Bell’s Nonlocality Can be Detected by the Violation of Einstein-Podolsky-Rosen Steering Inequality

Recently quantum nonlocality has been classified into three distinct types: quantum entanglement, Einstein-Podolsky-Rosen steering, and Bell’s nonlocality. Among which, Bell’s nonlocality is the strongest type. Bell’s nonlocality for quantum states is usually detected by violation of some Bell’s inequalities, such as Clause-Horne-Shimony-Holt inequality for two qubits. Steering is a manifestation of nonlocality intermediate between entanglement and Bell’s nonlocality. This peculiar feature has led to a curious quantum phenomenon, the one-way Einstein-Podolsky-Rosen steering. The one-way steering was an important open question presented in 2007, and positively answered in 2014 by Bowles et al., who presented a simple class of one-way steerable states in a two-qubit system with at least thirteen projective measurements. The inspiring result for the first time theoretically confirms quantum nonlocality can be fundamentally asymmetric. Here, we propose another curious quantum phenomenon: Bell nonlocal states can be constructed from some steerable states. This novel finding not only offers a distinctive way to study Bell’s nonlocality without Bell’s inequality but with steering inequality, but also may avoid locality loophole in Bell’s tests and make Bell’s nonlocality easier for demonstration. Furthermore, a nine-setting steering inequality has also been presented for developing more efficient one-way steering and detecting some Bell nonlocal states.

Unlike quantum entanglement and Bell's nonlocality, the research field of quantum steering has been sterile till 2007, when Wiseman, Jones, and Doherty 8 reformulated the idea and placed it firmly on a rigorous ground. Since then EPR steering has gained a very rapid development in both theories [9][10][11][12][13][14][15][16] and experiments [17][18][19][20][21][22][23][24][25][26] . Most research topics as well as research approaches in the field of Bell's nonlocality have been transplanted similarly to the field of EPR steering. For instance, steering inequalities have been proposed to reveal the EPR steerability of quantum states, very similar to the violation of Bell's inequalities reveals Bell's nonlocality.
According to ref. 8, entanglement, EPR steering and Bell's nonlocality are called by a joint name as " quantum nonlocality", which has an interesting hierarchical structure: quantum entanglement is a superset of steering, and Bell's nonlocality is a subset of steering. However, among the three types of quantum nonlocality, only steering can possess a curious feature of "one-way quantumness". Suppose Alice and Bob share a pair of two-qubit state, it is not hard to imagine that if Alice entangles with Bob, then Bob must also entangle with Alice. Such a symmetric feature holds for both entanglement and Bell nonlocality. However, the situation is dramatically changed when one turns to a novel kind of quantum nonlocality in the middle of entanglement and Bell nonlocality, the EPR steering. It may happen that for some asymmetric bipartite quantum states, Alice can steer Bob but Bob can never steer Alice. This distinguished feature would be useful for some one-way quantum information tasks, such as quantum cryptography. The "one-way EPR steering" or "asymmetric EPR steering" is an important "open question" first proposed by Wiseman et al. in ref. 8. Very recently, the question has been answered by Bowles et al. 15 , who presented a simple class of one-way steerable states in a two-qubit system with at least 13 projective measurements (a linear 14-setting steering inequality was given explicitly in the work). The inspiring result for the first time theoretically confirms quantum nonlocality can be fundamentally asymmetric. Later on, Bowles et al. investigated the one-way steering problem by presenting a sufficient criterion (being a nonlinear criterion) for guaranteeing that a two-qubit state is unsteerable 27 .
In this work, we focus on another curious quantum phenomenon raised by steering: Bell nonlocal states can be constructed from some EPR steerable states. Explicitly we present a theorem, showing that for any two-qubit state τ, if its corresponding state ρ is EPR steerable, then the state τ must be Bell nonlocal. Bell's nonlocality of the quantum state τ can be detected indirectly by the violation of steering inequality for the quantum state ρ. The novel result not only pinpoints a deep connection between EPR steering and Bell's nonlocality, but also sheds a new light to avoid locality loophole in Bell's tests and make Bell's nonlocality easier for demonstration. In addition, we also present a 9-setting linear steering inequality for developing more efficient one-way steering and detecting some Bell nonlocal states. We find that the new steering inequality can actually improve the result of ref.
15 by detecting the one-way steering with fewer measurement settings but with larger quantum violations, which would be helpful for the experimenters.

