About Factorization of Quantum States with Few Qubits

We study the factorization conditions of a wave function made up of states of two, three and four qubits and propose and analytical expression which can characterize entangled states in terms of the coefficients of the wave function and density matrix elements.


Introduction
In quantum mechanics, multiple dimensional or multiple particles systems are characterized by the tensor product of the Hilbert subspaces [1], where each subspace is associated to each element. It is well known [2] that when there is not interaction among these elements, the wave function is just the 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 [3]. 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 [4,5,6,7,8,9,10,11].

Factorized State
When a quantum systems is made up of several quantum subsystems, where the ith-subsystem is characterized by a Hamiltonian H i and a Hilbert space E i , corresponding a two-states and having a basis {|ξ i } ξi=0,1 , the Hilbert space is written as the tensorial product of each subsystem, E = E n ⊗ · · · ⊗ E 1 (for n-subsystems), The state in each subsystems is defined as a qubit in quantum computation and information theory [15] and is given by where {|0 , |1 } is the basis of the two-states Hilbert subspace E i . A general state |Ψ in the Hilbert space E can be written as where C ′ ξ s are complex numbers, and |ξ is an element of the basis of E, |ξ = |ξ n . . . ξ 1 = |ξ n ⊗ · · · ⊗ |ξ 1 , ξ k = 0, 1 k = 1, . . . , n.
For n = 2, one has a general state |Ψ ∈ E, Let us assume that this state can be written as with |ψ k , k = 1, 2 given by (1). Then, substituting (1) in (6)and equaling coefficients with (5), it follows that From these expression 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 [26] as a characterization of an entangled state For n = 3, a general state in the Hilbert space E is Assuming that this wave function can be written as |Ψ = ψ 3 ⊗ |ψ 2 ⊗ |ψ 1 with ψ k given by (1), and after some identifications (as before) and rearrangements, one gets These expression reflex a parallelism between the complex vectors (C 1 , C 3 , C 5 , C 7 ) and (C 2 , C 4 , C 6 , C 8 ), the vectors (C 1 , C 2 , C 5 , C 6 ) and (C 3 , C 4 , C 7 , C 8 ), and the vectors (C 1 , C 2 , C 3 , C 4 ) and (C 5 , C 6 , C 7 , C 8 ). In addition, they bring about the following eight independent relations If one of these expression is not satisfied, the wave function (10) represents an entangled state. Therefore, one can propose the following expression to characterize an entangled state For n = 4, a general state in the Hilbert space E is of the form Assuming this function can be expressed as |Ψ = |ψ 4 ⊗ |ψ 3 ⊗ |ψ 2 ⊗ |ψ 1 with |ψ k , k = 1, 2, 3, 4 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 [6,10,13,14]. 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 C j , with j = 1, . . . , 6 randomly generated. As we can see, the values obtained with the coefficients C ′ j s and with the density matrix elements ρ nm are the same. Figure 2 shows for the |W state, the possible values of C (3) (Ψ). As we can see, there are four possible maxima corresponding to the values C 2 = C 3 = C 5 = ±1/ √ 3 and four maxima corresponding to the values C 5 = 0, C 2 = C 3 = ±1/ √ 2, related to semi-factorized state |0 ⊗ (C 2 |01 + C 3 |10 ). Figure 3 shows the possible values of C (3) (Ψ) for the state The maximum value of C (3) (Ψ) is gotten for C 1 = C 8 = 1/ √ 2, as one would expect.
For a Hilbert space E generated by n-qubits, C (n) defines a continuous function C (n) : E → ℜ + with ℜ + = [0, +∞), c ≥ 1. Since the coefficients of the wave function defines a compact set on the real space ℜ 2n due to the relation i |C i | 2 = 1, the image of this compact set is a compact set in ℜ + [16]. Thus, a normalization factor is possible to introduce on this function to define any compact set [0, c], which it is not important.

