Maximal violation of Bell inequalities under local filtering

We investigate the behavior of the maximal violations of the CHSH inequality and Vèrtesi’s inequality under the local filtering operations. An analytical method has been presented for general two-qubit systems to compute the maximal violation of the CHSH inequality and the lower bound of the maximal violation of Vértesi’s inequality over the local filtering operations. We show by examples that there exist quantum states whose non-locality can be revealed after local filtering operation by the Vértesi’s inequality instead of the CHSH inequality.

are the Pauli matrices. For any two-qubit quantum state ρ, the maximal violation of the CHSH inequality (MVCI) is given by 14 Another effective Bell inequality for two-qubit system is given by the Bell operator 15 Vértesi where A i , B j , C ij and D ij are observables of the form σ ∑ α α α = x 1 3 with → = x x x x ( , , ) 1 3 the unit vectors. The maximal violation of Vértesi's inequality(MVVI) is lower bounded by the following inequality 16 The maximum on the right side of the inequality goes over all the integral area Ω × Ω of a state ρ admitting LHV model is upper bounded by 1.
The maximal violation of a Bell inequality above is derived by optimizing the observables for a given quantum state. With the formulas (3) and (5) one can directly check if a two-qubit quantum state violates the CHSH or the Vértesi's inequality. It has been shown that the maximal violation of a Bell inequality is in a close relation with the fidelity of the quantum teleportation 17 and the device-independent security of quantum cryptography 18 .
The maximal violation of a Bell inequality can be enhanced by local filtering operations 19 . In ref. 20, the authors present a class of two-qubit entangled states admitting local hidden variable models, and show that the states after local filtering violate a Bell inequality. Hence, there exist entangled states, the non-locality of which can be revealed by using a sequence of measurements.
In this manuscript, we investigate the behavior of the maximal violations of the CHSH inequality and Vértesi's inequality under local filtering operations. An analytical method has been presented for any two-qubit system to compute the maximal violation of the CHSH inequality and the lower bound of the maximal violation of Vértesi's inequality under local filtering operations. The corresponding optimal local filtering operation is derived. We show by examples that there exist quantum states whose nonlocality can be revealed after local filtering operation by Vértesi's inequality instead of the CHSH inequality.

Results
We consider the CHSH inequality for two-qubit systems first. Before the Bell test, we apply the local filtering operation on a state  . ρ is mapped to the following form under local filtering transformations 20,21 : is a normalization factor, and F A/B are positive operators acting on the subsystems respectively. Such operations can be a local interaction with the dichroic environments 22 .
For two-qubit systems, let be the spectral decompositions of F A and F B respectively, where U and V are unitary operators. Define that and X be a matrix with entries given by where  is locally unitary with ρ.
We have the following theorem.

Theorem 1:
The maximal quantum bound of a two-qubit quantum state ρ ρ is given by where τ′ 1 and τ′ 2 are the two largest eigenvalues of the matrix X † X/N 2 with X given by (8). The left max is taken over all B CHSH operators, while the right max is taken over all  that are locally unitary equivalent to ρ. See Methods for the proof of theorem 1. Now we investigate the behavior of the Vèrtesi-Bell inequality under local filtering operations. In ref. 16 we have found an effective lower bound for the MVVI by considering infinite many measurements settings, n → ∞ . Then the discrete summation in (4) is transformed into an integral of the spherical coordinates over the sphere ⊂ S R 2 3 . We denote the spherical coordinate of S 2 by (φ 1 , where X is defined by (8). X t stands for the transposition of X, and The maximization on the right side of the inequality goes over all the integral area Ω × Ω See Methods for the proof of theorem 2.

Remark:
The right hand sides of (9) and (10) depend just on the state σ which is local unitary equivalent to ρ. Thus to compare the difference of the maximal violation for ρ and that for ρ′ , it is sufficient to just consider the difference between σ and ρ′ .
Without loss of generality, we set According to the definition of δ k and η l in (7), one computes that Let σ = ( ) Then one has where O A and O B are 3 × 3 orthogonal operators. Define that → r and → s be three dimensional vectors with entries where τ′ 1 and τ′ 2 are the two largest eigenvalues of the matrix X † X/N 2 with X given by kl w k l

