Quantum Decoherence Calculator

Because 0.5 is the purity of the maximally mixed state for a two-level system: Tr(rho^2) is bounded below by 1/d, and d = 2 for a qubit. A fully decohered qubit is an even mixture of the two basis states, and no amount of additional dephasing can take it past that. Any model that reports a qubit purity below 0.5 -- or a purity of zero -- is not describing a physical density matrix.

Because T2 combines two independent decay channels -- energy relaxation (T1) and pure dephasing (T_phi) -- through 1/T2 = 1/(2*T1) + 1/T_phi. Even with zero pure dephasing (T_phi at infinity), the 1/(2*T1) term alone sets T2 = 2*T1; any additional dephasing can only shorten T2 further, never lengthen it past that ceiling.

Fidelity to the initial state decays as 1/2*(1 + e^(-t/T2)) -- one power of the coherence-decay factor. Purity decays as 1/2*(1 + e^(-2t/T2)) -- the SQUARE of that same factor, because purity is a trace of the density matrix squared. Purity therefore always falls at or below Fidelity for the same elapsed time, and both converge to 0.5 as Evaluation Time grows.

Temperature does not change T2 or the main Purity/Fidelity figures directly -- those are set purely by T1 and T_phi. Instead, Temperature drives a thermal occupation of the qubit's energy levels, n_bar, which stimulates additional relaxation. Thermal T1 is what T1 becomes once that stimulated relaxation is included: it is always shorter than or equal to the T1 you entered, and the gap only becomes significant once k_B*T approaches the qubit's transition energy h_bar*omega.

It's T2 divided by the Gate Time you enter -- roughly how many gate operations fit inside the qubit's coherence window. Use a realistic gate time: a superconducting single-qubit gate is typically 20-50 ns and a two-qubit gate 40-300 ns, while trapped-ion gates run microseconds to tens of microseconds.