Notice the difference in normalization with [34]. In [34], a density matrix is mapped to a vector by the Choi-Jamiołkowski isomorphism, so it needs additional normalization by $\mathrm{Tr}{\rho }^2$. In our definition of CoPs, we do not need any such normalization once the initial state is normalized as $\mathrm{Tr}\rho =1$.

For pure states, it is manifest from our definition of CoPs that they reduce to original notions of complexities.

The equality ${\mathcal{C}}_K[\rho (t)]=\mathcal{C}K\left(\right|\mathrm{\Psi }{\rho }^{\mathbb{I}}(t)⟩)$ for pure states follows from matching autocorrelation functions.

Note that even when the initial state is maximally entangled ($p\to 1$), it is only within the two-dimensional subspace. Thus, this does not mean a contradiction to the observation in [23].

Interestingly, the scaling of the autocorrelation function in our example ${\mathrm{sech}}^2(\alpha t)$ matches with that of the low-$T$ Sachdev-Ye-Kitaev model in the decoupling limit ($q\to 1$). Here, $\alpha$ corresponds to $\pi T$.

This is expected because the two-mode squeezing operation is equivalent to the Bogoliubov transformation, which relates the Rindler vacuum to the Minkowski vacuum.

We emphasize here again that the “energy” appearing in the Lloyd bound is not the expectation value of the actual Hamiltonian generating the time evolution. The modular energy we call here is the expectation value of a time-independent effective Hamiltonian, modeling the mixed state at each instance of time by a thermal state with a time-dependent temperature.

Note that the instantaneous purification of the TFD state equals itself.

Note that the sign of the evolution arises from the state evolution, contrasting with the Heisenberg operator evolution commonly considered in the Krylov operator complexity.

Note that the mapping between time and the temperature ${\tanh }^2(\alpha t)=\exp (-\beta (t)\mathrm{\Delta }E)$ agrees with that of the perturbation in the late time [22, 67].