Entanglement swapping for partially entangled qudits and the role of quantum complementarity

TL;DR

This work addresses how to distribute entanglement via entanglement swapping when the input pairs are partially entangled, by analyzing the protocol through complete complementarity relations (CCRs). It employs generalized Bell states for qudits and shows that the average entanglement shared by distant parties is bounded above by both the initial entanglement of a single pair and the product of the initial entanglements, with the CCRs tying these limits to local predictability and coherence. In the qubit case, the average distributed entanglement saturates the product bound, while in the qutrit case it equals half the product, leading to a conjectured tighter bound for general qudits: ⟨E⟩ ≤ (Eξ Eη)/(d−1). The results reveal that vanishing local coherence is sufficient for the analysis, clarify how CCRs constrain ESP, and have implications for entanglement distribution in quantum networks and quantum communication protocols, where local information and coherence are resources that get consumed to generate remote entanglement.

Abstract

We extend the entanglement swapping protocol (ESP) to partially entangled qudit states and analyze the process within the framework of complete complementarity relations (CCRs). Building on previous results for qubits, we show that the average distributed entanglement between two parties via ESP is bounded above by the initial entanglement of one of the input pairs, and also by the product of the initial entanglements. Notably, we find that using initial states with vanishing local quantum coherence is sufficient to capture the essential features of the protocol, simplifying the analysis. By exploring the cases of qubits and qutrits, we observe that the upper bound on the average distributed entanglement -- expressed in terms of the product of the initial entanglements -- can be improved, and we conjecture what this tighter bound might be. Finally, we discuss the role of quantum complementarity in the ESP and show how local predictability constrains the entanglement that can be operationally distributed via ESP.

Figures (3)

• Figure 1: Entanglement swapping protocol in space-time axis can be performed using three laboratories: Alice ($\mathcal{A}$), Bob ($\mathcal{B}$), and Charlie ($\mathcal{C}$). To emphasize the distribution of entanglement over long distances, we can consider two other laboratories: Darwin ($\mathcal{D}$) and Erin ($\mathcal{E}$). Darwin sends a pair of entangled quantons ($A$ and $C$) to Alice and Charlie , respectively. Erin, in turn, sends the pair of entangled quantons (C$^\prime$ and $B$) to Charlie and Bob , respectively. Now, Charlie has qubits from two pairs of entangled quantons: one pair shared with Alice and the other with Bob. Subsequently, Charlie performs a Bell-basis measurement (BBM) on the two qubits in his possession. As a result, Alice and Bob---who initially shared no entanglement---end up with a maximally entangled pair of quantons. The result of the Charlie measurement can be shared by the classical communication channel, represented by the double dashed lines between Charlie and Alice, to allow Alice and Bob to prepare an specific entangled state.
• Figure 2: Diagram illustrating a broad scenario involving a pure entangled state shared between the quantons $A$ and $B$. The wave-particle features of a quanton are fully characterized by a complete complementarity relation, expressed in Eq. \ref{['eq:CCR']}, where $\mathfrak{W}$ is the wave measure represented by quantum coherence, $\mathfrak{P}$ is the predictability measure, and $\mathfrak{E}$ is an entanglement monotone. The sum of these three quantities is equal to a constant $\alpha(d_A)$, which depends solely on the system's dimension. In general, the local quantities $\mathfrak{P}$ and $\mathfrak{W}$ may differ between the quantons, however the global quantity $\mathfrak{E}(|\Psi_{AB}\rangle)$ is the same for both systems $A$ and $B$.
• Figure 3: The probabilities $\Pr(\Phi_{0q}^{CC'})$ and $\Pr(\Phi_{1q}^{CC'})$, together with the post-BBM entanglements $E_{l_1} ( |\hat{\phi}_{0q}^{AB}\rangle )$ and $E_{l_1} ( |\hat{\phi}_{1q}^{AB}\rangle )$, as functions of the parameters $x$ and $y$ for the initial states given by Eq. \ref{['eq:initqubits']}.