Two-qubit trace-norm geometric discord: the complete solution

We present the complete solution of the problem of determination of trace-norm geometric discord for arbitrary two-qubit state. Final answer is achieved due to effective reduction of the problem to the study of critical points of certain mapping depending on projectors. Our results are illustrated on various, also new, families of two-qubit states and compared to already known special solutions.


I. INTRODUCTION
It is a common belief that question of the explicit quantitative characterization of correlations for a two-qubit systems can always be answered, but actually in many cases we land on a uncharted territory, where the explicit answer is not known [1][2][3]. One of these situations is the strict solution of the question of trace-norm quantum geometric discord D 1 for an arbitrary state of the two-qubit system. The two-qubit system is the simplest nontrivial compound system which we have at our disposal to study quantum correlations and all relevant tools invented to measure them [1]. The geometric quantum discord measured by means of the distance induced by the trace-norm seems to be one of the solid candidates. Contrary to the geometric discord defined by means of the Hilbert-Schmidt distance, it is well defined and satisfies requirements of the bona fide measure of quantum correlations. As it is well known, the Hilbert-Schmidt distance yields to an anomalous behavior of related to it geometric discord D 2 , but it has an attractive property, that it can be calculated relatively easy. The procedure for calculating geometric discord involves the minimization which for generic states for the trace-distance is rather challenging. As we have shown for qutrits and higher dimensional bipartite systems can hardly be operationally/analytically performed, except for selected families of states [4].
In the present work we show how explicitly the D 1 can be determined, we reveal the geometry of the problem of minimization, what in turn deepens the understanding of the intrinsic geometry of the two-qubit quantum state space. For the Hilbert-Schmidt distance quantum discord D 2 such a question has been answered in a compact form by Dakić et al. in 2010 [5], namely the explicit formula for the D 2 has been obtained. The ease of computation does not heal the nonphysical behavior of the D 2 under the local evolution of the system [6] and its non-contractibility under completely positive trace preserving mappings.
The trace-norm measurement-induced geometric discord is well defined correlation measure. For two qubits it is also equal to the standard geometric discord defined as the distance from a given state to the set of classical-quantum states [7,8] and it is more convenient to use quantity based on the disturbance induced by measurement. Such quantity will be simply called geometric quantum discord. However, its calculation is still rather challenging, and by now no explicit answer for arbitrary two-qubit state has been known. According to our best knowledge supported by the recent review on the quantum correlations and geometric quantum discord [3], prior to the present work only few answers were found for selected families of states: (a) for the Bell diagonal states (or states with maximally mixed marginals) [9], (b) for the X-shaped states [10], (c) for the states with the correlation matrix having only one non-zero singular value and arbitrary Bell vectors of the marginals -such class contains in particular so called quantum-classical states [10].
The objective of this study is to provide the complete solution of the problem of giving the explicit value of geometric quantum discord D 1 (ρ) for arbitrary two-qubit state ρ. We find the answer to the minimization problem which has nice geometric meaning. As an illustration, we analyse previously known partial solutions for mentioned above families of states and indicate mechanism making that these solutions were so much simpler. The paper is organized as follows. In Section 2, we briefly review basic notions and fix the relevant notation. Then, in Section 3, we prove the theorem giving characterization of the central formula for the trace-norm of the disturbance of the system by a local projective measurements. To guide the reader gently to the full solution, in Section 4, we compute the D 1 (ρ) for the relatively large family of states for which the minimization procedure is not necessary at all or is straightforward. The Section 5 is central for the final result, there we study the critical points of the trace-norm disturbance mapping and obtain that there are two possibilities: singular critical points and smooth critical points. As a preliminary application we apply new results to the states with maximally mixed marginals. In the Section 6 we prove the main result: the determination of the trace norm geometric discord in general case. Finally, the Section 7 will allow reader to make contact with known solutions and go beyond the family of the X-shaped states. We finish the paper with some conclusions and comments on the main findings and obstructions to their extensions to higher dimensional systems.

