Understanding Quantum Entanglement: Qubits, Rebits and the Quaternionic Approach

It has been recently pointed out by Caves, Fuchs, and Rungta that real quantum mechanics (that is, quantum mechanics defined over real vector spaces provides an interesting foil theory whose study may shed some light on just which particular aspects of quantum entanglement are unique to standard quantum theory, and which ones are more generic over other physical theories endowed with this phenomenon. Following this work, we discuss some entanglement properties of two-rebits systems, making a comparison with the basic properties of two-qubits systems, i.e., the ones described by standard complex quantum mechanics. We also discuss the use of quaternionic quantum mechanics as applied to the phenomenon of entanglement.


I. INTRODUCTION
The phenomenon of entanglement is one of the most characteristic non-classical features exhibited by quantum systems [6]. Quantum entanglement is the basic resource of several quantum processes as superdense coding [7], quantum teleportation [8] and quantum computation [6,9] studied by quantum information theory [10][11][12][13]. A state of a composite quantum system constituted by subsystems A and B is called "entangled" if it cannot be represented as a mixture of factorizable pure states.
The simplest systems exhibiting the phenomenon of entanglement are two-qubits systems. For these systems an explicit expression of the entanglement of formation, which is a natural quantitative measure of entanglement [14], has been found by Wootters [15].
The correspondent space of mixed two-qubits states in standard quantum mechanics is 15-dimensional. The amount of entanglement in this space has been established by Wootters [16][17][18][19].
For quantum mechanics defined over real vector spaces the simplest composite systems are two-rebits systems. An explicit expression for the entanglement of formation of arbitrary states of two-rebits has been obtained by Caves, Fuchs and Rungta [1]. Pure states of rebitssystems are described by normalized vectors in a two dimensional real vector space. The correspondent space of mixed two-rebits states is 9-dimensional (vis-à-vis 15 for 2 qubits).
The aim of the present work is to explore numerically, as well as conceptually, the entanglement properties of two-rebits systems [20], as compared to the usual two-qubits ones, so as to detect the differences between the two types of system. We pay particular attention to the distribution of states according to their degree of entanglement. We obtain, analytically the probability densities for finding pure states states with a given amount of entanglement E (or with fixed values of the so called concurrence squared C 2 ). The same is done for mixed states in numerical fashion.
The paper is organized as follows. In section II we review the general properties of twoqubits systems. Several quantities related to the entanglement of formation are investigated for both rebits and qubits in section III, with an emphasis on the differences and similitudes between both formalisms. In section IV we discuss how to follow a quaternionic approach to quantum entanglement. Finally, some conclusions are drawn in section V.

II. ENTANGLEMENT OF TWO-QUBITS SYSTEMS
As already stressed in the introduction, the two-qubits system is the simplest quantum mechanical system that exhibits the sort of "extra correlation" one calls entanglement between two parties. A representation of a qubit is given by the Bloch sphere. The poles correspond to classical bits |0 , |1 while any point on this unit sphere, given by a pair of angles φ, ψ, represents a general qubit cosφ|0 + e iψ sinφ|1 . Qubits constitute the essential new elements in the switch from Classical to Quantum Information Theory and entanglement the basic ingredient of many striking processes now under active investigation.
Here we use the concurrence C[ρ] and the entanglement of formation E[ρ] as quantitative measures of the amount of entanglement. Wootters' formula for the entanglement of formation of a two-qubits stateρ reads [15] where and the concurrence C is given by being the square roots, in decreasing order, of the eigenvalues of the matrixρρ, with The above expression has to be evaluated by recourse to the matrix elements ofρ computed with respect to the conventional product basis.
In this work we have generated all pure and mixed states of a two-qubits system according to the measure defined in [16,17,21].
The distribution of two-qubit states according to their degree of mixture can be obtained analytically adopting a geometric representation [22] for the eigenvalues ofρ as a regular tetrahedron T ∆ of side length 1, in R 3 , centered at the origin. There exists a mapping connecting these eigenvalues (p 1 , . . . , p 4 ) with the points r of the tetrahedron, so that we can relate the participation ratio R(ρ) = 1/T r(ρ 2 ) to the radius r =| r | of a sphere embedded within the tetrahedron T ∆ according to Thus, the states with a given degree of mixture lie on the surface of a sphere of radius r concentric with the tetrahedron T ∆ . For a completely description, see [22]. It is then easy to find out just how our states are so distributed, according to the degree of mixture R. This distribution is depicted in Figure 1. The dualistic nature of the concepts of entanglement and mixedness becomes apparent. For two-qubits systems one would expect that the states will tend to become less entangled as the degree of mixture increases. In fact, for R ≥ 3 (vertical line in Figure 1) all states are separable. It is important to stress here that this curve is the same no matter what quantities (rebits, qubits, or even quaterbits) are involved, although their respective entanglement properties are different.

