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Characterizing complex quantum systems is a vital task in quantum informationscience. Quantum tomography, the standard tool used for this purpose, uses awell-designed measurement record to reconstruct quantum states and processes.It is, however, notoriously inefficient. Recently, the classical signalreconstruction technique known as "compressed sensing" has been ported toquantum information science to overcome this challenge: accurate tomography canbe achieved with substantially fewer measurement settings, thereby greatlyenhancing the efficiency of quantum tomography. Here we show that compressedsensing tomography of quantum systems is essentially guaranteed by a specialproperty of quantum mechanics itself---that the mathematical objects thatdescribe the system in quantum mechanics are matrices with nonnegativeeigenvalues. This result has an impact on the way quantum tomography isunderstood and implemented. In particular, it implies that the informationobtained about a quantum system through compressed sensing methods exhibits anew sense of "informational completeness." This has important consequences onthe efficiency of data taking for quantum tomography, and enables us toconstruct informationally complete measurements that are robust to noise andmodeling errors. Moreover, our result shows that one can expand the numericaltool-box used in quantum tomography and employ highly efficient algorithmsdeveloped to handle large dimensional matrices on a large dimensional Hilbertspace. While we mainly present our results in the context of quantumtomography, they apply to the general case of positive semidefinite matrixrecovery.

Quantum Tomography Protocols with Positivity are Compressed Sensing Protocols