Quantum Zeno effect: a qutrit controlled by a qubit

TL;DR

This work addresses controlling quantum jumps in a three-level system (qutrit) by coupling to a two-level ancilla and applying the Quantum Zeno Effect to enable error-control-like shelving and deterministic dynamics. It develops a density-matrix formalism with Kraus operators and a least-action–based dynamical treatment to show how monitoring frequency drives transitions from coherent evolution to Zeno freezing, and demonstrates practical gate and communication protocols. The main contributions include a cNOT gate with the qutrit as control, dense coding and teleportation adapted to a qutrit-ancilla platform, and a Toffoli-like gate facilitated by the Kraus structure, together with explicit operator constructions for qutrit Pauli-like gates and Hadamard. The results highlight a path toward universal control of qudits via Zeno monitoring, with potential impact on quantum error correction, entanglement-based communication, and scalable higher-dimensional quantum computation.

Abstract

For a three-level system monitored by an ancilla, we show that quantum Zeno effect can be employed to control quantum jump for error correction. Further, we show that we can realize cNOT gate, and effect dense coding and teleportation. We believe that this work paves the way to generalize the control of a qudit.

Figures (8)

• Figure 1: A qutrit is interacting with an ancilla or detector. Ancilla$^{(3)}$ is a two-level system which monitors the state $\ket{3}$ of the qutrit with a coupling strength $J_3$. The transition frequencies of the three-levels are $\omega_{12}$ between states $\ket{1}$ and $\ket{2}$, $\omega_{23}$ between states $\ket{2}$ and $\ket{3}$ and $\omega_{13}$ between states $\ket{1}$ and $\ket{3}$.
• Figure 2: Figure shows the dynamics of the three-level system, the variation of its 8 variables plotted with time when the third level is being monitored. The initial conditions have been chosen as $x_1=x_3=x_5=x_7=0.3$, $x_2=x_4=x_6=0.5$ and $x_8=\sqrt{(4/3)^2-(\sum_i x_i)^2}$, where $i=1,2,\dots,7$. The Rabi frequencies of the three levels is chosen to be $\omega_{12}=0.6$, $\omega_{23}=1$ and $\omega_{13}=1.6$. The system is being monitored in three frequency ranges, (a) $\alpha_3=0.2$, (b) $\alpha_3=0.7$ and (c) $\alpha_3=1.7$. In fig. (a), there are usual coherent oscillation. In (b), the system begins to freeze fairly early at a particular state. In fig. (c) the Zeno regime has set in.
• Figure 3: For $\alpha_3=0.1$, the phase space dynamics of variable (a) $x_1$, (b) $x_2$, (c) $x_3$, (d) $x_4$, (e) $x_5$, (f) $x_6$, (g) $x_7$ and (h) $x_8$ is shown for chosen initial conditions $x_1=x_3=x_5=x_7=0.3$, $x_2=x_4=x_6=0.4$, $p_1=p_3=p_5=p_7=1$ and $p_2=p_4=p_6=p_8=0.5$ for a total time of $t=20$. Figure shows evolution of the system in phase space in the non-Zeno regime.
• Figure 4: For $\alpha_3=1.7$, the phase space dynamics of variable (a) $x_1$, (b) $x_2$, (c) $x_3$, (d) $x_4$, (e) $x_5$, (f) $x_6$, (g) $x_7$ and (h) $x_8$ is shown for the same initial conditions as in Fig. \ref{['fig:x_p-single_al_0.1']} for a total time of $t=20$. The figure shows that when the Zeno regime has completely set in, i.e., the detector frequency $\alpha_3>\omega_{12},\omega_{23}, \omega_{13}$, the dynamics is completely arrested at a particular point. Since the system is delocalised in the momentum coordinates and localised in the position coordinates, there is a saddle point. The qutrit, in the Zeno regime gets shelved at the critical point.
• Figure 5: A qutrit is interacting with two ancillae or detectors. Ancilla$^{(2)}$ (Ancilla$^{(3)}$) is a two-level system which monitors the state $\ket{2}$ ($\ket{3}$) of the qutrit with a coupling strength $J_2$ ($J_3$). The transition frequencies of the three-levels are $\omega_{12}$ between states $\ket{1}$ and $\ket{2}$, $\omega_{23}$ between states $\ket{2}$ and $\ket{3}$ and $\omega_{13}$ between states $\ket{1}$ and $\ket{3}$.