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Every sufficiently rich set of measurements on a fixed quantum system defines a statistical norm on the states of that system via the optimal bias that can be achieved in distinguishing the states using measurements from that set (assuming equal priors).
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J. M. Renes, R. Blume-Kohout, A. J. Scott, C. M. Caves, “Symmetric Informationally Complete Quantum Measurements”, arXiv[quant-ph]:0310075 (2003)
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A. Ambainis, J. Emerson, “Quantum t-designs: t-wise independence in the quantum world”, in Proc. 22nd Annual IEEE Conference on Computational Complexity (CCC’07), 129-140. arXiv[quant-ph]:0701126
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W. Matthews, A. Winter, “On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states”, Comm. Math. Phys. xxx
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