A minimal informationally complete measurement is s-tight if and only if its suitably rescaled vectors form an acute orthocentric simplex centered at the origin, equivalent to homothetic self-duality.
Beyond morphophoricity: $s$-tight IC measurements in geometric generalised probabilistic theories
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abstract
The analysed in this paper new class of $s$-tight IC measurements contains both morphophoric measurements, preserving the geometry of the states space, and tight IC measurements, introduced nearly 20 years ago by Scott in the quantum case as optimal for the task of linear quantum tomography. By looking at the mathematical side of these classes we discover their common feature, which is also preserved in the broader class of $s$-tight IC measurements: a particularly elegant form of the formula that can be seen as the generalised form of the Urgleichung known from the QBist approach to quantum theory. In particular, the tight IC measurements are identified as the ones for which this generalised Urgleichung takes an exceptionally simple form.
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From minimal informationally complete measurements to orthocentric simplices and back again
A minimal informationally complete measurement is s-tight if and only if its suitably rescaled vectors form an acute orthocentric simplex centered at the origin, equivalent to homothetic self-duality.