[Paper Review] Noise-tolerant testing of high entanglement of formation

Rotem Arnon-Friedman, Henry Yuen|arXiv (Cornell University)|Dec 26, 2017

Quantum Computing Algorithms and Architecture5 citations

TL;DR

This paper presents a family of non-local games $\{G_n\}$ that enable a classical user to certify high-dimensional entanglement of formation $\Omega(n)$ in noisy, uncharacterized quantum devices. The tests are noise-tolerant, meaning they remain effective even when entangled states are distributed over noisy channels, avoiding the need for fault-tolerant quantum technologies and overcoming limitations of prior robust self-testing methods that fail under fixed noise levels.

ABSTRACT

In this work we construct tests that allow a classical user to certify high dimensional entanglement in uncharacterized and possibly noisy quantum devices. We present a family of non-local games $\{G_n\}$ that for all $n$ certify states with entanglement of formation $Ω(n)$. These tests can be derived from any bipartite non-local game with a classical-quantum gap. Furthermore, our tests are noise-tolerant in the sense that fault tolerant technologies are not needed to play the games; entanglement distributed over noisy channels can pass with high probability, making our tests relevant for realistic experimental settings. This is in contrast to, e.g., results on self-testing of high dimensional entanglement, which are only relevant when the noise rate goes to zero with the system's size $n$. As a corollary of our result, we supply a lower-bound on the entanglement cost of any state achieving a quantum advantage in a bipartite non-local game. Our proof techniques heavily rely on ideas from the work on classical and quantum parallel repetition theorems.

Motivation & Objective

• To develop classical tests that can certify high-dimensional entanglement in uncharacterized and noisy quantum systems.
• To overcome the limitation of existing robust self-testing protocols, which fail under fixed noise levels despite high entanglement.
• To construct non-local games that remain effective even when entangled states are degraded by noise during distribution.
• To provide a noise-tolerant alternative to fault-tolerant quantum protocols for verifying complex quantum behavior in near-term devices.
• To establish a lower bound on the entanglement cost required for quantum advantage in bipartite non-local games.

Proposed method

• The authors construct a family of non-local games $\{G_n\}$ derived from any bipartite non-local game with a classical-quantum gap, ensuring they can certify high entanglement of formation.
• The method relies on a novel application of classical and quantum parallel repetition theorems to maintain soundness under noise.
• A key technical component is the use of a probabilistic method to identify subsets $S$ of questions where the winning probability remains high, even under noise.
• The analysis introduces a conditional entropy bound using von Neumann entropy, showing that the quantum system's uncertainty remains bounded under measurement conditions.
• The protocol conditions on high-winning-probability subsets $S$ to ensure that the resulting state is close to a highly entangled state in terms of entanglement of formation.
• The construction ensures that even noisy states like $\sigma^{\otimes n}$ (with fixed fidelity to EPR pairs) can pass the test with high probability, unlike prior self-testing methods.

Experimental results

Research questions

• RQ1 Can classical users certify high-dimensional entanglement in noisy, uncharacterized quantum devices without requiring fault-tolerant quantum error correction?
• RQ2 Do existing robust self-testing protocols fail under fixed noise levels, and if so, can this limitation be overcome?
• RQ3 Is it possible to design non-local games that remain effective when entangled states are transmitted over noisy channels?
• RQ4 What is the minimal entanglement cost required for a quantum state to achieve a quantum advantage in a bipartite non-local game?
• RQ5 Can parallel repetition techniques be adapted to provide noise-tolerant certification of entanglement in high-dimensional systems?

Key findings

• The proposed non-local games $\{G_n\}$ certify entanglement of formation $\Omega(n)$ for all $n$, meaning the entanglement scales linearly with system size.
• The tests are noise-tolerant: even when the shared state is $\sigma^{\otimes n}$ with $\sigma$ at a fixed fidelity $1-\nu$ to a Bell pair, the games can still be won with high probability.
• The protocol remains effective under fixed noise levels, unlike prior robust self-testing results that require noise to vanish as $n \to \infty$, making it suitable for near-term quantum experiments.
• The analysis establishes a lower bound on the entanglement cost of any state achieving a quantum advantage in a bipartite non-local game.
• The method achieves high confidence in entanglement certification using a probabilistic construction of subsets $S$ of questions, ensuring robustness under noise.
• The entropy-based analysis proves that the conditional quantum uncertainty remains bounded, enabling certification of high entanglement even when the overall winning probability is slightly reduced.