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Bipartite entanglement and control in multiqubit systems

The effect of the control on bipartite entanglement is discussed from a geometric viewpoint for a nuclear magnetic resonance (NMR) system as a model of the n-qubit control system. The Hamiltonian of the model is the sum of the drift and control Hamiltonians, each of which describes the interaction between pairs of qubits and between one of qubits and an external magnetic field, respectively. According to a bipartite partition, the Schroedinger equation for the NMR system is put in the matrix form. This paper gives a solution to the Schroedinger equation with the assumptions of small coupling among qubits and of constant controls. The solution is put in the form of power series in small parameters. In particular, in the case of the two-qubit NMR system, the drift and control Hamiltonians are shown to be coupled to work for entanglement promotion, by examining solutions to the Schroedinger equation in detail. The concurrence, a measure of entanglement, is evaluated along the solution for a small time interval in order to observe that the control effect appears, not at the first-order terms in t, but at the higher-order terms in t. The evaluated concurrence also suggests which control makes the two-qubit more entangled or less.


Introduction
Entanglement plays many roles in quantum computation and quantum information theory, and is a resource for quantum communication. Recently, the geometry of entanglement has been developed [9,[11][12][13][18][19][20][21][22]. The algebraic study of entanglement also has been made to seek for invariants that quantify multiqubit entanglement [4,10,26]. The geometry of entanglement in few-qubit systems was studied in [18,22] and in [20,21] in terms of the Hopf fibration and the complex projective spaces, respectively.
The first author [11] showed that the concurrence for the two-qubit system can be characterized as the coordinate of the factor space of the state space by the action of the local transformation group. He continued to study bipartite entanglement in multiqubit systems [12]. In particular, the sets of separable and maximally entangled states in bipartite entanglement are identified, and the concurrence is extended to a measure of bipartite entanglement in multiqubit systems. Further, the distance between the set of states of prescribed entanglement and the set of separable states is measured. The bipartite entanglement is measured also for the process of Grover's search algorithm [13]. For the bipartite entanglement, see also [9,19].
On the other hand, from a practical standpoint, it is of much importance to control quantum states in order to create entangled states. Many of papers on control problems for multiqubit systems deal with control systems on unitary groups [1][2][3]5,6,[15][16][17]23].
The theory of control on compact Lie groups has been established [14]. A sufficient condition for controllability of the control system on SU(N) was given in [3]. The Cartan decomposition of the Lie algebra and the symmetric space are techniques frequently used for studying control systems [15][16][17].
Two notions as to controllability of multiqubit systems were defined and studied by Albertini and D'Alessandro [1,2], which are "pure state controllability" and "equivalent state controllability". In [2], equivalent state controllability was shown to be equivalent to pure state controllability. In the case of multiqubit systems lying in electro-magnetic fields, they [1] gave a necessary and sufficient condition for the system to be (pure state) controllable. The condition is described in terms of the graph associated with the interactions of qubits, which is related with the Lie algebraic structure of the system.
Their method is based on the control theory on compact Lie groups [6,14]. As will be pointed out in Appendix B, their model Hamiltonian is a bit different from that adopted in this article.
The aim of this article is to study entanglement and control of nuclear magnetic resonance (NMR) multiqubit systems from a geometric viewpoint. Since the notion of entanglement cannot be defined in terms of the whole transformation group, the control theory on compact Lie groups will not fit in with the study of entanglement and control.
The state space of a finite-dimensional quantum system is a finite complex vector space with a constraint required for the probability density. In this article, the state space is a linear space of complex matrices with the constraint from the probability density.
