Probing the geometry of two-qubit state space by evolution

We derive an explicit expressions for geometric description of state manifold obtained from evolution governed by a three parameter family of Hamiltonians covering most cases related to real interacting two-qubit systems. We discuss types of evolution in terms of the defining parameters and obtain relevant explicit description of the pure state spaces and their Remannian geometry with the Fubini-Study metric . In particular, there is given an analysis of the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. For families of states resulting from the unitary evolution, we characterize a degree of entanglement using the squared concurrence as its measure.

In particular, there is given an analysis of the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. For families of states resulting from the unitary evolution, we characterize a degree of entanglement using the squared concurrence as its measure.

Introduction
The precise geometric description of the full state space of quantum system is crucial in studying its physical properties [1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18]. Specially, for compound systems, where characterization beyond the general property of convexity, for multilevel-systems, gets rather involved, even for bipartite systems. In general, such a quantum state space cannot be expected to form a smooth manifold. On the other hand, there is an option to focus on distinguished subsets of states of quantum system, namely, on orbits generated by unitary evolutions defined by physically relevant Hamiltonians. Such a focus has been proven fruitful from various perspectives like, the control theory [19,20], the quantum brachistochrone problem [21,22], a time-optimal evolution [5,6,7,8,20,23,24], or the question of Zermelo navigation [25,26,27].
The geometry of the set of quantum states obtained as result of a family of unitary evolutions depending on a set of parameters can be naturally studied with the use of the Fubini-Study metric [2,4,28,29,30,31,32], where the dimension of such obtained Riemannian manifold is equall to the number of parameters. Obviously, the details of such orbits depend on applied Hamiltonian and selected initial state. Most of interactions of a two-qubit systems can be described by the generalized Heisenberg type interaction Hamiltonian containing anisotropic terms, which is conventionally put into the following physical form suitable e.g. for studying quantum dot systems [33] where S j are spins of j = 1, 2 of subsystems, κ DM is the Dzyaloshinski-Moriya vector controlling anti-symmetric part and Θ is symmetric, traceless 3 × 3 matrix. However, for further considerations, to describe manifolds of states, we shall use the explicit σ-matrix notation. Above interaction Hamiltonian is covered by the general form of a nonlocal Hamitonian for two-qubit system h ij are real nad σ j , j = 1, 2, 3 are Pauli matrices. As it is known [34], such a Hamiltonian can be transformed into the diagonal form The full Hamiltonian contains additionaly a local term H loc = H 1 ⊗1+1⊗H 2 , with H a , a = 1, 2 being on-qubit Hamiltonians. For simplicity, in the present work we shall fix two-qubit local Hamiltonian in the form of the coupling of both systems to an external magnetic field along third axis i.e.
In the following we shall consider unitary evolutions generated by fourparameter family of Hamiltonians In the next Section we get the explicit parametrizations of sets quantum states generated by evolution of selected initial states and obtain relevant manifolds of dimension depending on the initial states. Then we shall describe Riemannian geometry of obtained manifolds introducing the Fubini-Study metrics for each case. Then we study changes of geometry resulting from small perturbation in the original Hamiltonian by switching on an additional weak magnetic field along the first-axis. Furthermore, in the Section 5, we discuss the characterization of entanglement for each of the obtained manifolds using the squared concurrence.
Let us classify the possible parametrizations as follows: C1. For η 1 = η 2 = 0 and η 3 = 0, η 4 = 0, the state (12) takes the form It is easy to see that this state depends only on parameters c + and satisfies the following periodic condition C2. For η 3 = η 4 = 0 and η 1 = 0 or η 2 = 0 we obtain the states (7) or (8) which depend only on parameters φ with periodic condition C3. For η 3 = η 4 = 0 and non-zero η 1 , η 2 , the family of states is defined by the parameters ω and φ as follows with the following periodic conditions C4. For η 1 = 0 or η 2 = 0 and η 3 = 0 or η 4 = 0 the family of states is defined by two parameters where c = 2c 3 + (−1) j c + + (−1) l+1 ω. Here l = 1, 2, j = 3, 4. The states satisfy the following periodic conditions C5. If η 1 , η 2 are non-zero, and η 3 = 0 or η 4 = 0 then the family of states is defined by three parameters Here c = 2c 3 + (−1) j c + . In this case the states satisfy the following periodic conditions C6. For η 1 = 0 or η 2 = 0, and nonvanishing η 3 , η 4 the family of states is defined by three parameters Here c = 2c 3 + (−1) l+1 ω. In this case we have the following periodic conditions C7. In the general case, when all parameters η 1 , η 2 , η 3 and η 4 are non-zero, we have the state defined by expression (12) with the following periodic conditions Analyzing above cases we can conclude that all obtained quantum state manifolds are closed. However, in the first two cases quantum state manifold is one-parametric, in the third and fourth cases it is two-parametric, in the fifth and sixth case the manifold is defined by three parameters, and in the last case we have the four-parameter manifold. Let us study the Fubiny-Study metric of these manifolds,

