Figure 2

Open quantum systems. (a) Quantum noise on a state-preserving quantum process. For zero noise intensity p _noise = 0, χ _D (depolarization), χ _AD (amplitude damping), and χ _PD (phase damping) are identified as genuinely quantum, as an identity unitary transformation. α and β for all the noise processes monotonically decrease with an increase in the noise intensity p _noise. These noise processes are identified as reliably close to the target state-preserving process if their α and β are greater than certain thresholds as marked with \(\bullet \) and \(\blacktriangle \), respectively. See the property (P2). (b) Non-Markovian dynamics. Since α and β monotonically decrease with time for Markovian dynamics, the non-Markovianity of χ _expt can be measured by integrating the positive derivative of α or β with respect to time: \({h}_{q}({\rm{\Delta }}t)\equiv {\int }_{\mathrm{0;}\dot{q} > 0}^{{\rm{\Delta }}t}\dot{q}dt\), for q = α, β. As shown in Fig. 1b, we consider a system that is coupled to an environment with a state \(p|0\rangle \langle 0|+\mathrm{(1}-p)|1\rangle \langle 1|\) via a controlled-Z-like interaction \(H=\mathrm{1/2}{\sum }_{i,j=0}^{1}{(-\mathrm{1)}}^{i\cdot j}|ij\rangle \langle ij|\) and depolarized with a rate γ. For example, we have \({h}_{\alpha }\mathrm{(15)}\sim 0.86\) for p = 0.5 and γ = 0.015. (i)-(iii) illustrate the invalidation of \({\alpha }_{{\chi }_{{\rm{expt}}}}={\alpha }_{{\chi }_{2}{\chi }_{1}}\). Such detection is more sensitive than the existing non-Markovianity quantifiers, such as the Breuer-Laine-Piilo (BLP)⁷⁵ and Rivas-Huelga-Plenio (RHP)⁷⁶ measures. For example, for γ = 0.25 and p = 0.1, we find that \({\alpha }_{{\chi }_{{\rm{expt}}}}\ne {\alpha }_{{\chi }_{2}{\chi }_{1}}\) when t < 1.1, whereas they certify the dynamics as Markovian. The certifications by the BLP and RHP measures are detailed in ref.⁷⁷ Indeed, our method is finer than the BLP and RHP measures for all the settings of γ and p considered therein.