Entangled *N*-photon states for fair and optimal social decision making

The study addresses the Competitive Multi-Armed Bandit (CMAB) problem, where multiple users simultaneously choose among probabilistic resources (machines) with unknown success probabilities and must balance maximizing total rewards with fair distribution among users. Traditional decision-making and MAB frameworks emphasize exploration–exploitation for single users, but many real-world applications (traffic, telecom bandwidth, energy grids) require simultaneous multi-user choices and fairness. Prior physical decision-making implementations with photonic systems demonstrated efficiency in single-user MAB. This work proposes using entangled N-photon polarization states to coordinate multiple users without communication, aiming to maximize total obtainable rewards while ensuring instantaneous fairness between users by constraining only local measurement-basis rotations. The central research question is how to design N-photon entangled states and player strategies that are device-independent and lead to globally optimal and fair outcomes in CMAB with two choices.

The paper situates the work across decision-making research (psychology, management, reinforcement learning, and dynamic resource allocation) and the classical MAB framework. Prior physical/photonic approaches to MAB include chaotic lasers, near-field coupled quantum dots, and single-photon sources, demonstrating scalable single-user decision making. Entanglement has been explored in quantum game theory for deterministic payoff matrices (Nash equilibria) and in quantum machine learning. The authors’ prior work showed entangled-photon pairs can efficiently allocate rewards in a two-user CMAB. For fairness metrics, socioeconomics and telecommunications commonly use Gini, Hoover, Theil, and especially the Jain index due to desirable axiomatic properties. The present study integrates these domains by formulating fairness and performance metrics and extending entanglement-based coordination to N users.

Problem setup and rules: CMAB with N≥2 users and two machines (A,B). In each turn i, each user j selects one machine; machine k yields reward x_{ik}∈{0,1} with unknown probability. Rules: (i) success reward per machine is 1; (ii) if k users select the same successful machine, each receives 1/k; (iii) users know reward amounts and N but not machine success probabilities; (iv) no inter-user communication. Formalism: Define user reward r_{ij} via selection indicator ε_{ijk}∈{0,1} with Σ_k ε_{ijk}=1. Accumulated user reward R_j(n_t)=Σ_i r_{ij}; total accumulated reward R(n_t)=Σ_j R_j(n_t). Optimality for total reward requires each machine be selected by at least one user per turn. Fairness metric: Jain index I_J = (Σ_j R_j)^2 / (N Σ_j R_j^2), with range [1/N,1], continuity, symmetry, and scale invariance. The study focuses on instantaneous (passive) fairness—equal expected rewards in the next trial—sufficient for long-term fairness. Performance-integrated metric: Define machine-available reward X_k(n_t)=Σ_i x_{ik}, total available X= X_A+X_B. A pondered index I_p in [0,1] assesses fairness and efficiency jointly and equals 1 iff the allocation is fair and all available rewards are captured; conceptually, it weights fairness by the fraction of total obtainable reward realized. Quantum-state design principles: Seek N-photon polarization-entangled states satisfying: (a) invariance of measurement probabilities under simultaneous rotation of all users’ polarization bases (device independence), (b) symmetry under permutation of users, and (c) elimination of terms where all users select the same choice, to maximize total reward. N=2 case: Use Bell antisymmetric state |ψ_2⟩= (|HV⟩−|VH⟩)/√2, which is invariant under simultaneous rotations of user bases. Users can only rotate local half-waveplates with single-qubit rotation r(θ)=[[cosθ,−sinθ],[sinθ,cosθ]]. With this state, fairness remains perfect for all angle pairs; total reward depends only on relative basis angle. Realignment algorithms (N=2): Two strategies compared for basis alignment from random initial angles: (i) prior incremental adjustment by one user in small steps; (ii) proposed memory-based random exploration triggered when conflict rate exceeds a threshold (both users can detect conflicts from shared rewards per Rules). Simulations: 100 random initial angle pairs (0–360° in 5° steps), 20 repetitions, memory size 8 reward events, conflict threshold 2, angle increment 5° for the incremental method; evaluate I_p over time. N=3 derivation: Impose global rotation invariance on density matrix ρ_3 under R_3(θ)=R(θ)⊗R(θ)⊗R(θ). Let |ψ_3⟩= Σ amplitudes over basis states excluding |HHH⟩ and |VVV⟩ (forbidden by constraint (c)). From invariance conditions applied to the canonical basis: α_1+α_2+α_3=0; α_i=±i β_i; equal magnitudes |α_i|=|β_i|=1/√6. Solutions involve cubic roots of unity, yielding a family such as |ψ_3⟩= (1/√6)[|HHV⟩+i|VVH⟩+e^{i2π/3}(|HVH⟩+i|VHV⟩)+e^{i4π/3}(|VHH⟩+i|HVV⟩)]. N=4 derivation: Separate symmetric split terms (two users per machine) and asymmetric split terms (three vs one). For symmetric state |S⟩_4=Σ_{6 perms} c_m |HHVV⟩-type terms, invariance yields constraints: c_1+c_2+c_3=0; paired equalities c_1=c_6, c_2=c_5, c_3=c_4; equal magnitudes |c_m|=1/√6; yielding |S⟩_4=(1/√6)(|HHVV⟩+|VVHH⟩+z(|HVHV⟩+|VHVH⟩)+z^2(|HVVH⟩+|VHHV⟩)) with z=±i/√2. For asymmetric state |A⟩_4 with coefficients over |HHHV⟩, …, |HVVV⟩, constraints: a_1+…+a_4=0; a_k=−b_{5−k}; equal magnitudes |a_k|=|b_k|=1/√8; leaving a relative phase parameter φ to define a family |A⟩_4(φ) combining pairs of mirror terms with phase e^{iφ}. Simulation protocol: To explore performance landscapes, fix one user’s angle (via rotation-invariance) and discretize others’ angles in 5° steps; evaluate total reward, fairness, and I_p over 1000 trials, averaging over 20 repetitions. For realignment in N=3 and N=4, test from 100 random initial angle configurations, repeat 20 times, periodically estimate I_p after blocks of events. Also analyze stability by comparing rewards of moving vs passive users. Security and verification: Propose Bell-type and device-independent verification protocols analogous to QKD to certify the N-photon source; provide a Mathematica script to test symmetry and rotation invariance of candidate N-photon states. N=5 state derivations and further simulations are detailed in Supplementary Information.