The reason for the choice of the present state space is that the entanglement measure like the concurrence is easy to treat on this state space. The state space admits the two-sided action of the local transformation group U(2 ℓ ) ⊗ U(2 m ), where m and ℓ are determined by the bipartite partition of the multiqubit (C 2 ) ⊗n = (C 2 ) ⊗ℓ ⊗ (C 2 ) ⊗m .
As will be stated later, the entanglement of the multiqubit system is invariant by the action of the local transformation group. Hence, in order to make the multiqubit more entangled, the control is required to move the state in the direction transverse to the orbit of the local transformation group. However, for NMR systems to be treated in this article, the control vectors generated by the control Hamiltonian are tangent to the orbit mentioned above, so that they would not make the system more entangled, if no coupling were made with other vector fields. Since the total Hamiltonian of the NMR system is the sum of the control and drift Hamiltonians, one may expect the coupling between them to occur. The objectives in this article are to set up the control problem on the above-mentioned state space and to show, by solving the Shrödinger equation with constant controls, that the control and drift Hamiltonian are coupled together to give rise to vector fields transverse to the group orbit for the entanglement promotion of the multiqubit system. The organization of this paper is as follows: Section 2 contains a geometric setting up of n-qubit systems. According to the bipartite partition (C 2 ) ⊗n = (C 2 ) ⊗ℓ ⊗ (C 2 ) ⊗m with ℓ+m = n, the state space M is viewed as the space of 2 ℓ ×2 m normalized matrices C, on which the local transformation group G = U(2 ℓ ) ⊗ U(2 m ) acts. The action of G leaves invariant the bipartite entanglement of the n-qubit. Linear operators on the n-qubit system are mapped to vector fields on the state space M. Lie brackets among those vector fields are given explicitly. In Section 3, the Hamiltonian of an NMR model is given, which is split into two operators, the drift HamiltonianĤ d and control HamiltonianĤ c . TheĤ d andĤ c describe the interactions between pairs of n qubits and between one of qubits and an external magnetic field, respectively. In addition, a solution to n-qubit NMR systems with a certain type of drift Hamiltonians is given with the assumption that the coupling constants between qubits are small enough. The solution is put in the form of power series in small parameters related to the coupling constants. It will be shown that the vector fields associated with the drift and control Hamiltonians are coupled actually. In Section 4, the coupling between those vector fields is extensively studied on the two-qubit system to observe that entanglement is actually promoted. Discussion runs as follows: A solution C(t) of the Schrödinger equation in the matrix form is obtained as a power series in the small coupling constant.
The solution is examined in detail to show that the vector fields associated with the control and drift Hamiltonians are coupled indeed to induce vector fields transverse to the G-orbit. The effect of the control on entanglement is verified by estimating the difference between the concurrences of the solutions for the controlled and uncontrolled systems for a small time interval in the case that the initial state is a diagonal matrix Λ = diag (λ 1 , λ 2 ). The result is that the effect of the control is so slow that it emerges at the third-order term in t. Concluding remarks are made in Section 5, in which the Cartan decomposition method is related to the present method. Appendix A gives a list of Lie brackets among vector fields related with the drift and control Hamiltonians for the two-qubit NMR system. Appendix B contains the proof of the controllability of the n-qubit NMR system. The method for proof is different from that in [1].
2 Geometric setting for bipartite entanglement
A vector |Ψ is said to be entangled if it is not separable.
According to the bipartite decomposition (C 2 ) ⊗n = (C 2 ) ⊗ℓ ⊗ (C 2 ) ⊗m , a state vector |Ψ can be rewritten in the form where A = (j 1 , · · · , j ℓ ) and B = (j ℓ+1 , · · · , j n ). This expression gives rise to the isomorphism of (C 2 ) ⊗n to the vector space C 2 ℓ ×2 m of 2 ℓ × 2 m complex matrices, (2.1) The C 2 ℓ ×2 m is endowed with the inner product given by tr(C * 1 C 2 ) for C 1 , C 2 ∈ C 2 ℓ ×2 m . Since |Ψ is normalized, the corresponding matrix C = ι(|Ψ ) is subject to the condition tr(C * C) = 1. Thus, the state space for the n-qubit system is defined to be which is diffeomorphic with the (2 n+1 − 1)-dimensional sphere. The state space M becomes a Riemannian manifold equipped with the standard metric where the tangent space to M at C is identified with

