By Dave Morris Bacon

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**Sample text**

2. Given some control, what can be done? In this chapter, we introduce the notion of control of a quantum system. The role of control which does not cause decoherence is emphasized. Various formalisms are developed to understand when such non-decohering control is possible. This formalism is then applied to the case of control of a qubit when coupled via a JaynesCummings Hamiltonian to a coherent state of the electromagnetic field. Finally, work regarding what can be done with a given amount of control is reviewed with the role of the Lie algebraic structure being emphasized.

5, the trace norm between two density matrices is a good measure of the absolute distinguishability of the density matrices. For this example we calculate that 1 D(ρ(0), ρ(t)) = Tr|δρ| = |α||β||1 − cos(2λt)|. 5) The best measurement to distinguish the initial state from the decohered state at time t will produce measurements probabilities whose absolute difference will differ by |α||β||1 − cos(2λt)|. If one is thinking about using this qubit for some sort of computation, then we see that the computation will err with a probability of at least this value.

7) 33 and tracing over the system we find that this implies α=0 ρα [Aα , |a a|] + [HA , |a a|] = 0. 8) This must hold for all ρS (0). The ρα , α = 0 form a convex set in the vector space 2 Rd −1 where d is the dimension of the system Hilbert space. This convex set contains the origin (which corresponds to ρS (0) = d1 I) and an open ball of dimension d2 − 1 around the origin[225]. This in turn implies that each of the terms in the expansion of Eq. 8) must vanish separately [Aα , |a a|] = 0 [HA , |a a|] = 0.

### Decoherence, control, and symmetry in quantum computers by Dave Morris Bacon

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