^{1}

^{1}

^{1}

^{*}

Under amplitude damping channel, the dependence of the entanglement on the initial states and , which reduce to four orthogonal Bell states if we take the parameter of states are investigated. We find that the entanglements for different initial states will decay along different curves even with the same acceleration and parame-ter of the states. We note that, in an inertial frame, the sudden death of the entanglement for will occur if , while it will not take place for for any α. We also show that the possible range of the sudden death of the entanglement for is larger than that for . There exist two groups of Bell state here we can’t distinguish only by concurrence.

In the theory of quantum information, entanglement, a very subtle phenomenon, has been investigated many years since it was first brought to light by Einstein, Podolsky and Rosen [

It is well known that the Bell state is a concept in quantum information science and represents the simplest possible examples of entanglement. And there are four orthogonal Bell states

where indicate Minkowski modes described by Alice and described by Rob, respectively. Sibasish Ghosh showed that it is not possible to discriminate between any three Bell states if only a single copy is provided and if only local operations and classical communication are allowed [

where. can degrade into the Bell states and into if we take, respectively. Then, we can find that the behavior of the entanglement will be greatly influenced by initial states, but we can only distinguish the initial states (or) from (or).

In this paper, we will investigate the dependence of the entanglement on the initial states which reduce to four orthogonal Bell states under amplitude damping channel. We will show that the entanglements for different initial states will decay along different curves even with the same acceleration and parameter of the states, and the possible range of the sudden death of the entanglement for 1 is larger than that for 2.

This paper is structured as follows. In Section 2 we will study the concurrence when both of the qubits under amplitude damping channel using the initial state. In Section 3 we will consider the concurrence when both of the qubits under the same environment by taking the state. Our work will be summarized in last section.

We first study the entanglement for initial states. We assume two observers, Alice who stays stationary has a detector only sensitive to mode and Rob who moves with a uniform acceleration has a detector which can only detect mode, share a entangled initial state at the same point in Minkowski spacetime. We can use a two-mode squeezed state to expend the Minkowski vacuum from the perspective of Rob[

On account of Rob is causally disconnected from region II, and tracing over the states in region II, we obtain

We now let both Rob and Alice interact with a amplitude damping environment [

Equation (5) shows that if the system stays both it and its environment will not change at all. Equation (6) indicates that if the system stays the decay will exist in the system with probability P, and it can also remain there with probability (1 – P).

If the environment acts independently on Alice’s and Rob’s states, the total evolution of these two qubits system can be expressed as [

where are the Kraus operators

where, P_{A} is the decay parameter in Alice’s quantum channel and P_{R} is the decay parameter in Rob’s quantum channel, and P_{i} is a parameter relating only to time. Under the Markov approximation, the relationship between the parameter P_{i} and the time t is given by [15,19], where is the decay rate. We must note that here we just consider the local channels [

where are square roots of the eigenvalues of the matrix, with is the “spin-flip” matrix for the state (5). So, we obtain the concurrence as a function of α, r and P

Due to the concurrence is just depended on and, we can’t distinguish the initial states described by with or.

Now, we consider the other initial state. Using the same method as mentioned above we obtain its density matrix

and the evolved state for

Thus, the concurrence is

From which we know that we can’t distinguish the initial states described by with or , too.

By comparing Equations. (10) and (13), we can see that there are obvious differences between and. Especially, we find that and for Bell states in an inertial frame.

But if, we have for any r and α, which means that the two groups of the initial states will be equivalent without the effect of environment.

To learn the behavior of the entanglement intuitively, we plot the concurrence for different initial states and with different parameters in

From

If the parameters r, α and P in Equation (10) satisfy the relation

we have, and if the parameters r, α and P in Equation (13) meet

we obtain. Using Equations (14) and (15) (See

And for the states, the sudden death of entanglement can happen only when

It is obviously that the possible range of the sudden death of the entanglement for is larger than that for. If, whatever r is, the disappear of the entanglement for will be earlier than that for.

Above discussions reveal some different behaviors of concurrences for the initial states and when both subsystems are coupled to noise environment. Thus, the entanglement is dependent to the initial states under the amplitude damping channel.

This work was supported by the National Natural Science Foundation of China under Grant No. 11175065, 10935013; the SRFDP under Grant No. 20114306110003; PCSIRT, No. IRT0964; the Hunan Provincial Natural Science Foundation of China under Grant No. 11JJ7001; and the Construct Program of the National Key Discipline.