III. ENTANGLEMENT PROPERTIES OF TWO-REBITS SYSTEMS. A COMPARISON
In the space of real quantum mechanics we can represent rebits on the Bloch sphere. The poles correspond to classical bits |0 , |1 , but the sphere reduces itself now to a maximum unit circle, described by just one parameter φ. We have cosφ|0 + e iψ sinφ|1 → cosφ|0 +sinφ|1 .
Entanglement can also be described in such a context with suitable modifications. Caves, Fuchs, and Rungta's (CFR) formula for the entanglement of formation of a two-rebits state ρ is given by (1), (2) using the concurrence C[ρ] = | tr(τ ) | = | tr(ρ σ y ⊗ σ y ) |, which has to be evaluated using the matrix elements of ρ computed with respect to the product basis, For a two-rebits state the entanglement of formation is completely determined by the expectation value of one single observable, namely, σ y ⊗ σ y . On the contrary, it has been recently proved that there is no observable (not even for pure states) whose sole expectation value will yield enough information to as to determine the entanglement of a two-qubits state [23]. As shown in [20], there are mixed states of two rebits with maximum entanglement (that is, with C 2 = 1) within the range 1 ≤ R ≤ 2. This is clearly in contrast to what happens with two-qubits states, because only pure states (R = 1) have maximum entanglement.
The measure of mixedness R introduced above can be used in the description of the entanglement phenomenon for two-rebits systems. Also, in order to explore numerically the properties of arbitrary two-rebits states, it is necessary to introduce an appropriate measure µ on the space S R of general two-rebits states. Such a measure is needed to compute volumes within the space S R , as well as to determine what is to be understood by a uniform distribution of states on S R [16,17].
An arbitrary (pure and mixed) state ρ of a (real) quantum system described by an N-dimensional real Hilbert space can always be expressed as the product of three matrices, The relationship between the amount of entanglement and the purity of quantum states of composite systems has been recently discussed in the literature [16][17][18][19]. The amount of entanglement and the purity of quantum states of composite systems exhibit a dualistic relationship. As the degree of mixture increases, quantum states tend to have a smaller amount of entanglement. In the case of two-qubits systems, states with a large enough degree of mixture are always separable [16]. To study the relationship between entanglement and mixture in real quantum mechanics, we compute numerically the probability P (E) of finding a two-rebits state endowed with an amount of entanglement E. In Figure 2 we compare (i) the distribution associated with two-rebits states with (ii) the one, associated with twoqubits states, recently obtained by Zyczkowski et al. [16,24]. The distribution P (E) or P (C 2 ) for pure two-rebits states can be obtained analytically.
Let us write a pure two-rebits state in the form The states (| φ i , i = 1, . . . , 4) are the eigenstates of the operator σ y ⊗ σ y . The four real numbers c i constitute the coordinates of a point lying on the three dimensional unitary hypersphere S 3 (which is embedded in R 4 ). We now introduce on S 3 three angular coordinates, φ 1 , φ 2 , and θ, defined by c 1 = cos θ cos φ 1 , c 2 = cos θ sin φ 1 , In terms of the above angular coordinates, the concurrence of the pure state | Ψ is given by Using (8) and (9) one deduces that the probability density P (C 2 ) of finding a pure two-rebits state with a squared concurrence C 2 is given by The distribution is to be compared with the one obtained for pure states of two-qubits systems, which is (analytically) found to be [24] P ( Both distributions are compared in Figure 3. Figure 3a depicts the one for qubits, while Fig. 3b shows the one for rebits. The distribution remains finite, in the case of qubits, for all C 2 . In the case of rebits it presents a sharp peak at the origin, an then saturates to 1/2 at C 2 = 1. The general conclusion that one draws from Figures 2 and 3 is that the curves representing the distributions P (E) and P (C 2 ) associated with (i) pure states and (ii) arbitrary states do not differ, in the case of two-rebits states, as much as they do in the case of two-qubits states.
We can determine analytically which is the maximum entanglement E m of a two-rebits state compatible with a given participation ratio R. Since E is a monotonic increasing function of the concurrence C, we shall find the maximum value of C compatible with a given value of R. In order to solve the ensuing variational problem (and bearing in mind that C =| σ y ⊗ σ y | ), let us first find the state that extremizes Tr(ρ 2 ) under the constraints associated with a given value of σ y ⊗ σ y , and the normalization of ρ. This variational problem can be cast in the fashion δ Tr(ρ 2 ) + β σ y ⊗ σ y − αTr(ρ) = 0, (12) where α and β are appropriate Lagrange multipliers.
After some algebra, and expressing the expectation value of σ y ⊗ σ y in terms of the parameter β, one finds that the maximum value of C 2 compatible with a given value of R is given by

