Spin-Polarized Initialization and Readout for Single-Qubit State Tomography

Abstract

We propose a theoretical protocol for reconstructing the density matrix of a single-electron spin qubit using spin-polarized transport. The system consists of a quantum dot coupled to ferromagnetic reservoirs and subject to a magnetic field lying in the $xy$ plane of the Bloch sphere. Spin-dependent tunneling events measured along the $x\pm$, $y\pm$, and $z\pm$ quantization axes give rise to probability distributions that encode the quantum state of the qubit. The open-system dynamics are described using a Lindblad master equation, which captures the time evolution of the spin under continuous coupling to the reservoirs. By counting tunneling events for four different magnetic alignments, we formulate a scheme for reconstructing the full density matrix of the qubit. The resulting simulation data are analyzed using machine-learning techniques to process the measured probability distributions and infer the corresponding density matrix elements. The proposed model enables complete access to the open-system density matrix, including both population probabilities and relative phase information. Successful state reconstruction demonstrates the validity and robustness of the approach, highlighting its applicability to experimentally accessible spin-transport platforms.

Figures (4)

• Figure 1: Schematic of the open quantum system considered in this work. A single-level quantum dot is tunnel coupled, with rate $\Gamma_0$, to two ferromagnetic drain electrodes labeled left (L) and right (R). These drain leads perform spin-selective detection, allowing electrons polarized along the $\delta\pm$ directions, with $\delta = x, y, z$, to tunnel out of the quantum dot. The upper lead is included only for illustration purposes and indicates a possible source electrode used to initialize the quantum dot with a spin-polarized electron, prepared in the spin $+$ (i.e., spin $\uparrow$ along the $z$ axis) state.
• Figure 2: Tunneling events counting as a function of time with spin alignment along (a) $+$, (b) $-$, (c) $x+$, and (d) $y+$. Parameters: $\Omega=1 \mathrm{\mu eV}$ (242MHz), $\Gamma_0=0.05 \Omega$$=$$0.05 \mathrm{\mu eV}$ (12.1MHz).
• Figure 3: Estimated density matrix elements $\rho^{\mathrm{o}}_{++}$, $\rho^{\mathrm{o}}_{--}$, $\rho^{\mathrm{o}}_{x+ x+}$, and $\rho^{\mathrm{o}}_{y+ y+}$ obtained via approximations in Eqs. (\ref{['pxp_approx']})–(\ref{['pzm_approx']}). These four elements allow determination of the off-diagonal components shown in Fig. \ref{['fig4']}. Parameters: $\Omega=1 \mathrm{\mu eV}$ (242MHz), $\Gamma_0=0.05 \Omega$$=$$0.05 \mathrm{\mu eV}$ (12.1MHz).
• Figure 4: Reconstructed density matrix elements of Eq. (\ref{['rho_reduced']}). Dashed lines represent the corresponding evolution for a closed system, while solid lines show the density matrix obtained from the machine-learning model trained with data from Fig. \ref{['fig3']}. Panels (a) and (b) show the diagonal elements, whereas panels (c) and (d) display the real and imaginary parts of the off-diagonal coherence. Parameters: $\Omega = 1~\mu\mathrm{eV}$ ($242~\mathrm{MHz}$) and $\Gamma_0 = 0.05\,\Omega = 0.05~\mu\mathrm{eV}$ ($12.1~\mathrm{MHz}$).