PHASE RETRIEVABILITY OF SUPER OPERATORS AND MEASUREMENTS

This paper investigates the conditions under which quantum super operators and measurements can uniquely recover information about quantum states of bounded rank, a problem known as phase retrieval. The authors aim to generalize existing results from frame theory and quantum information—which typically focus on pure (rank-one) states and quantum channels—to a broader class of operators and higher-rank states, while also establishing the stability of these recovery processes.

The authors define "Omega-phase retrievability" for a subset of operators (specifically positive semidefinite operators of rank at most k) and general super operators. They leverage tools from quantum tomography and frame theory to characterize when these operators are injective on the specified subset. By identifying that phase retrievability is equivalent to the ability to distinguish between pairs of operators with orthogonal supports, the authors derive criteria for injectivity and analyze the stability of the reconstruction process using the trace norm and the Bures-Wasserstein distance.

The study establishes several key results:

• It proves that a super operator is phase-retrievable on rank-k operators if and only if it can distinguish between pairs of such operators that have orthogonal images.
• It demonstrates that phase retrievability is always Lipschitz stable with respect to the trace norm, and provides a generalized stability result for the Bures-Wasserstein distance using Fuchs-van de Graaf inequalities.
• It characterizes phase retrievability for Hermitian-preserving super operators using conditions analogous to the classical complement property from frame theory.
• It confirms that there exist quantum channels that are phase-retrievable for rank-k operators but not for rank-(k+1) operators, highlighting the increased complexity of higher-rank state recovery.