Introduction

Quantum mechanics is inherently nonlocal. After performing local measurements on a composite quantum system, non-locality, which is incompatible with local hidden variable theory1 can be revealed by Bell inequalities. The non-locality is of great importance both in understanding the conceptual foundations of quantum theory and in investigating quantum entanglement. It is also closely related to certain tasks in quantum information processing, such as building quantum protocols to decrease communication complexity2,3 and providing secure quantum communication4,5. We refer to ref. 6 for more details.

To determine whether a quantum state has non-locality, it is sufficient to construct a Bell inequality7,8,9,10,11,12,13 which can be violated by the quantum state. For two qubits systems, Clauser-Horne-Shimony-Holt have presented the famous CHSH inequality7.

Let denote the Bell operator for the CHSH inequality,

with Ai and Bj being the observables of the form and respectively, i, j = 1, 2,

are the Pauli matrices. For any two-qubit quantum state ρ, the maximal violation of the CHSH inequality (MVCI) is given by14

where τ1 and τ2 are the two largest eigenvalues of the matrix T†T, T is the matrix with entries , α, β = 1, 2, 3, † stands for transpose and conjugation. For a state admitting local hidden variable (LHV) model, one has .

Another effective Bell inequality for two-qubit system is given by the Bell operator15 Vértesi

where Ai, Bj, Cij and Dij are observables of the form with the unit vectors.

The maximal violation of Vértesi’s inequality(MVVI) is lower bounded by the following inequality16. For arbitrary two-qubit quantum state ρ, we have

where . The maximum on the right side of the inequality goes over all the integral area with and . Here the maximal value 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 teleportation17 and the device-independent security of quantum cryptography18.

The maximal violation of a Bell inequality can be enhanced by local filtering operations19. 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 with . ρ is mapped to the following form under local filtering transformations20,21:

where is a normalization factor, and FA/B are positive operators acting on the subsystems respectively. Such operations can be a local interaction with the dichroic environments22.

For two-qubit systems, let and be the spectral decompositions of FA and FB 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 is given by

where and are the two largest eigenvalues of the matrix X†X/N2 with X given by (8). The left max is taken over all BCHSH 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 . We denote the spherical coordinate of S2 by (ϕ1, ϕ2). A unit vector can be parameterized by x1 = sin ϕ1 sin ϕ2, x2 = sin ϕ1 cos ϕ2, x3 = cos ϕ1. For any , we denote .

Theorem 2: For two-qubit quantum state ρ′ given by (6), we have

where X is defined by (8). Xt stands for the transposition of X, and . The maximization on the right side of the inequality goes over all the integral area with and .

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

with x, y ≥ 0. According to the definition of δk and ηl in (7), one computes that

Let . Set , , and . We have and , where

Then one has xkl = (CWD†), where W is a 4 × 4 matrix with entries . Let and where OA and OB are 3 × 3 orthogonal operators. Define that and be three dimensional vectors with entries and respectively. And let . One can further show that

and

where , , and . Numerically, one can parameterize OA and OB and then search for the maximization in theorem 1. For the lower bound in theorem 2, we refer to ref. 16.

Corollary: For two-qubit Werner state23 , with , one computes . Then by using the symmetric property of the state, (15) and (16), together with theorem 1, we have

where and are the two largest eigenvalues of the matrix X†X/N2 with X given by

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 . By using the positive partial transposition criteria one has that 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.

Figure 1: For r = 0.3, both the CHSH inequality and Vértesi’s inequality fail to detect the non-locality of for the whole parameter region of p.
figure 1

After local filtering, non-locality is detected for 0.6291 ≤ p ≤ 0.7 (by Theorem 2) and 0.6164 ≤ p ≤ 0.7 (by Theorem 1) respectively.

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Case 2: Set p = 0.7050 and r = 0.0400. The MVCI of 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 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 criteria24, one checks that 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 admits LHV models for −0.5 ≤ p ≤ −0.3090.

First we rewrite as a convex combination of singlet and separable states,

where and q = −p. According to ref. 25, with a visibility of , the correlations of measurement outcomes produced by measuring the observables and on the singlet state can be simulated by an LHV model in which the hidden variable is biased distributed with probability density

With probability , Alice and Bob can share the biased distributed variable resource and output and , respectively. With probability 1 − q, Alice outputs a = ±1 with probability , and Bob outputs ±1 with probability . Then we can simulate the correlations produced by measuring obesrvables A and B on ,

which can be given by the following LHV model,

where . Explicitly,

where , , , .

Therefore the state 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.

Figure 2: The MVCI of (dashed line) v.s. the MVCI after Local filtering (solid line).
figure 2

f(p) stands for the MVCI. Note that the classical bound of the CHSH inequality is 2.

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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, 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,

where . Since ρ and are local unitary equivalent, they must have the same value of the maximal violation for CHSH inequality.

We have that

In deriving the fourth equality in (28) we have used the double cover relation between the special unitary group SU(2) and the special orthogonal group SO(3): for any given unitary operator U, , where the matrix O with entries Oij belongs to SO(3)26,27.

Finally, one has that

and

By noticing the orthogonality of the operator OB we have that the eigenvalues of (T′)†T′ and X†X/N2 must be the same, which proves theorem 1.

We can further obtain theorem 2 by substituting (29) into (5).

Additional Information

How to cite this article: Li, M. et al. Maximal violation of Bell inequalities under local filtering. Sci. Rep. 7, 46505; doi: 10.1038/srep46505 (2017).

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