Simultaneous Estimation of Dimension, States and Measurements: Rigidity Considerations

The Gram matrix of the experimentally prepared states and the performed measurements determines the associated density matrices and POVM (projective operator valued measure) elements uniquely up to simultaneous rotations of all density matrices and POVM elements. In [Stark, <a href="/abs/1209.5737">arXiv:1209.5737</a> (2012)] we showed how the state-measurement Gram matrix can be estimated efficiently, and we have seen that in the generic case involving full-rank density matrices and full-rank POVM elements, the measurement data never suffices to uniquely determine the state-measurement Gram matrix. In this paper, we first provide a sufficient criterion to test, which additional assumptions (such as the projectiveness of some measurements) suffice to guarantee the uniqueness of the state-measurement Gram matrix up to discrete symmetries. Subsequently, inspired by the non-uniqueness of the Gram matrix in situations involving full-rank matrices, we examine more closely scenarios involving projective, non-degenerate measurements and their pure post-measurement states. We find that these experiments can be identified on the sole basis of the collected measurement data. Moreover, the Gram matrix is determined uniquely and can be read off directly from the measurement data.