Weak Values with Remote Postselection and Shared Entanglement

We propose a new protocol for the weak measurement of any observable with remote pre and postselections. We show that if two parties share a pure entangled state, then by using local operations and classical communication they can preselect and postselect at distant locations leading to the weak value of an observable as a shift in the pointer of the apparatus at one location in the process of the weak measurement. This can be achieved with either sharing of a pure maximally or non-maximally entangled state. We generalize the protocol for realizing the weak value of any observable with remote pre and postselection of mixed states. Finally, we show how the weak value is modified in the remote pre and postselection setting if Alice and Bob share a mixed entangled state.