Estimating Bell Diagonal states with separable measurements

TL;DR

This work addresses the problem of estimating Bell diagonal two-qubit states in quantum networks where only local operations and classical communication are available. It compares Bayesian mean estimation against direct inversion and maximum-likelihood estimation, deriving a quantum Cramér-Rao bound-based risk limit and providing closed-form analyses for LOCC-relevant measurements, notably Pauli parity checks. The study shows that Bayesian mean estimation yields lower average risk with meaningful uncertainty quantification, and that Pauli parity checks can closely approach optimal discrimination within the LOCC constraint, with numerical results validating the analytical findings. The results have practical implications for robust entanglement characterization in networked quantum information tasks and open pathways to extensions to higher-dimensional or multipartite settings with appropriate LOCC measurement strategies.

Abstract

Quantum network protocols depend on the availability of shared entanglement. Given that entanglement generation and distribution are affected by noise, characterization of the shared entangled states is essential to bound the errors of the protocols. This work analyzes the estimation of Bell diagonal states within quantum networks, where operations are limited to local actions and classical communication. We demonstrate the advantages of Bayesian mean estimation over direct inversion and maximum-likelihood estimation, providing analytical expressions for estimation risk and supporting our findings with numerical simulations.

Figures (5)

• Figure 1: Bell diagonal states can be geometrically represented by interpreting vector $\bm{t}$ of Eq. \ref{['eq:pauli_rep']} as a point in $\mathbb{R}^3$Horodecki1996. All such states lie within a tetrahedron, formed as the convex hull of four vertices corresponding to the Bell states, $\Phi^+, \Phi^-, \Psi^+$, and $\Psi^-$. Within this tetrahedron, the set of separable states forms a shaded octahedron, satisfying the condition $\abs{t_1} + \abs{t_2} + \abs{t_3} \leq 1$. The completely mixed state is located at the center of the tetrahedron.
• Figure 2: Average risk in terms of the Hilbert-Schmidt distance over a uniform prior for Bell state measurements plotted against the number of measurements $N$ for Bayesian mean estimation and direct inversion. The solid lines correspond to the analytical results of Eqs. \ref{['eq:avg_risk_bsm_di']} and \ref{['eq:avg_risk_bsm_b']}. Each data point corresponds to the average over $1000$ samples. For each sample, the true state $\rho_0$ is drawn uniformly and $N$ Bell state measurements are simulated. To approximate the posterior distribution in the Bayesian mean estimation, the state space is discretized with $10^4$ states.
• Figure 3: Average risk in terms of the Hilbert-Schmidt distance over a uniform prior for Pauli parity checks plotted against the number of measurements $N$ for Bayesian mean estimation and direct inversion. The solid lines correspond to the analytical results of Eqs. \ref{['eq:avg_risk_dirinv']} and \ref{['eq:avg_risk_dirinv_ordered']} and green dashed line represents the upper bound of Eq. \ref{['eq:bound_BME_sep']}. Each data point corresponds to the average over $1000$ samples. For each sample, the true state $\rho_0$ is drawn uniformly and $N$ Pauli parity check measurements are simulated. To approximate the posterior distribution in the Bayesian mean estimation, the state space is discretized with $10^4$ states.
• Figure 4: Average risk in terms of the Hilbert-Schmidt distance over a uniform prior plotted against the number of measurements $N$ for different types of estimators and different sets of measurements. The the risk of different measurements is compared for a fixed estimator: (a) Bayesian mean estimator, (b) maximum-likelihood estimator, and (c) direct inversion estimator. (d)-(g) Measurements are fixed to compare the estimators: (d) Bell state measurement, (e) parity checks, (f) MUB measurement, and (g) Pauli measurement. Each data point corresponds to an average of 4000 samples. The Pauli parity checks, Pauli measurements, and mutually unbiased basis measurements were conducted such that each of the three Pauli parity checks, five MUB measurements, and nine Pauli measurements was measured equally many times. To approximate the posterior distribution in the Bayesian mean estimation, the state space is discretized with $10^4$ states.
• Figure 5: Average risk for Bayesian mean estimation over a uniform prior plotted against the number of measurements $N$ for different types of measurements and different loss functions. The plot displays the numerical results of Fig. \ref{['fig:figure4']} for the Hilbert-Schmidt distance alongside the results obtained when defining the loss function with the infidelity. The same sample size and discretization as in Fig. \ref{['fig:figure4']} are used.

Theorems & Definitions (13)

• Lemma 3.1
• proof
• Lemma 3.2: Direct inversion, Bell state meas.
• proof
• Corollary 3.2.1
• Lemma 3.3: BME, Bell state meas.
• proof
• Lemma 3.4: Direct inversion, parity checks
• proof
• Lemma 3.5: Direct inversion, ordered parity checks