Classical Bandit Algorithms for Entanglement Detection in Parameterized Qubit States

TL;DR

The paper reframes entanglement detection among multiple two-qubit states as a classical 'bad arm identification' problem within a new $(m,K)$-quantum MAB framework. It defines a parameterized state class $\mathcal{F}$ and a witness-based sufficiency criterion $S_{\mathcal{E}}(\rho)$, enabling conclusive entanglement detection without full state tomography in many cases. Two MAB policies are developed: Modified Successive Elimination (SEA) and a lil'HDoC variant, with theoretical PAC guarantees and (near) optimal sample complexities under fixed-confidence settings. Numerical experiments on depolarized Bell states and arbitrary two-qubit states demonstrate substantial measurement reduction while preserving reliable entanglement identification, highlighting the potential of classical ML techniques to augment quantum entanglement diagnostics.

Abstract

Entanglement is a key resource for a wide range of tasks in quantum information and computing. Thus, verifying availability of this quantum resource is essential. Extensive research on entanglement detection has led to no-go theorems (Lu et al. [Phys. Rev. Lett., 116, 230501 (2016)]) that highlight the need for full state tomography (FST) in the absence of adaptive or joint measurements. Recent advancements, as proposed by Zhu, Teo, and Englert [Phys. Rev. A, 81, 052339, 2010], introduce a single-parameter family of entanglement witness measurements which are capable of conclusively detecting certain entangled states and only resort to FST when all witness measurements are inconclusive. We find a variety of realistic noisy two-qubit quantum states $\mathcal{F}$ that yield conclusive results under this witness family. We solve the problem of detecting entanglement among $K$ quantum states in $\mathcal{F}$, of which $m$ states are entangled, with $m$ potentially unknown. We recognize a structural connection of this problem to the Bad Arm Identification problem in stochastic Multi-Armed Bandits (MAB). In contrast to existing quantum bandit frameworks, we establish a new correspondence tailored for entanglement detection and term it the $(m,K)$-quantum Multi-Armed Bandit. We implement two well-known MAB policies for arbitrary states derived from $\mathcal{F}$, present theoretical guarantees on the measurement/sample complexity and demonstrate the practicality of the policies through numerical simulations. More broadly, this paper highlights the potential for employing classical machine learning techniques for quantum entanglement detection.

Figures (4)

• Figure 1: A phase diagram representing the region of damping and depolarizing parameters, $r$ and $p$, respectively, where the damped-depolarized Bell state has negative or positive partial transpose.
• Figure 2: Average number of samples v/s $\delta$ for the $(1,K)$-quantum MAB problem
• Figure 3: Average stopping time v/s $\delta$ for the $(m,K)$-quantum MAB problem
• Figure 4: Entanglement Detection ratio v/s $\delta$ for the $(1,K)$-quantum MAB problem for arbitrary quantum states

Theorems & Definitions (27)

• Definition 1: Entanglement Witness
• Definition 2: $\delta$-PC
• Definition 3: $(\lambda,\delta)$-PAC
• Definition 4
• Remark 1
• Proposition 5
• Proposition 6
• Lemma 7
• proof
• Lemma 8