[Paper Review] Constant-sized correlations are sufficient to robustly self-test maximally entangled states with unbounded dimension

Honghao Fu|arXiv (Cornell University)|Nov 4, 2019

Quantum Mechanics and Applications9 references4 citations

TL;DR

This paper demonstrates that constant-sized correlations—specifically of size Θ(r²) for r ∈ {2, 3, 5}—can robustly self-test maximally entangled quantum states of unbounded local dimension. By leveraging Slofstra’s embedding construction and exploiting the algebraic structure of prime numbers with small primitive roots, the authors design a correlation that enforces a unitary conjugation relation UOU† = Or, enabling self-testing of high-dimensional maximally entangled states with fixed question and answer sets regardless of dimension.

ABSTRACT

We show that for any prime odd integer $d$, there exists a correlation of size $Θ(r)$ that can robustly self-test a maximally entangled state of dimension $4d-4$, where $r$ is the smallest multiplicative generator of $\mathbb{Z}_d^\ast$. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. Vol. $7$, ($2019$)). Since there are infinitely many prime numbers whose smallest multiplicative generator is at most $5$ (M. Murty The Mathematical Intelligencer $10.4$ ($1988$)), our result implies that constant-sized correlations are sufficient for robust self-testing of maximally entangled states with unbounded local dimension.

Motivation & Objective

• To minimize the size of question and answer sets required for self-testing maximally entangled states of high local dimension.
• To show that self-testing of maximally entangled states with unbounded local dimension is possible using only constant-sized correlations.
• To exploit number-theoretic properties of primes with small primitive roots (r ∈ {2, 3, 5}) to construct such correlations.
• To extend the self-testing framework beyond qubit and low-dimensional systems to arbitrarily large dimensions with fixed measurement settings.
• To establish robustness of the self-testing protocol under small statistical perturbations in observed correlations.

Proposed method

• Construct a bipartite quantum correlation Pp,r using a binary linear system Âx = 0 with Θ(r) equations and variables, derived from group-theoretic properties of multiplicative generators in Zp*.
• Use Slofstra’s embedding procedure to map the linear system into a non-local game with constant-sized question and answer sets for each prime p with smallest primitive root r.
• Enforce the unitary conjugation relation UOU† = Or via a perfect correlation in the linear system, which acts as a quantum constraint on the shared state.
• Design a quantum strategy using controlled-unitary operations and entangled ancilla states to realize the correlation, where the shared state is shown to be locally isometric to a maximally entangled state of dimension p−1.
• Apply representation-theoretic techniques to show that any strategy achieving the perfect correlation must implement observables satisfying the required anti-commutation and conjugation relations.
• Prove robustness by bounding the distance between the actual state and the target maximally entangled state in terms of statistical deviation ε, yielding an error bound of O(r^p ε^{1/8}).

Experimental results

Research questions

• RQ1 Can self-testing of maximally entangled states with unbounded local dimension be achieved using only constant-sized question and answer sets?
• RQ2 What number-theoretic properties of primes enable the construction of such constant-sized correlations?
• RQ3 Is it possible to self-test high-dimensional maximally entangled states via correlations that do not scale with the local dimension?
• RQ4 Can the unitary conjugation relation UOU† = Or be used as a structural constraint to enforce self-testing of high-dimensional entangled states?
• RQ5 How robust is the self-testing protocol under small deviations from the ideal correlation?

Key findings

• There exists a family of correlations Pp,r of size Θ(r²) that self-test a maximally entangled state of local dimension p−1 for any odd prime p with smallest primitive root r.
• For r ∈ {2, 3, 5}, the size of both the question and answer sets remains constant across all primes p in the infinite set D of primes with smallest primitive root r.
• The construction relies on a linear system Âx = 0 that enforces the unitary conjugation relation UOU† = Or, which is key to inducing the desired entanglement structure.
• The self-testing protocol is robust: if the observed correlation deviates by ε from Pp,r, the actual state is within O(r^p ε^{1/8}) of the target maximally entangled state under local isometry.
• The result implies that constant-sized correlations suffice for self-testing of maximally entangled states with unbounded local dimension, due to the infinitude of primes with primitive roots in {2, 3, 5}.
• The protocol can be extended to self-test the tensor product of the target state with two EPR pairs, though self-testing of individual observables O₁ and O₂ remains open due to differing isometries.