Entanglement summoning from entanglement sharing

TL;DR

The paper advances the theory of entanglement summoning under relativistic and network-constrained causality by connecting dynamical entanglement tasks to entanglement sharing schemes. It proves a complete iff condition for bidirected causal graphs via the complement graph $G_C^c$ (no odd cycles) and provides a broad set of sufficient conditions for general mixed causal graphs by constructing associated ESSs and translating graph constraints into causal-graph criteria. The work leverages monogamy constraints in ESS and graph-partition results (two-clique and two-quasi-clique) to map summoning feasibility to well-understood graph properties, and proposes a concrete protocol that reduces summoning to an ESS realization in the mixed case. These results illuminate how entanglement structure can be engineered in quantum networks with relativistic constraints and guide protocol design for time-sensitive entanglement distribution.

Abstract

In an entanglement summoning task, a set of distributed, co-operating parties attempt to fulfill requests to prepare entanglement between distant locations. The parties share limited communication resources: timing constraints may require the entangled state be prepared before some pairs of distant parties can communicate, and a restricted set of links in a quantum network may further constrain communication. Building on earlier work, we continue the characterization of entanglement summoning. We give an if and only if condition on entanglement summoning tasks with only bidirected causal connections, and provide a set of sufficient conditions addressing the most general case containing both oriented and bidirected causal connections. Our results rely on the recent development of entanglement sharing schemes.

Figures (6)

• Figure 1: Graphs representing examples of entanglement summoning problems. The vertices $D_i$ represent network or spacetime locations where requests for entanglement may be received. Edges represent directed communication links. a) A simple example where all the communication links are singly-directed. b) An example where all of the communication links are bidirected.
• Figure 2: The simplest uncharacterized entanglement summoning problem. Removing the single-directed edge results in an impossible task, while making that edge bidirected results in a solvable task.
• Figure 3: A causal graph corresponding to a "two-out" scenario.
• Figure 4: On the left is a bidirected entanglement summoning tasks involving $5$ diamonds, and on the right is the corresponding access-pair graph as described by the construction in protocol \ref{['protocol:bidirected']}.
• Figure 5: Examples of unachievable bidirectional entanglement summoning tasks. Both causal graphs contain odd cycles in their complement.

Theorems & Definitions (49)

• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 5
• Theorem 6
• Definition 7
• Definition 8
• Definition 9
• Theorem 10