Noncooperative LOCC Protocols in Quantum Networks

• Noncooperative LOCC protocols are defined by local quantum operations with only classical communication, ensuring no global entanglement coordination.
• They enable robust information locking through locally indistinguishable ensembles, requiring full party collaboration to unlock encoded data.
• These protocols underpin decentralized quantum networks, where individual user strategies yield equilibrium effects and impact resource allocation.

Noncooperative LOCC (Local Operations and Classical Communication) protocols constitute a distinct paradigm where spatially separated quantum subsystems perform only local quantum manipulations, communicating solely by classical channels, without global entanglement coordination or, in certain protocols, without any pre-shared entanglement. These protocols are central to questions of quantum information locking, distributed quantum communication, and resource allocation in multiparty settings where coordination is fundamentally limited or incentive-aligned. Two primary application regimes of noncooperative LOCC protocols are information locking with locally indistinguishable (LI) state ensembles (Goswami et al., 2023) and quantum networks with decentralized user control (Shao et al., 17 Dec 2025). Noncooperative LOCC both restricts information extraction in composite quantum systems and can lead to counterintuitive effects in resource-limited quantum networks.

1. Formal Definition and Fundamental Principles

A noncooperative LOCC protocol is defined by two key constraints:

• Only local quantum operations (unitaries, projective measurements, generalized measurements) and classical communication between spatially separated parties (nodes, users, or labs) are permitted; no global quantum operations are allowed.
• There is no global entanglement-sharing or, in some protocols, no pre-shared entanglement is assumed. Resource sharing, path-selection, and all protocol strategies are determined independently by each participant, without central coordination.

In network contexts, this setting is modeled by associating each edge in a graph $G=(V,E)$ with a supply of bipartite quantum states (pure Bell states or mixed Werner states) and allowing users (Alice–Bob pairs) to independently select paths for entanglement swapping. Nash equilibrium behavior arises, as no user can unilaterally change strategies to improve individual payoff (end-to-end entanglement fidelity) (Shao et al., 17 Dec 2025). In information locking, noncooperative LOCC captures the inability of up to $m-1$ of $m$ parties to perfectly distinguish among global orthogonal states encoded into the multipartite system (Goswami et al., 2023).

2. Information Locking via Locally Indistinguishable Ensembles

In multipartite distribution, a classical message $i\in\{1,\ldots,N\}$ is encoded into an orthogonal set of pure states $\{|\psi_i\rangle\}$ on $(\mathbb{C}^2)^{\otimes m}$. The ensemble is locally indistinguishable (LI) if no LOCC protocol—no matter how collaboration proceeds—can perfectly identify the label $i$ when the message is uniformly random and less than all $m$ parties cooperate. This is a robust form of information locking: classical information can be encoded so that only the cooperation of all $m$ parties can unlock the message, and noncooperative LOCC (even with $m-1$ collaborating) provides no access beyond random guessing among possibilities.

A family of such ensembles is constructed as follows for even $m\ge4$:

$$S^m_1 = \Big\{ |0^{\otimes m}\rangle \pm |1^{\otimes m}\rangle,\, |1 0 \dots 0\rangle + |0 1 \dots 1\rangle,\, \ldots,\, |0\dots 01\rangle + |1\dots10\rangle \Big\}$$

This set contains $N = m+2$ elements, generalizing the GHZ-type superpositions. Any coalition of up to $m-1$ parties faces an embedded three-state Bell subspace in the remaining bipartition, which cannot be perfectly distinguished by any LOCC protocol. The best success probability is then strictly sub-unity, concretely bounded by $1/3$, so the message remains locked (Goswami et al., 2023).

3. Entanglement-Assisted Unlocking and Resource Efficiency

Although maximal information locking is enforced under noncooperative LOCC, perfect recovery of the encoded label is rendered possible if the parties pre-share a modest supply of bipartite entanglement. Specifically, pairing the $m$ parties into $m/2$ disjoint pairs and providing each pair with a single Bell pair enables teleportation regroupings. This strategy reconstructs local subsystems of dimension $4$ (two qubits each) so that the entire ensemble $S^m_1$ becomes LOCC-distinguishable in the $4^{\otimes (m/2)}$ partition.

The associated entanglement cost is $E(S^m_1) = m/2$ Bell pairs for even $m$, yielding linear scaling in party number. This cost is strictly lower than that required for "fully bipartition-LI" sets ($E(S^m_2)\ge m-1$), where indistinguishability is enforced across every bipartition; in such cases, any regrouping carries a higher entanglement expense.

