Adaptive Negativity Estimation via Collective Measurements

Abstract

This paper explores an efficient method for entanglement quantification in two-qubit and qubit-qutrit quantum systems based upon the framework of collective measurements in conjunction with machine learning. We introduce an adaptive measurement procedure in which measurement settings are dynamically adjusted based on prior measurement outcomes aiming to optimize the inference precision given a limited number of these measurement settings. The procedure makes use of the Long Short-Term Memory networks to recurrently process collective measurements on two copies of the investigated states. Obtained results demonstrate the tangible benefits of the adaptive measurements in comparison to previously described non-adaptive strategies.

Figures (6)

• Figure 1: The conceptual diagram of a two-copy collective measurement and subsequent negativity prediction by artificial neural network (ANN) model with Long Short-Term Memory (LSTM) cells. Two instances of the system under investigation, denoted as $\hat{\rho}$, are measured simultaneously. During the measurement, one subsystem from each instance undergoes a local projection using operators $\hat{\Pi}_{x_i}$ and $\hat{\Pi}_{y_i}$, while the other two subsystems are non-locally projected using the $\hat{\Pi}_{\mathrm{Bell}}$ operator. The results of these measurements, denoted as $P_{x_{i}y_{i}}$, are sequentially used as inputs for the ANN, which then after each measurement estimates the negativity $\hat{\mathcal{N}}_{i}$ of $\hat{\rho}$ and recommends the most suitable local measurements for the next iteration $\hat{\Pi}_{x_{i+1}}$ and $\hat{\Pi}_{y_{i+1}}$.
• Figure 2: Topology of the artificial neural network for the negativity $\hat{\mathcal{N}}_{i}$ estimation and for the prediction of the next set of local measurements $\hat{\Pi}_{x_{i+1}}$, $\hat{\Pi}_{y_{i+1}}$. The LSTM cell states and hidden states are 128 32-bit integers long while the size of the hidden fully connected layers is 256.
• Figure 3: Comparison of the ANN models for two-qubit systems. Performance in terms of the $\mathcal{L}^{(1)}$ error for all combinations of adaptive/non-adaptive and greedy/last approaches is visualized as a function of the number of $n$ iterations.
• Figure 4: Histograms showing the distribution of paired values of the true negativity $\mathcal{N}$ and its estimate by the models $\hat{\mathcal{N}}$ in two-qubit systems: (a) the non-adaptive strategy with $n=5$ iterations, (b) the adaptive $\emph{last}$ approach with $n=5$ iterations and (c) the adaptive $\emph{last}$ approach with $n=10$ iterations. Note that ideally only diagonal elements, where $\mathcal{N} = \hat{\mathcal{N}}$, should have non-zero values.
• Figure 5: Comparison of the ANN models for qubit-qutrit systems. Performance in terms of the $\mathcal{L}^{(1)}$ error for all combinations of adaptive/non-adaptive and greedy/last approaches is visualized as a function of the number of $n$ iterations.