Results
Bell's Nonlocal states can be constructed from EPR steerable states. It is well-known that quantum nonlocality possesses an interesting hierarchical structure (see Fig. 1). EPR steering is a weaker nonlocality in comparison to Bell's nonlocality. Here we would like to pinpoint a curious quantum phenomenon directly connecting these two different types of nonlocality. We find that Bell's nonlocal states can be constructed from some EPR steerable states, which indicates that Bell's nonlocality can be detected indirectly through EPR steering (see Fig. 2), and offers a distinctive way to study Bell's nonlocality. The result can be expressed as the following theorem.
Theorem 1: For any two-qubit state τ AB shared by Alice and Bob, define another state B AB B AB being the reduced density matrix at Alice's side, and µ = 1 3 . If ρ AB is EPR steerable, then τ AB is Bell nonlocal.
Proof. The implication of the theorem is that, the EPR steerability of the state ρ AB determines Bell's nonlocality of the state τ AB . Namely, the nonexistence of local hidden state (LHS) model for ρ AB implies the nonexistence of LHV model for τ AB . We shall prove the theorem by proving its converse negative proposition: if the state τ AB has a LHV model description, then the state ρ AB has a LHS model description.
Suppose τ AB has a LHV model description, then by definition for any projective measurements A for Alice and B for Bob, one always has the following relation , Πâ n A is the projective measurement along the n A -direction with measurement outcome a for Alice, Πb n B is the projective measurement along the n B -direction with measurement outcome b ( , ) and P ξ denote some (positive, normalized) probability distributions. Let the measurement settings at Bob's side be picked out as x, y, z. In this situation, Bob's projectors ξ ξ ξ ξP a A n P a A P n P P a A n P a A P n P

