Random dimension reduction and learning symmetric properties of quantum states

Abstract:We describe a procedure that simultaneously reduces the dimensions of many, potentially distinct quantum states while preserving properties invariant under the tensor power action of an isometry. This serves as a black-box method to replace the dimension with the maximum rank in the sample complexity of learning symmetric properties, even those depending on multiple input states. Combined with full state tomography, random dimension reduction yields improved upper bounds for estimating distances, fidelities, and relative entropies between pairs of states. Expressing the action of this procedure through the Choi--Jamiołkowski isomorphism reveals an intimate connection with the recently introduced random purification channel by Tang, Wright, and Zhandry. With this new perspective, we obtain a simple description of the rank-dependent version of the random purification channel and give an end-to-end analysis of sample-optimal tomography without requiring a reference to the Schur transform or Schur polynomials. Finally, we prove that there does not exist a random purification channel that simultaneously purifies copies of multiple, potentially different input states. Hence, the procedure is related to, but distinct from, random purification.

arXiv:2606.23592 [quant-ph]

(or arXiv:2606.23592v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.23592 Focus to learn more arXiv-issued DOI via DataCite

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