The algorithms discussed in Chapter 21 rely on having isolated systems to store information. In practical applications this is not possible, and the environment interacts with the systems causing decoherence. Suppose the data is contained in the state |x〉, and the environment is described by |E〉. The initial state of the entire system is described by the tensor product of the states |X〉⊗|E〉, which evolves according to some unitary operation U. The state |x〉 evolves according to the unitary operation U x which describes the algorithm. In classical error correction codes, all that needs to be corrected are bit flips (see Chapter 10). In the quantum case errors such as bit flips, phase changes and rotations complicate the error correction techniques. Since arbitrary errors in an encoding of information cannot be corrected, only certain types of errors are assumed to occur (this was also the case in classical error correction, where an upper bound in the number of bits that could be flipped in a transmission was assumed). The types of errors depend on the implementation. For example, suppose the types of errors (which we assume are distinguishable due to an encoding) are described by the unitary basis E 1,...E n , so that all errors are a linear combination [138] E = al.E 1 + ... + a n E n ,...E † E = I.