Quantum Bit Strings and Prefix-Free Hilbert Spaces

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We give a mathematical framework for manipulating indeterminate-length quantum bit strings. In particular, we define prefixes, fragments, tensor products and concatenation of such strings of qubits, and study their properties and relationships. The results are then used to define prefix-free Hilbert spaces in a more general way than in previous work, without assuming the existence of a basis of length eigenstates. We prove a quantum analogue of the Kraft inequality, illustrate the results with some examples and discuss the relevance of prefix-free Hilbert spaces for lossless compression.