Bath Dynamical Decoupling with a Quantum Channel

Abstract

Bang-bang dynamical decoupling protects an open quantum system from decoherence due to its interaction with the surrounding bath/environment. In its standard form, this is achieved by strongly kicking the system with cycles of unitary operations, which average out the interaction Hamiltonian. In this paper, we generalize the notion of dynamical decoupling to repeated kicks with a quantum channel, which is applied to the bath. We derive necessary and sufficient conditions on the employed quantum channel and find that bath dynamical decoupling works if and only if the kick is ergodic. Furthermore, we study in which circumstances CPTP kicks on a mono-partite quantum system induce quantum Zeno dynamics with its Hamiltonian cancelled out. This does not require the ergodicity of the kicks, and the absence of decoherence-free subsystems is both necessary and sufficient. While the standard unitary dynamical decoupling is essentially the same as the quantum Zeno dynamics, our investigation implies that this is not true any more in the case of CPTP kicks. To derive our results, we prove some spectral properties of ergodic quantum channels, that might be of independent interest. Our approach establishes an enhanced and unified mathematical understanding of several recent experimental demonstrations and might form the basis of new dynamical decoupling schemes that harness environmental noise degrees of freedom.

Figures (3)

• Figure 1: Bath dynamical decoupling and Zeno Hamiltonian suppression with the ergodic qubit channel $\mathcal{E}^\updownarrow$ introduced in Example \ref{['ex:ergodic_mixing']}(ii). (a) The worst-case purity of the reduced Choi-Jamiołkowski state, $\min\mathtt{P}(\Lambda_1^\mathrm{DD})$, for 100 generators $\mathcal{H}=[H,{}\bullet{}]$ constructed from randomly sampled Hamiltonians $H\in\mathcal{B}(\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ with $\|H\|_\infty=1$ for the bath dynamical decoupling. (b) The worst-case distance for the Zeno Hamiltonian suppression, $\max\Vert \Lambda(\mathcal{E}_\mathrm{Z}^\updownarrow)-\Lambda((\mathcal{E}^\updownarrow)^n)\Vert_1$, for 100 generators $\mathcal{H}=[H,{}\bullet{}]$ constructed from randomly sampled Hamiltonians $H\in\mathcal{B}(\mathbb{C}^2)$ with $\|H\|_\infty=1$. In both panels, $t=1$.
• Figure 2: Bath dynamical decoupling and Zeno Hamiltonian suppression with the qubit dephasing channel $\mathcal{E}^\mathrm{d}$ introduced in Example \ref{['ex:ergodic_mixing']}(vi). (a) The purity $\mathtt{P}(\Lambda_1^\mathrm{DD})$ of the reduced Choi-Jamiołkowski state for the Hamiltonian $H=Z\otimes Z\in\mathcal{B}(\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ (blue) and the average purity $\mathbb{E}[\mathtt{P}(\Lambda_1^\mathrm{DD})]$ over 100 randomly sampled Hamiltonians $H\in\mathcal{B}(\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ normalized as $\|H\|_\infty=1$ (red), for the bath dynamical decoupling. The total evolution time is fixed at $t=1$. The former (blue) is constant at $\mathtt{P}(\Lambda_1^\mathrm{DD})\approx0.59$, while the latter (red) saturates to $\mathbb{E}[\mathtt{P}(\Lambda_1^\mathrm{DD})]\approx0.85$. The bath dynamical decoupling does not work. (b) The worst-case distance for the Zeno Hamiltonian suppression, $\max\Vert \Lambda(\mathcal{E}_\mathrm{Z}^\mathrm{d})-\Lambda(\mathcal{E}^\mathrm{d})\Vert_1$, for 100 randomly sampled Hamiltonians $H\in\mathcal{B}(\mathbb{C}^2)$ normalized as $\|H\|_\infty=1$. The total evolution time is fixed at $t=1$.
• Figure 3: Bath dynamical decoupling and Zeno Hamiltonian suppression with the two-qubit channel $\mathcal{E}^{\mathbb{1}/2}$ introduced in Example \ref{['example:DFSS']}(i), which admits a decoherence-free subsystem on the first qubit. (a) The purity $\mathtt{P}(\Lambda_0^\mathrm{DD})$ of the reduced Choi-Jamiołkowski state for the Hamiltonian $H=Z\otimes Z\otimes\mathbb{1}\in\mathcal{B}(\mathbb{C}_0^2\otimes\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ (blue) and the average purity $\mathbb{E}[\mathtt{P}(\Lambda_0^\mathrm{DD})]$ over 100 randomly sampled Hamiltonians $H\in\mathcal{B}(\mathbb{C}_0^2\otimes\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ normalized as $\|H\|_\infty=1$ (red), for the bath dynamical decoupling. The total evolution time is fixed at $t=1$. The former (blue) is constant at $\mathtt{P}(\Lambda_0^\mathrm{DD})\approx0.59$, while the latter (red) saturates to $\mathbb{E}[\mathtt{P}(\Lambda_0^\mathrm{DD})]\approx0.91$. The bath dynamical decoupling does not work. (b) The distance $\Vert \Lambda(\mathcal{E}_\mathrm{Z}^{\mathbb{1}/2})-\Lambda(\mathcal{E}^{\mathbb{1}/2})\Vert_1$ for the Hamiltonian $H=Z\otimes\mathbb{1}\in\mathcal{B}(\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ (blue) and the average distance $\mathbb{E}[\Vert \Lambda(\mathcal{E}_\mathrm{Z}^{\mathbb{1}/2})-\Lambda(\mathcal{E}^{\mathbb{1}/2})\Vert_1]$ over 100 randomly sampled Hamiltoanians $H\in\mathcal{B}(\mathbb{C}_1^2\otimes\mathbb{C}_2^2)$ normalized as $\|H\|_\infty=1$ (red), for the Zeno Hamiltonian suppression. The total evolution time is fixed at $t=1$. The former (blue) is constant at $\Vert \Lambda(\mathcal{E}_\mathrm{Z}^{\mathbb{1}/2})-\Lambda(\mathcal{E}^{\mathbb{1}/2})\Vert_1\approx1.68$, while the latter (red) approaches $\mathbb{E}[\Vert \Lambda(\mathcal{E}_\mathrm{Z}^{\mathbb{1}/2})-\Lambda(\mathcal{E}^{\mathbb{1}/2})\Vert_1]\approx0.55$. The Zeno Hamiltonian suppression does not work.

Theorems & Definitions (23)

• Definition 1: Quantum channel
• proof
• Definition 3: Recurrences and fixed points
• proof
• proof
• Definition 6: Decoherence-free subsystem
• Example 7: Decoherence-free subsystem
• Definition 8: Ergodic and mixing quantum channels
• Example 9: Ergodic and mixing quantum channels
• Proposition 10: Spectral structure of ergodic quantum channel