Bipartite entanglement
In this subsection, we make a review of a measure of bipartite entanglement introduced by the first author [11,12].
The separability condition of a state vector |Ψ can be expressed as rank CC * = 1, where C is the matrix corresponding to |Ψ . Since the 2 ℓ × 2 ℓ matrix CC * is positive semi-definite together with tr(CC * ) = 1 and ℓ ≤ m, the condition of rank CC * = 1 is equivalent to det(I 2 ℓ − CC * ) = 0, where I 2 ℓ denotes the 2 ℓ × 2 ℓ identity matrix. The first author [11,12] showed that the quantity serves as a measure of bipartite entanglement, and as an extension of the concurrence which is well-known as a measure of entanglement in two-qubit system. In fact, one has F (C) = det(CC * ), the square of the concurrence, for the two-qubit with ℓ = m = 1.
The quantity F (C) is non-negative, and vanishes if and only if C is separable. Further, the F takes the maximal value (1−1/2 ℓ ) 2 ℓ if and only if all eigenvalues of CC * coincide.
If F attains the maximal value at C, the matrix C ∈ M or the corresponding state vector |Ψ is said to be maximally entangled with respect to the bipartite decomposition where g ∈ U(2 ℓ ) and h ∈ U(2 m ). This action is isometric with respect to (2.2). We remark here that the map ι is G-equivariant, namely, ι −1 (gCh T ) = (g ⊗ h)ι −1 (C). As is easily verified, the function F is invariant under the G-action, so that the G-action We now consider the quotient space M/G. An arbitrary matrix C ∈ M is decomposed into the product where 0 is the 2 ℓ × (2 m − 2 ℓ ) zero matrix, and λ 2 1 , · · · , λ 2 2 ℓ are the eigenvalues of CC * . In the following, we assume that λ 1 ≥ · · · ≥ λ 2 ℓ ≥ 0. The quantities λ 1 , · · · , λ 2 ℓ are called the singular values of C. The expression (2.5) is equivalent to the Schmidt decomposition of a state vector |Ψ . From this decomposition, the quotient space M/G can be identified with The sets of separable and maximally entangled states project through the natural projection M → M/G ; C → (λ 1 , · · · , λ 2 ℓ ) to the points (1, 0, · · · , 0) and (1/ √ 2 ℓ , · · · , 1/ √ 2 ℓ ) ∈ R 2 ℓ , respectively, both of which lie on the boundary of the quotient M/G.
The state space M is stratified into strata, according to G-orbit types [12], which are determined by the isotropy subgroups G C := {g⊗h ∈ G | gCh T = C}. The isotropy subgroup and the orbits are already studied for four-qubit systems in detail, and for multiqubit systems in brief [12]. We now summarize the results in a refined form. (1) If CC * is non-singular, the isotropy subgroup G C is isomorphic to where the product U(m 1 ) × · · · × U(m N ) is viewed as a subgroup of U(2 ℓ ), and the where V 2 ℓ (C 2 m ) denotes the Stiefel manifold of orthonormal 2 ℓ frames in C 2 m . 1 (1-1) In the generic case that singular values of C are all distinct and non-zero, G C and O C take, respectively, the form (1-2) For a maximally entangled state C, G C and O C become, respectively, (2) If CC * is singular, the group G C is isomorphic to (2.8) and the orbit O C is diffeomorphic to (2-1) In particular, if C is separable, G C and O C are, respectively, of the form

Vertical and horizontal subspaces
This subsection deals with the infinitesimal action of G = U(2 ℓ ) ⊗ U(2 m ). For ξ ∈ u(2 ℓ ) and η ∈ u(2 m ), the fundamental vector field associated with ξ ⊗ I 2 m + I 2 ℓ ⊗ η ∈ g = Lie(G) is determined through X ξ⊗I 2 m +I 2 ℓ ⊗η (C) := d dt t=0 e tξ Ce tη T = ξC + Cη T . (2.10) The vertical subspace V C is defined to be the tangent space T C O C at C to the orbit O C , and formed by the fundamental vector fields evaluated at C, The dimension of V C is given by dim V C = 2 2ℓ + 2 2m − 1 − dim G C . In particular, from The horizontal subspace H C of T C M is defined to be the orthogonal complement of V C with respect to the metric (2.2). A vector X ∈ T C M is horizontal, if and only if X, ξC + Cη T = 0 for any ξ, η ∈ u(2 ℓ ), so that the horizontal subspace at C ∈ M is expressed as if C is maximally entangled.

