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Nonlocal Measurements via Quantum Erasure

Non-local observables play an important role in quantum theory, from Bell inequalities and various post-selection paradoxes to quantum error correction codes. Instantaneous measurement of these observables is known to be a difficult problem, especially when the measurements are projective. The standard von Neumann Hamiltonian used to model projective measurements cannot be implemented directly in a non-local scenario and can, in some cases, violate causality. We present a scheme for effectively generating the von Neumann Hamiltonian for non-local observables without the need to communicate and adapt. The protocol can be used to perform weak and strong (projective) measurements, as well as measurements at any intermediate strength. It can also be used in practical situations beyond non-local measurements. We show how the protocol can be used to probe a version of Hardy's paradox with both weak and strong measurements. The outcomes of these measurements provide a non-intuitive picture of the pre- and post-selected system. Our results shed new light on the interplay between quantum measurements, uncertainty, non-locality, causality and determinism.