Editing Quantum decoherence (section)

===Rotational decoherence===
Consider a system of ''N'' qubits that is coupled to a bath symmetrically. Suppose this system of ''N'' qubits undergoes a rotation around the <math>|{\uparrow}\rangle \langle{\uparrow}|, |{\downarrow}\rangle \langle{\downarrow}|</math> <math>\big(|0\rangle \langle0|, |1\rangle \langle1|\big)</math> eigenstates of <math>\hat{J_z}</math>. Then under such a rotation, a random [[Phase factor|phase]] <math>\phi</math> will be created between the eigenstates <math>|0\rangle</math>, <math>|1\rangle</math> of <math>\hat{J_z}</math>. Thus these basis qubits <math>|0\rangle</math> and <math>|1\rangle</math> will transform in the following way:

: <math>|0\rangle \to |0\rangle, \quad |1\rangle \to e^{i\phi} |1\rangle.</math>

This transformation is performed by the rotation operator

: <math>R_z(\phi) =
\begin{pmatrix} 
 1 & 0 \\ 
 0 & e^{i\phi} 
\end{pmatrix}
.</math>

Since any qubit in this space can be expressed in terms of the basis qubits, then all such qubits will be transformed under this rotation. Consider the <math>j</math>th qubit in a pure state <math>\vert\psi_{j}\rangle\langle\psi_{j}\vert</math> where <math>|\psi_{j}\rangle = a |0\rangle + b |1\rangle</math>. Before application of the rotation this state is:

:<math> \rho_{j,\text{init}} =
\begin{pmatrix}
 |a|^2 & ab^{\ast} \\
 a^{\ast}b & |b|^2 
\end{pmatrix}</math>.

This state will decohere, since it is not ‘encoded’ with (dependent upon) the dephasing factor <math>e^{i\phi}</math>. This can be seen by examining the [[density matrix]] averaged over the random phase <math>\phi</math>:

: <math>\rho_{j} = \mathbb{E}[
R_{z}(\phi)
\vert\psi_{j}\rangle
\langle\psi_{j}\vert
R_{z}^{\dagger}(\phi)
]
= \int\limits_{-\infty}^{\infty} R_z(\phi) |\psi_{j}\rangle \langle\psi_{j}|R_z^\dagger(\phi) \;P(\text{d}\phi)</math>,

where <math>P(\cdot)</math> is a [[probability measure]] of the random phase, <math>\phi</math>. Although not entirely necessary, let us assume for simplicity that this is given by the [[Gaussian distribution]], ''i.e.'' <math>P(\text{d}\phi) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}(\frac{\phi}{\sigma})^{2}}\,\text{d}\phi</math>, where <math>\sigma</math> represents the spread of the random phase. Then the density matrix computed as above is

:<math> \rho_{j} =
\begin{pmatrix}
 |a|^2 & ab^{\ast}\,e^{-\frac{1}{2}\sigma^{2}} \\
 a^{\ast}b\,e^{-\frac{1}{2}\sigma^{2}} & |b|^2 
\end{pmatrix}</math>.

Observe that the off-diagonal elements—the coherence terms—decay as the spread of the random phase, <math>\sigma</math>, increases over time (which is a realistic expectation). Thus the density matrices for each qubit of the system become indistinguishable over time. This means that no measurement can distinguish between the qubits, thus creating decoherence between the various qubit states. In particular, this dephasing process causes the qubits to collapse to one of the pure states in <math>\{
\vert 0\rangle \langle 0\vert,
\vert 1\rangle \langle 1\vert
\}</math>. This is why this type of decoherence process is called '''collective dephasing''', because the ''mutual'' phases between ''all'' qubits of the ''N''-qubit system are destroyed.