Universal dynamical decoupling: Two-qubit states and beyond

Uhrig’s dynamical decoupling pulse sequence has emerged as a universal and highly promising approach to decoherence suppression. So far, both the theoretical and experimental studies have examined single-qubit decoherence only. This work extends Uhrig’s universal dynamical decoupling from one-qubit to two-qubit systems and even to general multilevel quantum systems. In particular, we show that by designing appropriate control Hamiltonians for a two-qubit or a multilevel system, Uhrig’s pulse sequence can also preserve a generalized quantum coherence measure to the order of $1+O(T^{N+1})$ with only $N$ pulses. Our results lead to a very useful scheme for efficiently locking two-qubit entangled states. Future important applications of Uhrig’s pulse sequence in preserving the quantum coherence of multilevel quantum systems can also be anticipated.

Figure 1

Expectation value of the coherence measure ${\mathcal{P}}_{|\Psi \rangle }$, denoted $F(t)$, as a function of time, with $|\Psi \rangle$ being the nonentangled state $|\uparrow \uparrow \rangle$ of a two-qubit system. The bath responsible for the decoherence is modeled by the three-spin system detailed in the text. The bottom curve is without any control and the decoherence is significant. The middle curve is calculated from a control Hamiltonian intuitively based on two independent qubits. The top solid curve represents significant decoherence suppression due to our two-qubit UDD control Hamiltonian described by Eq. (19). All variables are in dimensionless units.

Figure 2

Same as in Fig. 1, but for ${\mathcal{P}}_{|\Psi \rangle }$ associated with the Bell state defined in Eq. (26). The smooth dashed curve represents significant decoherence without control. The drastically oscillating dashed curve is calculated from an intuitive single-qubit-based control Hamiltonian, showing strong population transfer from the initial state to other two-qubit states. The top, solid curve represents significant decoherence suppression due to our two-qubit UDD control sequence in Eq. (32). All variables are in dimensionless units.

Figure 3

The time-averaged distance $D$ between the actual density matrix from that of a completely locked Bell state, for $c=T/100$ and $c=T/1000$, versus the number of UDD pulses. The initial state is the same as in Fig. 2. All variables are in dimensionless units.

Figure 4

Expectation value of the coherence measure ${\mathcal{P}}_{|\Psi \rangle }$, denoted $F(t)$, as a function of time, with $|\Psi \rangle$ being one basis state of a three-level system. The central system is coupled with a bath modeled by four other three-level subsystems. The bottom curve represents significant decoherence without decoherence control. The top two curves represent decoherence suppression based on the control operator constructed in Eq. (42), for $N=2$ and $N=10$. All variables are in dimensionless units.