In quantum mechanics, multiple dimensional or multiple particles systems are characterized by the tensor product of the Hilbert subspaces  , where each subspace is associated to each element. It is well known  that when there is no interaction among these elements, the wave function is just the tensor product of the wave functions of each element; that is, the tensor product of the wave function associated to each element determines the non-interacting characteristic of the elements in a quantum system. If interaction occurs at some time among these elements, this tensor product disappears, and the wave function becomes entangled  . So, it is necessary to point out that if the wave function is not factorized, the wave function is entangled. In this way, the characterization of factorization is somewhat equivalent to the characterization of entanglement. In this paper, we will follow this line of ideas to determine the characterization of an entangled state  -  .
2. Factorized State
When a quantum systems is made up of several quantum subsystems, where the ith-subsystem is characterized by a Hamiltonian and a Hilbert space , corresponding a two-states and having a basis , the Hilbert space is written as the tensorial product of each subsystem, (for n- subsystems), The state in each subsystems is defined as a qubit in quantum computation and information theory  and is given by
where is the basis of the two-states Hilbert subspace . A general state in the Hilbert space can be written as
where are complex numbers, and is an element of the basis of ,
A full factorized state in this space is
where is given by (1).
For , one has a general state ,
where we have chosen to use decimal notation for the coefficients. Let us assume that this state can be written as
with given by (1). Then, substituting (1) in (6) and equaling coefficients with (5), it follows that
From these expressions one obtains a single condition for factorization,
Thus, if this condition is not satisfied, the state (5) represents an entangled state. So, one can use the following known expression  as a characterization of an entangled state
For , a general state in the Hilbert space is
where, we have chosen again decimal notation for the coefficients. Assuming that this wave function can be written as with given by (1), and after some identifications (as before) and rearrangements, one gets
These expression reflex a parallelism between the complex vectors and , the vectors and , and the vectors and . In addition, they bring about the following eight independent relations
If one of these expressions is not satisfied, the wave function (10) represents an entangled state. Therefore, one can propose the following expression to characterize an entangled state
For , a general state in the Hilbert space is of the form
Assuming this function can be expressed as with given by (1), and after some identifications and rearrangements, one gets
expressing similar parallelism we mentioned before. Each row gives us 28 relations, having a total of 112 possible relations, and from these relations, one can get the following 36 independent conditions
Again, if one of these expression fail to happen, (15) will represents an entangled state. Thus, one can propose the following expression to characterize an entangled state made up of 4-qubits basis
As we can see from these examples, the number of conditions needed to characterize a factorized state (or entangled state) grows exponentially with the number of qubits. So, characterization of an entangled state for n-qubits in general becomes a very hard work. Now, in terms of the density matrix elements, one could have the characterization of entangled states made up of 2, 3 and 4 qubits as
For other considerations of entanglement multiqubit entanglement see     . Figure 1 below shows the values of the expressions (14) and (19) for 50 entangled states made up of 3-qubits basis and with values , with randomly generated. As we can see, the values obtained with the coefficients and with the density matrix elements are the same. Figure 2 shows for the state,
the possible values of . As we can see, there are four possible maxima corresponding to the values and four maxima corres-
Figure 1. for arbitrary entangled state. Diamonds (density matrix) and circles (coefficients).
Figure 2. for the state such that .
ponding to the values , , related to semi-factorized state . Figure 3 shows the possible values of for the state
The maximum value of is gotten for , as one would expect.
For a Hilbert space generated by n-qubits, defines a continuous function with , . Since the coefficients of the wave function defines a compact set on the real space due to the relation , the image of this compact set is a compact set in  . Thus, a normalization factor is possible to introduce on this function to define any compact set , which it is not important.
3. Dynamical Consideration
Following Lloyd’s idea  , consider a linear chain of nuclear spin one half, separated by some distance and inside a magnetic in a direction , , and making and angle with respect this linear chain. Choosing this angle such that , the dipole-dipole interaction is canceled, the Larmore’s frequency for each spin is different, with the gyromagnetic ratio. The magnetic moment of the nucleus is related with its spin through the relation , and the interaction energy between the magnetic field and magnetic moments is . If in addition, one has first and second neighbor Ising interaction, the Hamiltonian of the system is just   .
Figure 3. for the state such that .
where is the number of nuclear spins in the chain (or qubits), and are the coupling constant of the nucleus at first and second neighbor. Using the
basis of the register of N-qubits, with , one has that . Therefore, the Hamiltonian is diagonal on this basis,
and its eigenvalues are
Consider now that the environment is characterized by a Hamiltonian and its interacting with the quantum system with Hamiltonian . Thus, the total Hamiltonian would be , where is the part of the Hamiltonian which takes into account the interaction system-environment, and the equation one would need to solve, in terms of the density matrix, is  
where is the density matrix which depends on the system and environment coordinates. The evolution of the system is unitary, but it is not possible to solve this equation due to a lot of degree of freedom. Therefore, under some approximations and tracing over the environment coordinates  , it is possible to arrive to a Lindblad type of equation   for the reduced density matrix ,
where are called Kraus’ operators. This equation is not unitary and Markovian (without memory of the dynamical process). This equation can be written in the interaction picture, through the transformation with , as
where is the Lindblad operator
with . The explicit form of Lindblad operator is determined by the type of environment to consider  at zero temperature. So, the operators can be (for dissipation) for the model independent with the environment. In this case, each qubit of the chain acts independently with the environment, and one has local decoherence of the system. The Lindblad operator is
where and are the ascend and descend operators such that
where is defined as . The solutions of the equations are
where are given by
In our case, we have three qubits space , and our parameter in units are
the time is normalized by the same factor of , and we include in this study the entangle state
Figure 4 shows the behavior of the entangled states and as a function of time when this entangled state interact with the environment. Purity behavior, , is also shown. The system starts as a pure entangled state, it
Figure 4. and Purity for the entangled state , , and .
evolves in a mixed state and finishes in the pure ground state , after sharing energy with the environment. The states and behave more robust than the state , grows since other entangled stated con- tribute to this function. In contrast, starting with the entangled state , there are not other entangled states which make contribution to the function in the dynamics, and one sees an exponential decay.
We have studied the full factorization of a state made up of up to 4-qubits basic states. We have seen that there is an indication that the number of conditions to characterize a factorized state grows exponentially with the number of qubits. For two, three and four basic qubits, we showed the conditions in order to have a factorized state, and if any of one of these conditions fails, one gets instead an entangled state. Therefore, an entangled state is also characterized by the complement of each of this conditions, and the resulting expression has been denoted by . This non-negative function expressed in terms of the coefficients of the wave function or in terms of the density matrix elements represents a measurement of the entanglement of any wave function made up of basic n-qubits . Using this function, we study the decay of entangled states , , and due to interaction with environment, and we noticed a great different behavior of the function , indicating some type of robustness behavior of the states and . The main reason for this different behavior is that the entangled states and contain the ground state , which is the final state in the dynamics.
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