Erik Woodhead

Quantum communication is often investigated in scenarios where only the dimension of Hilbert space is known. However, assigning a precise dimension is often an approximation of what is actually a higher-dimensional process. Here, we introduce and investigate quantum information encoded in carriers that nearly, but not entirely, correspond to standard qudits. We demonstrate the relevance of this concept for semi-device-independent quantum information by showing how small higher-dimensional components can significantly compromise the conclusions of established protocols. Then we provide a general method, based on semidefinite relaxations, for bounding the set of almost qudit correlations, and apply it to remedy the demonstrated issues. This method also offers a novel systematic approach to the well-known task of device-independent tests of classical and quantum dimensions with unentangled devices. Finally, we also consider viewing almost qubit systems as a physical resource available to the experimenter and determine the optimal quantum protocol for the well-known Random Access Code.

Performing a measurement on one half of an entangled pair of systems remotely prepares the other half in an ensemble of quantum states. Now consider any set of ensembles that mix to the same density operator. The Schrödinger--HJW theorem states that one can remotely prepare any chosen ensemble from this set by a choice of measurement performed on one half of a fixed entangled state. We generalise this result to show that any set of ensembles can be remotely prepared if one is allowed to post-select on the outcome of the preparing measurement. The probability that the remote preparation is successful is then lower bounded in terms of the distance between the ensembles. In a prepare-and-measure quantum key distribution protocol the distance between the ensembles represents the amount of information that is leaked about the choice of ensemble, e.g. the choice of basis in the BB84 protocol. Using our generalised result, we can prove the security of such protocols from only the fundamental assumption of how much information is leaked about the choice of ensemble/basis, i.e. from bounded basis-dependency.