Monitoring of the Process of System Information Broadcasting in Time

One of the problems of quantum physics is how a measurement turns quantum, noncopyable data, towards copyable classical knowledge. We use the quantum state discrimination in a central system model to show how its evolution leads to the broadcasting of the information, and how orthogonalization and decoherence factors allow us to monitor the distance of the state in question to the one perfectly broadcasting information, in any moment of time not just asymptotically. We illustrate this in the spin-spin model where the distance is shown to be typically small and provide the related time scales.

Figure 1

Plot of $\overline{B^2}$ (colored surface), and $\overline{{|\gamma |}^2}$ (green surface) as functions of the initial environmental parameters ${\lambda }_{+}$ and $\beta$, ${\forall }_j{\lambda }_j={\lambda }_{+}\wedge {\beta }_j=\beta$; the observed and unobserved macrofractions have both 100 spins.

Figure 2

The upper bound on $\frac{1}{2}{\parallel {\varrho }^{(fM)}(t)-{\varrho }_{SBS}^{(fM)}(t)\parallel }_1$ obtained from the Corollary 1 using the optimal measurements and averaged over the Haar distribution of Euler angles, the uniform for the couplings, and the Hilbert-Schmidt for the eigenvalues. We take equal size $N/2$ of the observed and unobserved macrofractions. In the pessimistic case the initial central state has ${\sigma }_{\pm }=|{\sigma }_{+-}|=1/2$ yielding the bound $\gamma (t)+B(t)$ shown on the plot. For small macrofractions ($N/2=30$, 50) the bound is substantially larger than 0 and does not imply an SBS formation.