• N=2: The antisymmetric Bell state |ψ_2⟩=(|HV⟩−|VH⟩)/√2 yields device-independent behavior: fairness remains 1 for all local basis angles; total reward depends only on relative angle. Modified index I_p mirrors total reward due to perfect fairness. The proposed random realignment algorithm converges to optimal configurations faster than the incremental method across 100 random initial angle pairs. • N=3: A family of rotation-invariant, symmetric states exists with phases given by cubic roots of unity; an explicit solution provided. Performance landscapes show both total reward and fairness vary with angle combinations; additional optimal configurations arise at relative rotations of π/3 and 2π/3. Realignment: at least two users must adapt; with two or three adapting, average I_p exceeds ~0.98 after 10,000 events, while with only one adapting it saturates near ~0.95. Active users are never disadvantaged; passive users receive lower or equal rewards, demonstrating evolutionary stability of the adaptation strategy. • N=4: Two families derived—symmetric split |S⟩_4 and asymmetric split |A⟩_4(φ). Fairness is near-perfect (>0.995) for essentially all angle configurations, especially for asymmetric states; variations in I_p are driven by conflict rates (total reward). For |S⟩_4, optimal configurations are sparse; typically at least three users must tune to reach optimality. For |A⟩_4(φ), I_p is maximized along planes (for φ=0 or π) or lines (for φ=±π/2) in the 3D angle space when one user is fixed; two users are sufficient to recover optimality even if others are fixed. For φ=0 [π], any single user can reach an optimal configuration by scanning its angle (optimal planes intersect every axis-parallel line). • General observations: Even N achieve maximum fairness regardless of user angle combinations, while odd N do not. Optimal-state phases empirically involve complex Nth roots of unity. A verification algorithm can test candidate N-photon states for symmetry and rotation invariance. N=5 optimal state expressions are given in the Supplementary Information. • Practicality: The approach is compatible with existing multi-photon entanglement generation (6–10 photons demonstrated) and with device-independent verification akin to QKD networks.

The findings demonstrate that appropriately engineered N-photon polarization-entangled states enable simultaneous maximization of total obtainable reward and fairness in CMAB with two choices, without inter-user communication. By imposing rotation invariance and permutation symmetry (and excluding all-on-one-choice terms), local optimization by users (simple basis rotations guided by observed conflicts) leads to global optimality and fair allocation. For N=2, perfect fairness is guaranteed independent of alignment; for higher N, the structure of the entangled state governs the geometry of optimal configurations and the number of users who must adapt. The proposed random exploration realignment converges quickly and is evolutionarily stable: users are incentivized to participate in alignment because active behavior yields equal or higher rewards. Device independence and compatibility with Bell-type checks support secure, decentralized resource allocation without relying on a trusted central coordinator beyond the entangled source, and known QKD-style verification can certify the source. These results illustrate a quantum advantage in coordinating competitive decisions under uncertainty, offering a pathway to quantum-assisted resource allocation in networks.

The study provides theoretical principles and numerical validation for using polarization-entangled N-photon states to solve CMAB tasks with two choices for up to five users, achieving both optimal total reward capture and fair distribution. It derives explicit optimal states for N=2–4 (and N=5 in Supplementary), reveals a parity-dependent fairness property (even N exhibit guaranteed fairness), and establishes a simple, decentralized realignment algorithm that converges from misalignment without communication. A verification script and discussion of device-independent certification support practical deployment. Future work includes deriving a general formula for arbitrary N, extending to more than two choices, analyzing robustness to noise and detector inefficiencies experimentally, optimizing realignment protocols, and implementing large-N experimental demonstrations with integrated photonics.

• Scope limited to two machines; extension to multiple choices is not derived here. • No closed-form general solution for arbitrary N; explicit forms provided up to N=5 (with N=5 in Supplementary). • Simulations assume idealized detection (100% efficiency) and perfect state preparation; practical imperfections are only qualitatively addressed (post-selection/coincidence). • The pondered index formula is conceptually defined; empirical evaluation relies on simulations with discretized angle spaces (5° steps), which may miss fine-grained optima. • Realignment analyses focus on specific algorithms and thresholds; convergence rates may vary with different parameters and noise. • Security and verification discussions rely on adapting known QKD techniques; scalability and efficiency for large N remain constrained in practice.