Gr\"obner-Shirshov bases for categories

In this paper we establish Composition-Diamond lemma for small categories. We give Gr\"obner-Shirshov bases for simplicial category and cyclic category.Comment: 20 page

Perturbational approach to the quantum capacity of additive Gaussian quantum channel

For a quantum channel with additive Gaussian quantum noise, at the large input energy side, we prove that the one shot capacity is achieved by the thermal noise state for all Gaussian state inputs, it is also true for non-Gaussian input in the sense of first order perturbation. For a general case of $n$ copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive.Comment: 5 page

Quantum correlations in the collective spin systems

Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.Comment: 8 pages, 4 figure

Spin-Orbit Coupling and Tunneling Current in a Parabolic Quantum Dot

We propose a novel approach to explore the properties of a quantum dot in the presence of the spin-orbit interaction and in a tilted magnetic field. The spin-orbit coupling within the quantum dot manifest itself as anti-crossing of the energy levels when the tilt angle is varied. The anti-crossing gap has a non-monotonic dependence on the magnitude of the magnetic field and exhibits a peak at some finite values of the magnetic field. From the dependence of the tunneling current through the quantum dot on the bias voltage and the tilt angle, the anti-crossing gap and most importantly the spin-orbit strength can be uniquely determined