Localized Entanglement Purification

Abstract

Entanglement purification protocols are fundamental primitives in quantum communication, enabling the distillation of high-quality entanglement using only local operations and classical communication. For large multipartite states, however, existing purification schemes typically require substantial resources and become progressively inefficient as system size increases. We introduce a new type of multipartite entanglement purification, Localized Entanglement Purification (LEP), a family of protocols that purify entanglement at the level of network regions rather than globally. By exploiting spatial noise asymmetries, LEP reduces resource consumption and enables scalable purification strategies for larger quantum systems.

Figures (11)

• Figure 1: Illustration of the TCP protocol. Multiple noisy copies of an $N$-linear graph state with an initial fidelity $F_0$. The arrows denote the multilateral (MCNOT) directions associated with the sub-protocols $P_1$ and $P_2$, while the color shading distinguishes the two color sets. After measurement and post-selection, copies failing the purification step are discarded, whereas successful copies with increased fidelity $F_j$ are retained and used in subsequent rounds. After several purification rounds, a single copy with the highest achievable fidelity $F_f$ remains.
• Figure 2: Illustration of the Localized Purification (LEP) protocol. Example for $N$-qubit linear cluster state, with panel (a) demonstrating the LEP approach, where a target qubit $T$ is selected along with its neighboring qubits set ${T_N}$, while all other qubits in set ${T_0}$ remain inactive during MCNOT operation. The direction of the operation is determined by the target qubit $\mu_T$, and the auxiliary state is constructed from the corresponding qubit $\nu_{T}$ and the neighboring qubits $\boldsymbol{\nu}_{T_N}$. Varying position of the target qubit $T$ enables the identification of the optimal LEP strategy, as shown in (b). Starting from an initial fidelity $F_0$, all candidate auxiliary operations are virtually evaluated, and the one that maximizes the resulting fidelity (e.g., $F_2$) is selected and applied. This procedure is then iteratively repeated.
• Figure 3: Fidelity vs Resources: $\boldsymbol{Z}$-noise on qubit 1. An 8-qubit linear cluster state subjected to Pauli-Z noise on qubit 1 (leaf) with $p_{z}^{(1)} = 0.7$, with no additional initial noise or gate noise ($p_{w} = p_{g} = 1.0$), while showcasing the effect of different purification strategies like TCP and LEP ($S-0$, $S-1$, and $S-5$). We limit the maximum number of resources to $10^9$; any value exceeding this is declared unsuccessful.
• Figure 4: Fidelity vs Resources: $\boldsymbol{Z}$-noise on qubit 1, 3, and 6. An 8-qubit linear cluster state is subjected to Pauli-Z noise on qubits: 1 ($p_{z}^{(1)} = 0.81$),3 ($p_{z}^{(3)} = 0.9$), and 6 ($p_{z}^{(6)} = 0.85$), with initial white noise $p_w = 0.95$ and gate noise $p_g = 0.998$, with application of different purification strategies such as: LEP (strategies $S-0$, $S-1$, and $S-5$) and TCP.
• Figure 5: Fixed Target Fidelity ($F_T$ = 0.90) for an 8-node linear cluster state. We consider this case to be subjected to asymmetric noise, where $Z$ noise with strength $p_z$ acts on qubits 1 and 6, while the white-noise parameter $p_w$ is varied simultaneously. The figure compares different LEP-based strategies, namely S-$\alpha$ and C-$\alpha$, and TCP. The S-$\alpha$ applies the auxiliary protocol with $\alpha$ pre-purification rounds. The combined strategy C-$\alpha$ consists of applying strategy $S$ once or twice, and selecting the option that yields the best performance relative to the TCP strategy. Sub-figures (a) and (b) show results for fixed target fidelity $F_T$ = 0.90. Sub-figure (a) presents outcomes of different strategies for $F_T$ = 0.90. For each noise configuration, we identify the strategy that achieves the fixed target fidelity $F_T=0.90$ with the least amount of resource consumption as the winning strategy. If the initial fidelity is already equal to or exceeds the fixed target fidelity, the result is labeled as same. The amount of resources used for $F_T=0.90$ per TCP (orange) and LEP (pink) strategies is displayed in (b).