Construction	Number of States	Entanglement Cost for Unlocking
$S^m_1$ (GHZ-type)	$N = m+2$	$E(S^m_1) = m/2$
$S^m_2$ (full-LI)	Variable	$E(S^m_2) \ge m-1$

The entanglement-cost saving $\Delta E = (m-1) - (m/2) = (m-2)/2$ grows linearly with $m$, substantiating resource efficiency in the correct ensemble design (Goswami et al., 2023).

4. Algorithmic and Measurement Pathways in Noncooperative LOCC

Attempted noncooperative LOCC unlocking follows structured yet fundamentally limited steps:

1. Coalition of up to $m-1$ parties pools local subsystems, performing optimal joint measurement on their collective system.
2. The classical measurement outcome is broadcast to remaining isolated parties.
3. Remaining parties apply optimal local measurements conditioned on the received classical data.

Due to the invariant presence of LI substructures—such as the Bell-like block in the one-versus-$(m-1)$ bipartition—no sequence of LOCC can surpass random assignment among the LI possibilities. Entanglement-assisted strategies alter the resource structure, specifically via teleportation and subspace projection, to break this indistinguishability, but crucially require only classical communication post-regrouping (Goswami et al., 2023).

Within networked environments, noncooperative LOCC is formalized via independent path assignments by user-pairs, entanglement swapping, and resource (Bell pair/entanglement copy) allocation at each step, always constrained by local resource availability and without a global optimizer (Shao et al., 17 Dec 2025).

5. Noncooperative LOCC in Quantum Networks: The Quantum Selfish Routing Effect

In decentralized quantum networks, noncooperative LOCC protocols—where user-pairs simultaneously and independently choose quantum paths—produce a quantum version of the classical selfish routing problem. The protocol is modeled as a Nash or Wardrop equilibrium:

• Each user optimizes for individual end-to-end fidelity $F_r$, computed via the effective Werner parameter after entanglement swapping, purification, and dilution.
• System-level outcomes can defy monotonicity: more initial entanglement resource may lower everyone's achievable fidelity, due to resource dilution from concentrated user flow on "better" links and subsequent overuse of high-quality edges.

A critical phenomenon arises: removing a high-traffic edge (even a resource-rich Bell state link) can, counterintuitively, increase the fidelity for all users—directly mirroring Braess’s paradox in classical networks. This effect is absent in fully cooperative scenarios, where increased entanglement can always be ignored or locally optimized for overall fidelity monotonicity (Shao et al., 17 Dec 2025).

6. Implications, Performance Scaling, and Mitigation Strategies

The noncooperative LOCC setting imposes several practical and theoretical constraints:

• Non-monotonic scaling: As network size increases, maximum possible fidelity improvements from the removal ("pruning" or "tolling") of an edge can exceed 4–5%. Thus, in large-scale networks, simply increasing the supply of entangled pairs or links does not guarantee better distributed entanglement.
• Dependence on mixedness: The paradoxical effect is maximized in networks with a 50:50 mix of pure Bell and mixed Werner edges and vanishes in all-pure networks.
• The price of anarchy: Quantitative bounds relate equilibrium performance to the global optimum, with explicit objective functions for the Wardrop equilibrium and the centralized global design (Shao et al., 17 Dec 2025).

Mitigation approaches include edge pruning, toll-based incentives, Pareto-superior optimization profiles (better-than-Nash optima), and centrally enforced fairness constraints. These strategies can recover near-global-optimal fidelity in a decentralized equilibrium by controlling resource allocation or deterring overuse of critical links.

7. Outlook, Comparison, and Design Principles

Noncooperative LOCC protocols clarify the operational linchpin between local indistinguishability for information locking and decentralized resource usage in quantum networks. They expose a fundamental design principle: impart sufficient local indistinguishability to ensure robust information locking or fair access, while avoiding global indistinguishability that would drive up entanglement costs or exacerbate resource allocation inefficiencies.

A plausible implication is that, in both quantum cryptography and network engineering, noncooperative LOCC settings must be carefully harnessed to navigate the trade-off between security (locking power), efficiency (entanglement cost), and performance scalability (Nash equilibrium dynamics). This domain remains critical for the robust deployment of distributed quantum information systems (Goswami et al., 2023, Shao et al., 17 Dec 2025).