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By Heinrich J. Stüttgen

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25) The coefficients a and b are known as probability amplitudes since, should a measurement of 0 be done, lal 2 (IW) gives the probability that: (i) the measurement outcome will be Ao (Ad, and (ii) the qubit will be left in the state 10) (11)) immediately after the measurement. As the measurement outcome must yield one of the eigenvalues of 0 with certainty lal 2 + IW = 1, and 17;b) is said to be normalized. In quantum computing, one uses n qubits to form a quantum register. 26) where is , ...

119) is satisfied and eq. EbIJ) both equal Aab. ({=) We are given the code Cq and the validity of eqs. 120). ges {E a 17")} of the CB state 17"), and let {Iv~)} be an orthonormal basis for Vi. ine suc~ a subspace for each of the CB states. Eq. 120) indicates that Vi and V j (7" #- J) are orthogonal subspaces. In the event that the direct sum EliV i is a proper subset of H~\ we denote its orthogonal complement by 0. Let {10 k ) } be an orthonormal basis for 0. Then the set of states {Iv~), 10k ) } constitutes an orthonormal basis for H~'.

Ges {E a 17")} of the CB state 17"), and let {Iv~)} be an orthonormal basis for Vi. ine suc~ a subspace for each of the CB states. Eq. 120) indicates that Vi and V j (7" #- J) are orthogonal subspaces. In the event that the direct sum EliV i is a proper subset of H~\ we denote its orthogonal complement by 0. Let {10 k ) } be an orthonormal basis for 0. Then the set of states {Iv~), 10k ) } constitutes an orthonormal basis for H~'. 122) It follows immediately that R is trace-preserving: t 6 6+ L RtRr = L IOk)(Okl k r = I +L r L Iv~)(v~1 i , since the set of states {Iv~), 10 k ) } is an orthonormal basis for H~'.

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