartite entanglement purification protocol built upon hashing and upon the guessing random additive noise decoding (GRAND) approach recently devised for classical error correction codes. Our protocol offers substantial advantages over existing hashing protocols, requiring fewer qubits for purification, achieving higher fidelities, and delivering better yields with reduced computational costs. We provide numerical and semi-analytical results to corroborate our findings and provide a detailed comparison with the hashing protocol of Bennet et al. Although that pioneering work devised performance bounds, it did not offer an explicit construction for implementation. The present work fills that gap, offering both an explicit and more efficient purification method. We demonstrate that our protocol is capable of purifying states with noise on the order of 10% per Bell pair even with a small ensemble of 16 pairs. The work explores a measurement-based implementation of the protocol to address practical setups with noise. This work opens the path to practical and efficient entanglement purification using hashing-based methods with feasible computational costs. Compared to the original hashing protocol, the proposed method can achieve some desired fidelity with a number of initial resources up to one hundred times smaller. Therefore, the proposed method seems well-fit for future quantum networks with a limited number of resources and entails a relatively low computational overhead.
The future of quantum repeater networking will require interoperability between various error-correcting codes. A few specific code conversions and even a generalized method are known, however, no detailed analysis of these techniques in the context of quantum networking has been performed. In this paper we analyze a generalized procedure to create Bell pairs encoded heterogeneously between two separate codes used often in error-corrected quantum repeater network designs. We begin with a physical Bell pair and then encode each qubit in a different error-correcting code, using entanglement purification to increase the fidelity. We investigate three separate protocols for preparing the purified encoded Bell pair. We calculate the error probability of those schemes between the Steane [[7,1,3]] code, a distance-3 surface code, and single physical qubits by Monte Carlo simulation under a standard Pauli error model and estimate the resource efficiency of the procedures. A local gate error rate of ${10}^{\ensuremath{-}3}$ allows us to create high-fidelity logical Bell pairs between any of our chosen codes. We find that a postselected model, where any detected parity flips in code stabilizers result in a restart of the protocol, performs the best.
Entanglement distillation, a fundamental building block of quantum networks, enables the purification of noisy entangled states shared among distant nodes by local operations and classical communication. Its practical realization presents several technical challenges, including the storage of quantum states in quantum memory and the execution of coherent quantum operations on multiple copies of states within the quantum memory. In this work, we present an entanglement purification protocol via quantum communication, namely a carrier-assisted entanglement purification protocol, which utilizes two elements only: i) quantum memory for a single-copy entangled state shared by parties and ii) single qubits travelling between parties. We show that the protocol, when single-qubit transmission is noiseless, can purify a noisy entangled state shared by parties. When single-qubit transmission is noisy, the purification relies on types of noisy qubit channels; we characterize Pauli channels such that the protocol works for the purification. We address this limitation by using multiple carrier qubits, and show that for any non-entanglement-breaking Pauli channel, the protocol's fixed-point fidelity approaches unity as the number of carriers increases. Our results significantly reduce the experimental overhead required for distilling entanglement: the practical advantage is demonstrated through parameters directly related to the capability of entanglement purification, such as noise in quantum memory, local measurements, channel use, and entanglement fidelity. We envisage that the protocol would make long-distance pure entanglement closer to a practical realization.
The determination of many special types of quantum states has been studied thoroughly, such as the generalized $|\text{GHZ}\ensuremath{\rangle}$ states, stochastic local operations and classical communication equivalent $|W\ensuremath{\rangle}$ states, and Dicke states. In this paper, we study another special entanglement state, that is, stabilizer states, which play an important role in quantum error correcting codes, multipartite purification, and so on. We show that all n-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of n independent generators of the corresponding stabilizer formalisms.