# Subcontractor: Consider a two level system with Hamiltonian

**Question description**

Consider a two level system with Hamiltonian

H = E_0 |E_0> + E_1 |E_1>

where |E> is Dirac notation for a quantum state E with eigenstates

|E_0> = 1/sqr(2) ( |0> + |1> )

and

|E_1> = 1/sqr(2) ( |0> - |1>)

(a) Consider a quantum state which starts as |0> at t= 0. Compute the minimal amount of time it takes the system to reach the orthogonal state |1>. Express the answer in terms of the expectation value and/or the standard deviation of the energy.

(b) Consider now an arbitrary quantum system in an arbitrary pure state at t=0. Derive a lower bound on the amount of time it takes for the systems to evolve from the original state to an orthogonal state as a function of the expectation value of the energy.

(c) If we interpre the question in the initial two level system as an abstract realization of a quantum (logical) gate transformation called NOT acting like | 0 > --> | 1 > , and | 1 > --> | 0 >, we can think of the conclusion in (a) as saying that quantum mechanics limits the velocity at which such operation can be performed. Along the same lines, how can we interpret the result in questions (b)?

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