AB AB
We now turn to study the EPR steerability of ρ AB . After Alice performs the projective measurement on her qubit, the state ρ AB collapses to Bob's conditional states (unnormalized) as  . If there exist some specific measurement settings of Alice, such that Eq. (5) cannot be satisfied, then one must conclude that the state ρ AB is steerable (in the sense of Alice steers Bob's particle).
Suppose there is a LHS model description for ρ AB , then it implies that, for Eq. (5) one can always find the solu- The solutions are given as follows: A where is the 2 × 2 identity matrix, σ σ σ σ = ( , , ) x y z is the vector of the Pauli matrices, and the hidden state ρ ξ has been parameterized in the Bloch-vector form, with which is the Bloch vector for density matrix of a qubit. It can be checked that | | ≤ ξ  r 1, and this ensures ρ ξ being a density matrix.
In the following, we provide two examples for the theorem, showing that Bell's nonlocality of quantum states can be detected indirectly by the violations of some steering inequalities.
without Bell's inequality. Based on the theorem, it is equivalent to detect the EPR steerability of the following two-qubit state The state (10) is nothing but the Werner state 28 with the visibility equals to 1/ 3, its steerability can be tested by using the steering inequality proposed in ref. 17 as Here  N is the steering parameter for N measurement settings, and C N is the classical bound, with = + .  C (1 5 )/6 0 5393 6 . The maximal quantum violation of the steering inequality is = .  1/ 3 0 5774 6 max  , which beats the classical bound.
Remark 2.-In a two-qubit system, Bell's nonlocality is usually detected by quantum violation of the Clause-Horne-Shimony-Holt inequality 29 . Bell's nonlocality is the strongest type of nonlocality, due to this reason Bell-test experiments have encountered both the locality loophole and the detection loophole for a very long time 30 . As a weaker nonlocality, EPR steering naturally escapes from the locality loophole and is correspondingly easier to be demonstrated without the detection loophole 19,20 , as stated in ref. 17: "because the degree of correlation required for EPR steering is smaller than that for violation of a Bell inequality, it should be correspondingly easier to demonstrate steering of qubits without making the fair-sampling assumption [i.e., closing the detection loophole]". Indeed, the steerability of the Werner state has been experimentally detected in ref. 17 by the steering inequality (11). Our result shows that the EPR steerability of the state ρ AB determines Bell's nonlocality of the state τ AB , thus may shed a new light to realize a loophole-free Bell-test experiment through the violation of steering inequality.
Example 2.-The theorem naturally provides a steering-based criterion for Bell's nonlocality, which is expressed as follows: given an EPR steerable two-qubit state ρ AB , if the matrix Scientific RepoRts | 6:39063 | DOI: 10.1038/srep39063 AB k It is worth to mention that the steering inequality (11) is applicable to show Bell's nonlocality of τ AB for some parameters α′ , β′ , γ′ . Here we would like to show that the similar task can be done by other new steering inequalities. In the following, we present a 9-setting linear steering inequality as AB k k k k 3 with β, γ, t k being the real coefficients, and û, v the unit vectors. Obviously, under LUT, the state ρ AB is said to be symmetric if and only if β = γ and =û v. Let one consider a simple situation with t 1 = t 2 = t 3 = − α, and = =û v (0, 0, 1), then he obtains the two-qubit state ρ AB as in Eq. (13). In such a case, if ρ AB is a one-way steerable state, then one must have β ≠ γ.
In ref. 15, the authors have chosen β = and used the SDP program to numerically prove that the state ρ AB is a one-way steerable state (with at least 13 projective measurements): for α ≤ 1/2, the state ρ AB is unsteerable from Bob to Alice, while for α > .
∼ 0 4983 the state is steerable from Alice to Bob when Alice performs 14 projective measurements. An explicit 14-setting steering inequality has been also proposed to conform the one-way steerability, although for α = 1/2, the quantum violation is tiny (only 1.0004). The inspiring result for the first time confirms that the nonlocality can be fundamentally asymmetric. However, the tiny inequality violation as well as the 14 measurement settings give rise to the difficulty in experimental detection. To advance the study of unidirectional quantum steering, here we present a more efficient class of one-way steerable states by choosing with α ∈ [0, 1]. The state ρ AB (α) is entangled for α > 0.3279. With the help of the SDP program, we found that in the range α .
< ≤  0 4846 1/2, the state ρ(α) is one-way steerable within 10-setting measurements, thus this is more efficient than the previous result in ref. 15 (For the detail derivation of more efficient one-way EPR steering see Supplementary Materials). Furthermore, we can extract an explicit 9-setting steering inequalities (16) based on the SDP program. It can be verified directly that, for the state ρ AB (1/2), the quantum violation of 9-setting inequality (16) 15 , the amount of violation is much larger but achieved with fewer measurements. To our knowledge, we do not know whether the quantum violation by inequality (16) could be observed with current quantum technology. However, we believe that this result would be interesting and helpful for both theoretical and experimental physicists.

Discussion
In this work, we have presented a theorem showing that Bell nonlocal states can be constructed from some EPR steerable states. This result not only offers a novel and distinctive way to study Bell's nonlocality with the violation of steering inequality, but also may avoid locality loophole in Bell's tests and make Bell's nonlocality easier for demonstration. An interesting and inverse problem is whether one can construct some steerable states τ AB from some Bell nonlocal state ρ AB , because Bell's nonlocality has been researched more deeply in theoretical aspect, so that people can conveniently study steering via known criteria of Bell's nonlocality. Furthermore, an explicit 9-setting linear steering inequality has also been presented for detecting some Bell nonlocal states and developing more efficient one-way steering. This result allows one to observe one-way EPR steering with fewer measurement setting but with larger quantum violations. We hope experimental progress in this direction could be made in the near future.

Methods
Verification of equation (8). Let    By comparing Eqs (19) and (20), it is easy to see that Eq. (8) holds. Thus, if there is a LHV model description for τ AB , then there is a LHS model description for ρ AB . This completes the proof.