II. PRELIMINARY NOTIONS
In the description of two-qubit systems we follow general formulation of d -level quantum systems introduced in [4], specified to the case d = 2.
A. Two qubits Qubit is a 2-level quantum system. The corresponding Hilbert space equals to C 2 and the observables are given by hermitian elements of full matrix algebra M 2 (C). As a basis in M 2 (C) we use standard Pauli matrices σ 1 , σ 2 , σ 3 and the identity matrix 1 1 2 . In the following we will use the notation Then from the properties of Pauli matrices it follows that n, σ m, σ = n, m 1 The set E 2 of all states of 2 -level system can be parametrized as follows where ||n|| ≤ 1. So E 2 is given by the unit ball in R 3 and the pure states correspond to the unit sphere ||n|| = 1. Consider now two qubits A and B. It is convenient to parametrize the set of states of composite system as follows where x, y ∈ R 3 and K = (K jk ) is the correlation matrix. Notice that The parametrization (II.3) is chosen in such a way that the marginals tr A ρ and tr B ρ are given by the vectors x and y as in (II.2). Let us discuss now the adjoint representation of the group SU(2). LetÛ ∈ SU(2) and define 3 × 3 matrix U by The matrix U is real and orthogonal. In this way to eachÛ ∈ SU(2) there corresponds U ∈ SO(3). In contrast to general case, when d = 2 such obtained group is exactly equal to the group SO (3). Consider now the local transformations of the state of two qubits The corresponding vectors x, y and the correlation matrix K transform as follows where U, V are the adjoint representations ofÛ,V respectively. Since in the case of qubits we can use the full group of orthogonal transformations to diagonalize correlation matrix, any two-qubit state is locally equivalent to the state with diagonal K.

B. Trace-norm geometric discord
Let ρ be a state of bipartite system AB. When we perform local measurement on the subsystem A, the state ρ may be disturbed due to such measurement. The trace-norm (one-sided) measurement induced geometric discord is defined as the minimal disturbance induced by projective measurement P A on subsystem A, computed using the trace distance in the set of states. It can be compared with the standard geometric discord equal to the distance from a given state to the set of classicalquantum states [5]. At it was already stated in the Introduction, in the case of qubits these two notions coincide, and it is more convenient to use the quantity based on the disturbance induced by the measurement which will be simply called trace -norm geometric discord. The formal definition is as follows [9] D 1 (ρ) = min where ||A|| 1 = tr |A|.
In the case of qubits, the local projective measurement P A is given by the one -dimensional projectors P 1 , P 2 on C 2 , such that P 1 + P 2 = 1 1 2 , P j P k = δ jk P k and P A = P ⊗ id, where One -dimensional projectors P k can be always chosen as Define now a real orthogonal projector P on R 3 If P 0 denotes such projector given by (II.7), where we take P 0 1 and P 0 2 , then P 0 = diag(0, 0, 1) and Define also orthogonal complements to P 0 and P M 0 = 1 1 3 − P 0 , M = 1 1 3 − P (II.10) Notice that in this case the projectors M run over the whole set of projectors with dimension 2.
Now we compute the disturbance of the state (II.3) caused by measurement P A . We have where e k , k = 1, 2, 3 are the vector of the canonical basis of R 3 . So where Q ′ is obtained from Q by taking vector x ′ = Ux and matrix K ′ = UKV T instead of x and K.

III. TRACE NORM OF DISTURBANCE S(M )
In this section we find the elegant formula for the trace norm of the disturbance S(M ) (see [10] for the other version of such formula).
Theorem III.1 The trace norm of the disturbance S(M ) of the state (II.3) is given by the formula where || · || denotes the euclidian norm in R 3 and E = adj K is the adjunct matrix of the correlation matrix K (i. e. the transpose of its cofactor matrix).
Proof: We start the proof of this theorem with the formula for Q(M ) = S(M )S(M ) ⋆ . By a direct computation we obtain where F k are the generators of SO(3), given by which satisfy covariance relations To further simplify (III.2) consider locally equivalent state for which the correlation matrix is diagonal. So, for a proper transformations U 0 , V 0 ∈ SO(3), we have and I 0 is diagonal Let E 0 = adj(I 0 ) be the adjunct matrix of I 0 . Then . The formula (III.4) follows since For any orthogonal transformation V define the mapping τ V : M 2 (C) → M 2 (C) as follows Now using the identity (III.4) we can transform the formula (III.2) to obtain Observe that where vector v is given by the third column of the matrix V . Similarly Let us now introduce the matrix This matrix has a block diagonal form with blocks The spectral analysis of R 0 (M ) is easy since characteristic polynomials for the blocks are given by where the +(−) sign corresponds to the lower (upper) block. One can also check that the vectors I 0 M x 0 and E 0 v are orthogonal, The formula (III.8) can be further simplified and we arrive at Now applying the inverse to the transformation (III.3), we finally obtain the formula (III.1).

