Entanglement Sharing Schemes (ESS)

• Entanglement Sharing Schemes (ESS) are quantum frameworks that define access structures, allowing specific parties to locally recover maximally entangled states.
• ESS distinguishes between known-partner and unknown-partner regimes, with each setting imposing unique constraints on the distribution and recoverability of entanglement.
• By using threshold protocols and stabilizer constructions, ESS enables efficient designs for quantum networks and cryptographic primitives.

Entanglement Sharing Schemes (ESS) formalize the constrained distribution of entanglement in multipartite quantum systems, distinguishing which pairs or collections of parties can locally recover maximally entangled states and which cannot. Unlike traditional classical secret sharing or quantum secret sharing protocols, ESS shifts focus to the selective redistributability of entanglement as a quantum resource, with utility in cryptographic primitives, quantum networks, and distributed quantum computing. ESS incorporates both structural questions—what kinds of pairwise or multipartite access structures can be realized—and algorithmic questions—how to engineer or construct states realizing these constraints efficiently, with or without the use of stabilizer codes, and in both the “known partner” and “unknown partner” regimes (Khanian et al., 25 Sep 2025).

1. Formal Definition and Structural Principles

An Entanglement Sharing Scheme (ESS) distributes a multipartite quantum state across $n$ parties, such that some disjoint pairs of subsets (authorized pairs) can each locally apply a recovery map to extract a fixed maximally entangled state (e.g., an EPR pair), while other pairs (unauthorized pairs) are unable to do so, even with arbitrary local operations. Let $\mathcal{S} = (\mathcal{A}, \mathcal{U})$ be the pair access structure, where $\mathcal{A}$ lists authorized pairs $\{T_i, T_j\}$ (disjoint subsets of parties), and $\mathcal{U}$ the forbidden pairs.

For an authorized pair $\{T_i,T_j\}$, there exist CPTP maps

$$\mathcal{N}^{ij}_{T_i\rightarrow a},\ \mathcal{N}^{ij}_{T_j\rightarrow b}$$

such that

$$\mathcal{N}^{ij}_{T_i}\otimes \mathcal{N}^{ij}_{T_j}\left(\Psi_{T_iT_j}\right)=\Psi^+_{ab}$$

with $\Psi^+_{ab}$ denoting a maximally entangled state of fixed local dimension. For unauthorized pairs, no such recovery to high-fidelity maximally entangled states is possible. In many constructions, the constraint is further sharpened so that reduced states on unauthorized pairs are separable.

ESS schemes come in two major flavors, based on the information available to the parties:

• Known-partner case: Each subset knows in advance which distinct partner it must entangle with.
• Unknown-partner case: Parties may not know with whom they are to distill entanglement; requirements become more stringent due to the monogamy of entanglement.

The ESS framework is motivated by both practical needs—regulating which network participants can dynamically share entanglement—and by foundational questions about the distribution and monogamy of quantum correlations in complex systems (Khanian et al., 25 Sep 2025).

2. Known-Partner and Unknown-Partner ESS

Known-partner case

In this regime, the parties seeking to recover entanglement know the identity of their partner at the moment of recovery. Many classical-like access structures can be realized. For example, a $(2,2,5)$ threshold ESS encodes entanglement such that any two disjoint subsets of two out of five subsystems can recover an EPR pair, while all smaller subsets cannot. An explicit state for the $(2,2,5)$ case is

$$\ket{\Psi}_{ABCDE} = \sum_{k,s} \ket{k}_A\,\ket{k+s}_B\,\ket{k+2s}_C\,\ket{k+3s}_D\,\ket{k+4s}_E$$

with arithmetic modulo 5 (Khanian et al., 25 Sep 2025).

Unknown-partner case

When the partner is not known in advance, monogamy of entanglement imposes severe constraints. Not every access structure possible in the known-partner scenario can be realized here. Structural graph-theoretic restrictions (such as the prohibition of odd-length cycles in the access graph) emerge. If the authorized-pair graph is bipartite and passes certain linear (support-based and commutation-like) constraints, an ESS can be constructed; otherwise, the structure is forbidden (Khanian et al., 25 Sep 2025).

Differences between these regimes are exemplified by the impossibility of realizing the $(2,2,5)$ threshold ESS in the unknown-partner case, though it is possible in the known-partner case.

