Existence of a robust optimal control process for efficient measurements in a two-qubit system

TL;DR

The paper addresses efficient verification of two-qubit entanglement without full tomography by proving that a unitary transformation can map any initial state to a final state whose $ZZ$ expectation value equals the initial concurrence. It combines a rigorous mathematical framework for density-matrix controllability with an optimal-control design (cost functional and Euler–Lagrange equations) and a GRAPE/Krotov-inspired gradient algorithm to realize the transformation. The study demonstrates that the two-qubit system is controllable under $\mathfrak{su}(4)$, remains robust to drift, and yields accurate concurrence verification with low measurement overhead in numerical experiments (typical ~5% error). The findings offer a practical, low-depth entanglement verification protocol suitable for industrial-scale QC and lay groundwork for generalization to larger systems and other entanglement measures.

Abstract

The verification of quantum entanglement is essential for quality control in quantum communication. In this work, we propose an efficient protocol to directly verify the two-qubit entanglement of a known target state through a single expectation value measurement. Our method provides exact entanglement quantification using the currencence measure without performing quantum state tomography. We prove the existence of a unitary transformation that drives the initial state of a two-qubit system to a designated final state, where the trace over a chosen observable directly yields the concurrence of the initial state. Furthermore, we implement an optimal control process of that transformation and demonstrate its effectiveness through numerical simulations. We also show that this process is robust to environmental noise. Our approach offers advantages in directly verifying entanglement with low circuit depth, making it suitable for industrial-scale quality control of entanglement generation. Our results, presented here, provide mathematical justification for our earlier computational experiments.

Figures (2)

• Figure 1: Scatter plot for the concurrence of the initial state and the $\sigma_z\otimes\sigma_z$ measurement on the final state compared with the line $x=y$.
• Figure 2: Histogram for the relative error between the concurrence of the initial state and the $\sigma_z\otimes\sigma_z$ measurement on the final state.

Theorems & Definitions (16)

• Definition 2.1: Dynamical Lie Algebra Dalessandro_2022
• Theorem 2.1: Existence of the unitary transformation
• proof
• Corollary 2.1.1: Robust controllability
• proof
• Theorem 2.2: Unitary equivalence for two Hermitian matrices Horn_1985
• Theorem 2.3
• proof
• Corollary 2.3.1
• proof