IV. THE QUATERNIONIC APPROACH TO QUANTUM ENTANGLEMENT
The quaternionic space H constitutes a generalization of the complex space C which, in turn, generalizes the real space R. Each step of this chain is possible by introducing new quantities: i 2 = −1 from R to C and j 2 = k 2 = −1 from C to H, with suitable commutation laws for the three quantities i, j, k.
A general quaternion φ and its associated commutation algebra are written in the following fashion with φ i ∈ R and ij = −ji = k, jk = −kj = i, ki = −ik = j.
This "natural extension" of complex numbers that yields quaternions cannot be generalized any further. Thus, if we give up the property of commutativity, the most general algebra that can be used in quantum mechanics is the quaternionic one [25].
A. Entanglement for pure states of two "quaterbits" systems The definition of entanglement in Quaternionic Quantum Mechanics (QQM) for pure states does not differ from the standard one. Given a pure state |ψ of a composite bipartite system, the entanglement is obtained via the von Neumann entropy of the marginal density matrix associated to subsystem A by tracing over the subsystem B:ρ A =Tr B |ψ AB ψ| of the total density matrixρ = |ψ AB ψ|, or, vice versa,ρ B =Tr A |ψ AB ψ|. Thus, E(ρ) = S(ρ A ) = S(ρ B ).
In the case of quaternions we face a higher dimensionality and, therefore, we need more parameters to describe the stateρ. Additionally, in using the kets and bras notation of Dirac's we must keep in mind the quaternions' non-commutativity rules. For the sake of simplicity, let us suppose that a pure state is written as The statistical matrixρ = |ψ AB ψ| reads withρ † =ρ since C 1 C 2 = C 2 C 1 = C 2 C 1 . Entanglement then is only a function of N(C 1 ) 2 = C 2 10 + C 2 11 + C 2 12 + C 2 13 . Identifying x ≡ C 2 10 + C 2 11 and y ≡ C 2 12 + C 2 13 , we plot E(ρ) in Fig.  4, together with E(ρ) for the three Quantum Mechanics' versions we are dealing with here.

B. Extension to general mixed states
The full analytical study of entanglement in the framework of Quaternionic Quantum Mechanics requires careful consideration of the algebra of states and operators for these "hyper-numbers". If one wishes to discuss how to carry out statistical studies and how entanglement-related properties are distributed over the space of all (pure and mixed) states, one notices that in this case the dimensionality of the problem for a general 4 x 4 matrix is substantially higher (3+4 * 6=27) than for the complex (3+2 * 6=15) or the real (3+1 * 6=9) cases. The ensuing statistical properties become clearly non-trivial and some substantial effort is required.
Let us merely list here the basic ingredients needed for a complete description of the statistical properties of quaternionic statesρ: i) to build and correctly parametrize the unitary transformations defined over the quaternionic Hilbert space. ii) to specify the form of the Haar measure for the concomitant space of unitary transformations. Notice that the measure on the simplex is exactly the same as for complex or real systems. For mixed statesρ, the distributions associated to quantities that depend only on the eigenvalues of a statistical operator do have the same form in the real or the complex cases, for they only depend on the simplex. The results of the corresponding numerical study will be published elsewhere.

V. CONCLUSIONS
We have explored numerically the entanglement properties of two-rebits systems. A systematic comparison has been established between many statistical properties of twoqubits and two-rebits systems. We paid particular attention to the relationship between entanglement and purity in both quantum mechanical frameworks. We have also determined numerically the probability densities P (E) of finding (i) pure two-rebits states and (ii) arbitrary two-rebits states, endowed with a given amount of entanglement E or concurrence squared C 2 . In particular, we determined analytically the maximum possible value of the concurrence squared C 2 of two-rebits states compatible with a given value of mixedness R.
As for the probability of finding states with a given amount of entanglement, the difference between mixed and pure sates is much larger for qubits than for rebits. Also, we have sketched the manner in which the quaternionic formalism could be applied to the study of quantum entanglement.

ACKNOWLEDGMENTS
This work was partially supported by the DGES grants PB98-0124 (Spain), and by CONICET (Argentine Agency). Fig. 1-Probability (density) distribution for finding a stateρ with a given participation ratio R. States (two-qubits) with R ≥ 3 are always separable.