3. Access Structure Characterizations and Stabilizer Constructions

For the stabilizer state setting, ESS admits a complete characterization in the known-partner scenario. The two principal necessary and sufficient conditions are:

1. Monotonicity: If $\{T_1, T_2\}$ is authorized and $T_1 \subseteq T_3$, $T_2 \subseteq T_4$, then $\{T_3, T_4\}$ is also authorized.
2. Rank condition: For each minimal authorized pair $A=\{T_1,T_2\}$, the associated matrices $M(A)$ and $\tilde{M}(A)$ (constructed from the stabilizer generators and supports) must satisfy

$$\operatorname{rank} M(A) < \operatorname{rank} \tilde M(A)$$

(Khanian et al., 25 Sep 2025).

For the unknown-partner case, necessary conditions are:

• The authorized-pair graph must be bipartite (no odd cycles).
• A refined vector-based compatibility constraint, encoding the possibility of choosing local recovery operations with supports restricted to authorized partner overlaps, and commutation relations corresponding to intended EPR recovery. These are formalized via support conditions and equations of the form

$$\operatorname{supp}(u_{pq}) \subseteq T_p\cap T_q\ , \quad z_0\cdot u_{pq}=c$$

for vectors $u_{pq}$ representing local Pauli recovery operations and $z_0$ encapsulating commutation information.

Non-stabilizer constructions may, in specific cases, realize access structures forbidden to stabilizer codes, evidencing a strict hierarchy (Khanian et al., 25 Sep 2025).

4. Efficient Construction of Threshold ESS

Threshold ESS generalize classical and quantum secret sharing to the deliberate distribution of entanglement. The canonical constructions proceed via “lifting” classical secret sharing schemes:

• For a $((r,r,3r-1))$ threshold ESS, a Shamir polynomial of degree $r-1$ over $\mathbb{F}_D$ (with randomness vector $\vec a$ and secret $s$)

$F_{\vec a}(x)=a_0+a_1x+\cdots+a_{r-1}x^{r-1}+s$

is used to coherently superpose over all randomness and secret values:

$$\ket{\textrm{state}} = \sum_{\vec{a},s}\ket{F_{\vec a}(1)}\otimes\cdots\otimes\ket{F_{\vec a}(3r-1)}$$

(Khanian et al., 25 Sep 2025).

Any two subsystems of $r$ parties can together recover $s$ and reconstruct the entangled pair, while subsets of insufficient size yield a separable state.

• More generally, a $((p,q,p+2q-1))$ threshold ESS uses quantum Reed–Solomon codes: the access structure allows pairs of $p$ and $q$ subsystems to recover maximum entanglement, subject to code properties ensuring no smaller sets have recovery power.

These constructions are efficient in the sense that each party’s share need only match the output local dimension of the EPR recovery.

5. Applications and Limitations in Quantum Network Tasks

ESS clarify the fundamental constraints of entanglement distribution in quantum networks, particularly in time-sensitive or resource-constrained environments. A highlighted application is to entanglement summoning: distributing entanglement on demand in a network given stringent round and locality constraints.

A central result is the impossibility of perfect deterministic entanglement summoning protocols in certain network topologies. For instance, in a pentagon network where only nearest-neighbor communication rounds are allowed, the pairwise access structure for on-demand EPR creation would contain an odd cycle—a configuration shown to be precluded for unknown-partner ESS (Khanian et al., 25 Sep 2025). This addresses and resolves an open problem in the literature.

Additionally, ESS have implications for anonymous conference key agreement, quantum secret sharing, and other network cryptographic primitives where controlling which parties can distill entanglement is paramount.

6. Generalizations, Open Problems, and Practical Directions

Several generalizations and open questions are raised by the ESS framework:

• The characterization of access structures for general (non-stabilizer) ESS remains conjectural; some structures may only be realized approximately or with non-maximal entanglement.
• Guaranteeing a fixed security gap (i.e., a lower bound on the inability of unauthorized pairs to approximate the target entangled states) is an open technical challenge, especially as the authorized access structure becomes dense.
• Practical deployment in quantum repeater networks, digital key distribution, and network coding would benefit from further analysis of the trade-off between access structure flexibility, efficiency of construction, and noise robustness.

In the stabilizer context, sufficient and necessary conditions for both the known and unknown-partner regimes are now available, making threshold and other monotone access structures algorithmically tractable. For large-scale and heterogeneous networks, hybrid classical-quantum or non-stabilizer approaches remain an area for continued theoretical development (Khanian et al., 25 Sep 2025).

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In summary, Entanglement Sharing Schemes constitute a rigorous formalism for regulating, quantifying, and engineering the recoverability of quantum entanglement in complex multipartite systems. ESS serve not only as a unifying language for several quantum information tasks but also as a foundational tool for the design and control of future quantum networks.