quantum-optimal-transport-entanglement

name	quantum-optimal-transport-entanglement
description	Bipartite entanglement measurement via minimal quantum Wasserstein distance to separable states — Lipschitz dual formulation, entanglement witness connection, and experimental detection framework.
category	quantum

Quantum Optimal Transport Entanglement Measure

Context

Based on arXiv:2606.04969 (Shao, Chen, He, Jun 2026). Proposes a bipartite entanglement measure defined as the minimal order-1 quantum Wasserstein distance from a state to the set of separable states.

Core Methodology

1. Define entanglement measure: E(ρ) = min_{σ separable} W_1(ρ, σ) — the minimal quantum Wasserstein distance to the separable set
2. Verify axioms: Owing to universal data-processing inequality of Wasserstein metric, satisfies all fundamental entanglement axioms within a single geometric framework
3. Lipschitz dual formulation: Derive explicit lower bounds for pure and mixed states using the dual variational problem
4. Two-qubit sharp constant: Establish sharp constant for two-qubit systems
5. Haar-random expected value: Compute expected entanglement value for Haar-random pure states
6. Witness connection: Prove that any negative witness expectation certifies a lower bound on E; dual variational bound equals maximal violation by Lipschitz-1 witness
7. Subadditivity and trace-distance estimates: Establish bounds on local observables and point toward large-deviation conjectures

Implementation Steps

1. Define the quantum Wasserstein distance W_1 on the state space
2. Identify the set of separable states as the reference set
3. Solve the minimization problem: min_{σ ∈ Separable} W_1(ρ, σ)
4. Use the Lipschitz dual formulation to compute lower bounds efficiently
5. For experimental detection: construct Lipschitz-1 witnesses and measure their expectation values
6. The maximal witness violation provides a certified lower bound on entanglement

Key Results

• Single geometric framework satisfies all entanglement measure axioms
• Lipschitz dual gives computable lower bounds for both pure and mixed states
• Sharp constant established for two-qubit systems
• Quantitative connection to entanglement witnesses: negative witness → certified lower bound on E
• Natural subadditivity and trace-distance estimates

Pitfalls

• Computing the exact minimum over separable states is generally hard; use dual bounds in practice
• The Wasserstein metric definition depends on the choice of underlying geometry
• Large-deviation conjectures remain open — current results provide bounds, not exact asymptotics

Verification

• For known entangled states (Bell states), verify E > 0
• For separable states, verify E = 0
• Check that the measure does not increase under LOCC (local operations and classical communication)
• Compare witness-based lower bounds against known entanglement measures (concurrence, negativity)

Activation

• quantum optimal transport, entanglement measure, Wasserstein distance, Lipschitz witness, separable states
• 量子最优传输, 纠缠度量, Wasserstein距离, Lipschitz见证