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Abstract: Quantum state tomography aims to produce a complete classical description of the state of a quantum system: a prohibitive task requiring exponential resources. Recent works have taken a different approach. They build an efficient classical description of the state, the so-called classical shadow, that can accurately capture properties of interest such as few-body observables, but is a poor approximation to the entire density matrix. The protocol is simple to implement and predicts observables much more efficiently and accurately than other techniques. It has accordingly developed into an important experimental tool.

The original local shadow protocol does not apply to systems with fundamental number conservation laws, such as cold atomic systems. This is a serious shortcoming. We propose and analyze a new local shadow tomographic protocol adapted to such systems. Our ``All-Pairs'' protocol is simple to implement, tomographically complete, and efficiently invertible. Our analytic treatment relies on the representation theory of the permutation group. We provide a reference implementation and demonstrate the efficiency on simulated data.