Effect of Wigner rotation on estimating the unitary-shift parameter of a relativistic spin-1/2 particle

We obtain the accuracy limit for estimating the expectation value of the position of a relativistic particle for an observer moving in one direction at a constant velocity. We use a specific model of a relativistic spin-1/2 particle described by a Gaussian wave function with a spin down in the rest frame. To derive the state vector of the particle for the moving observer, we use the Wigner rotation that entangles the spin and the momentum of the particle. Based on this wave function for the moving frame, we obtain the symmetric logarithmic derivative (SLD) Cramér-Rao bound that sets the estimation accuracy limit for an arbitrary moving observer. It is shown that estimation accuracy decreases monotonically in the velocity of the observer when the moving observer does not measure the spin degree of freedom. This implies that the estimation accuracy limit worsens with the observer's increasing velocity, but it is finite even in the relativistic limit. We derive the amount of this information loss by the exact calculation of the SLD Fisher information matrix in an arbitrary moving frame.

Figure 1

Numerically calculated $(1-\xi )/2$ as a function of $m\kappa$ at $V=1$, 0.95, 0.5, and 0.1. The set of velocities $V$ is chosen differently to make the distance between the plots more even. The $m\kappa$, $\xi$, and $V$ are dimensionless in natural units.

Figure 2

Numerically calculated probability density $|{\psi }_{\uparrow }^{\mathrm{\Lambda }}{\left(x\right)|}^2$ for $\kappa =0.1{\text{eV}}^{-1}$ as a function of $x^1$ at $V=0.99$, 0.98, 0.9, 0.7, and 0.1. The distance $x^1$ is converted to SI units (nm).

Figure 3

Numerically calculated derivative of the probability density $|{\psi }_{\uparrow }^{\mathrm{\Lambda }}{\left(x\right)|}^2$ as a function of $x^1$ at $V=0.98$, 0.9, 0.7, and 0.1. The distance $x^1$ is converted to SI units (nm).

Figure 4

Numerically calculated ratio $\mathrm{\Delta }\left(V\right)$ as a function of $V$, the velocity of the moving observer at $m\kappa =0.1$, 0.5, 1.0, and 3.0.

Figure 5

Numerically calculated ratio ${\mathrm{\Delta }}_{\downarrow }\left(V\right)$ as a function of $V$, the velocity of the moving observer at $m\kappa =0.1$, 0.5, 1.0, and 3.0. (The scale of the vertical axis is different from that in Fig. 4.) The $m\kappa$ and $V$ are dimensionless in natural units.

Figure 6

Numerically calculated ratio ${\mathrm{\Delta }}_{\downarrow }\left(V\right)$ as a function of $m\kappa$ for the velocity of the moving observer at $V=1.0$, 0.95, 0.90, and 0.85.

Figure 7

Numerically calculated the sum of ${\left[{\left(J_{}^{\mathrm{\Lambda }}\right)}^{-1}\right]}_{11}+\mathrm{\Delta }\left(V\right)$ to illustrate this tradeoff relation for $V=0.1$, 0.3, 0.6, 0.8, 0.95, and 1.0.

Figure 8

Numerically calculated ratio ${\mathrm{\Delta }}_x\left(V\right)$ as a function of $V$, the velocity of the moving observer at $\kappa =0.1$, 0.5, 1.0, and 3.0. (The scale of the vertical axis is different from that in Fig. 4.)

Figure 9

Plot of $\kappa \eta /V$ as a function of $m\kappa$ at $V=0.95$, 0.7, and 0.1.