  Measurement uncertainty relations are lower bounds on the errors of any
approximate joint measurement of two or more quantum observables. The aim of
this paper is to provide methods to compute optimal bounds of this type. The
basic method is semidefinite programming, which we apply to arbitrary finite
collections of projective observables on a finite dimensional Hilbert space.
The quantification of errors is based on an arbitrary cost function, which
assigns a penalty to getting result $x$ rather than y, for any pair (x,y). This
induces a notion of optimal transport cost for a pair of probability
distributions, and we include an appendix with a short summary of optimal
transport theory as needed in our context. There are then different ways to
form an overall figure of merit from the comparison of distributions. We
consider three, which are related to different physical testing scenarios. The
most thorough test compares the transport distances between the marginals of a
joint measurement and the reference observables for every input state. Less
demanding is a test just on the states for which a "true value" is known in the
sense that the reference observable yields a definite outcome. Finally, we can
measure a deviation as a single expectation value by comparing the two
observables on the two parts of a maximally entangled state. All three error
quantities have the property that they vanish if and only if the tested
observable is equal to the reference. The theory is illustrated with some
characteristic examples.
