# Additional Phase factors in SU(2)

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1. Aug 31, 2015

### Meditations

I am curious as to the meaning of, and name given to the phase $\xi(t)$ which may be added as a prefix to the time evolution operator $\hat{U}(t)$. This phase acts to shift the energy of the dynamical phase $<{\psi(t)}|\hat{H}(t)|\psi(t)>$, since it appears in the Hamiltonian along the diagonal.

Specifically the time evolution operator can be expressed,
\nonumber
\hat{U}(t)=e^{-\imath\xi(t)}
\begin{pmatrix}
a+ib & c+id \\
-c+id & a-ib
\end{pmatrix},

and the related Hamiltonian is given by,
\nonumber
\hat{H}(t)=i\dot{\hat{U}}(t)\hat{U}^\dagger(t)=\dot{\xi}(t)\hat{\sigma}_{(1)}+\frac{H^x(t)}{2}\hat{\sigma}_{(x)}+\frac{H^y(t)}{2}\hat{\sigma}_{(y)}+\frac{H^z(t)}{2}\hat{\sigma}_{(z)},

where, $\hat{\sigma}_{(1)}$ is the identity and $\hat{\sigma}_{(a)}$ are the Pauli matrices for $a=x,y,z$, and
\begin{align} \nonumber
H^x(t)&=2(\dot{a}d-a\dot{d}+\dot{b}c-b\dot{c}), \\
\nonumber
H^y(t)&=2(\dot{a}c-a\dot{c}-\dot{b}d+b\dot{d}),\\
\nonumber
H^z(t)&=2(\dot{a}b-a\dot{b}+\dot{c}d-c\dot{d}).
\end{align}
The phase of interest $\xi(t)$ does not affect the dynamics of the qubit, since it is absent in the density matrix. However, as I understand this phase contributes to the global phase but is itself a different entity than the global phase, since the global phase is present with or without this term. So my question is, what is the meaning and role of this phase and does it have a name ?

Thanks in advance for any help/insight you can offer on this.

2. Aug 31, 2015

### Strilanc

That phase factor matters if the operation is conditional.

For example, if you have a 2-qubit quantum circuit and you hit the first qubit with a $U$ controlled by the second qubit (so it only applies in the parts of the superposition where the second qubit is ON), then the $U$'s "global" phase factor is now a relative phase factor and you can detect its effects.