Quantum steerability based on joint measurability

Occupying a position between entanglement and Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in recent years. Many criteria have been proposed and experimentally implemented to characterize EPR-steering. Nevertheless, only a few results are available to quantify steerability using analytical results. In this work, we propose a method for quantifying the steerability in two-qubit quantum states in the two-setting EPR-steering scenario, using the connection between joint measurability and steerability. We derive an analytical formula for the steerability of a class of X-states. The sufficient and necessary conditions for two-setting EPR-steering are presented. Based on these results, a class of asymmetric states, namely, one-way steerable states, are obtained.

In this work, we investigate the analytical formula for quantification of EPR-steering and obtain the necessary and sufficient condition of steerability for a class of quantum states. The asymmetric feature of EPR-steering is also investigated.

Setting up the stage
Consider a bipartite qubit system ρ AB shared by Alice and Bob with reduced density states ρ A and ρ B . Alice performs positive-operator-valued measures ( 1 The quantum state AB ρ is unsteerable from A to B if for all local POVMs, the state assemblages are all unsteerable. The quantum state ρ AB is steerable from A to B if there exist measurements in Alice's case that produce an assemblage that demonstrates steerability.
The corresponding local hidden state model and the joint measurement observables are connected through by the one-to-one mapping between the joint measurement problem and the steerability problem, whenever ρ B is invertible 27 . The steerability can be detected through the joint measurability of the observables. Two-setting steering scenario: Any two-qubit quantum state can be expressed by ρ is the correlation matrix. When Alice performs two sets of POVMs I n ( ( 1) )/2 (sin cos , sin sin , cos ), , are jointly measurable [35][36][37] , namely, ), (1) gives rise to the condition for Alice to steer Bob's state. If Bob performs two sets of POVMs n i Π κ| → on his system to steer Alice's state, the corresponding condition can be similarly written by changing in (1). However, it is generally quite difficult to address condition (1) and obtain explicit conditions to judge the steerability for an arbitrary given two-qubit state. For Bell-diagonal states, a necessary and sufficient condition of steerability has been derived from the relations between steerability and the joint measurability problem 31 . In the following, we study the steerability of any arbitrary given two-qubit states. We present analytical steerability conditions for classes of two-qubit X-state.

Results
Steerability of two-qubit states. First, based on the jointly measurable condition (1) → κ κ for the two-setting steering scenario, we define the steerability of two-qubit states AB ρ by the following , Due to the relationship between the joint measurements and steerability, local hidden states which are all possible sets of four measurements satisfying the marginal constraints for any two jointly measurable observables → [35][36][37] . The steering radius R( ) AB ρ 24 can be calculated by optimizing → z and Z. In the following, we analytically calculate the steerability S for some X-states X ρ . We define a class of two-qubit X-states to be zero-states zero ρ if the X-states ρ X satisfy the condition that the maximum points (stationary points) of S 1 belong to the zero points of S 2 with respect to the measurement parameters α i and β We obtain the following results: Theorem. For the zero-states zero ρ , the analytical formula of the steerability is given by When S 0, > the optimal measurements that give rise to maximal S are σ x and σ y if ∆ > ∆ ∆ max{ , ,0}, and σ y and σ z if max{ , ,0} ∆ > ∆ ∆ . The proof is given in the supplementary material. It is obvious that any X-state with = t 0 3 belongs to ρ zero , e.g., a a 00 1 11 For the Bell-diagonal state, interestingly, the steerability S is given by the non-locality characterized by the maximal violation of the CHSH inequality. Let  CHSH denote the Bell operator for the CHSH inequality 38 , , are unit vectors. Thus, the the maximal violation of the CHSH inequality is given by 39  (3), we find that the steerability of Bell-diagonal state is given by For t 0 3 ≠ , we give the explicit conditions of the zero states in the supplementary material. In the following, we present the maximum value of the steerability S for a given N of ρ zero .

Corollary 1:
For zero-states ρ zero with given N, = − i.e., zero ρ has the following form, The following corollary gives the conditions at which we obtain the minimal value of S for a given N.

