Introduction to Quantum Message Space

According to Landauer's Principle the annihilation (and presumably the creation) of one bit costs at least kT ln(2) much energy. This seems to imply that serial bit transmission is impossible; however, a complete quantum theory of communication should permit Alice to send Bob a message of arbitrary length, unknown to Bob in advance. One is lead to consider the Hilbert state space ∞ n=0 C 2 ⊗n , where C 2 is conventional qubit space, but the obvious operator which appends a 0 (say) to a string x (i.e., sends | x to | x0) is no longer unitary on this space because it is not onto. The remedy proposed in this report is to use ℓ 2 (FG(2)) for the quantum message space (QMS), where FG (2) is the free group on the bit symbols {0,1}; the "anti-bits" 0 −1 and 1 −1 , represented by 2 and 3 respectively, are introduced, and thus the conservation laws are retained. After developing the basic QMS constructs we describe the message-length observable N , noting that most quantum operators used hitherto in quantum information theory commute with N. In QMS the operation which appends a string to a message is implemented by a right translation operator. We review the harmonic analysis on the free group FG (2) and the decomposition of the right regular representation into irreducible representations of FG (2). This decomposition is implemented by the spectral analysis of the operator A , which is left convolution by (| 0 +| 1 +| 2 +| 3)/ 4. This analysis yields a family of projectors which commute with the extended qubit operations implemented by the right regular representation and therefore can be used to construct quantum operators for quantum computation on QMS.