Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension

Honghao Fu

doi:10.22331/q-2022-01-03-614

Abstract

Let p be an odd prime and let r be the smallest generator of the multiplicative group Zp&#x2217;. We show that there exists a correlation of size &#x0398;(r2) that self-tests a maximally entangled state of local dimension p&#x2212;1. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. (2019)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set {2,3,5} (D.R. Heath-Brown The Quarterly Journal of Mathematics (1986) and M. Murty The Mathematical Intelligencer (1988)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.

Highlights

Let p be an odd prime and let r be the smallest generator of the multiplicative group Z∗p
The fact that the verifier only interacts classically with the unknown device makes self-testing a powerful tool for applications in quantum cryptography and computational complexity theory
Our work aims at minimizing the sizes of the question and answer sets of a correlation that can self-test a maximally entangled state with large local dimension

Summary

Preliminaries

We introduce our notations and bipartite quantum correlations. The EPR pair is denoted by. A quantum projective measurement strategy for a nonlocal scenario ([nA], [nB], [mA], [mB]) is a tuple (|ψ ∈ HA ⊗ HB,{{Pi(k)|k ∈ [mA]}|i ∈ [nA]}, {{Q(jl)|l ∈ [mB]}|j ∈ [nB]}), where HA and HB are Hilbert spaces of arbitrary dimension, {{Pi(k)|k ∈ [mA]}|i ∈ [nA]} and {{Q(jl)|l ∈ [mB]}|j ∈ [nB]} are two sets of projective measurements on HA and HB respectively. Note that the tensor product structure indicates that the two parties cannot communicate with each other, which is the reason why we say such scenario is nonlocal This quantum strategy induces the bipartite quantum correlation. The proof of the lemma can be found in the proof of Theorem 4 of [6] and in Section 3 of [28] so we omit it here Another correlation that we are interested in is the correlation that gives the maximal violation of the cot(μ)-weighted CHSH inequality defined in Equation (2). Please refer to [26]

Group theory background

Constructing the linear system

Revisiting the self-testing property of Pμ

The self-testing correlation

Conclusion and discussions