Geometry, robustness, and emerging unitarity in dissipation-projected dynamics

Quantum information can be encoded in the set of steady states (SSS) of a driven-dissipative system. Nonsteady states are separated by a large dissipative gap that adiabatically decouples them while the dynamics inside the SSS is governed by an effective, dissipation-projected, Hamiltonian. The latter results from the interplay between a weak driving and the fast relaxation process that continuously projects the system back to the SSS. This amounts to a different type of environment-induced quantum Zeno effect. We prove that the dissipation-projected dynamics is of geometric nature and that it is robust against different types of Hamiltonian and dissipative perturbations. Remarkably, in some cases an effective unitary dynamics can emerge out of purely dissipative interactions.

Figure 1

Robustness against dissipative errors. Distance from the exact evolution and the effective one as a function of $1/T.$ The unperturbed Lindbladian is of the form ${\mathcal{L}}_0={\sum }_{\alpha }{\gamma }_{\alpha }{\mathcal{L}}^{\alpha }$, where ${\mathcal{L}}^{\alpha }$ is generated by a collective Lindblad operator $S^{\alpha }={\sum }_{j=1}^N{\sigma }_j^{\alpha }$ with $N=4$. The control Hamiltonians are ${\mathcal{K}}^{\alpha }=-i[H^{\alpha },•]$ with $H^x=(3/2)({\sigma }_1^z{\sigma }_2^z+{\sigma }_2^z{\sigma }_3^z)+\mathbb{1}$ and $H^z=-(\sqrt{3}/2)({\sigma }_1^z{\sigma }_2^z-{\sigma }_2^z{\sigma }_3^z)+{\sigma }_1^z$. The effective dynamics generate the unitary gates $u^{\alpha }=exp(-i\theta {\sigma }^{\alpha })$ with an arbitrary angle $\vartheta$ up to an error $O\left(T^{-1}\right)$ (see also Ref. [20]). The same dynamics is obtained (up to an error $O\left(T^{-1}\right)$ with possibly a different prefactor) with an error on ${\mathcal{L}}_0$, which replaces $S^{\alpha }\to S^{\alpha }+T^{-1}{X}^{\alpha }$. For the numerical simulation we used $X^{\alpha }=g{\mathbf{\sigma }}_1·{\mathbf{\sigma }}_2{S}^z$. The plot is obtained by fixing ${\gamma }^{\alpha }=g=\vartheta =1$. The norm used is the maximum singular value of the maps realized as matrices over ${\mathcal{H}}^{\otimes 2}.$ The linear fit is obtained using the four most significant points.

Figure 2

Unitarity from dissipation: a unitary dynamics is generated out of a purely dissipative one. To highlight the unitary character we use the method described in Ref. [37]. For unitary dynamics one has $\rho \left(t\right)=e^{-it{H}_{\mathrm{eff}}}{\rho }_0{e}^{it{H}_{\mathrm{eff}}}={\sum }_{n,m}{e}^{-it(E_n-E_m)}{\left[{\rho }_0\right]}_{n,m}|n\rangle \langle m|$, with $H_{\mathrm{eff}}={\sum }_n{E}_n|n\rangle \langle n|$. In the figure we plot the real and imaginary part of $\langle 1\left|\rho \right(t\left)\right|2\rangle$. Left panel: two-qubit example generating the effective Hamiltonian $H_{\mathrm{eff}}={\sigma }^y\otimes \mathbb{1}$. The time evolution window is $t\in [T/100,10T]$ and $T=100$ in arbitrary units. Right panel: four-qubit example with perturbed collective noise. The effective Hamiltonian is given in Eq. (28). The time-evolution window is $[T/2,4T]$ with $T=100$.

Figure 3

Unitarity from dissipation: a unitary dynamics is generated out of a purely dissipative one. Here we plot real and imaginary part of the eigenvalues of the exact map ${\mathcal{E}}_T{\mathcal{P}}_0=exp\left[T({\mathcal{L}}_0+\frac{1}{T}{\mathcal{L}}_1+\frac{1}{T^2}{\mathcal{L}}_2)\right]{\mathcal{P}}_0$ for different $T$. The examples are the same as in Fig. 2, i.e., two- (four-) qubit example on the left (right) panel. By increasing $T$ the eigenvalues tend to go on the unit circle. Left panel: $T\in [1,1000]$, the eigenvalues approach $e^{\pm 2i},1$, consistent with Eq. (26). Right panel: $T\in [5,600]$. The eigenvalues of the effective Hamiltonians (28) are $\pm 8,0$. Consequently the eigenvalues of ${\mathcal{E}}_T{P}_0$ approach $e^{\pm 16i},e^{\pm 8i},1$.