τλ

Abstract

This demo shows a qubit driven by a fixed-width Gaussian pulse. The left curve is excited-state probability, the dashed gray line is the normalized pulse envelope, and the right-axis curve is a dimensionless quantum-estimation diagnostic for dephasing. In that sense the example is a compact version of the standard quantum-Fisher-information story for a single qubit [1, 2]. The two sliders control pulse amplitude and dephasing strength in units set by the fixed pulse width.

Method

We evolve a Bloch vector with

\dot{\mathbf m}=\mathbf\Omega(t)\times\mathbf m,\quad \Omega_x(t)=A e^{-(t-t_0)^2/(2\sigma^2)},

plus a dephasing map at each timestep.

For

\rho_\theta=\frac{1}{2}\left(I+\mathbf r_\theta\cdot\boldsymbol\sigma\right),

the QFI for parameter $\theta$ is

F_Q(\theta)=\left|\partial_\theta\mathbf r\right|^2 +\frac{\left(\mathbf r\cdot\partial_\theta\mathbf r\right)^2}{1-\left|\mathbf r\right|^2},

and we use $\theta=\gamma_\phi$ with finite difference

\partial_{\gamma_\phi}\mathbf r(t)\approx \frac{\mathbf r(t;\gamma_\phi+\delta\gamma)-\mathbf r(t;\gamma_\phi)}{\delta\gamma}.

The qubit Bloch-vector form of the QFI and the general estimation viewpoint follow standard references such as [1, 2].

The right-axis curve is

\tilde F_Q(t)=Q_{\gamma_\phi}(t)=\gamma_\phi^2 F_Q(t),

with two trajectories at $\gamma_\phi$ and $\gamma_\phi+\delta\gamma$ .

References

[1] Samuel L. Braunstein and Carlton M. Caves. “Statistical distance and the geometry of quantum states,” Physical Review Letters 72 (22), 3439--3443 (1994). DOI: 10.1103/PhysRevLett.72.3439.
[2] Matteo G. A. Paris. “Quantum estimation for quantum technology,” International Journal of Quantum Information 7 (supp01), 125--137 (2009). DOI: 10.1142/S0219749909004839.