Applications
In the following we discuss the applications of local filtering. First we show that a state which does not violate the CHSH and the Vértesi's inequalities could violate these inequalities after local filtering. Consider the following density matrix for two-qubit systems: where − 0.3104 ≤ p ≤ 0.7 to ensure the positivity of  1 . By using the positive partial transposition criteria one has that  1 is separable for − 0.3104 ≤ p ≤ 0.3104.
Case 1: Set r = 0.3. It is direct to verify that both the CHSH inequality and Vértesi's inequalities fail to detect the non-locality for the whole region − 0.3104 ≤ p ≤ 0.7. After filtering, non-locality can be detected for 0.6291 ≤ p ≤ 0.7 (by Theorem 2) and 0.6164 ≤ p ≤ 0.7 (by Theorem 1) respectively, see Fig. 1.
Case 2: Set p = 0.7050 and r = 0.0400. The MVCI of  1 is 1.994 without local filtering and 1.9988 after local filtering, which means that the CHSH inequality is always satisfied before and after local filtering. The lower bound (5) for 1  is computed to be less than one, implying the non-locality can not be detected by the lower bound for MVVI derived in ref. 16 without local filtering. However, by taking x = y = 1.1, a = c = 0.1671, b = d = 1.1096, from Theorem 2 we have the maximal violation value 1.0005 which is larger than one. Therefore, after local filtering the state's non-locality is detected.
Next we give an example that a state admits local hidden variable model (LHV) can violate the Bell inequality under local filtering. Consider two-qubit quantum states with density matrices of the following form: According to the positivity of a density matrix, we have − 0.5 ≤ p ≤ 0.3090. By using the positive partial transposition criteria 24 , one checks that  2 is entangled for − 0.5 ≤ p ≤ 0.3090. The quantum state satisfies the CHSH inequality for the whole parameter region.
We first show that the state 2  admits LHV models for − 0.5 ≤ p ≤ − 0.3090. First we rewrite  2 as a convex combination of singlet and separable states,  which can be given by the following LHV model,  Therefore the state  2 admits LHV model for − 0.5 ≤ p ≤ − 0.309. However, after local filtering, non-locality (violation of the CHSH inequality) is detected for − 0.5 ≤ p ≤ − 0.4859, see Fig. 2.
Remark: In ref. 17 Horodeckis have presented the connection between the maximal violation of the CHSH inequality and the optimal quantum teleportation fidelity: which means that any two-qubit quantum state violating the CHSH inequality is useful for teleportation and vice versa. Acín et al. have derived the relation between the maximal violation of the CHSH inequality and the Holevo quantity between Eve and Bob in device-independent Quantum key distribution (QKD) 18 : where h is the binary entropy. From our theorem, B B C HSH CHSH | | ρ max can be enhanced by implementing a proper local filtering operation from smaller to larger than 2, which makes a teleportation possible from impossible, or can be improved to obtain a better teleportation fidelity. The proper (optimal) local filtering operation can be selected by the optimizing process in (9) together with the double cover relationship between the SU(2) and SO (3). For application in the QKD, Eve can enhance the upper bound of Holevo quantity by local filtering operations which makes a chance for attacking the protocol.

Discussions
It is a fundamental problem in quantum theory to recognize and explore the non-locality of a quantum system. The Bell inequalities and their maximal violations supply powerful ability to detect and qualify the non-locality. Furthermore, the constructing and the computation of the maximal violation of a Bell inequality is in close relationship with quantum games, minimal Hilbert space dimension and dimension witnesses, as well as quantum communications such as communication complexity, quantum cryptography, device-independent quantum key distribution etc. ref. 6. A proper local filtering operation can generate and enhance the non-locality. We have investigated the behavior of the maximal violations of the CHSH inequality and the Vértesi's inequality under local filtering. We have presented an analytical method for any two-qubit system to compute the maximal violation of the CHSH inequality and the lower bound of the maximal violation of Vértesi's inequality under local filtering. We have shown by examples that there exist quantum states whose nonlocality can be revealed by local filtering operations in terms of the Vértesi's inequality instead of the CHSH inequality.

Methods
Proof of Theorem 1 and Theorem 2. The normalization factor N has the following form, In deriving the fourth equality in (28) we have used the double cover relation between the special unitary group SU (2)  By noticing the orthogonality of the operator O B we have that the eigenvalues of (T′ ) † T′ and X † X/N 2 must be the same, which proves theorem 1.