Susumu Osawa | Searcharxiv

arXiv2017

Considerations in the Time-Energy Uncertainty Relation from the Viewpointt of Hypothesis Testing

It is difficult to discriminate between the state of initial time $t_0$ and that of time $t_0+Δt$. The task of discriminating between the two states can be interpreted as the hypothesis testing problem as to a state $ρ$ deciding whether $ρ$ is $ρ_{t_0}$ or$ρ_{t_0+Δt}$ We show the process of optimization of this hypotheses testing and the condition that discrimination between the two states become difficult even when optimized test was executed. In addition, We consider time-enegy uncertainty relation from the viewpoint of optimized hypotheses testing. Optimization consists of two steps. The first step is to optimize the decision rule based on the outcome of the given measurement by the Neyman-Pearson theorem. The second step is to select optimum measurement in order to maximize the power of test. The optimum measurement is proved to be $\{ |0\rangle \langle 0|,1-|0\rangle \langle 0|\}$ and the power of test in the optimized test holds $γ_{Max}=1-\exp(-\frac{2n}{\hbar^2}Δt^2ΔH^2)+o(Δt^2)$ when $Δt \ll 1$ satisfies,where n is the number of the data. It will be impossible to discriminate between the two states when the power of this test declines. This condition becomes $\frac{2n}{\hbar^2}Δt^2ΔH^2\ll 1$ in the optimized hypotheses testing. It is remarkable that the optimum measurement is composed of the operators $Q_0$ and $1-Q_0$ in the proof of time-energy uncertainty relation. Namely, the time-energy uncertainty relation presents the time interval in which even the optimum measurement hardly detect modification of the initial state .

quant-ph↗

arXiv2017

Mass Renormlization in the Nelson Model

The asymptotic behavior of the effective mass $m_{\rm eff}(Λ)$ of the so-called Nelson model in quantum field theory is considered, where $Λ$ is an ultraviolet cutoff parameter of the model. Let $m$ be the bare mass of the model. It is shown that for sufficiently small coupling constant $|α|$ of the model, $m_{\rm eff}(Λ)/m$ can be expanded as $m_{\rm eff}(Λ)/m= 1+\sum_{n=1}^\infty a_n(Λ) α^{2n}$. A physical folklore is that $a_n(Λ)\sim [\log Λ]^{(n-1)}$ as $Λ\to \infty$. It is rigorously shown that $$0<\lim_{Λ\to\infty}a_1(Λ)<C,\quad C_1\leq \lim_{Λ\to\infty}a_2(Λ)/\logΛ\leq C_2$$ with some constants $C$, $C_1$ and $C_2$.

math-ph↗

arXiv2001

Numerical Experiments on The Capacity of Quantum Channel with Entangled Input States

The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to aspecial case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum channel capacity.

quant-ph↗