surements Zij (ij = 12, 23, 13) on the individual blocks U= A, B, C, yielding measurement outcomes (Zij)u = +1. The key conclusion is that the Shor code can be implemented to correct both phase-flip and bit-flip errors on any qubit of the general form + B|1). The powerful error-correction capability of the Shor code does not end here. Indeed, we may conceive of this code as being able to correct a true continuum of error events, while simply using the discrete set of the above-described syndrome operators! To show, this recall from Chapter 23 that a noisy quantum channel can be described by a trace-preserving quantum operation E of the form (24.34) which is referred to as its operator-sum representation. In this definition, PM = /VM) (VM is the originator message density matrix, which most generally is an nth 24.3 Calderbank-Shor-Steine (CSS) codes tensor product of symbol states Pi = lqi) (qil selected from a qubit symbol alphabet {|qi}}. The operators Uk, called channel-operator elements, are responsible for vari- ous sources of noise impairment. For single qubits (pi), the operator elements take the form: U1 = JPIX for the bit-flip channel, U2 = JP2Z for the phase-flip channel, = JP3Y for the bit-phase-flip channel, the noiseless channel, with {Pn} representing the corresponding probability distribution. We may define Uki as the operator of noise-type k acting on the qubit i in the Shor codeword 19). Thus, we have: (24.35) with {Pni} being the corresponding probability distribution. We observe that through the weighted action of the operator I, X, Y, Z, the above-defined quantum channel defines a continuous qubit transformation onto the Bloch sphere. As we have seen, the Shor code makes it possible to perform syndrome diagnosis and corrections onto any discrete, single-error type, as caused by the operator elements Uki. Therefore, any qubit passed through this channel, and corrupted by an "error continuum, " can be effectively restored in its full original integrity. This remarkable property has no counterpart