Dynamical consideration
Following Lloyd's idea [17], consider a linear chain of nuclear spin one half, separated by some distance and inside a magnetic in a direction z, B(z) = (0, 0, B 0 (z)), and making and angle θ with respect this linear chain. Choosing this angle such that cos θ = 1/ √ 3, the dipole-dipole interaction is canceled, the Larmore's frequency for each spin is different, ω k = γB 0 (z k ) with γ the gyromagnetic ratio. The magnetic moment of the nucleus µ k is related with its spin through the relation µ k = γS k , 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 [18,19] where N is the number of nuclear spins in the chain (or qubits), J and J ′ are the coupling constant of the nucleus at first and second neighbor. Using the basis of the register of N-qubits, {|ξ N , . . . , ξ 1 } with ξ k = 0, 1, one has that S z k |ξ k = (−1) ξ k |ξ k /2. Therefore, the Hamiltonian is diagonal on this basis, and its eigenvalues are Consider now that the environment is characterized by a Hamiltonian H e and its interacting with the quantum system with Hamiltonian H s . Thus, the total Hamiltonian would be H = H s + H e + H se , where H se 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 [20,21] i where ρ t = ρ t (s, e) 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 [22], it is possible to arrive to a Lindblad type of equation [23,24] for the reduced density matrix ρ(s) = tr e (ρ t ), where V i 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 transformatioñ ρ = U ρU † with U = e iHst/ , as whereL(ρ) is the Lindblad operator The explicit form of Lindblad operator is determined by the type of environment to consider [25] at zero temperature. So, the operators can be V i = S − i (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 whereS + k andS − k are the ascend and descend operators such thatS ± k = U S ± k U † = S ± k e ±iΩ k t , whereΩ k is defined asΩ k = w k + J (S z k+1 + S z k−1 ) + J ′ (S z k+2 + S z k−2 ).. The solutions of the equations are ρ 11 (t) = ρ 11 (0) + ρ 22 (0) + ρ 33 (0) + ρ 44 (0) + ρ 55 (0) + ρ 66 (0) + ρ 77 (0) + ρ 88 (0) −(ρ 55 (0) + ρ 66 (0) + ρ 77 (0) + ρ 88 (0))e −γ1t − (ρ 33 (0) + ρ 44 (0) + ρ 77 (0) + ρ 88 (0))e −γ2t −(ρ 22 (0) + ρ 44 (0) + ρ 66 (0) + ρ 88 (0))e −γ3t + (ρ 77 (0) + ρ 88 (0))e −(γ1+γ2)t +(ρ 66 (0) + ρ 88 (0))e −(γ1+γ3)t + (ρ 44 (0) + ρ 88 (0))e −(γ2+γ3)t − ρ 88 (0)e −(γ1+γ2+γ3)t ; ρ 22 (t) = (ρ 22 (0) + ρ 44 (0) + ρ 66 (0) + ρ 88 (0))e −γ3t − (ρ 66 (0) + ρ 88 (0))e −(γ1+γ3)t Figure 4 shows the behavior of the entangled states |W and |GHZ as a function of time when this entangled state interact with the environment. Purity behavior, tr(ρ 2 ), is also shown. The system starts as a pure entangled state, it evolves in a mixed state and finishes in the pure ground state (|000 ), after sharing energy with the environment. The states |GHZ and |Ψ 1 behave more robust than the state |W , C ρ grows since other entangled stated contribute to this function. In contrast, starting with the entangled state |W , there are not other entangled states which make contribution to the function C

Conclusions
We have studied the full factorization of an state made up of up to 4-qubits basic states. We have seen that there is an indication that the number of conditions to characterizes 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 this 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 C (n) (n=2, 3,4). This non negative function expressed in terms of the coefficients of the wave function or in terms the density matrix elements represents a measurement of the entanglement of any wave function made up of basic n-qubits (n = 2, 3, 4). Using this function, we study the decay of entangled states |W , |GHZ , and |Ψ 1 due to interaction with environment, and we noticed a great different behavior of the function C ρ , indicating some type of robustness behavior of the states |GHZ and |Ψ 1 . The main reason for this different behavior is that the entangled states |GHZ and |Ψ 1 contain the ground state |000 , which is the final state in the dynamics.