IV. SOLVING THE MINIMIZATION PROBLEM IN THE SIMPLE CASE
To compute the geometric discord of a given state, we have to find minimum of the mapping M → ||S(M )|| 1 . In general it is a hard problem and its general solution will be described in the next sections. Now, let us consider a class of two-qubit states for which the solution of this problem is straightforward. This class (denoted by E 0 ) contains the states (II.3) with arbitrary Bloch vectors x, y and the correlation matrix of the form where V 0 is some orthogonal matrix and t is a real parameter. From general properties of the two-qubit states it follows that t belongs to the interval |t| ≤ 1, but the actual value of t depends on the choice of the matrix V 0 and vectors x, y. To apply the formula (III.1), notice that Now the formula (III.1) gives and minimal value is achieved for such M which projects on x ⊥ in R 3 . Thus we have Theorem IV.1 For every state ρ ∈ E 0 , geometric discord is given by Characterization of the class E 0 is not an easy task. Partial information we can obtain considering locally equivalent states with diagonal correlation matrix. Therefore let us consider the correlation matrix K of the form In the following we restrict our analysis to I 1 = 1 1 3 and I 2 = diag(1, −1, 1). In the first case we obtain the family of states where a, b ∈ R, w, z ∈ C. Similarly, in the second case, we have Notice that the relation between parameters a, b, w, z and Bloch vectors x, y is as follows and To obtain more specific information, we must restrict the number of parameters. The simplest is the case when Bloch vectors equal to zero vector. Then (IV.1) contains the one -parameter family of Werner states On the other hand, (IV.2) contain the family of isotropic states Isotropic states can be obtained as a mixture of maximally mixed state ρ ∞ = 1 4 1 1 4 and maximally entangled state Ψ + given by On the other hand, the Werner statesρ W with reversed parametrization (t → −t) can be obtained as he mixture of ρ ∞ and the state Φ − , where So both states are locally equivalent. More interesting is the case of states with non-trivial Bloch vectors. For simplicity we consider the vectors of the form In this case the properties of the states (IV.1) and (IV.2) differ significantly. In particular, ρ 1 is positive -definite when On the other hand, the positivity region for ρ 2 is given by the conditions

A. Formulation of the problem
In this section we start to analyze the real problem of finding minimum of the trace norm ||S(M )|| 1 in general case. As a first step we will study critical points of the mapping M → ||S(M )|| 1 . Since the projector P = 1 1 3 − M is one dimensional, there is a unit vector v ∈ R 3 such that P = P v . The vector v is given by the third column of the matrix V ∈ SO(3) relating P and P 0 (II.9).
To study critical points of trace norm of disturbance it is useful to consider the auxiliary function g defined on the unit sphere S 2 ⊂ R 3 with values in R 2 . The function is defined as follows and Notice that using the functions (V.2) and (V.3), the formula (III.1) can be rewritten as In the following we will consider the mapping g as a function of two-dimensional projectors M or vectors v ∈ S 2 .
The formulas (V.2) and (V.3) can be further simplified if we introduce the following operators and where x is a Bloch vector. Now we obtain and From the equation (V.6) we obtain in particular that where λ 1 ≤ λ 2 ≤ λ 3 are the eigenvalues of real non-negative matrix L + . The relation (V.8) allows to obtain the first general result concerning the value of geometric discord D 1 (ρ). Namely we have From the relation (V.8) we can also derive well known exact formula for Hilbert -Schmidt norm geometric discord D 2 (ρ) [5]. By definition Obviously, for any two-qubit state The right hand side of (V.9) in some cases gives not only the lower bound but the exact value of discord D 1 . It can happen for such states ρ for which D 1 (ρ) = D 2 (ρ). Below we show it for an explicit family of states. Consider the states [11] For the family (V.10), we have The order of eigenvalues of (V.11) depends on θ. For θ ∈ [0, π/4] sin 4 θ ≤ sin 2 θ cos 2 θ and for such θ On the other hand, one can check that so in this case we obtain only the lower bound. Now for θ ∈ (π/4, π/2] sin 2 θ cos 2 θ ≤ sin 4 θ and for such θ which equals to the value of D 1 (ρ θ ).

