April 2017

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Author:
Reinhard Werner
Body:

Uncertainty relations come in two kinds: The more familiar type, called preparation uncertainty, captures the impossibility of preparing states that are sharp for two given observables, whereas measurement uncertainty provides quantitative bounds to the errors in any attempt to jointly measure the two. The quantitative theory requires, for each observable involved, a cost function for errors, usually a power of a metric on the outcome space. This induces a transport metric on probability distributions, which is the basis for comparing the marginal of a joint measurement to the ideal reference observables. At the same  time the distance of a probability distribution from the set of point measures is a generalized variance used to quantify sharpness of preparation.  Thus measurement and preparation uncertainty can be compared directly. It turns out that preparation uncertainty (or rather a convex hull of it) is smaller than measurement uncertainty, for any finite set of observables and any choice of error cost functions. I will also discuss some algorithms to obtain optimal bounds for both kinds of uncertainty, and present the results in typical cases.   

 
 
 
 
 
 
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