I’ve written a new paper showing how to teleport a state which is post-selected. This combines three of my favourite topics: teleportation, post-selection, and the reality of an a non-local quantum state. My final result gives a simplified measurement for checking that a shared entangled state is really “in” that state at a given time, regardless of how far apart the parts of the state may be. I’ll explain what this means in a little more detail below.
First, Teleportation starts with the question: “How can I send a quantum state from one place to another?” This is tricky as quantum states tend to decohere, i.e. become irretrievably noisy in transmission, the further they are sent, so we prefer sending classical bits instead. Since the uncertainty principle prevents us from measuring the quantum state to convert it into classical bits, it seems we can’t avoid the decoherence. Bennett et. al. showed that we can in fact send it perfectly using classical communication, pre-shared entangled states, and local operations.
Second, Post-Selection is conditioning on a particular final state of a system. In classical deterministic physics, conditioning on a particular final state is equivalent to conditioning on some particular initial state. However in quantum mechanics post-selection tells us more. For example if a particle is initially prepared at a particular position, and later we condition on a final measurement of its momentum being a particular value, we can predict that any intermediate measurement of position/momentum would give the initial/final values, apparently letting us know the position and momentum of a particle at the same time.
My main result was to show that a post-selected state can be teleport ed from one place to another. That is, if we have two systems where a post-selection is applied to A but we would like instead it to be applied to B, we can make it happen with certainty using entanglement, classical communication and local operations. Since a post-selection is usually evolved backwards in time to see what it tells us about intermediate time measurements, this can be called teleportation of a state evolving backwards in time.
Finally, I addressed the question of how one can verify that an entangled state of two spatially separated systems really is what we think it is at a fixed time. If we cannot, it would make us query whether an entangled state is the right description for our system after all. Now the usual way you check a state of one system is by doing a Von Neumann measurement. However it is impossible to perform this instantaneously for two spatially separated systems as such procedure would allow faster than light communication. Instead Vaidman showed that one could perform an instantaneous verification measurement, which checks but destroys the state, using a large amount of shared entanglement and later classical communication and processing. My contribution is to show how to perform such a verification measurement on a set of pre and post-selected systems using exponentially less entanglement than the earlier protocols. This gives us a simplified demonstration that the systems really are described by that state at a given time.