B. Singular critical points
We see that the problem of finding minimum of the mapping M → ||S(M )|| 1 is relatively easy when it is achieved on the set Each critical point v 0 ∈ D is called a singular critical point in contrast to the smooth critical point i.e. critical point belonging to the set S 2 \ D. When the minimum is achieved at some singular critical point, the minimal value of trace norm ||S(M )|| 1 is given by the minimum of the function g 1 . To study the properties of the set D, consider another representations of the functions g 1 and g 2 . Let L − = KK T − W x . One can check that and where we dropped the term proportional to 1 − ||v|| 2 . Using (V.13) and (V.14) one can show that the set (V.12) can be discrete or it can be equal to the whole sphere S 2 . It follows from the following Lemma, which can be proved by a direct computation.
Remark V.1 One can check that in the discrete case, the set D can contain only one or two projectors, if we identify the vectors v and −v.
For the further analysis it is crucial to have a convenient criterion that enables to decide whether the value of D 1 (ρ) is achieved at a critical point belonging to the set D or S 2 \ D. To obtain such criterion, consider the function For any t * > 0 consider the function And finally, define the family of regions Ω t * , t * ≥ 0 such that Ω 0 = Ω and for t * > 0 Suppose now that the function g 1 restricted to D achieves at some vector v * ∈ D its absolute minimum. Let Ran g be the range of the function g. The values of D 1 (ρ) is achieved at v * ∈ D if and only if So we obtain Proposition V.1 Let the function g 1 restricted to D achieves its absolute minimum at the vector v * ∈ D. The value of D 1 (ρ) is achieved at this vector, if and only if the following conditions are satisfied for all v ∈ S 2 , and for all v ∈ S 2 such that g 1 (v) ≤ g 1 (v * ).
Remark V.2 Notice that if the conditions (V.16) and (V.17) are not satisfied, then the value g 1 (v * ) gives the upper bound for D 1 (ρ) i.e.
Now we determine the minimal value g 1 (v * ). To this end, consider the following cases: The solutions of (V.15) are given by Note that λ 2 ≥ 0. The equality (V.19) is valid in any orthonormal basis. But if we choose the basis of ordered and normalized eigenvectors of L − , then is also valid and we obtain 3. λ 1 = λ 2 = λ 3 . In this case D = S 2 and the formula (V.19) is valid.
So we have the following where int denotes the intermediate value and λ 1 , λ 2 , λ 3 are the eigenvalues of the matrix L − .

Remark V.3
Notice that if the Bloch vector x is zero vector, then the function g 1 is constant on the set D and equals to 2 int {λ 1 , λ 2 , λ 3 }, where λ 1 , λ 2 , λ 2 are the eigenvalues of the matrix KK T .

C. Smooth critical points
Smooth critical points v 0 ∈ S 2 \ D satisfy Applying differential analysis to the set S 2 \ D one can find all smooth critical points of the mapping M → ||S(M )|| 1 . Let with the constrain ||v|| = 1 and where λ is a Lagrange multiplier. For the smooth critical points one obtains To find critical points of g we look for the possible solutions of the equation A vector v ∈ S 2 is a solution of (V.23) if there exists a real number µ such that v is a solution of the above eigenvector problem.
Applying the above results one can prove the following Theorem: i.e. v is the solution of (V.23) if and only if On the other hand, the information about the Lagrange multiplier ω can be recovered from the equation (V.25) that leads to

