Symmetry Breaking and Error Correction in Open Quantum Systems

Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system—a Lindbladian—a unitary symmetry can be imposed in a “weak” or a “strong” way. We characterize the possible ${\mathbb{Z}}_n$ symmetry-breaking transitions for both cases. In the case of ${\mathbb{Z}}_2$, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correction.

Figure 1

(a) Phase diagram for the model in Eq. (3) with one-photon loss ${\kappa }_1$, ${\kappa }_d=0$, in the thermodynamic limit $\lambda /{\kappa }_2\to \infty$. Integers indicate the dimension of the steady-state manifold. (b) Analytical expression for the dissipative gap (red line) and numerical spectrum (black dots) in the unbroken phase for ${\kappa }_2={\kappa }_d=0$, ${\kappa }_1/\omega =2$. The dissipative gap closes as the phase boundary at $\lambda /\omega =\sqrt{5}/2\approx 1.1$ is approached. (c) Strong transition: decay rate of the four modes with the longest lifetime; two modes are always pinned to zero and the other two are degenerate (textured colors indicate twofold degeneracy of all modes in this subfigure). A 2-to-4 transition occurs near $\lambda /\omega =0.5$ (red dashed line) in the limit $N\to \infty$, in agreement with the phase diagram. ${\kappa }_1=0$, $\lambda /{\kappa }_2=N$, ${\kappa }_d/\omega =0.01$. (d) Weak transition: decay rate of the two modes with the longest lifetime. Full lines emphasize a lack of exact twofold degeneracy present in (c). A 1-to-2 transition is observed. $\lambda /{\kappa }_2=N$, ${\kappa }_1/\omega =0.02$, ${\kappa }_d/\omega =0.01$.

Figure 2

(a) Fidelity of the initial and final states for the quench protocol given in the main text with $\lambda /{\kappa }_2=N$, ${\kappa }_d=0$, ${\tau }_q\lambda =10$, $F({\rho }_i,{\rho }_f)=\mathrm{Tr}[\sqrt{\sqrt{{\rho }_i}{\rho }_f\sqrt{{\rho }_i}}{]}^2$. Quenches to the strong-broken phase (black dots) have a fidelity that tends to one in the thermodynamic limit, while quenches to the strong-unbroken phase (red dots) do not. (b) Same parameters as in (a) with $\lambda /\omega =2$; the fidelity tends to one exponentially fast in $N$. (c) A dephasing error ${\kappa }_d/\lambda =0.03$, $\omega =0$, $\lambda /{\kappa }_2=N$, ${\tau }_q\lambda =10$; again the fidelity is exponentially close to one. (d) Purity of $M$ [see Eq. (5)] for different quench times with the same parameters as in (b) and $N=15$. The dashed line is the timescale set by the dissipative gap ${\tau }_g={\mathrm{\Delta }}_g^{-1}$ of ${\mathcal{L}}_0+{\mathcal{L}}^{\prime }$. (${\mathrm{\Delta }}_g$ is the decay rate of the longest-lived excitation above the four steady-state solutions.) Short quenches keep the system approximately pure, while long quenches evolve the system to a mixed NS steady state. Errors are correctable in both cases. For all figures, $c_e=1/\sqrt{2}$, $c_o=i/\sqrt{2}$.