The Lie algebra of linear vector fields on M
In discussing control problems, we need to calculate Lie brackets of vector fields on In particular, we have to work with the Lie brackets of vector fields of the form where A ∈ C 2 ℓ ×2 ℓ and B ∈ C 2 m ×2 m do not need to be Hermitian or skew Hermitian at Since this bracket is verified to satisfy the Jacobi identity, the linear space C 2 ℓ ×2 ℓ ⊗ C 2 m ×2 m is endowed with a Lie algebraic structure. This Lie algebra is isomorphic to In fact, f is shown to be bijective and homomorphic, f Now we are in a position to state the following proposition: This means that the correspondence A ⊗ B → X A⊗B gives rise to an anti-isomorphism Corollary 2.3. In the two-qubit system with ℓ = m = 1, one has Here the σ j , j = 1, 2, 3, denote the Pauli matrices defined as Proof of Proposition 2.2 . Let ϕ t A⊗B denote the flow generated by the vector field X A⊗B on C 2 ℓ ×2 m . We identify (C 2 ) ⊗n with the space C 2 n of 2 n -dimensional column vectors, and the isomorphism ι : (C 2 ) ⊗n → C 2 ℓ ×2 m with an isomorphism of C 2 n to C 2 ℓ ×2 m . We note here that the following diagram commutes, This means that By the definition of the Lie bracket together with (2.14), we obtain We now apply (2.13) for the two-qubit system. We take, for instance, Other Lie brackets will be found in Appendix A.

Controls in NMR systems
Sections 3 and 4 deal with control problems in nand 2-qubit systems, respectively.
In this paper, we adopt a nuclear magnetic resonance (NMR) system as a quantum computation model from n-qubit systems [16].
The total Hamiltonian operatorĤ of the NMR system is the sum of two operators, whereĤ d andĤ c are called the drift and control Hamiltonians, and expressed aŝ respectively. Here, J αβ ∈ R are coupling constants determining the strength of the interactions between the α-th and β-th qubits. The v for j = 1, 2, 3, and α = 1, · · · , n, with I the 2 × 2 identity matrix. The controls v (α) 1 and v (α) 2 are assumed to be piecewise constant functions in t.
The Schrödinger equation for the NMR system is expressed as where the natural unit system has been adopted, so that the Plank constant is equal to one. Since we are interested in bipartite entanglement, we have to put the Schrödinger equation in the form of matrix equation. According to the isomorphism (C 2 ) ⊗n ∼ = C 2 ℓ ×2 m with n = ℓ + m, the vector fields associated with the drift and control Hamiltonians are expressed, respectively, as and where the symbols with square brackets like [σ As is easily verified, −iH d and −iH c are vector fields on M. In what follows, we will refer to −iH d (C) and −iH c (C) as the drift and control vector fields, respectively.

A solution to the NMR system
We show that Eq. (3.5) can be solved with the assumption that the coupling constants are small enough and that the controls are constant. In view of (3.4), we may put the drift and control vector fields in the form, N, are constant Hermitian matrices, and where ξ 1 ∈ su(2 ℓ ), ξ 2 ∈ su(2 m ) are constant controls. The Schrödinger equation we are to solve iṡ Since the coupling constants ε k are sufficiently small, we may suppose that the solution can be expanded into a power series in ε k , k = 1, · · · , N, By substituting this expansion into (3.7) and comparing the left and right hand-sides, we obtain the series of differential equationṡ To solve the above equations, we introduce new unknown functions by Q n (t) = (e −tξ 1 ⊗ e −tξ 2 ) · C n (t). Then Q n (t) with n ∈ (Z ≥0 ) N prove to satisfy the equationṡ 2 )Q n−e k for n = 0.
These equations for Q n are inductively integrable. The first (N+1) of solutions are given by Q 0 (t) = Q 0 (0) and In order to write down solutions in a compact form, we introduce N linear operators Then, in terms of these operators, Q e k (t), k = 1, · · · , N, take the form where Q 0 (0) in the right-hand side is viewed as a constant function.
We proceed to, say, Q e 1 +e 2 (t). The equation for Q e 1 +e 2 (t) is expressed aṡ and integrated to yield In a similar manner, the Q n (t) is inductively integrated and expressed as where |m| := m 1 + · · · + m N , and where T K : Hence, we have obtained the following.
is put in the form of power series in ε k , k = 1, · · · , N,

10)
where we have used the fact that C(0) = n ε n C n (0) = n ε n Q n (0).
If the control is piecewise constant in t, the continuation along time of solutions of the form (3.10) with respective constant controls will yield a solution to (3.7).