D. First application: states with maximally mixed marginals
General analysis of the equations (V.23) or (V.25) will be presented in the next section, here we consider the first application to the states with maximally mixed marginals (MMM states) i.e. such states ρ that In this case the Bloch vectors x and y are zero vectors. Let K be the correlation matrix of MMM state ρ. Using the local transformation one can bring the matrix KK T to the diagonal form diag (i 2 1 , i 2 2 , i 2 3 ) with i 2 1 ≥ i 2 2 ≥ i 2 3 . Now the equation (V.15) for the set D has a form so the general form of P v satisfying (V.26) depends on the degeneracy of the eigenvalues of G µ . Consider first the case when i 2 1 > i 2 2 > i 2 3 . Suppose that µ / ∈ {i 2 1 , i 2 2 , i 2 3 }, then the eigenvalues of G µ are non-degenerate and the only possible solutions of (V.26) correspond to the vectors In this section we use the equation (V.25) to solve the main problem, namely to find the value of geometric discord in general case. It is convenient to consider the matrix (V.24) with respect to ordered basis of orthonormal eigenvectors of the operator L − (we will assume that λ 1 ≥ λ 2 ≥ λ 3 ). When we also neglect the parts which commute with the projectors P v , we obtain the matrix In the following we will also use the matrices Λ = Λ − ( λ 1 − λ 2 ) 1 1 3 and By a direct computation one can find the representation of µ along the critical points of the function g, described by the condition To find the critical points of the mapping M → ||S(M )|| 1 we use the condition (V.22), which now reads To obtain the result, let us assume that the equation (VI.1) has a non-empty set of solutions. Then we get that But this representation can not be valid outside the set D and we obtain the contradiction with our assumption. Let now the equation (VI.2) has the non-empty set of solutions. Then we get and after simplifications we obtain Similarly, if x 3 = 0, the critical points never lay on the big circle v 3 = 0 with v 1 v 2 = 0, unless λ 1 + λ 3 > 2 λ 2 and the following condition is satisfied The additional informations of this kind can simplify the determination of quantum discord in some situations. Nevertheless, we will not discussing this improvements here and focus on the direct method of finding the absolute minimum of the mapping M → ||S(M )|| 1 .
Now we can go to the solution of the main problem. This is a problem of minimizing the quadratic form µ, restricted to the big circles. The calculations are elementary but tedious, so we do not present all details. 1. Let us start with v 1 = 0. One can find Inserting the minimum µ min one gets To describe the absolute minimum of the function (VI.3), we need to consider auxiliary functions. The complexity of formulae below mainly follows from the presence of the absolute value in (VI.3). Let us define functions p(θ) and r(θ), as follows. The common domain of p(θ) and r(θ) is the closed interval [−π, π]. To define these functions we need the gap angle ϕ gap ∈ (0, π/2), determined by the equation Moreover, let σ : [−π, π] → [−1, 1] be any function such that and O gap be the open set Now we are ready to define Finally, let θ * and θ * * be the angles defined below in two steps. First, let If x 1 x 3 = 0 we pass to the one -sided limits x 1 → 0 ± or x 3 → 0 ± (or both) in the formulas (VI.4) and (VI.5). The left and right limits can only differ by sign, but this has no importance as the functions p(θ) and r(θ) give the same result in these situations.
To obtain the upper bound for quantum discord, define and Then one can show that The above analysis leads to the following result.
Theorem VI.1 In the case of non-degenerate spectrum of the matrix KK T − W x , the value of trace -norm geometric discord is given by with µ * , µ * * given by (VI.6) and (VI.7) respectively.

B. Degenerate spectrum of L −
When there is any degeneracy in the spectrum of L − , the analysis simplifies significantly. From the above discussion we obtain:

VII. EXAMPLES
In this section we present the applications of Theorems VI.1 and VI. In the non-degenerate case we must apply Theorem VI.1 and the computation is more involved. Observe that in the case of MMM states where obviously µ * * = λ 2 . To show that µ * also equals to λ 2 one checks that θ * = 0. Then p(θ * ) = cos 2 ϕ gap = λ 1 − λ 2 λ 1 − λ 3 so by (VI.6), µ * = λ 2 . Thus It is well known that up to the local equivalence, every pure state has the form The corresponding correlation matrix and Bloch vector are given by an we can apply Theorem VI.2 to obtain On the other hand, the measure of entanglement defined by negativity, gives the same value.

D. X-states
Two -qubit X -state has the X -shaped form We can always take off -diagonal matrix elements ρ 14 and ρ 23 to be positive [10]. Then the correlation matrix K is diagonal and the Bloch vector x has only third component non-zero. So the matrix KK T is diagonal and commutes with the operator W x . It follows that the matrix L − has also diagonal form and the basis of eigenvector of L − coincides with the canonical basis.

