All Projective Measurements Can be Self-tested

We show that every real-valued projective measurement can be self-tested from correlations. To achieve this, we develop the theory of post-hoc self-testing, which extends existing self-tested strategies to incorporate more measurements. A sufficient and computationally feasible condition for a projective measurement to be post-hoc self-tested by a given strategy is proven. Recent work by Man{\v{c}}inska et al. [arxiv:2103.01729] showed that a strategy containing $d+1$ two-output projective measurements and the maximally entangled state with the local dimension $d$ is self-tested. Applying the post-hoc self-testing technique to this work results in an extended strategy that can incorporate any real-valued projective measurement. We further study the general theory of iterative post-hoc self-testing whenever the state in the initial strategy is maximally entangled, and characterize the iteratively post-hoc self-tested measurements in terms of a Jordan algebra generated by the initial strategy.