Disti-Mator: an entanglement distillation-based state estimator

TL;DR

The paper introduces Disti-Mator, a state-estimation toolbox that extracts Bell-diagonal parameters of noisy entangled states directly from measurement statistics of probabilistic two-way distillation protocols, avoiding separate tomography when distillation is required. It provides explicit inversions for Werner and Bell-diagonal states, establishes Hoeffding-based concentration bounds and sample complexities, and demonstrates robustness to SPAM and depolarizing noise. The approach is particularly advantageous in high-fidelity regimes and offers potential real-time network monitoring and verification for quantum networks. By integrating estimation into the distillation workflow, the work advances practical state certification and resource-efficient management in NISQ-era quantum information processing.

Abstract

Minimizing both experimental effort and consumption of valuable quantum resources in state estimation is vital in practical quantum information processing. Here, we explore characterizing states as an additional benefit of the entanglement distillation protocols. We show that the Bell-diagonal parameters of any undistilled state can be efficiently estimated solely from the measurement statistics of probabilistic distillation protocols. We further introduce the state estimator `Disti-Mator' designed specifically for a realistic experimental setting, and exhibit its robustness through numerical simulations. Our results demonstrate that a separate estimation protocol can be circumvented whenever distillation is an indispensable communication-based task.

Figures (5)

• Figure 1: The distillation-based estimation. Several copies of noisy entangled state $\rho$ are prepared and shared between two parties. Both parties jointly perform one of the shown entanglement distillation protocols, which, given some probability of success, results to a higher-quality distilled state: (a) the distillation protocol proposed by Bennett et al.Bennett96(1) (which we call Distillation-(a)); (b) a modification of Bennett et al.'s protocol, where the first copy is locally measured in the $X$ basis (Distillation-(b)); (c) the distillation protocol proposed in Deutsch et al.Deutsch96 (Distillation-(c)). After repeated applications of the protocols, we obtain a set of measurement statistics $\{[t]{\hat{p}^{(i)}}\}$ which can be post-processed via our proposed state estimator to generate an estimation $\hat{\rho}$ of the prepared noisy state.
• Figure 2: Number of i.i.d. Werner states consumed for parameter estimation via a distillation experiment. Here, $N^{(1)}$ is the number of i.i.d. Werner state pairs $\rho_w^{\otimes 2}$ required to estimate the Werner parameter $w$ with failure probability bound $\delta = 10^{-2}$, for different values of the error threshold $\epsilon_w$. The solid curves correspond to the expected total number of $\rho_w$ consumed with an estimation via a noiseless distillation protocol, $N_{\mathrm{consumed}}= (2-p^{(1)})N^{(1)}$, leaving $p^{(1)}N^{(1)}$ distilled states for further use. The dashed horizontal lines correspond to the total number of Werner states consumed to estimate using tomography. The dash-dotted curves correspond to the distillation protocol in the presence of depolarizing noise, where the average depolarization $1-S = 1-\exp(-1/4)$, and the expected $N_{\mathrm{consumed}}= (2 - p_S^{(1)})N^{(1)}$. Whenever $w$ is close to zero, we observe that an estimation via distillation consumes fewer overall resources than tomography.
• Figure 3: Number of i.i.d. Bell-diagonal states consumed for parameter estimation via a noiseless distillation experiment. Here, we assume that all distillation protocols are equally allocated with $N^{(i)}$ i.i.d. Bell-diagonal state pairs $\rho^{\otimes2}$. We compare this with conventional state tomography, where we assume that the joint measurements are also equally allocated with states $\rho$, totalling $N_{\mathrm{tom}}$ (see Methods \ref{['M_c']}). The dashed line describes the collection of Werner states. We set the error bound $\epsilon_i= 10^{-2}$ for all $i$, and we impose $\delta=10^{-2}$ when estimating the trace distance $D(\hat{\rho}(\hat{\mathbf{q}}),\overline{\rho}(\mathbf{q}))$ either via distillation or tomography. While tomography consumes all $N_{\mathrm{tom}}$ of the states, the distillation-based estimation is expected to consume $N_{\mathrm{consumed}} =\sum_{i\in\{1,2,3\}} (2-p^{(i)})N^{(i)}$ of the states, leaving $\sum_{i\in\{1,2,3\}} p^{(i)}N^{(i)}$ distilled states for further use. Whenever $q_1$ is close to one, we observe that an estimation via distillation consumes fewer resources than tomography (up to about $60\%$ fewer resources near $\ket*{\Phi^+}$).
• Figure 4: Parameter estimation for Werner states in a simulated noisy distillation experiment, with the following noise parameters: ${T_{A,B}^{\text{dpo},\text{dph}}}$ are the characteristic times for the depolarizing and dephasing channels acting on the first prepared state such that $[t]{t/T_{A,B}^{\text{dpo},\text{dph}}=\Delta t_{\text{geom}}(p_g)/100}$, where $t_{\text{geom}}(p_g)$ is drawn from a geometric distribution with Bernoulli success probability equal to the $\rho_w$-generation rate of $p_g = 0.2$; $y_{A,B}=0.01$ are the CNOT depolarizing parameters; $[t]{\eta_{A,B}^{Z}= 0.99}$ are the non-error probabilities of the measuring devices. We set the error bound as $\epsilon_w = 10^{-2}$. (a) Estimation with $N^{(1)} = 10^5$$\rho_w^{\otimes2}$ Werner pairs; (b) with $N^{(1)} = 10^6$ pairs. For both cases, the empirical success probabilities $\hat{p}^{(1)}$ are shown alongside the red dashed curves representing the expected behavior. The inset plots show the deviation of the estimation from the true $w$. (c) The failure probability bound $\delta$ for each simulation. The solid curves indicate the expected bound given a noiseless distillation, while the dashed curves indicate the expected bound for a noisy distillation with the given noise parameters.
• Figure 5: Parameter estimation for Bell-diagonal states in a simulated noisy distillation experiment, with $N^{(i)} = 2\times 10^5$ the number of $\overline{\rho}^{\otimes2}$ pairs for each distillation protocol, and with the following noise parameters: characteristic times $[t]{T_{A,B}^{\text{dpo},\text{dph}}}$ of depolarizing and dephasing channels acting on the first prepared state such that $[t]{\Delta t/T_{A,B}^{\text{dpo},\text{dph}}=t_{\text{geom}}(p_g)/100}$, where $t_{\text{geom}}(p_g)$ is drawn from a geometric distribution with Bernoulli success probability equal to the $\overline{\rho}$-generation rate of $p_g = 0.2$; $m_{A,B}=0.01$ are the depolarizing parameters for the local $\pm\pi/2$ rotations; $y_{A,B}= 0.01$ are the CNOT depolarizing parameters; ${\eta_{A,B}^{Z,X}= 0.99}$ are the non-error probabilities of the measuring devices. We set $q_3=q_4=(1-q_1-q_2)/2$, and the error bound $\epsilon_i=10^{-2}$ for all $i$. The dashed lines describe the collection of Werner states. (a) Trace distance $D(\hat{\rho}(\hat{\mathbf{q}}),\overline{\rho}(\mathbf{q}))$ between the estimation $\hat{\rho}(\hat{\mathbf{q}})$ and the expected state in Bell-diagonal form $\overline{\rho}(\mathbf{q})$. We observe that the estimation is close to the expected $\mathbf{q}$ whenever $q_1$ is close to one. (b) The failure probability bound $\delta$ of the trace distance exceeding $\epsilon_T = 3\times10^{-2}$.

Theorems & Definitions (1)

• proof