Optimizing Quantum State Transformation Under Locality Constraint

TL;DR

This work tackles transforming bipartite quantum states under locality constraints by introducing a gradient-based framework that optimizes local CPTP maps via a complex Stiefel-manifold parameterization of Kraus operators. It develops two complementary strategies: a deterministic state-to-state transformation that enables entanglement distillation of weakly entangled states by preprocessing toward R-states for EPL, and a probabilistic local transformation that directly maximizes fidelity to a Bell state through post-selection, saturating known theoretical bounds. The approach provides a versatile tool for distributed quantum information tasks, offering practical pathways to improve distillation and state conversion when only local operations are feasible. The methods are demonstrated on entanglement-distillation scenarios with low fully entangled fraction (FEF), and the framework is applicable to broader quantum-control problems where locality constraints are essential.

Abstract

In this paper, we present a general numerical framework for both deterministic and probabilistic quantum state transformations, under locality constraints. For a given arbitrary bipartite initial state and a desired bipartite target state, we construct an optimized local quantum channel that transforms the initial state into the target state with high fidelity. To achieve this goal, local quantum channels are parametrized on a complex Stiefel manifold and optimized using gradient-based methods. We demonstrate that this approach significantly enhances entanglement distillation for weakly entangled states via two complementary strategies: optimized local state transformation and probabilistic local transformation. These results establish our method as a powerful and versatile tool for a broad class of quantum information processing tasks.

Figures (8)

• Figure 1: After evaluating the gradient vector $G$ at the point $S$, we project it onto one of two subspace coordinates of the tangent space $T_S V_k(\mathbb{C}^n)$, either $S_A \otimes \delta S_B$ or $\delta S_A \otimes S_B$. These subspaces allow variation in only one subsystem's parameters, effectively constraining the optimization to local updates.
• Figure 2: First, we transform each pair of qubits from state $\rho^{\mathrm{int}}_{AB}$ to state $\rho^\prime_{AB}$ using the mapping $\Lambda_A \otimes \Lambda_B$. These mappings are chosen such that the state $\rho^\prime_{AB}$ closely approximates an R-state. Then, as depicted, we apply the EPL protocol, CNOT operation followed by measurement, on two pairs of qubits $\rho'_{AB} \otimes \rho'_{AB}$.
• Figure 3: Output fidelity and the corresponding success probability plotted as functions of the parameter $p$ in the R-state, which serves as the target state. For each value of $p$, the initial state $\rho_{AB}^{*}$ is first transformed via an optimized channel to approximate the target R-state $\rho_{R}(p)$. The EPL distillation protocol is then applied to the transformed state, and both the fidelity of the resulting state and the probability of success are subsequently calculated.
• Figure 4: Comparison of the FEF for 50 randomly generated 2-qubit states before and after processing. The black solid line represents the FEF of the initial states, the red dotted line shows the FEF after applying the EPL protocol directly to the initial states, and the purple dash–dotted line corresponds to the FEF obtained after applying the optimized local maps followed by the EPL protocol. As shown, approximately $75\%$ of the processed states achieve an FEF exceeding $0.5$.
• Figure 5: Starting from the point $S_0$ on the manifold $V_l(\mathbb{C}^{n})$, we follow the trajectory $\gamma(t)$ to the optimal point $S_f$ and evaluate the output fidelity $F(S_t)$ and success probability $P(S_t)$ for each intermediate channel associated with the points $S_t$ along the path.