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I see $T_1$ error mentioned in many experimental papers (for example, page $10$ of this paper mentions such a decay.)

How do I model $T_1$ decay theoretically? More concretely, here's my question. Say I have a single qubit density matrix $\rho$, and a noise channel $\mathcal{N}$ that models $T_1$ decay. How does $\mathcal{N}(\rho)$ look like explicitly?

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Relaxation time $T_1$ describes the strength of amplitude damping$^1$ by specifying the mean lifetime$^2$ of the $|1\rangle$ state. More precisely, $\mathcal{N}$ has a Kraus representation $$ E_0=\begin{bmatrix}1&0\\0&\sqrt{1-\gamma}\end{bmatrix}\quad E_1=\begin{bmatrix}0&\sqrt{\gamma}\\0&0\end{bmatrix} $$

where $\gamma=1-e^{-\frac{t}{T_1}}$. Explicitly$^3$, $$ \begin{align} \mathcal{N}\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)&=\begin{bmatrix}a+d\gamma&b\sqrt{1-\gamma}\\c\sqrt{1-\gamma}&d-d\gamma\end{bmatrix}\\ &=\begin{bmatrix}a+d(1-e^{-\frac{t}{T_1}})&be^{-\frac{t}{2T_1}}\\ce^{-\frac{t}{2T_1}}&de^{-\frac{t}{T_1}}\end{bmatrix}. \end{align} $$ In the geometric picture of the Bloch sphere, the channel appears as a contraction pulling the sphere towards $|0\rangle$. In fact $|0\rangle$ is the channel's sole fixed point.


$^1$ We assume here that the bath absorbing energy from the qubit is so cold that at thermal equilibrium the qubit is in the ground state $|0\rangle$. If bath's temperature is higher and hence the equilibrium state has non-zero population in the $|1\rangle$ state, then the appropriate model is the generalized amplitude damping channel. See section $8.3.5$ in Nielsen & Chuang for details.
$^2$ The inverse $\frac{1}{T_1}$ is the decay rate and $T_1\ln 2$ is the half-life, see exponential decay for more details.
$^3$ Note that amplitude damping does not only reduce the occupation of the $|1\rangle$ state, but also acts like a dephasing channel on the off-diagonal elements, albeit at half the rate. This is a mathematical necessity since otherwise the output would not necessarily be positive semi-definite. This is why in more general noise models with both amplitude and phase damping we have $T_2\leqslant 2T_1$ where $T_2$ describes the exponential decay of the off-diagonal elements.

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