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Entanglement swapping theory and beyond

Zhaoxu Ji, Huanguo Zhang

TL;DR

This work develops a comprehensive, measurement-based and density-matrix formalism for entanglement swapping across pure, mixed, and chained scenarios in $d$-level and qubit systems. It provides explicit constructions for generalized pure states, maximally entangled states, $X$-states, and CAT/GHZ-type resources, including detailed formulas and normalization factors for both single swaps and swapping chains. The results are established via mathematical induction and alternative density-matrix derivations, unifying disparate swapping schemes under a common framework. The findings have direct implications for scalable quantum networks and repeaters by enabling systematic design of entanglement distribution protocols with provable properties.

Abstract

We focus on the general theory of entanglement swapping, including entanglement swapping of pure and mixed states. We also study the theory of entangled swapping chains, which can be regarded as an application of entangled swapping. For maximally entangled states, we consider the entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis. We further consider the entanglement swapping between d-level maximally entangled states, which is realized by performing a joint measurement on the particles that contain the first particle of the selected entangled states and the particles without involving the first particle in other entangled states. For the entanglement swapping of general pure state, we generalize the case of two bipartite general pure states to multi-state cases. Besides, we propose the entanglement swapping between bipartite general pure states and maximally entangled states. For entanglement swapping chains, we propose the entanglement swapping chains for maximally entangled states, which is realized by performing joint measurements on multiple particles in each state, we use mathematical induction to prove the results of entanglement swapping chains of maximally entangled states and that of general pure states. Moreover, we study the entanglement swapping and entanglement swapping chains of mixed states, including X states and mixed maximally entangled states. Finally, we provide a new proof for our previous work [2022, Physica A, 585, 126400].

Entanglement swapping theory and beyond

TL;DR

This work develops a comprehensive, measurement-based and density-matrix formalism for entanglement swapping across pure, mixed, and chained scenarios in -level and qubit systems. It provides explicit constructions for generalized pure states, maximally entangled states, -states, and CAT/GHZ-type resources, including detailed formulas and normalization factors for both single swaps and swapping chains. The results are established via mathematical induction and alternative density-matrix derivations, unifying disparate swapping schemes under a common framework. The findings have direct implications for scalable quantum networks and repeaters by enabling systematic design of entanglement distribution protocols with provable properties.

Abstract

We focus on the general theory of entanglement swapping, including entanglement swapping of pure and mixed states. We also study the theory of entangled swapping chains, which can be regarded as an application of entangled swapping. For maximally entangled states, we consider the entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis. We further consider the entanglement swapping between d-level maximally entangled states, which is realized by performing a joint measurement on the particles that contain the first particle of the selected entangled states and the particles without involving the first particle in other entangled states. For the entanglement swapping of general pure state, we generalize the case of two bipartite general pure states to multi-state cases. Besides, we propose the entanglement swapping between bipartite general pure states and maximally entangled states. For entanglement swapping chains, we propose the entanglement swapping chains for maximally entangled states, which is realized by performing joint measurements on multiple particles in each state, we use mathematical induction to prove the results of entanglement swapping chains of maximally entangled states and that of general pure states. Moreover, we study the entanglement swapping and entanglement swapping chains of mixed states, including X states and mixed maximally entangled states. Finally, we provide a new proof for our previous work [2022, Physica A, 585, 126400].

Paper Structure

This paper contains 19 sections, 7 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Basic theory of entanglement swapping
  3. Basics of pure states entanglement swapping
  4. Basics of mixed state entanglement swapping
  5. Measurement operators
  6. Pure state entanglement swapping
  7. Entanglement swapping for generalized pure states
  8. Entanglement swapping for maximally entangled states
  9. Entanglement swapping of d-level maximally entangled states
  10. Entanglement swapping of 2-level maximally entangled states
  11. Entanglement swapping between generalized pure states and maximally entangled states
  12. Mixed state entanglement swapping
  13. Entanglement swapping of $X$ states
  14. Entanglement swapping of mixed maximally entangled states
  15. Entanglement swapping chains
  16. ...and 4 more sections

Key Result

Lemma 1

Suppose that there are $n$ arbitrary pure states containing $m_r$ particles each, denoted as $\left\{ \left\vert \psi_r \right\rangle \right\}_{r=1}^n$, then their tensor product, $\otimes_{r=1}^n \left\vert \psi_r \right\rangle$, can always be expressed in the following form: where $\left\vert l_{ij} \right\rangle$ are single-particle states.

Theorems & Definitions (16)

  • Lemma 1
  • Proof
  • Lemma 2
  • Proof
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Proof
  • Theorem 3
  • Proof
  • ...and 6 more