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See however below. Thirdly, it is absolutely crucial to appreciate that the mere fact that at some time t two states are eﬀectively “decohered”, that is that in the relevant Qubits, Cbits, Decoherence, Quantum Measurement and Environment 19 representation the oﬀ-diagonal elements ρ12 = ρ21 of the reduced system density matrix are 1, by no means implies that no eﬀects of phase coherence between 1 and 2 can ever be seen at any subsequent time. ) The reason is quite a subtle one: in many cases of practical interest, even though a high degree of entanglement of our system with a second system occurs, that entanglement is often “adiabatic” in nature and thus reversible.
6) to its value for q = ± q0 . 1), with Cα given by q0 C˜α , the sum over α taken only over the low-energy (ωα < ωc ) oscillators and the eﬀective tunneling matrix element ∆ a function of ωc . ) For a (relatively) rigorous formulation of the above argument, and a precise prescription for computing the quantity ∆(ωc ), see Section 2 and Appendix A of ref. . 13) c − ω) ˜ where J(ω) may be obtained, as described above, from experiments conducted in the classical regime. e. of the “spin” components σi ) can be obtained by a variety of techniques, since the oscillators can be formally integrated out of the problem; a particularly convenient technique is the two-state functional integral method used in ref.
From now on, when discussing the case of a quantum computer, Qubits, Cbits, Decoherence, Quantum Measurement and Environment 21 I shall treat the whole N -qubit system rather than the individual bits as the “system” and discuss the question of how far that system as a whole does or does not get entangled with anything else in the universe. ) Note that once we go beyond a single qubit, not only the pseudo magnetic ﬁelds felt by the individual qubits, but also the strength of their interactions, may be operators with respect to the environment variables, and thus lead to decoherence.