Controllability of the NMR systems
From Eq. (3.6), we observe that the control vector field are vertical, so that the measure defined in (2.3) does not change along this vector field. Hence, the control would not make the NMR system entangled without the coupling with the drift vector field. With this observation in mind, we gain insight into (3.10) to understand how the coupling occurs. We first look into the operators T k defined in (3.9). Writing out the integrand of (3.9) in the form of a power series in s, we obtain 2 ]] T + · · · ds.
By applying Proposition 2.2, we can put the above equation in the form ]](F (s)) ]](F (s)) + · · · ds. (3.11) Since the expansion (3.11) means that the drift and control vector fields are coupled indeed, and since the operator T K is the composition of these T k , Eq. (3.10) shows that the drift and control vector fields work together to give rise to the solution. If the coupling between the drift and control vector fields generates vector fields with non-vanishing horizontal components, the NMR system will get more entangled by controls.
Incidentally, if the NMR system is controllable, then any initial state can be trans- is isomorphic to su(2 n ). For the time being, we are interested in the coupling between the drift and control vector fields, and postpone the controllability problem to Appendix B, in which we will give a necessary and sufficient condition for controllability.
However, it is of great help to apply the controllability theorem to the NMR system.
To state the theorem, we need to introduce the notion of spin graph. Here, the spin graph associated with a given drift HamiltonianĤ d is defined to be the un-oriented graph that has n nodes and edges joining the nodes labeled as α and β such that J αβ = 0, where nodes and edges represent spin-1 2 particles and interactions among spin- 1 2 particles such that J αβ = 0, respectively. In terms of the spin graph, the controllability theorem is stated as follows (see Theorem B.8): The NMR system is controllable, if and only if the associated spin graph is connected. In view of this, we may set some of J αβ in (3.4) to be zero. We here take otherwise. (3.12) Then, the associated spin graph is connected, so that the NMR system with these coupling constants is controllable. The drift vector field in (3.4) is now expressed as In this case, one has N = 2, and where ξ α , ξ β ∈ span R {iσ 1 , iσ 2 }.

Controls in two-qubit NMR systems
In this section, we deal with the two-qubit case. Our system is expressed as where J > 0 is the coupling constant between the two qubits, and where ξ 1 , ξ 2 are constant controls taking values in span R {iσ 1 , iσ 2 }. In comparison to (3.7), this equation is easier to solve, and we can investigate solutions in detail to understand how controls make the NMR system more entangled.

Two-qubit NMR systems without control
To understand why one needs controls to make the NMR systems entangled, we look into the uncontrolled system with ξ 1 = ξ 2 = 0 in (4.1). If ξ 1 = ξ 2 = 0, Eq.  We note in addition that F (C(t)) = (1 − cos Jt)/8 in this case. In contrast with this, if D(C(0)) = 0, then C(0) is separable, and the state C(t) is always separable because of 0 ≤ F (C(t)) ≤ D(C(0)) = 0. An example of such an initial state is This initial state never gets entangled without controls. From these facts, we observe that the separable states 1 2 (|0 + |1 ) ⊗ (|0 + |1 ) and e iθ |0 ⊗ |0 are of different nature with respect to the drift Hamiltonian.