Corollary 2:
For zero-states zero ρ with given CHSH value N, S obtains the minimal value when a 0 3 = and b 0 3 = or 2 . The proofs of Corollary 1 and Corollary 2 are given in the supplementary material. In Fig. 1, we give a description for the boundaries of the steerability S for a given value of N. From Fig. 1, we observe that for any given N with ≤ ≤ N 0 2 , the lower bound of S is always 0 and the upper bound of S is always less than 2 (light blue), and for N 2, > the lower bound of S is always greater than 0, and the upper bound of S is always 2 (dark blue). For zero-states , zero ρ the steering radius R( ) zero ρ can be obtained when Alice measures her qubit along the directions σ x and σ , y or σ x and , z σ or σ y and σ . z Indeed, from the construction of joint measurements 35 , when Alice measures her qubit along the directions of σ x and σ , z the local hidden states can be expressed as follows σ σ µ ; 1  3  2  1   2   3  3  3  3  2   3  33  2   2  3  2  1   2   3  3  3  3  2   3  33  2 It is not easy to calculate ρ r( ) zero xz and ρ r( ) zero yz analytically. We give the analytical results for R( ) zero ρ for some special states in the following.
Asymmetric two-setting EPR-steering. Different from Bell-nonlocality and quantum entanglement, EPR-steering has the asymmetric property of one-way EPR steering: Alice may steer Bob's state but not vice versa. The demonstration of asymmetric steerability has practical implications in quantum communication networks 40 . Until now, only a few asymmetric steering states have been found 24,[32][33][34] . In this work we present a class of asymmetric steering states of the form ρ X 0 in (5). If Alice performs measurements on her qubit, the steerability is given by ρ = . If Bob performs measurements on his qubit, the related steerability is given by the following which is equal to zero as long as . Therefore, when c b 0 3 3 < < − and → − b 1, 3 Alice can always steer Bob's state, but Bob can never steer Alice's state (see Fig. 2 for the asymmetric EPR-steering for b 0 999 3 = − . ). We note that Alice can always steer Bob's state, but Bob can not steer Alice's state.  In the following subsection, we investigate the geometric features of the asymmetric steering statex 0 ρ in terms of the steering ellipsoid 41 . The steering ellipsoid of X 0 ρ when Alice performs POVMs is quite different from that when Bob performs POVMs. The centre of the steering ellipsoid ε B for Alice performing POVMs on her qubit is 2 , which goes to (0,0, 1) − when → − b 1, and the volume of the steering ellipsoid ε B is given as follows which goes to π + c (1 ) 3 3 2 when b 1 3 → − . The steering ellipsoid is also tangent to the Bloch sphere. In this case the ellipsoid shows some peculiar features, i.e., when b 1 3 → − and c 0 3 → , the ellipsoid B ε is nearly 0, but Alice can still steer Bob; however, when → − b 1 3 and → − c b 3 3 , the ellipsoid ε A is almost the entire Bloch sphere, but Bob can not steer Alice.
As a special case of ρ , . The state has the following form, From the theorem, we obtain the following when Alice measures her qubit, The sufficient and necessary condition in the two-setting steering scenario is 1/(2 ) η χ > − for Alice to steer Bob's state. The corresponding optimal measurements are σ x and σ . y If Bob measures his qubit, the steerability is given by the following The sufficient and necessary condition for Bob to steer Alice's state is η χ > + 1/(1 ). The related optimal measurements are σ x and σ .
y The asymmetric property in quantum steering given by this example is shown in Figs 3 and 4. The steering radius is 1 . θ W V is a zero state. From our theorem, we know that when Alice performs measurements on her qubit, σ . This state is always steerable for Alice except when V 1/2 = . When Bob performs two projective measurements on his qubit, we have the following From our theorem, the analytical results of steerability can be obtained for more detailed zero states, and the asymmetric property of steering can be readily studied. In the following, we give two examples of symmetric two-setting EPR-steering.

Discussion
Based on the one-to-one correspondence between EPR-steering and joint measurability, we have investigated the steerability for any two-qubit system in the two-setting measurement scenario. The steerability we introduced is invariant under local unitary operations. The analytical formula for steerability has been derived for a class of X-states, and the sufficient and necessary conditions for two-setting EPR-steering have been presented. For general two-qubit states, it has been shown that the lower and upper bounds of steerability are explicitly connected to the non-locality of the states given by the CHSH values of maximal violation. Moreover, we have also presented a class of asymmetric steering states by investigating steerability with respect to the measurements from Alice's and Bob's sides. Our strategy might also be used to study the quantification of steerability for multi-setting scenarios, in particular, for three-setting scenarios for which the joint measurability problem of three qubit observables has already been investigated 42,43 . Our method might also be used in continuous variable steering, temporal and channel steering, for which the steerability of the state assemblages or the instrument assemblages can be connected to the incompatibility problems of the quantum measurement assemblages 44,45 . Hence, the steerability of the quantum states or the quantum channels might also be studied based on the corresponding measurement incompatibility problems.