Measurement-device-independent Schmidt number certification of all entangled states

TL;DR

The paper investigates certifying entanglement dimensionality via Schmidt number. It first shows fundamental limits of fully device-independent certification using Bell-nonlocal games, including states whose Schmidt number cannot be certified under local projective measurements, and extends this limitation to certain POVMs. It then proves that measurement-device-independent certification using semiquantum nonlocal games with trusted inputs can certify the Schmidt number for all bipartite states, by constructing games from Schmidt-number witnesses. An explicit $d=3$, $r=2$ example demonstrates how to realize such a game and certifies $SN>2$ for isotropic states with certain parameters. The results offer a practical route for robust high-dimensional entanglement certification and point toward future work on broader DI approaches and experimental implementations.

Abstract

Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilisation. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: it cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell-nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with Schmidt number exceeding $r$, there exists a semiquantum nonlocal game that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.

Figures (3)

• Figure 1: Illustration of the nested subset structure of the sets $\mathcal{S}_r \subseteq \mathcal{D}(\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B})$ consisting of states with Schmidt number at most $r$, where $1 \leq r \leq d$ and $d=\min\{d_A,d_B\}$. The outermost set $\mathcal{S}_d$ corresponds to all states. $W_r$ is a witness operator that detects states with Schmidt number greater than $r$: its expectation value is positive semidefinite for all states lying right to it and strictly negative for any state lying left to it.
• Figure 2: A schematic setup for the certification of higher Schmidt number states in a standard Bell scenario with classical inputs and outputs.
• Figure 3: A schematic setup for the certification of states with a higher Schmidt number in a measurement-device-independent framework using trusted quantum inputs and classical outputs.

Theorems & Definitions (8)

• Theorem 1
• proof
• Lemma 1
• Lemma 2
• Lemma 3
• Corollary 1
• Theorem 2
• proof