Encoding and decoding in complementary bases with quantum gates

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Abstract
The total information content of a composite system consisting of k qubits can either be completely encoded in a specific computational basis, or alternatively it can be partially encoded in a number of different bases. In that case the information encoded in a complete set of mutually complementary bases is again k bits. Using only two single-qubit gates and the controlled-NOT gate, one can implement coding and decoding in such a complete set. The total information content is then invariant under the particular choice of a complete set of mutually complementary bases. For maximally entangled states the k bits of information are not encoded into the k qubits separately but only into their joint properties.

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