1. Introduction

In the fundamental level nature requires quantum description rather than classical one. Now a days our interest spontaneously go to the entanglement in many bodyquantum systems. It is known to everybody that entanglement is a characteristic feature and powerful resource of quantum mechanics and is the basic ingredient of the foundation of the theory. To find the states which are useful for quantum information tasks, characterization of entangled states is of our great interest, this is called separability problem. This is one of the unsolved problems of quantum mechanics. Quantum inseparability first recognized by Einstein, Podolsky and Rosen[1] and Schrödinger[2,14] is one of the most astonishing features of quantum formalism. Recently people give their deep concentration to find out separability criteria in some particular states basically pure states and dynamical development of experimental methods and non-locality of quantum mechanics together inspire people to find out on the inseparability problems in quantum mechanics. Now there available several criterions to detect quantum entanglement of composite quantum states. But in the mixed states there no rigorous criteria for finding whether a state is entangled or not. Here we want to confined ourselves in positive partial transpose (PPT) criteria[4], computable cross norm or realignment criteria[5] and an other computable separability criteria[5]. According to Peres and Horodechi[3],PPT is a strong sufficient condition for detecting entanglement for 2×2 and 2×3 systems. On the way of searching some powerful criterion which will perform better than PPT criterion, some desirable results for detecting entanglement [6,7]. In this paper, we have performed a comparative study of these three criterions for a most general state of a composite quantum bipartite system.

2. Definitions and Explanations

Bipartite Pure States:

Let A and B be two non-interacting systems and and be their respective Hilbert spaces. The Hilbert space of the composite system is the tensor product and . If the system A is in the state and system B is in the state , then the composite system is in the state . The state of a composite system white can be represented in this form are called seperable states or product states. If there are states which can not be represented in this form are called entangled states. Let a projector on a vector is a pure state if the local subsystems are pure states. For a pure state can be represented as

Where and are the basis for and respectively, and where min(dim, dim ). This decomposition of is called Schmidt decomposition and’s are called Schmidt coefficients.

Bipartite Mixed states[8-10,18,19]:

Let be respectively the states of the systems and . Then combined state of their joint system can be written as

A necessary and sufficient condition of the mixed state AB to be separable is that it can be represented as a convex combination of the product of projectors on local states as

Otherwise, the mixed state is said to be entangled state.

3. Partial Transposition Criterion[11,12]

A bipartite density matrix can be expressed as

Where and and and are orthogonal bases of and respectively.

The partial transposition with respect to system A, is defined as

If , that is the eigen values of are non-negative then the state has positive partial transpose(PPT). Otherwise the state has non-positive partial transposition(NPT).

Detection of entanglement:

If any one of the eigen values of partial transposition of AB be negative then the state is entangled.

4. Crossnorm or Realignment Criteria

For a bipartite density matrix

The realignment operation R can be defined as[15-17]

Detection of entanglement:

If the sum of all the eigen values of R be less than or equal to 1 then the state is separable otherwise it is entangled.

5. A Computable Separability Criterion

We consider Hilbert spaces and of the systems A and B respectively. Then there is a correspondence between states and Hilbert-Schmidt operators

According to the rule :

if be decomposition of in terms of orthogonal bases and of and respectively. Then T is given by

Conversely, if for some orthogonal bases and of and respectively,then

Theorem: If and be finite dimentional Hilbert spaces and KA =HS()Cⁿ and KB=HS() C^m be the spaces of Hilbert-Schmidt operators on and respectively.

Then there exists a one to one correspondence between Hilbert-Schmidt operators THS() and Hilbert-Schimdt operators

Detection of entanglement:

We know that, for a finite dimentional Hilbert space be a density operator then is separable imples the sum of squares of the eigen values .

6. Results in Details

We consider a most general state

For this state, we are going to test where it is entangled and where separable using different types of entanglement detection criteria.

At first for partial transpose criterion, the state is

Taking partial transpose on the party B, we have

Eigen values of this matrix are given by

For the state to be separable , we have

And

Since, the physical range of separability of is

That is when , the state is entangled.

Now, we test for cross norm or realignment criterion, the realignment operator R of the state

is given by

Now theeigen values are obtained as

Sum of all the eigen values is

The is separable when ,

but the physical range of p is which implies that the state is separable for any value of p.

Now for another computable separability criterion, we consider the same state

We write this state in matrix form as

Therefore, we have

The eigen values of the above matrix are

p ²,p²,4,p²

Now, adding the absolute values of the square root of all eigen values, we arrive at

Hence we see that for all physical values of p, the state is entangled.

In short, we represent the above results in tabular form as

Let us consider a state

where

Table 1. Representation of different separability criterian for the warner state

For normality, we have

the state becomes,

The realignment operator R can be written as

Sum of all the eigen values of the above matrix is

To find Min

,

where and we assume that

Then Min

and Max

The state must be separable if

The state must be entangled if

Partial Traspose Criteria:

Taking Partial transpose on the party B, we get

Now the eigen values of the above matrix are

All of this eigen values are non-negative if

Therefore the state must be separable if

Which includes the separablelity condition for realignment criteria that is

and .

7. Conclutions

Now we come to the conclusion about the detection of entanglement of a general state using three strong entanglement detection criteria, we see that through the state is entangled when formed, PPT criteria shows that when p lies on[0, ], the state is separable and entangled on (]. But realignment criteria shows that the state is always separable whatever be the value of p. And the other criteria shows that the state is always entangled for any value of p. These conflicting results do not help us to reach to any concrete decision which one is stronger without any hesitations. So we can decide that PPT criteria is always strong criteria than others in and systems. Here PPT criteria exhibit more explicit and clear range of separability with respect to the other two methods. So in this paper through a detailed numerical search of the well known Warner state, it is clearly established that PPT is the reliable excellent criteria in the and systems.