  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.
