Quantum information theory is concerned with the storage, transmission, and manipulation of information that is represented in the degrees of freedom of quantum systems. These degrees of freedom are described relative to an external frame of reference. The lack of a requisite frame of reference imposes restrictions on the types of states quantum systems can be prepared in and the type of operations that can be performed on quantum systems. This thesis is concerned with the communication between two parties that lack a shared frame of reference. Specifically, I introduce a protocol whereby the parties can align their respective frames of reference, and a protocol for communicating quantum information in a reference frame independent manner.
Using the accessible information to quantify the success of a reference frame alignment protocol I propose a new measure-the alignment rate-for quantifying the ability of a quantum state to stand in place of a classical frame of reference. I show that for the case where Alice and Bob lack a shared frame of reference associated with the groups G=U(1) and G=Z_M (the finite cyclic group of M elements), the alignment rate is equal to the regularized, linearized G-asymmetry. The latter is a unique measure of the frameness of a quantum state and my result provides an operational interpretation of the G-asymmetry that was thus far lacking. In addition, I show that the alignment rate for finite cyclic groups of more than three elements is super-additive under the tensor product of two distinct pure quantum states. The latter is, to my knowledge, the first instance of a regularized quantity that exhibits super-additivity.
In addition, I propose a reference-frame-independent protocol for communicating quantum information in the absence of a shared frame of reference associated with a general finite group G. The protocol transmits m logical qudits using r+m physical qudits prepared in a specific state that is reference-frame invariant. Measuring the first r qudits allows one to infer the unitary correction that is required to retrieve the remaining m qudits with perfect fidelity. Moreover, the number of ancillary qudits, r, is finite and depends only the group G associated with the requisite frame of reference. I show that the number of single and two-qubit gates required to encode and decode m logical qudits into m+r physical qudits scales linearly with m and the number of group elements \lvert G\rvert. Furthermore, the number of single and two-qubit gates required per logical qudit m is constant allowing for a more efficient implementation than the best currently known reference frame independent protocols.