Behaviours of solutions in the two-qubit NMR system
We turn to the controlled system (4.1) with ξ 1 and ξ 2 non-vanishing constants. Since the associated spin graph is connected, the two-qubit NMR system is controllable, so that any separable state can get entangled. This implies that the coupling between the drift and control vector fields must occur.
We solve the Schrödinger equation (4.1) as a power series of the coupling constant J > 0. As in the previous section, we assume that the J is small enough, and that the solution C(t) of (4.1) can be expanded into a power series in J as C(t) = ∞ n=0 J n C n (t). Substituting this into (4.1) brings about the series of differential equations for C n , (4.6) These equations can be integrated inductively, like (3.8). In fact, from the above equations, differential equations for e −tξ 1 C n (t)e −tξ T 2 are easily obtained and integrated to give where we have used the assumption that C(0) = ∞ n=0 J n C n (0). The following proposition is easy to prove. Proposition 4.1. The power series (4.7) uniformly converges to a function P (t), with which the solution of the NMR system (4.1) is given by C(t) = e tξ 1 P (t)e tξ T So far we have found the solution as the power series (4.7) in the coupling constant J. We now analyze the solution in order to see in detail that the coupling between the drift and control vectors actually occurs to make the NMR system more entangled.
We denote the n-th order term in J of (4.7) by P n (t). Then, the solution is expressed as C(t) = ∞ n=0 J n e tξ 1 P n (t)e tξ T 2 . This explains how the controls ξ 1 and ξ 2 contribute to entanglement promotion of the system. The zeroth-order term does not make the system more entangled because of F (e tξ 1 P 0 (0)e tξ T 2 ) = F (C(0)). Contrarily, the higherorder terms in J are expected to make the NMR system entangled in general. We look into (4.7) in detail. The first-order term P 1 (t) of (4.7) is put in the form (T 1 C(0))(t).
Then, as is seen from (3.11), P 1 (t) is expressed as (4.8) This shows that the drift and control vector fields are coupled indeed. We note that ] by the Jacobi identity and that [X ξ 1 ⊗I , X I⊗ξ 2 ] = 0.
By a similar computation, the second-order term P 2 (t) of (4.7) is written out as We formally denote the first and second terms of the right-hand side of the above equa- , respectively. The first term t 2 2! (−X iσ 3 ⊗σ 3 ) 2 comes from the Taylor expansion of the flow ϕ t −iσ 3 ⊗σ 3 with respect to t. It is inductively shown that the n-th order term P n (t) is expanded into a power series of t with the lowest-order term t n n! (−X iσ 3 ⊗σ 3 ) n , which comes from the Taylor expansion ϕ t −iσ 3 ⊗σ 3 (C(0)) = ∞ n=0 t n n! (−X iσ 3 ⊗σ 3 ) n | C(0) . Hence, we have If t is sufficiently small, the solution C(t) of (4.1) allows of the approximation of the form where ϕ Jt −iσ 3 ⊗σ 3 (C(0)) is the solution of the uncontrolled system dC/dt = −JX iσ 3 ⊗σ 3 with the initial state C(0).

Bases of vertical and horizontal subspaces
So far we have shown that the drift and control vector fields are coupled. We now wish to show that the coupling generates vector fields with non-vanishing horizontal components. To this end, we have to look into vertical and horizontal vector fields in detail.
Since the subspaces V C and H C depend on the stratum to which C belongs, we make a brief review of a stratification of the state space M for the two-qubit system after [12]. According to singular values λ 1 (C) ≥ λ 2 (C) of C ∈ M, the state space M of the two-qubit system is stratified into three G-invariant strata as (4.10) The strata M 0 and M 2 are the sets of separable and maximally entangled states, respectively.
We now deals with bases of the subspaces V C and H C . We take up a typical matrix Λ = diag (λ 1 , λ 2 ) with λ 1 ≥ λ 2 , to which all the matrices with the same singular values are translated by the U(2) ⊗ U(2) action. Let us start with the vertical subspace V Λ .
A basis of V C are formed from that of V Λ by the differential map (g ⊗ h) * .
From these propositions, we observe that when Λ = I/ √ 2 ∈ M 2 ∼ = U(2), the vertical subspace V Λ is identified with the space u(2) of 2 × 2 skew Hermitian matrices, and that the horizontal subspace H Λ with Λ = I/ √ 2 is identified with the space isu (2) of traceless Hermitian matrices. Further, the basis vectors of H Λ given in Proposition 4.4 (iii) are alternatively expressed as Remark. The vector fields X iσ 2 ⊗σ 1 and X iσ 1 ⊗σ 1 are generated, by taking the Lie brackets among X iσ 3 ⊗σ 3 , X iσ 1 ⊗I and X iI⊗σ 2 , (4.14) respectively. The first and second equations of the above imply that, if doubly coupled, the drift and control vector fields generates a horizontal vector at C = Λ ∈ M 1 and another horizontal vector at C = Λ ∈ M 0 , respectively.
The O(J 2 t 3 )-term is also vertical at Λ, as is shown by The O(Jt 3 )-term in (4.9) is significant for entanglement promotion. The vector are expressed as are vertical. In contrast with this, from Proposition 4.4 and Remark after it, it turns out that the tangent vector [X ξ 1 ⊗I , [X I⊗ξ 2 , −X iσ 3 ⊗σ 3 ]] Λ contains the horizontal components X iσ 2 ⊗σ 1 (Λ), X iσ 1 ⊗σ 2 (Λ). In particular, if ξ 1 and ξ 2 are taken as ξ 1 = iσ 1 and ξ 2 = iσ 2 , respectively, the tangent vector in question is horizontal, This shows that the drift vector field X iσ 3 ⊗σ 3 and the control vector fields X ξ 1 ⊗I , X I⊗ξ 2 are coupled to generate a horizontal vector at Λ. We note further that the horizontal vector emerges at the third-order term in t, if Λ = I/ √ 2.
We wish to evaluate the entanglement to confirm that the two-qubit system gets more entangled. As our entanglement measure and the concurrence are equivalent for the two-qubit system on account of F (C(t)) = | det C(t)|, we here use the concurrence as a measure. From Eq. (4.9), the concurrence is approximately evaluated where we have used the following facts; (i) F (e −tξ 1 C(t)e −tξ T 2 ) = F (C(t)), (ii) [X ξ 1 ⊗I , −X iσ 3 ⊗σ 3 ] and [X I⊗ξ 2 , −X iσ 3 ⊗σ 3 ] are vertical at Λ if Λ is not the maximally entangled state ] are vertical at Λ, and Since the derivative of the measure F is given by one has Hence, substituting this into (4.17), we have the following estimate of the concurrence, for a sufficiently small t > 0. It is to be noted that the effect of controls on the concurrence emerges actually at the third-order term in t. We have to note that for . This means that the state Λ = diag (1, 0) gets entangled slowly by the controls.
We turn to the case that the initial state C(0) is not diagonal. Let us be reminded of the fact that when the C(0) is diagonal, the O(Jt 2 )-term in (4.9) is vertical. However, if C(0) is not diagonal, the O(Jt 2 )-term is not vertical, so that Eqs. (4.15) and (4.18) are put together to yield the contribution to entanglement promotion by Re e −iθ (c 01 + c 10 )(z 1 c 00 + z 2 c 11 ) , (4.20) where z α = x α + iy α for α = 1, 2, and where e iθ is defined by det C = e iθ | det C| = e iθ F (C). From (4.20), the O(Jt 2 )-term identically vanishes for arbitrary ξ 1 and ξ 2 if c 00 = c 11 = 0 or c 01 + c 10 = 0, but it takes a non-zero value in general, if c 01 + c 10 = 0 and (c 00 , c 11 ) = (0, 0).

