Batch Entanglement Detection in Parameterized Qubit States using Classical Bandit Algorithms

Bharati K, Vikesh Siddhu, Krishna Jagannathan

Abstract: Entanglement is a key property of quantum states that acts as a resource for a wide range of tasks in quantum computing. Entanglement detection is a key conceptual and practical challenge. Without adaptive or joint measurements, entanglement detection is constrained by no-go theorems~\citep{tomography2016no-go}, necessitating full state tomography. Batch entanglement detection refers to the problem of identifying all entangled states from amongst a set of $K$ unknown states, which finds applications in quantum information processing. We devise a method for performing batch entanglement detection by measuring a single-parameter family of entanglement witnesses, as proposed by \citet{mintomography}, followed by a thresholding bandit algorithm on the measurement data. The proposed method can perform batch entanglement detection conclusively when the unknown states are drawn from a practically well-motivated class of two-qubit states $\mathcal{F}$, which includes Depolarised Bell states, Bell diagonal states, etc. Our key novelty lies in drawing a connection between batch entanglement detection and a Thresholding Bandit problem in classical Multi-Armed Bandits (MAB). The connection to the MAB problem also enables us to derive theoretical guarantees on the measurement/sample complexity of the proposed technique. We demonstrate the performance of the proposed method through numerical simulations and an experimental implementation. More broadly, this paper highlights the potential for employing classical machine learning techniques for quantum entanglement detection.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

We express our sincere gratitude to the Area Editor and all reviewers for their constructive comments, which have been instrumental in refining our paper. While many suggestions were incorporated during the rebuttal phase, we have implemented further improvements in this camera-ready version, particularly to enhance readability. A summary of these changes are as follows:
1. **Streamlining Sections 2 and 3:** Following the consensus among reviewers to condense these sections, we have retained the concise structure from the revised manuscript. To further improve readability, we have introduced descriptive paragraph subtitles (e.g., "Single-parameter Witness family" in Section 2.1.2).
2. **Revised Notation (-correct):** As suggested by Reviewer VTy7, we have renamed the notation -PC to **-correct** to avoid confusion with -PAC. The definition has been updated in Section 2.2 (pp. 4–5), and the terminology has been unified throughout the manuscript.
3. **Clarifications on Model and Notation:** Addressing Reviewer muTT’s comments, we have clarified the Reward model in Sections 3.1 and A.2, as well as the definition of the notation $\mathcal{F}$ in Section 4.
4. **Benchmarking with Tomography:** In response to the suggestion to compare our method with tomography for Bell Diagonal States (BDS), we have added the baseline to **Figure 4**. To contextualize this, we included a paragraph titled "Benchmarking" in Section 6.4 (p. 14). We also added paragraph titles to Section 8.1 to improve the organization of the discussion.
5. **Copy Complexity of BDS Tomography:** As suggested by Reviewer AdQr, we have added **Appendix A**, which summarises the copy complexity details for tomography in BDS to compare with the complexities derived in Theorems 10 and 13.

Assigned Action Editor: ~Arya_Mazumdar1

Submission Number: 5355