The Fubini-Study metric of quantum state manifolds
The Fubini-Study metric is defined by the infinitesimal distance ds between two neighbouring pure quantum states |ψ(ξ µ ) and |ψ(ξ µ + dξ µ ) [4] where ξ µ is a set of real parameters which define the state |ψ(ξ µ ) . The components of the metric tensor g µν have the form where γ is an arbitrary factor which is often chosen to have value of 1, √ 2 or 2 and As we have previously noted, the states (12) are defined by four real parameters. Using definition (26) we obtain the components of the metric tensor with respect to parameters (ω, φ, c 3 , c + ) where . From the explicit form of the metric tensor we see, that in the case of the magnetic field switched off, one of the parameters disappears (φ = 0) and the the manifold becomes flat. It is the result of the reciprocal commutativity of the interaction terms in the Hamiltonian (6). It is worth noting, that if c 1 = c 2 and c 3 = αc + /2 than φ = π/2 and we obtain the metric of the two-parameter manifold as in [18] where α is some real number that determines the anisotropy of the system. If α = 1 then we obtain the Fubini-Study metric of the quantum state manifold of isotropic Heisenberg model [17]. The metric (28) can be redued to the diagonal form with the use of the new parameters after the following transformation where Let us additionally assume that Then in these new parameters the metric (28) takes the following form It is evident that the ratio between the parameters of the initial state has the influence on the components of the metric tensor. For instance if η 3 = η 4 then η − 34 = 0 and g c ′ + c ′ + takes the maximal value for the specific initial state. Let us analyze in detail the geometry of the manifold defined by above metric for the cases considered in the previous Section: 1. In the first case the manifold is defined by the parameter c + ∈ [0, π] and metric tensor is reduced to g c + c + component with η + 34 = 1. This is the metric of the circle of the radius γ 1 − η − 34 2 /2.

2.
In the second case the manifold is defined by parameter φ ∈ [0, 2π] and metric tensor is reduced to g φφ with η + 12 = 1. This metric also describes the circle of the radius γ/2.

4.
Here we have also two-parametric manifold defined by parameters φ ∈ [0, 2π], c ∈ [0, 2π] and described by the following metric tensor As we can see that components of the metric tensor do not depend on the parameters φ and c. This means that manifold is flat. Taking into account periodic conditions (17) we conclude that it is a torus.

5.
In the fifth case the manifold is three-parametric and defined by the parameters θ ∈ [0, π], φ ′ ∈ [0, 2π], c ′ ∈ [0, 2π]. In the diagonal form the metric tensor components g (0) θθ , and g φ ′ φ ′ are defined by expression (32) and other component takes the form where c ′ is related to the parameter c from (20) by the following formula The manifold which we obtain here is the product of the sphere of radius γ η + 12 /2 in parameters θ, φ ′ and of the circle of radius 2γ η + 12 |η j | in parameter c ′ .
6. In the case C6 we obtain a manifold with the metric tensor in the diagonal form and the component g is defined by the expression (32). Therefore we obtain a three-parameter manifold defined by To diagonalize this metric we use the following transformation where c is defined as for the state (22). So, this manifold can be expressed by circle of radius γ|η l |/2 in parameter φ and torus in parameter c ′ , c ′ + . 7. In the general case the metric is defined by expression (28) or (32). This manifold consists of two submanifolds, namely, the sphere of radius γ η + 12 /2 in parameters θ ∈ [0, π], φ ′ ∈ [0, 2π] and torus in parameter
5 Entanglement characterization of two-qubit quantum state manifolds M |ψ (0) In the present section, using the squared concurrence as an entanglement measure, we shall study the entanglement of states belonging to the manifolds obtained in the Section 2. The concurrence of a pure state of bipartite twolevel system is defined as follows [35,36] where a, b, c and d are defined by expression The squared concurrence for state (12) takes the form Let us calculate the squared concurrence for the families of states discussed in the previous sections: 1. In the case C1 the concurrence takes the form For η 3 = |η 3 | and η 3 = |η 3 |e iχ , where χ ∈ [0, 2π]. we obtain the maximally entangled state if c + = 1/4 [(2n + 1)π − 2χ], where n ∈ Z.
2. The squared concurrence in the case C2 takes simple form 3. For the C3 family of states the manifold is defined by two parameters. The entanglement of the states is described by the following expression Similarly as in the previous case C1 we put η 1 = |η 1 | and η 2 = |η 2 |e iχ . As we can see, regardless of the initial state the maximally entangled state we obtain when φ = 0 and ω = 1/4 [(2n + 1)π − 2χ].
5. Fort the C5-family of states, to simplify the calculations, we analyze the case when η 1 = η 2 and we put η 1 = |η 1 |, η j = |η j |e iχ . The squared concurrence takes finally the form The conditions for peparation of maximally entangled states are given in Table 2.

Conclusions
The geometric characterization of the state manifold of quantum system is of great value, but for compound systems such task becomes very complex when addressed in general setting.
In the present work we have studied quantum state manifolds obtained by means of the unitary evolution defined by large family of physically interesting Hamiltonians. Despite the knowledge of the whole set of the two-qubit quantum state space it is important to know what manifolds lying inside this set can be reached using the evolution governed by the realistic Hamiltonians. The geometry of such obtained quantum state spaces is of Riemannian type defined by the Fubini-Study metrics depending on initial conditions and parameters entering the definition of the families of Hamiltonians. We have given the classification of possible state manifolds and thoroughly discussed the explicit description of two-qubit unitary orbits generated by physically relevant Hamiltonians. The relevant Fubini-Study metrics were obtained with the use of explicit parametrizations.
It is worth noting, that we also studied the question how obtained geometries are modified by the noncommutative linear perturbation term included into the original Hamiltonian. We describe its influence on the scalar curvature of the relevant state spaces. In some cases the answer turns out to be nontrivial.
As an important physical characterization of the considered systems we have studied the degree of entanglement of states for all obtained quantum state spaces and we have provided conditions for obtainig maximally entangled state in each case, where the concurrence is used as an entanglement monotone.