[Paper Review] Self-testing quantum systems of arbitrary local dimension with minimal number of measurements

Shubhayan Sarkar, Debashis Saha+2 more|arXiv (Cornell University)|Sep 27, 2019

Quantum Mechanics and ApplicationsPhysics and Astronomy35 references50 citations

TL;DR

This paper presents the first self-testing protocol for maximally entangled qudit states of arbitrary local dimension d using only two measurements per party, based on the violation of a single d-outcome Bell inequality. It achieves device-independent certification of the state and measurements, enabling unbounded randomness expansion with just one random bit for measurement choice.

ABSTRACT

Bell nonlocality as a resource for device independent certification schemes has been studied extensively in recent years. The strongest form of device independent certification is referred to as self-testing, which given a device certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit-qudit entangled states based on violation of a single $d$-outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, $\log_2d$ perfect random bits, while it requires only one random bit to encode the measurement choice.

Motivation & Objective

• To address the open problem of self-testing qudit-qudit entangled states using a single d-outcome Bell inequality.
• To minimize the number of measurements required for self-testing, achieving the theoretical minimum of two per party.
• To establish a device-independent certification scheme for maximally entangled states |φ⁺_d⟩ of arbitrary local dimension d.
• To enable unbounded randomness expansion using only one random bit for measurement choice.
• To generalize self-testing protocols beyond qubit systems and reduce experimental complexity.

Proposed method

• Proposes a self-testing protocol based on the SATWAP Bell inequality, a d-outcome generalization of CHSH.
• Uses the correlation picture via Fourier-transformed expectation values ⟨A(k)_x B(l)_y⟩ to analyze quantum correlations.
• Derives a set of algebraic equations from the Tsirelson bound of the SATWAP inequality to characterize optimal quantum realizations.
• Employs unitary transformations to relate the optimal observables to the d-dimensional CGLMP measurements.
• Proves that maximal violation of the SATWAP inequality uniquely characterizes the maximally entangled state |φ⁺_d⟩ and specific measurement settings up to local unitaries.
• Demonstrates that outcomes of maximally violating measurements are perfectly random, enabling unbounded randomness expansion.

Experimental results

Research questions

• RQ1 Can a self-testing statement for maximally entangled qudit states be achieved using only a single d-outcome Bell inequality?
• RQ2 Is it possible to achieve self-testing with the minimal number of two measurements per party for qudit systems?
• RQ3 What is the structure of quantum correlations that achieve the Tsirelson bound of the SATWAP Bell inequality for arbitrary d?
• RQ4 How can maximal violation of a d-outcome Bell inequality be used to certify both the state and measurements in a device-independent way?
• RQ5 Can such a protocol enable unbounded randomness expansion with minimal resource overhead?

Key findings

• The paper establishes the first self-testing protocol for the maximally entangled state |φ⁺_d⟩ using only two measurements per party.
• Maximal violation of the SATWAP Bell inequality uniquely certifies the state |φ⁺_d⟩ and specific measurement operators up to local unitaries.
• The protocol enables unbounded randomness expansion, generating log₂d perfect random bits from a single random bit for measurement choice.
• The optimal quantum realizations of the SATWAP inequality are shown to be equivalent to d-dimensional CGLMP measurements via unitary transformations.
• The derived algebraic equations from the Tsirelson bound uniquely determine the state and measurement operators, proving the self-testing statement.
• The outcomes of the maximally violating measurements are perfectly random, confirming the protocol's utility for quantum randomness generation.