Complete Optimal Convex Approximations of Qubit States under $B_2$ Distance

We consider the optimal approximation of arbitrary qubit states with respect to an available states consisting the eigenstates of two of three Pauli matrices, the $B_2$-distance of an arbitrary target state. Both the analytical formulae of the $B_2$-distance and the corresponding complete optimal decompositions are obtained. The tradeoff relations for both the sum and the squared sum of the $B_2$-distances have been analytically and numerically investigated.


I. INTRODUCTION
Quantifying correlations among multipartite systems is one of the most important problems in quantum theory. However, most correlation measures become notorious difficult to calculate with the increasing partite and dimension. An alternative way to deal with the problem is to consider the distance of a given state to the so called free states in resource theory. For example, entanglement is considered as the minimal distance of a given state to the set of separable states in quantum systems [1][2][3][4]. The quantum discord is regarded as the minimal distance of a given state to classically correlated states [5]. And quantum coherence can be quantified by the optimal convex approximation of the given state to the reference orthogonal base [6].
While convexity is a very important property in mathematics and has been studied for long time, several related recent developments in quantum information have stimulated new interest in this topic [7,8]. The problem of optimal approximation to an unavailable quantum channel or state by the available channels or states was considered in [9,10] recently. It was shown that the optimally approximated distance has an natural operational interpretation. It can quantify the least distinguishable channel (state) from the given convex set to the target channel (state). The trace distance measure of coherence can be regarded as convex approximation to the target state with respect to a fixed base of the system, where the fixed base can be either orthogonal or nonorthogonal [11][12][13][14][15]. In Ref. [10], the author considered the the B 3 -distance, the distance from a target qubit state to the convex approximation of bases containing the eigenstates of all Pauli matrices. The optimal convex approximation on the B 3 -distance has been obtained.
In this work, we focus on B 2 -distance, the distance corresponding to the convex approximation of bases containing the eigenstates from one of the pairs of Pauli matrices. We investigate all the optimal convex decompositions for the desired quantum state. The paper is organized as follows. In II, we calculate the B 2 -distance in eight different cases, with the parameter regions achieving each optimal approximation explicitly given. In III, we study tradeoff relations for both the sum and the square sum of the B 2 -distance.

II. THE PAULI B2−DISTANCE OF QUBIT STATE
For an equal priori probability of two given quantum states ρ and ρ 0 , the optimal discrimination between them can be quantified by the following probability p discr (ρ, ρ 0 ), where ρ 1 denotes the trace norm of ρ, ρ 1 = T r ρ † ρ = i √ r i , r i are the eigenvalues of ρ † ρ.
This optimal convex approximation provides the worst probability of discriminating the desired state ρ from any of the available states i p i ρ i . For any other figure of merit that quantifies the distance between quantum states, Solving the above equations, we can obtain the complete analytical solutions to the optimal convex approximation ρ opt of ρ. The S(ρ opt ) of ρ with respect to B ′ 2 is given as i) For a ≥ k a(1 − a) cos φ, we have which is attained at where ii) For a < k a(1 − a) cos φ, we have the the optimal convex approximation distance which is attained with Denote A ′ 2 = {p 0 ρ 0 + p 2 ρ 2 }, with p 0 , p2 given by Eq. (7). Then A ′ 2 contains all the optimal states achieving the distance D B ′ 2 (ρ) in Eq. (6). Therefore S(ρ opt ) is given by S(ρ opt ) = A ′ 1 A ′ 2 , which is the set of optimal states that gives rise to the optimal convex approximations.
Next we consider the optimal convex approximation of ρ with respect to B , we have i) For a ≥ k a(1 − a) sin φ, the optimal convex approximated distance is given by The with the optimal probability weights are given by ii) For a < k a(1 − a) sin φ, we have the optimal convex approximated distance with the optimal probability weights given by Let A ′′ 2 = {p 0 ρ 0 + p 4 ρ 4 } be the set of states with p 0 and p 4 given by Eq. (11). Then S(ρ opt ) is given by For the optimal approximation of ρ with respect to the basis in B ′′′ 2 , we have i) For 1/2 ≥ k a(1 − a)(sin φ + cos φ), the optimal convex approximated distance has the form with the optimal probability weights where t is given by (13), contains all the optimal states achieving the distance D B ′′′ 2 (ρ) in Eq. (12).
In Fig. 1, we plot the distance D B ′ 2 (ρ) for fixed parameters of k and φ. One can see that for the fixed value φ = π 4 , Fig.1(a) shows that the optimal distance D B ′ 2 (ρ) increases with k and decreases with the parameter a. Fig.1(c) shows the interface such that the region above the surface corresponds to the case i), namely, a ≥ k a(1 − a) cos φ; and the region below the surface is the case ii), a < k a(1 − a) cos φ. In Fig. 2 and 3, the distances D B ′′ 2 (ρ) and D B ′′′ 2 (ρ) with the fixed values are also plotted, respectively. The corresponding interface is plotted in Fig.2(c) (Fig.3(c)): the region above the surface corresponds to the case a ≥ k a(1 − a) sin φ (1/2 ≥ k a(1 − a)(sin φ + cos φ)), the region below the surface is the case a < k a(1 − a) sin φ (1/2 < k a(1 − a)(sin φ + cos φ)), respectively.  1.(b)]. The interface of the regions of the two cases i) and ii) is plotted in FIG. 1.(c), the region above the surface corresponds to the case a ≥ k a(1 − a) cos φ, the region below the surface is the case a < k a(1 − a) cos φ. (ρ)}. One can see from Fig.4 that, for fixed φ = π/4 and k = 4/5, min D B2 (ρ) is always nonzero for nonzero parameters a and k or φ, namely, all the three Pauli B 2 −distances are nonzero.
Concerning the tradeoff relations of the three Pauli B 2 −distances, for convenience, we denote By the numerical calculation, we obtain the tradeoff relation among D B  3.(b)]. The interface of the regions of the two cases i) and ii) is plotted in FIG. 3.(c), the region above the surface corresponds to the case 1/2 ≥ k a(1 − a)(sin φ + cos φ), the region below the surface is the case 1/2 < k a(1 − a)(sin φ + cos φ).   Fig.5 shows all the parameter regions of a, k, φ such that the three B 2 −distances are achieved. These regions completely characterize all the optimal convex approximations of a sate ρ w.r.t. B 2 −distance.
It has been shown that, for a given state, the three optimal distances to the bases in B   . Therefore, we have where τ = 2 √ 3 is the triple constant given in the uncertain relations in [21,22]. From formulae (4), (8) and (12), we immediately get that in region 1, our Pauli B 2 −distances is in accordance with the uncertainty relation.

IV. CONCLUSION
In summary, we have shown that a qubit mixed state ρ can be approximated by a number of effectively available pure states spanned by the eigenstates of the Pauli matrices. It is well known that correlation limits the extractable information [16][17][18][19], where one does want to minimize the probability of discrimination. The advantage of our results is that we presented the complete set of optimal decompositions of a given state. In [10] for a given state, only one particular optimal decomposition has been elegantly derived, in which p 3 and p 5 are chosen to be zero. Hence, basically it is the minimal distance with respect to four of six eigenvectors of the Pauli matrices. As a simple example, consider the following mixed qubit state, ρ = 1/2 1/5 1/5 1/2 . One can verify that D B ′