Concluding remarks
We have discussed how the control and drift vector fields are coupled to make It would be of help to state the relation of our theory to the Cartan decomposition of the Lie algebra. As for the two-qubit systems, the associated Lie algebra is su(4), of which the Cartan decomposition is given by The subalgebra k generates vertical tangent vectors, and some of elements in p span the horizontal subspace at Λ = diag (λ 1 , λ 2 ). For example, if λ 1 = λ 2 , one has a horizontal vector X iσ 1 ⊗σ 2 (Λ) (see Props. 4.3 and 4.4). However, the vectors associated with p are not always horizontal. As is seen from (4.16), X iσ 2 ⊗σ 3 (Λ) and X iσ 1 ⊗σ 3 (Λ) are vertical if Λ = I/ √ 2. In [27], as to non-local operation, they state that k can be viewed as the local part in su(4) and p as the non-local part. In this sense, the horizontal vectors are of non-local nature, and the horizontal subspace makes the non-locality quite sharp.
This is because the entanglement gets promoted most efficiently in the direction of horizontal vectors.
We comment on the motivation behind this study, which is related to the decomposition method for the analysis of quantum control problems on compact Lie groups. The decomposition method is of much use to design a control in order to construct a desired unitary operator [15][16][17], and then has been developed [7,8]. In these situations, the control (or local operation) works instantaneously, and the drift (or non-local) Hamiltonian is assumed to be negligible while the control works. Under this assumption, the time evolution operator U t = e −itĤ can be decomposed into the product of the local and non-local operators, where L 0 , · · · , L N ∈ SU(2 ℓ ) ⊗ SU(2 m ) are generated by the control Hamiltonian, and where τ 1 , · · · , τ N > 0 are time intervals for which the control Hamiltonian vanishes.
In this view, the bipartite entanglement is not concerned by the coupling between the drift and control Hamiltonians in the time evolution. Our motivation is to consider the coupling in the evolution without the above assumption, and to investigate the relation of the coupling to the bipartite entanglement. The work [24] by Romano shares the motivation with us.
In [24], the decomposition of the unitary group SU(4) is used to study entanglement magnification. The time evolution U t is decomposed into U t = L t A t K t , where L t , K t ∈ SU(2) ⊗ SU(2) and A t ∈ H with H the Cartan subgroup of SU(4). The author says that L t is irrelevant, and claims that the non-local operator A t makes a contribution to entanglement magnification together with a help of K t . The decomposition method is sharp to study the entanglement in the evolution, because the evolution operator is completely separated into the "entangling part" A t and the two local operators L t and K t . But, the process of entanglement promotion does not explicitly appear in the decomposition, since not only A t but also L t , K t take in the coupling between the drift and control Hamiltonians. In fact, the A t , L t and K t are all unknown until the time evolution operator U t is integrated in an explicit form.
On the other hand, the method of this article is represented in a familiar form with bipartite entanglement, and explains how the coupling between the drift and control Hamiltonians occurs in the time evolution. See (3.10), (3.11), (4.7) and (4.9). In particular, the right hand-side of (4.9) is not discussed in the context of (5.1). In a comparison to [24], our method allows one not to solve the time evolution of the system, although it is not sharp in the sense that the terms in (4.9) contain vertical (or local) components.
The formulas, (3.10) together with (3.11), and (4.9) also show how the effect of the controls spreads among qubits in the dynamical system. In contrast with this, the proof of the controllability of the NMR system would show the spread of the controls among qubits in a "static" situation. For the sake of self-containedness, we give the proof of the controllability of our NMR system (see Theorem B.8) in a method different from that in [1]. As is pointed out in App. B, the NMR system is controllable if and only if the associated spin graph is connected.
For comparison's sake, we here quote the drift and control Hamiltonians studied in [1], which are given bŷ respectively, where M αβ , N αβ , P αβ ∈ R are coupling constants and r α ∈ R are the gyromagnetic ratio of the α-th spin-1 2 particle (or qubit). In our case of (B.2), it is assumed that each of spin-1 2 particles can be stimulated by the only two components of the magnetic field. In contrast with this, in the case of (B.4), all the spin-1 2 particles are assumed to be stimulated simultaneously by all the three components of the magnetic field with possibly different gyromagnetic ratios. While our drift Hamiltonian (B.1) contains only σ 3 factors, but theirs (B.3) has all the σ j factors. In [1], they prove that the NMR system with the HamiltonianĤ ′ 0 + 3 j=1 u jĤ ′ j is controllable if and only if the associated spin graph is connected.
Our objective in what follows is to show that our NMR system is controllable if and only if the associated spin graph is connected. The main point is to show that the Lie algebra generated by the operators −iĤ µ , µ = 0, 1, · · · , 2n, is equal to su(2 n ). In this respect, what we have to do for proof is the same as that in [1]. However, as our Hamiltonian is different from that in [1], the method for proof should be different from that in [1].

