State tomography of a qubit through scattering of a probe qubit

We discuss the state tomography of a fixed qubit (a spin-1/2 target particle), which is in general in a mixed state, through one-dimensional scattering of a probe qubit off the target. Two strategies are presented by making use of different degrees of freedom of the probe, i.e. spin and momentum. Remarkably, the spatial degree of freedom of the probe can be utilized for optimizing the tomographic schemes.

Figure 1

A probe qubit $X$ is sent with a fixed wave number $k$ from the left to a target qubit $A$ fixed at $x=0$, scattered by the delta-shaped potential produced by the target qubit $A$ with a spin-spin interaction of the Heisenberg type, and detected on the right or on the left with spin-sensitive detectors.

Figure 2

The coefficients in the transmission probability $P^t$ in Eqs. (14, 15, 16) as functions of the incident wave number $k$.

Figure 3

The coefficients in the reflection probability $P^r$ in Eqs. (17, 18, 19) as functions of the incident wave number $k$.

Figure 4

$J\left(\Omega \right)$ in Eq. (29) (solid), $\lambda \left(\Omega \right)$ in Eq. (30) (dashed), and $\mu \left(\Omega \right)$ in Eq. (35) (dotted), as functions of the incident wave number $k$ of the probe $X$, which are to be minimized by tuning $k$ for the optimizations of the two approaches of strategy I. $J^{1/3}\left(\Omega \right)$ is actually plotted instead of $J\left(\Omega \right)$ and ${\mu }^{1/2}\left(\Omega \right)$ instead of $\mu \left(\Omega \right)$ to better compare them with $\lambda \left(\Omega \right)$. $J\left(\Omega \right)$ is minimum at $1/\Omega ={\hslash }^{2}k/mg=\sqrt{1+\sqrt{127}}/2\simeq 1.98$, $\lambda \left(\Omega \right)$ at $\sqrt{3}\simeq 1.73$, and $\mu \left(\Omega \right)$ at 2.01 (indicated by dots).

Figure 5

(Color online) Contour plot of $I^{1/3}\left({\Omega }_1,{\Omega }_2\right)$ computed with Eq. (40) as a function of $k_1$ and $k_2$. Only the contours in the range $I^{1/3}\left({\Omega }_1,{\Omega }_2\right)\le 20$ are shown. $I^{1/3}\left({\Omega }_1,{\Omega }_2\right)$ takes its minimum value $\simeq 8.74$ at $\left(k_1,k_2\right)\simeq \left(1.51,5.13\right)$ in the unit $mg/{\hslash }^2$ (indicated by a dot).