A mathematical foundation for self-testing: Lifting common assumptions

TL;DR

This paper develops a rigorous mathematical framework for self-testing in quantum mechanics, removing common assumptions and identifying fundamental limitations in the current understanding.

Contribution

It proves a general theorem to eliminate typical assumptions in self-testing, enabling more assumption-free results and highlighting cases where assumptions are necessary.

Findings

Most existing self-testing results can be generalized without assumptions.

Identified a quantum correlation that requires assumptions to be a self-test.

First example of a correlation not realizable with projective measurements on full Schmidt rank states.

Abstract

In this work we study the phenomenon of self-testing from the first principles, aiming to place this versatile concept on a rigorous mathematical footing. Self-testing allows a classical verifier to infer a quantum mechanical description of untrusted quantum devices that she interacts with in a black-box manner. Somewhat contrary to the black-box paradigm, existing self-testing results tend to presuppose conditions that constrain the operation of the untrusted devices. A common assumption is that these devices perform a projective measurement of a pure quantum state. Naturally, in the absence of any prior knowledge it would be appropriate to model these devices as measuring a mixed state using POVM measurements, since the purifying/dilating spaces could be held by the environment or an adversary. We prove a general theorem allowing to remove these assumptions, thereby promoting most existing self-testing results to their assumption-free variants. On the other hand, we pin-point situations where assumptions cannot be lifted without loss of generality. As a key (counter)example we identify a quantum correlation which is a self-test only if certain assumptions are made. Remarkably, this is also the first example of a correlation that cannot be implemented using projective measurements on a bipartite state of full Schmidt rank. Finally, we compare existing self-testing definitions, establishing many equivalences as well as identifying subtle differences.