Ignorance is strength: Quantum measurements in a non-local world

In quantum theory we often use the term 'observable' interchangeably with 'Hermitian operator'. However, not all Hermitian operators correspond to quantities that can be `observed' in the usual quantum mechanical sense, i.e a projective measurement.  Causality, the finite speed of light, and measurement-disturbance relations, impose constraints on the types of non-local observables that can be measured without additional resources such as communication (i.e time) and/or entanglement.  Unfortunately, there are no generic methods to identify and quantify the types of resources necessary to measure a specific non-local observable while maintaining ignorance about the local properties.  I will present a number of cases where the resource requirements are surprising, and present one probabilistic protocol for an instantaneous measurement of an important class of observables that correspond to interaction terms in a Hamiltonian. I will then show how we are currently using this protocol in an optical implementation of the quantum pigeonhole paradox, effectively creating photon-photon interactions and how results related to quantum measurements can be applied to problems in quantum control.