We propose a scheme of multipartite entanglement distillation driven by a complementary pair of stabilizer measurements to distill directly a wider range of states beyond the stabilizer code states (such as the Greenberger-Horne-Zeilinger states). We make our idea explicit by constructing a recurrence protocol for the 3-qubit state [formula: see text]. Noisy states resulting from typical decoherence can be directly purified in a few steps, if their initial fidelity is larger than a threshold. For general input mixed states, we observe distillations to hierarchical fixed points, i.e., not only to the state but also to the 2-qubit Bell pair, depending on their initial entanglement.
We propose a scheme of multipartite entanglement distillation driven by a complementary pair of stabilizer measurements to distill directly a wider range of states beyond the stabilizer code states (such as the Greenberger-Horne-Zeilinger states). We make our idea explicit by constructing a recurrence protocol for the 3-qubit state [formula: see text]. Noisy states resulting from typical decoherence can be directly purified in a few steps, if their initial fidelity is larger than a threshold. For general input mixed states, we observe distillations to hierarchical fixed points, i.e., not only to the state but also to the 2-qubit Bell pair, depending on their initial entanglement.
Entanglement distribution is a critical task in quantum networks. Since the distributed entanglement can suffer from noise in the channel, it is necessary to develop methods that distill higher quality entanglement from the shared noisy entangled states. In this work, we propose a protocol to distill multi-qubit Greenberger-Horne-Zeilinger (GHZ) states among the nodes of a network using quantum error correcting codes. The method builds upon a Bell state distillation protocol by Wilde et at. (2007) that we recently generalized to 3-qubit GHZ states. The key technical result that enables our protocol shows how measuring a Pauli operator, or in general a set of code stabilizers, on one subsystem of a multipartite GHZ state affects the other subsystems. The design and analysis of the protocol is driven by the stabilizer formalism for measurements, and we provide discussions to elucidate the steps of the protocol. A similar approach can be applied to distill other multipartite entangled states as long as they are stabilizer states.
We introduce an entanglement distillation protocol that utilizes an arbitrary [[n,k,d]] stabilizer code to convert n raw copies of an N-qubit Greenberger-Horne-Zeilinger (GHZ) state into k logical copies in the presence of Pauli noise. Our explicit formulation of the scheme on a star network topology is efficiently scalable to arbitrary N, requires only local operations, and minimizes the impact of errors on idle physical qubits, enabling the distribution of high-fidelity entangled logical states between any number of quantum processors. We report the results of numerical experiments using the 5-qubit code, toric code, and [[144,12,12]] bivariate bicyle code to protect against independent single-qubit depolarizing noise on all qubits. Our encoding scheme improves the final GHZ state fidelity relative to a single bare GHZ state sent over the same noisy channel for depolarizing rates up to or above 6% for all codes tested. Additionally, we use the stabilizer formalism to prove that our protocol can be applied to any N-qubit Calderbank-Shor-Steane (CSS) stabilizer state, from which the GHZ state emerges as a special case. This result explicitly establishes the working principle of previous stabilizer-based Bell pair and GHZ state distillation schemes, and generalizes such protocols beyond the traditionally considered states. We also show that some stabilizer codes can distill any stabilizer state with our protocol.
In this work, we study how to distill two typical multipartite entangled noisy states, the amplitude-damped W state and the amplitude-damped Greenberger-Horne-Zeilinger state. Unlike the currently existing multipartite entanglement purification schemes, our scheme is a single-copy-based scheme, i.e., only one noisy state is needed in each distillation round, and the complicated multilateral controlled-not (cnot) operations are replaced by measurements of local positive-operator-valued measures (POVMs). The results show that local POVM measurements on one copy of the noisy state can project the initial state onto a new state with higher entanglement and nonlocality. In this scheme, although the final state can violate the multipartite Bell inequality, the corresponding initial noisy state does not violate it, which means that this distillation scheme can reveal the hidden genuine multipartite nonlocality of these initial states. For the case of the amplitude-damped W state, if the POVM parameters are appropriately chosen, the fidelity of the output state can approach 1.0 for all the initial states with fidelity bigger than zero. So this single-copy-based multipartite entanglement distillation scheme has a wide range of distillable fidelities, and it is much simpler than the two-copy- or multiple-copy-based schemes. Furthermore, if the POVM measurement is carried out only on some member qubits rather than all of them, the distillation scheme can succeed too, which will greatly decrease the implementation complexity of the scheme. So this multipartite entanglement distillation scheme may be implementable in some physical system.
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.
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.
The distribution of high-quality Greenberger-Horne-Zeilinger (GHZ) states is at the heart of many quantum communication tasks, ranging from extending the baseline of telescopes to secret sharing. They also play an important role in error-correction architectures for distributed quantum computation, where Bell pairs can be leveraged to create an entangled network of quantum computers. We investigate the creation and distillation of GHZ states out of nonperfect Bell pairs over quantum networks. In particular, we introduce a heuristic dynamic programming algorithm to optimize over a large class of protocols that create and purify GHZ states. All protocols considered use a common framework based on measurements of nonlocal stabilizer operators of the target state (i.e., the GHZ state), where each nonlocal measurement consumes another (nonperfect) entangled state as a resource. The new protocols outperform previous proposals for scenarios without decoherence and local gate noise. Furthermore, the algorithms can be applied for finding protocols for any number of parties and any number of entangled pairs involved.