B.4 A necessary and sufficient condition for controllability
We now consider the case where the spin graph S is disconnected. We assume that the graph S is broken up into two disjoint subgraphs, S 1 and S 2 , each of which is connected. Suppose that S 1 and S 2 have nodes 1 to r, and r + 1 to n, respectively.
According to the decomposition, S = S 1 ∪ S 2 , of the spin graph, the drift Hamiltonian is also decomposed into the sum of two terms, Since S 1 is connected, in the same procedure as taken in proving Prop. B.6, the Lie algebra L 1 generated from −iĤ ′ 0 and {−iĤ α , −iĤ n+α } α=1,··· ,r proves to be L 1 = su(2 r ) ⊗ I ⊗(n−r) . In the same manner, the operators −iĤ ′′ 0 and {−iĤ r+α , −iĤ n+r+α } α=1,··· ,n−r generate the Lie algebra L 2 = I ⊗r ⊗ su(2 n−r ). Since two operators each of which is in the respective Lie algebras L 1 and L 2 commute, the Lie algebra generated by the whole operators {−iĤ µ } µ=0,1,··· ,2n splits into the direct sum L 1 ⊕ L 2 , which is a subalgebra of su(2 n ) but not equal to the whole su(2 n ). We may generalize this fact to the case where the spin graph is broken up into more than two disjoint subgraphs, but the generalization is easy to perform. So far we have shown the following Proposition B.7. If the spin graph associated with the drift Hamiltonian is disconnected, the NMR system