Does the energy of a qubit change when it undergoes a collapse?

Question

Can the state of a qubit affect its energy? Please elaborate on your answer

Answer 1

Depending on your physical implementation, it may be that the two basis states are at different energies. Thus, when the state collapses, the energy changes. However, the average energy remains constant so you cannot use this for energy generation or similar.

For example, assume we have a qubit with a Hamiltonian $$ H=E|1\rangle\langle 1|. $$ For any state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, we have an average energy $$ \bar E=\langle\psi|H|\psi\rangle=E|\beta|^2. $$ If we measure this qubit in the standard, $Z$ basis, we get:

• answer $0$ with probability $|\alpha|^2$. Afterwards, the system is in the state $|0\rangle$ and has energy 0.
• answer $1$ with probability $|\beta|^2$. Afterwards, the system is in the state $|1\rangle$ and has energy $E$.

So, you can see that the average energy is $|\alpha|^2\times 0+|\beta|^2\times E=E|\beta|^2$, which is unchanged from the original state. Sometimes you gain energy, sometimes you lose energy, but on average it stays the same.