Fig. 4: Breakdown of monotonic relationship between quantum entropy production and fluctuations in quantum heat for dimensions greater than two.

Here, we choose Hamiltonians with uniformly gapped spectra, i.e., \({E}_{k+1}^{(1)}-{E}_{k}^{(1)}=\hslash {\omega }_{1}\). The probability spectra for the states \(\tilde{\rho }(\Theta )\) are chosen to be nondegenerate, but concentrated around \(\left|{\psi }_{1}^{\Theta }\right\rangle\) and \(\left|{\psi }_{d}^{\Theta }\right\rangle\). For d = 2, p = (0.9, 0.1), while for d = 3, p = (0.49, 0.04, 0.47). a Variance in quantum heat and average quantum entropy production as a function of Θ defined in Eq. (29). For d = 2, both \({\rm{Var}}({Q}_{{\rm{qu}}}(\Theta ))\) and \(\langle {s}_{{\rm{irr}}}^{{\rm{qu}}}(\Theta )\rangle\) monotonically increase as Θ → 1. For d = 3, however, while \(\langle {s}_{{\rm{irr}}}^{{\rm{qu}}}(\Theta )\rangle\) monotonically increases with Θ, \({\rm{Var}}({Q}_{{\rm{qu}}}(\Theta ))\) takes a maximum value at Θ ≈ 0.8, after which it decreases. b Here we choose the initial states \(\tilde{\rho }(\Theta )\) with Θ = 0.3, and evaluate \({\rm{Var}}({Q}_{{\rm{qu}}}(\Theta ,t))\) and \(\langle {s}_{{\rm{irr}}}^{{\rm{qu}}}(\Theta ,t)\rangle\) for the states \({e}^{t{\mathcal{L}}}(\tilde{\rho }(\Theta ))\) with \({\mathcal{L}}\) defined in Eq. (33). For d = 2, both \({\rm{Var}}({Q}_{{\rm{qu}}}(\Theta ,t))\) and \(\langle {s}_{{\rm{irr}}}^{{\rm{qu}}}(\Theta ,t)\rangle\) monotonically decrease with t, while for d = 3, \({\rm{Var}}({Q}_{{\rm{qu}}}(\Theta ,t))\) takes its maximum value at t ≈ 1.