E. The class containing quantum-classical states
Now we apply our results to the class of states with arbitrary Bloch vector x and correlation matrix having only one non-zero singular value. Such class in particular contains quantum -classical states [10]. We can always assume that One can check that possible critical points lay on the big circle v 2 = 0 or are given by (0, ±1, 0). By a direct inspection one gets that the minimal value can be achieved only on the circle v 2 = 0. So we are left with one dimensional minimization problem. This can be done by a direct analysis (see [10]), but we apply our general result. To find value of D 1 (ρ) we need only µ * and µ * * which take the form with θ * ∈ (ϕ gap , ϕ gap ) and θ * * ∈ (−π/2, −ϕ gap ) ∪ (ϕ gap , π/2). After a direct but quite lengthy computations one finds that the absolute minimum is given by µ * and the value of discord reads Notice that this formula is in a full agreement with the result obtained in [10].

VIII. SUMMARY
Let us now summarize the results reported in the article. First, we have obtained new compact and elegant formula for the trace norm of the disturbance S(M ) of the general two-qubit state. We want to emphasize, that to obtain this result we have used the qubit structure of subsystems in the crucial way. In the general qudit case we were able only to find simplified formula for the square of the modulus of disturbance and only for the special classes of states [4]. It seems that the computation of trace norm of S(M ) in general case is a highly non-trivial or even not possible at all. Then, we have applied the formula for ||S(M )|| 1 to compute the value of geometric discord in the situation when the minimization procedure is not needed at all or is straightforward. This is the case of the class of states with arbitrary Bloch vectors and the correlation matrix proportional to some orthogonal matrix in three dimensions i.e. K = t V 0 . In particular, for such states with zero Bloch vectors, ||S(M )|| 1 = |t| and D 1 (ρ) = |t|. The presence of non-zero Bloch vectors only slightly changes the trace norm and the minimization can be done in elementary way. This again must be contrasted with the general qudit case, where the similar results are valid only for vanishing Bloch vectors and the classes of Werner and isotropic states [4]. As a first step towards the computation of D 1 (ρ) in general case, we considered critical points of the mapping M → ||S(M )|| 1 . To solve this problem, we introduce the function g defined on the unit sphere S 2 ⊂ R 3 with values in R 2 . Any critical point of the mapping M → ||S(M )|| 1 is a critical point of the function g and the properties of this function can be used to isolate singular and smooth critical points. It is remarkable that the information about critical points is encoded in the spectrum of the matrix L − = KK T − W x . One of the most important results of our study says that when this spectrum is non-degenerate, the critical points are located on big circles in S 2 . It means that the problem of finding geometric discord reduces to minimization of the quadratic form restricted to big circles. The complete solution of this problem is rather involved and it simplifies significantly, when any degeneracy of the spectrum of L − is present.
We apply the Theorems VI.1 and VI.2 to the cases already discussed in the literature (Bell diagonal, X-shaped and quantumclassical states) as well as new examples beyond the class of X-states. It is also worth to emphasize that in particular cases, methods based on partial results can be more efficient then the direct application of the general result, what is explicitly shown by the discussions of MMM states in Sections V.D and VII.B.
Having the complete answer for the two-qubit system we are able to estimate possible obstacles in finding general explicit formula for the trace-distance quantum discord for qudits. As we have already stressed, the geometry of the two-qubit quantum state space is exceptional and the set of geometrical tools which is at our disposal is very reach. Let us recall here the most effective one, that any correlation matrix can be diagonalized by means of local transformations, due to the accordance of the dimensions of relevant spaces and groups i.e. SO(3) ≃ Adj(SU (2)). For the next higher level system of two-qutrits d = 3 we have that Adj(SU(3))) is a small subgroup of SO (8), too small to allow diagonalization of a generic two-qutrit correlation matrix by means of local transformations. For higher dimensional cases this effects only aggravates, what means that using local orbit of the set of diagonal correlation matrices we can cover only a part of the quantum state space. The solution of the minimization problem for some families of two-qudit states is given in our previous works [4,12].
The strict analytical general formulas characterising quantum correlations valid for all or sufficiently wide class of quantum states of compound systems, in general are not known and only selected cases are described in the literature. Such explicit knowledge would be of value that hardly can be overestimated, as it gives insight into the structure of the quantum states, but on the other hand due to the complexity of this space, computation of physically important measures of quantum correlations is so hard task. We hope that results obtained in the present paper will fill the gap in the knowledge on measures of quantum correlations of compound systems for the most fundamental physical system of two qubits.