Mathematics Colloquia and Seminars

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Description

The Positive Operator-Valued Measure (POVM) is the part of a quantum measuring instrument that reveals information about the input state being measured.  The remainder of a measuring instrument is a unitary transformation which determines the post-measurement state conditioned on the registered data.  By performing multiple measurements sequentially in time, one can generate new kinds of POVMs. The rules for how POVMs are generated are tricky.  To understand them, one must overcome two essential features:  First, new information always appears on the opposite side of the post-measurement unitary from the POVM element degrees of freedom.  Second, the product of two positive operators is not generally positive.  It turns out these are precisely the features of a non-Euclidean geometry! In this talk, I will explain how the post-measurement unitary is the transformation between the body frame of the observer and the space frame of the "POV Manifold'', how positive Kraus operators correspond to Levi-Civita connections in the POV Manifold, and finally how the contorsion tensor corresponds to a very important class of adaptive continuous measurements.  As examples, I will consider two adaptive measurements of non-commuting observables, one which keeps the Kraus operators positive (or "absolutely Luders”, one could say) and the other which keeps them solvable (or upper triangular if you prefer matrix language).  Wonderfully, these natural instances of adaptive measurement correspond to the curvature-free connections that align with the Cartesian frames of the usual Poincare models, namely the ball and half-space models of the hyperboloid.