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.

Figure 1 Hierarchical structure of quantum nonlocality. Bell’s nonlocality is the strongest type of quantum nonlocality. If a state possesses EPR steerability or Bell’s nonlocality, then the state must be entangled. EPR steering is a form of nonlocality intermediate between entanglement and Bell nonlocality. Full size image

Figure 2 Illustration of detecting Bell’s nonlocality through EPR steering. If a state ρ AB violates a steering inequality, then it implies that ρ AB possesses the EPR steerability. Traditionally, Bell’s nonlocality of the two-qubit state τ AB is revealed by violations of Bell’s inequality. Based on Theorem 1, Bell’s nonlocality of the state τ AB can be detected through EPR steerability of the state ρ AB , and the relation between ρ AB and τ AB is given in Eq. (1). Full size image

Theorem 1: For any two-qubit state τ AB shared by Alice and Bob, define another state

with , being the reduced density matrix at Alice’s side, and . 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

Here is the joint probability, quantum mechanically it is computed as , is the projective measurement along the -direction with measurement outcome a for Alice, is the projective measurement along the -direction with measurement outcome b for Bob (with a, b = 0, 1), , 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 are , , , respectively. Since the state τ AB has a LHV model description, based on Eq. (2) we explicitly have (with )

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

To prove that there exists a LHS model for ρ AB is equivalent to proving that, for any measurement and outcome a, one can always find a hidden state ensemble and the conditional probabilities , such that the relation

is always satisfied. Here ξ’s are the local hidden variables, ρ ξ ’s are the hidden states, and are probabilities satisfying and . 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 solutions of if Eq. (3) is valid. The solutions are given as follows:

where is the 2 × 2 identity matrix, 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 , and this ensures ρ ξ being a density matrix.

By substituting Eq. (6) into Eq. (5), we obtain

To prove the theorem is to verify the relation (8) is always satisfied if Eq. (3) is valid. The verification can be found in Methods.

Remark 1.— In Eq. (7), by requiring the condition be valid for any probabilities P(0|x, ξ), P(0|y, ξ), P(0|z, ξ) ∈ [0, 1], in general one can have . Generally, Theorem 1 is valid for any . In the theorem we have chosen the parameter μ as its maximal value , because the state τ AB is convexed with a separable state , the larger value of μ, the easier to detect the EPR steerability.

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.

Example 1.— For example, let us detect Bell’s nonlocality of the maximally entangled state (with τ AB = |Ψ〉 〈Ψ|)

without Bell’s inequality. Based on the theorem, it is equivalent to detect the EPR steerability of the following two-qubit state

with . The state (10) is nothing but the Werner state28 with the visibility equals to , its steerability can be tested by using the steering inequality proposed in ref. 17 as

with N = 6. Here is the steering parameter for N measurement settings, and C N is the classical bound, with . The maximal quantum violation of the steering inequality is , 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 inequality29. 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 time30. As a weaker nonlocality, EPR steering naturally escapes from the locality loophole and is correspondingly easier to be demonstrated without the detection loophole19,20, 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

is a two-qubit density matrix, then τ AB is Bell nonlocal.

Let us consider a two-qubit state ρ AB in the following form

By substituting the state ρ AB as in Eq. (13) into Eq. (12), then one obtains

with

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

here for convenient we have used the same notations as in ref. 15 (where (σ 1 , σ 2 , σ 3 ) is equivalent to (σ x , σ y , σ z )). The inequality are characterized by matrices {S, SA, SB} with real coefficients s ij , , and , and the local bound is L = 1 (see Supplementary Materials). The steering inequality (16) may have other particular application for improving the result ref. 15 by developing more efficient one-way steering, which we shall address in the coming section. But now we use it to detect Bell’s nonlocality.

For example, let α′ = 0.96, β′ = −1/5, γ′ = 1/6, ones finds that τ AB is a two-qubit state, and the steering inequality (16) is violated by the state ρ AB (with the violation value 1.0064), hence the Bell’s nonlocality of state τ AB can be revealed in this way indirectly by the steerability of the state ρ AB .

More efficient one-way EPR steering

Under local unitary transformation (LUT), any two-qubit state can be written in the following form ref. 31

with β, γ, t k being the real coefficients, and , the unit vectors. Obviously, under LUT, the state ρ AB is said to be symmetric if and only if β = γ and . Let one consider a simple situation with t 1 = t 2 = t 3 = −α, and , 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 , 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 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