Verification of entangled states under noisy measurements

TL;DR

This work addresses verifying entangled quantum states under noisy measurement readouts. It develops a systematic noisy QSV framework, establishes a necessary and sufficient distinguishability condition for uniquely identifying the target state, and introduces a symmetric hypothesis-testing verification algorithm. The authors provide both analytical and SDP-based methods to quantify noise effects, demonstrating a nearly $N \sim \epsilon^{-2}$ scaling of the required sample size under symmetric testing and extending verification feasibility to GHZ, stabilizer, and W states in noisy scenarios. The results offer practical, robust verification protocols suitable for current NISQ experiments, enabling reliable entanglement verification with imperfect devices.

Abstract

Entanglement plays an indispensable role in numerous quantum information and quantum computation tasks, underscoring the need for efficiently verifying entangled states. In recent years, quantum state verification has received increasing attention, yet the challenge of addressing noise effects in implementing this approach remains unsolved. In this work, we provide a systematic assessment of the performance of quantum state verification protocols in the presence of measurement noise. Based on the analysis, a necessary and sufficient condition is provided to uniquely identify the target state under noisy measurements. Moreover, we propose a symmetric hypothesis testing verification algorithm with noisy measurements. Subsequently, using a noisy nonadaptive verification strategy of GHZ and stabilizer states, the noise effects on the verification efficiency are illustrated. From both analytical and numerical perspectives, we demonstrate that the noisy verification protocol exhibits a negative quadratic relationship between the sample complexity and the infidelity. Our method can be easily applied to real experimental settings, thereby demonstrating its promising prospects.

Figures (5)

• Figure 1: Illustration of the noisy QSV with two types of errors. We consider the verification of a five-qubit stabilizer state under a noise model given in Section 5, where ${\lambda_0\sim0.9271}$ and ${\nu(\tilde{\Omega})\sim0.4341}$. The infidelity is chosen as ${\epsilon=0.01}$. The horizontal axis $f$ is the frequency of the number of passed measurements among ${N = 20000}$ measurements. The vertical axis is the kernel density function computed from $10000$ experiments. We plot the kernel density distribution for both the target state and the worst-case state. The type I and II errors correspond to the areas shaded in gray and pink respectively, which are dependent on the threshold frequency $f^{\prime}$ that we set, shown as the black vertical line.
• Figure 2: Noisy verification of a five-qubit stabilizer state. The simulated noisy strategy has the dominant eigenvalue ${\lambda_0\sim0.9271}$ and the second-largest eigenvalue ${\lambda_{1}\sim0.4930}$. Four curves are plotted for different confidence levels $\delta=0.01, 0.05, 0.1, 0.2$ (top to bottom).
• Figure 3: Numerical results on a three-qubit $W$ state noisy verification. The vertical axis labels the probability $p(\epsilon)$ of passing noisy nonadaptive strategy $\tilde{\Omega}_{W}$ and the horizontal axis denotes the infidelity $\epsilon$. The red line is the dominant eigenvalue of noisy strategy $\tilde{\Omega}_{W}$. The blue line represents the passing probability of the noisy strategy $\tilde{\Omega}_{W}$ for the $W$ state. The black line indicates the passing probability calculated for the worst-case scenario using the SDP in Equation 17.
• Figure 4: Simulation and theoretical results on a five-qubit stabilizer state noisy verification. The vertical axis labels the confidence level $\delta$ and the horizontal axis $g$ denotes the noisy amplitude. For convenience, the noise amplitude is chosen to be the same in all directions. The simulation settings are of sample complexity $N=20000$, infidelity $\epsilon=0.01$ with 10000 times repetition.
• Figure 5: Simulation and theoretical results on a five-qubit GHZ state noisy verification. The vertical axis labels the confidence level $\delta$ and the horizontal axis $g$ denotes the noisy amplitude. For convenience, the noise amplitude is chosen to be the same in all directions. The simulation experiment settings are of sample complexity $N=20000$, infidelity $\epsilon=0.01$ with 10000 times repetition.

Theorems & Definitions (6)

• Proposition 1
• Proposition 2
• proof
• proof
• proof
• proof