Another convex combination of product states for the separable Werner state

In this paper, we write down the separable Werner state in a two-qubit system explicitly as a convex combination of product states, which is different from the convex combination obtained by Wootters' method. The Werner state in a two-qubit system has a single real parameter and varies from inseparable state to separable state according to the value of its parameter. We derive a hidden variable model that is induced by our decomposed form for the separable Werner state. From our explicit form of the convex combination of product states, we understand the following: The critical point of the parameter for separability of the Werner state comes from positivity of local density operators of the qubits.Comment: 7 pages, Latex2e; v2: 9 pages, title changed, an appendix and a reference added, minor correction

The intercept/resend attack and the collective attack on the six-state protocol of the quantum key distribution

In this paper, we discuss security of the six-state quantum key distribution protocol against intercept/resend attacks and collective attacks. For the intercept/resend attacks, we evaluate a probability that the legitimate users, Alice and Bob, do not notice interference caused by the eavesdropper Eve and a probability that Eve obtains a secret bit sent by Alice. For the collective attacks, we estimate not only the above two probabilities but also Eve's Shannon information. In the assessment of the security against the collective attacks, imposing looser constraints upon Eve's strategies than Bechmann-Pasquinucci and Gisin, we reach substantially the same result as their study. In other words, although we let Eve's strategies satisfy lower symmetry than Bechmann-Pasquinucci and Gisin, both their work and ours attain the same conclusion, i.e. the identical, unique, and optimum strategy. This implies that we obtain Eve's best tactic by applying simpler conditions to it than Bechmann-Pasquinucci and Gisin, so that this result is novel. Finally, we consider a relation between the six-state protocol and the E91 scheme. We show that the intercept/resend attacks can always be described by hidden variable models. In contrast, we demonstrate that we cannot regard the collective attacks as the hidden variable theories if the disturbance is smaller than $1/3$.Comment: 24 pages, 2 eps figures, latex2

The Leggett-Garg inequalities and the relative entropy of coherence in the Bixon-Jortner model

We investigate the Leggett-Garg inequalities and the relative entropy of coherence in the Bixon-Jortner model. First, we analytically derive the general solution of the Bixon-Jortner model by a technique of the Laplace transform. So far, only a special solution has been known for this model. The model has a single state coupled to equally spaced quasi-continuum states. These couplings cause discontinuities in the time evolution of the occupation probability of each state. Second, using the analytical solution, we show that the probability distribution of the quasi-continuum states approaches the Lorentzian function in a period of time between the initial time and the first discontinuity. Third, we examine violation of the Leggett-Garg inequalities and temporal variation of the relative entropy of coherence in the model. We prove that both the inequalities and the relative entropy are invariant under transformations of the energy-level detuning of the single state.Comment: 22 pages, 7 eps figures, latex2e; v2: Section 1 has been rewritten; v3: minor corrections; v4: references added; v5: minor correction

Violations of the Leggett-Garg inequality for coherent and cat states

We show that in some cases the coherent state can have a larger violation of the Leggett-Garg inequality (LGI) than the cat state by numerical calculations. To achieve this result, we consider the LGI of the cavity mode weakly coupled to a zero-temperature environment as a practical instance of the physical system. We assume that the bosonic mode undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of the violation of the inequality for both systems prepared initially in the coherent state $|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. For the evaluation of the inequality, we choose the displaced parity operators characterized by a complex number $\beta$. We look for the optimum parameter $\beta$ that lets the upper bound of the inequality be maximum numerically. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the violation of the LGI in a specific range of three equally spaced measurement times (spacing $\tau$). Moreover, as we let $\tau$ approach zero, the optimized parameter $\beta$ diverges and the LGI reveals intense singularity.Comment: 29 pages, 14 eps figures, latex2e; v2: The title has been changed. We have improved Sect. 8 and added many references; v3: Equation (57) has been modifie