Entanglement-swapping measurements for deterministic entanglement distribution

TL;DR

This work identifies the exact structural conditions for bipartite entanglement-swapping measurements to yield deterministic entanglement distribution across all input states, up to local unitaries. By linking determinism to unbiased measurement operators and, when optimized, to complex Hadamard matrices, the authors provide a dimension-dependent classification of optimal, universal LU-deterministic swapping: a unique class for $d=2,3$, infinitely many for $d=4k$, and 72 inequivalent classes for $d=5$. They show that, in networks, using MEMs yields LU-equivalent end states independent of measurement outcomes and, for low dimensions, independent of swapping order. The results offer a postselection-free, symmetry-rich framework for scalable entanglement distribution with robustness to depolarizing noise under suitable conditions, and they connect optimal swapping to the rich structure of complex Hadamard matrices. These insights have practical implications for designing efficient quantum networks and motivate further exploration of mixed-input robustness and higher-order network topologies.

Abstract

Entanglement swapping is a key primitive for distributing entanglement across nodes in quantum networks. In standard protocols, the outcome of the intermediate measurement determines the resulting state, making the process inherently probabilistic and requiring postselection. In this work, we fully characterize those measurements under which entanglement swapping becomes deterministic: for arbitrary pure inputs, every measurement outcome produces local-unitarily equivalent states. We also show that an optimal measurement, maximizing a concurrence-type entanglement measure, is built from complex Hadamard matrices. For this optimal protocol, we provide a complete, dimension-dependent classification of deterministic entanglement-swapping measurements: unique in dimensions $d=2,3$, infinite for $d=4$, and comprising $72$ inequivalent classes for $d=5$. We further consider a general network with multiple swapping nodes and show that, for $d=2,3$ the resulting end-to-end state is independent of the order in which the repeaters perform the optimal measurements. Our results establish optimal entanglement-swapping schemes that are post-selection free, in the sense that they distribute entanglement across generic quantum network architectures without unfavorable measurement outcomes.

Figures (1)

• Figure 1: Entanglement swapping protocol. Alice and Bob share arbitrary pure entangled input pair $(| \tilde{\psi}{\rangle}, | \tilde{\phi}{\rangle})$ with the swapping node. The end parties and the node locally rotate the states through local unitaries $U_1^\dagger \otimes V^*_1 \otimes V^\dagger_2 \otimes U_2^\ast$ into the computational Schmidt basis, and subsequently perform the swapping measurement $M_{\mathsf N}=\{| \Gamma_i{\rangle}\!\langle \Gamma_i|\}_{i=1}^{d^2}$ on the node. Conditioned on outcome $i$, Alice and Bob are projected onto the state $| \eta_i{\rangle}=(A E_i B \otimes \mathds{1}) | \Phi{\rangle}_{\mathsf{AB}}/ \sqrt{p_i}$ with probability $p_i$.

Theorems & Definitions (49)

• Definition 1: Universal LU-determinism
• Lemma 1
• Theorem 1: Condition for Universal LU-Determinism
• Definition 2: Optimality
• Lemma 2: G-concurrence factorization
• Theorem 2: Phase-Conjugateion Classes of Complex Hadamard Matrices
• Remark 1: Order independence in low dimensions
• Theorem 3
• proof
• Theorem 4