Entanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension d , we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.
Entanglement distillation is a well-studied problem in quantum information, where one typically starts with
n noisy Bell pairs and distills
k Bell pairs of higher fidelity. While distilling Bell pairs is the canonical setting, it is important to study the distillation of multipartite entangled states because these can be useful for realizing distributed algorithms on quantum networks. In this paper, we study the distillation of GHZ states using quantum error correcting codes (QECCs). Using the stabilizer formalism, we begin by explaining the QECC-based Bell pair distillation protocol in arXiv:0708.3699, which relies particularly on the transpose symmetry between Alice's and Bob's qubits in Bell states. Extending this idea, we show that, given
n GHZ states, performing a matrix on Alice's qubits is equivalent to performing a "stretched" version of the transpose of the matrix on the qubits of Bob and Charlie. We call this mapping to the stretched version of the matrix the GHZ-map, and show that it is an algebra homomorphism. Using this property, we show that Alice projecting her qubits onto an
[[n,k]] stabilizer code implies the simultaneous projection of Bob's and Charlie's qubits onto an induced
[[2n,k]] stabilizer code. Guided by this insight, we develop a GHZ distillation protocol based on local operations and classical communication that uses any stabilizer code. Inspired by stabilizer measurements on GHZ states, we also develop a new algorithm to generate logical Pauli operators of any stabilizer code and use it in the protocol. Since quantum codes with finite rate and almost linear minimum distance have recently been discovered, this paper paves the way for high-rate high-output-fidelity GHZ distillation. We provide simulation results on the
5-qubit perfect code to emphasize the importance of the placement of a certain local Clifford operation in the protocol.
We present different algorithms for mixed-state multicopy entanglement distillation for pairs of qubits. Our algorithms perform significantly better than the best-known algorithms. Better algorithms can be derived that are tuned for specific initial states. These algorithms are based on a characterization of the group of all locally realizable permutations of the
4n possible tensor products of n Bell states.