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Entanglement distribution in quantum networks via swapping of partially entangled pure states

Henrique Guerra, Tailan S. Sarubi, Rafael Chaves, Jonas Maziero

Abstract

The entanglement swapping protocol (ESP) is a fundamental primitive for distributing quantum correlations across distant nodes in a quantum network. Recent studies have demonstrated that even when the involved qubit pairs are only partially entangled, it is still possible to concentrate and transmit entanglement via Bell-basis measurements. In this work, we extend these ideas to quantum networks with various topologies--including linear, star, and hybrid configurations--by analysing the application of the ESP to initially partially entangled pure states. We investigate how entanglement evolves under such protocols by considering the transformations of the initial states and evaluating the success probabilities for generating maximally entangled states at the output. Our results offer new insights into the dynamics of the entanglement distribution in quantum networks.

Entanglement distribution in quantum networks via swapping of partially entangled pure states

Abstract

The entanglement swapping protocol (ESP) is a fundamental primitive for distributing quantum correlations across distant nodes in a quantum network. Recent studies have demonstrated that even when the involved qubit pairs are only partially entangled, it is still possible to concentrate and transmit entanglement via Bell-basis measurements. In this work, we extend these ideas to quantum networks with various topologies--including linear, star, and hybrid configurations--by analysing the application of the ESP to initially partially entangled pure states. We investigate how entanglement evolves under such protocols by considering the transformations of the initial states and evaluating the success probabilities for generating maximally entangled states at the output. Our results offer new insights into the dynamics of the entanglement distribution in quantum networks.

Paper Structure

This paper contains 6 sections, 3 theorems, 61 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Entanglement swapping in quantum networks
  3. Linear networks
  4. Star networks
  5. Star-linear networks
  6. Final Remarks

Key Result

Theorem 1

Let $\ket {A}=\ket \eta ^{\otimes (x+1)}$ and $S^{\otimes (x-1)}\ket A = \ket{A_{x-1}} \otimes \ket {\eta}$. If $x-1$ is even and $\ket{A_{x-1}}$ is given by then $\ket{A_{x}}$ is equal to

Figures (8)

  • Figure 1: Schematic of the entanglement swapping protocol. The segment before the red dashed line represents the initial preparation of two entangled qubit pairs. The segment after the line illustrates the entanglement swap operation, performed by measuring one qubit from each pair in an entangled basis--specifically, the Bell basis in this case. After that, classical information is sent to the end nodes, that perform local operations to prepare a fixed maximally entangled state between systems that have never interacted directly.
  • Figure 2: Entanglement exchange process in a linear network. The dots represent qubits, and the qubits connected by edges are entangled. $A$, $B_1$, $B_2$, and $C$ represent nodes in the network, and we are interested in achieving quantum communication between nodes $A$ and $C$, which is enabled by sharing entangled qubits. Through successive entanglement swappings, the intermediate nodes $B_i$ enable nodes $A$ and $C$ to establish entanglement even though they have never interacted directly.
  • Figure 3: Schematic of the entanglement swapping protocol between two partially entangled $\ket{\eta_m},\ket{\eta_p}$ states, outputting an entangled state which is either $\ket{\eta_{m+p}}$ or $\ket{\eta_{m-p}}$. This procedure is what we will from now on refer to as the $S$ map, that acts upon four qubits and outputs two qubits. Whether the output is $\ket{\eta_{m+p}}$ or $\ket{\eta_{m-p}}$ can be determined by the measurements of second and third qubits. As we are not interested in what is the final state of a specific entanglement swap iteration, but rather in the possible states generated by it and its probabilities, we will not use that information.
  • Figure 4: Schematic of the entanglement swapping protocol between two partially entangled states $\ket{\xi_m},\ket{\eta_p}$, outputting an entangled state which is either $\ket{\xi_{m+p}}$ or $\ket{\xi_{m-p}}$. It is clear that only one qubit from $\ket{\xi_m}$ participates in the protocol.
  • Figure 5: Entanglement swapping process in star-shaped quantum networks. Initially, the central node $A$ shares entangled qubit pairs with the peripheral nodes $B_1, B_2, B_3$. After a joint measurement at $A$, the entanglement is transferred, establishing direct quantum connections among the peripheral nodes. This process can be generalized to $B_n$ peripheral nodes.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof