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arXiv:2402.16067 · Acta Scientiarum Mathematicarum

Log-majorization and matrix norm inequalities with application to quantum information

Fumio Hiai

Published Jun 14, 2024• 5 citations
arXivHTMLPDF
DOI
10.1007/s44146-024-00142-w

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This paper derives an inequality relating the p-norm of a positive 2 x 2 block matrix to the p-norm of the 2 x 2 matrix obtained by replacing each block by its p-norm. The inequality had been known for integer values of p, so the main contribution here is the extension to all values p >= 1. In a special case the result reproduces Hanner's inequality. As an application in quantum information theory, the inequality is used to obtain some results concerning maximal p-norms of product channels.
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We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a non-commutative analogue of a well-known problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r restricted isometry property (RIP). This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.