Maximal randomness from partially entangled states

TL;DR

This paper explores the maximum randomness extractable from partially entangled two-qubit states in a device-independent setting, showing that up to about 3.96 bits can be achieved with nonprojective measurements, but not the full four bits.

Contribution

It establishes bounds on the amount of randomness obtainable from partially entangled states using different measurement strategies in a device-independent framework.

Findings

Two bits of randomness are always achievable.

Maximum randomness with nonprojective measurements is between 3.58 and 3.96 bits.

Four bits of randomness cannot be attained with extremal qubit measurements.

Abstract

We investigate how much randomness can be extracted from a generic partially entangled pure state of two qubits in a device-independent setting, where a Bell test is used to certify the correct functioning of the apparatus. For any such state, we first show that two bits of randomness are always attainable both if projective measurements are used to generate the randomness globally or if a nonprojective measurement is used to generate the randomness locally. We then prove that the maximum amount of randomness that can be generated using nonprojective measurements globally is restricted to between approximately 3.58 and 3.96 bits. The upper limit rules out that a bound of four bits potentially obtainable with extremal qubit measurements can be attained. We point out this is a consequence of the fact that nonprojective qubit measurements with four outcomes can only be self-tested to a limited degree in a Bell experiment.