Maximal violation of Bell inequalities under local filtering
aa r X i v : . [ qu a n t - ph ] A p r Maximal violation of Bell inequalities under local filtering
Ming Li † , Huihui Qin ‡§ , Jing Wang † , Shao-Ming Fei § ♯ , and Chang-Pu Sun ♭ † College of the Science, China University of Petroleum,Qingdao 266580, P. R. China ‡ Department of Mathematics, School of Science, South China University of TechnologyGuangzhou 510640, P. R. China § Max-Planck-Institute for Mathematics in the Sciences,Leipzig 04103, Germany ♯ School of Mathematical Sciences, Capital Normal University,Beijing 100048, P. R. China ♭ Beijing Computational Science Research Center,Beijing 100048, P. R. China ∗ Correspondence to [email protected]
Abstract
We investigate the behavior of the maximal violations of the CHSH inequality andV` e rtesi’s inequality under the local filtering operations. An analytical method hasbeen presented for general two-qubit systems to compute the maximal violation of theCHSH inequality and the lower bound of the maximal violation of V´ e rtesi’s inequalityover the local filtering operations. We show by examples that there exist quantumstates whose non-locality can be revealed after local filtering operation by the V´ e rtesi’sinequality instead of the CHSH inequality. Quantum mechanics is inherently nonlocal. After performing local measurements on acomposite quantum system, non-locality, which is incompatible with local hidden variabletheory [1] can be revealed by Bell inequalities. The non-locality is of great importanceboth in understanding the conceptual foundations of quantum theory and in investigatingquantum entanglement. It is also closely related to certain tasks in quantum informationprocessing, such as building quantum protocols to decrease communication complexity [2, 3]and providing secure quantum communication [4, 5]. We refer to [6] for more details.1o determine whether a quantum state has non-locality, it is sufficient to construct aBell inequality [7–13] which can be violated by the quantum state. For two qubits systems,Clauser-Horne-Shimony-Holt have presented the famous CHSH inequality [7].Let B CHSH denote the Bell operator for the CHSH inequality, B CHSH = A ⊗ B + A ⊗ B + A ⊗ B − A ⊗ B , (1)with A i and B j being the observables of the form A i = P k =1 a ik σ k and B j = P l =1 b jl σ l respectively, i, j = 1 , σ = − ! , σ = ! and σ = i − i ! (2)are the Pauli matrices. For any two-qubit quantum state ρ , the maximal violation of theCHSH inequality (MVCI) is given by [14]max B CHSH |hB
CHSH i ρ | = 2 √ τ + τ , (3)where τ and τ are the two largest eigenvalues of the matrix T † T , T is the matrix withentries T αβ = tr [ ρ σ α ⊗ σ β ], α, β = 1 , ,
3. For a state admitting local hidden variable (LHV)model, one has max B CHSH |hB
CHSH i LHV | ≤ e rtesi B V = 1 n [ n X i,j =1 A i ⊗ B j + X ≤ i
We consider the CHSH inequality for two-qubit systems first. Before the Bell test, weapply the local filtering operation on a state ρ ∈ H = H A ⊗ H B with dim H A = dim H B = 2. ρ is mapped to the following form under local filtering transformations [19, 22]: ρ ′ = 1 N ( F A ⊗ F B ) ρ ( F A ⊗ F B ) † , (6)where N = tr [( F A ⊗ F B ) ρ ( F A ⊗ F B ) † ] is a normalization factor, and F A/B are positiveoperators acting on the subsystems respectively. Such operations can be a local interactionwith the dichroic environments [23].For two-qubit systems, let F A = U Σ A U † and F B = V Σ B V † be the spectral decomposi-tions of F A and F B respectively, where U and V are unitary operators. Define that δ k = Σ A σ k Σ A , η l = Σ B σ l Σ B (7)and X be a matrix with entries given by x kl = tr [ ̺δ k ⊗ η l ] , k, l = 1 , , , (8)where ̺ is locally unitary with ρ .we have the following theorem. Theorem 1:
The maximal quantum bound of a two-qubit quantum state ρ ′ = N ( F A ⊗ F B ) ρ ( F A ⊗ F B ) † is given by max B CHSH |hB
CHSH i ρ ′ | = max ̺ p τ ′ + τ ′ , (9)where τ ′ and τ ′ are the two largest eigenvalues of the matrix X † X/N with X given by (8).The left max is taken over all B CHSH operators, while the right max is taken over all ̺ thatare locally unitary equivalent to ρ . 3ee Methods for the proof of theorem 1.Now we investigate the behavior of the V` e rtesi-Bell inequality under local filtering oper-ations. In [20] we have found an effective lower bound for the MVVI by considering infinitemany measurements settings, n → ∞ . Then the discrete summation in (4) is transformedinto an integral of the spherical coordinates over the sphere S ⊂ R . We denote the spher-ical coordinate of S by ( φ , φ ). A unit vector ~x = ( x , x , x ) can be parameterized by x = sin φ sin φ , x = sin φ cos φ , x = cos φ . For any 0 ≤ a ≤ b ≤ π , we denoteΩ ba = { x ∈ S : a ≤ φ ( x ) ≤ b } . Theorem 2:
For two-qubit quantum state ρ ′ given by (6), we havemax B V |hB V i ρ ′ | ≥ max a,b,c,d N (cid:20) s ab s cd | Z Ω ba × Ω dc < ~x, X~y > dµ ( ~x ) dµ ( ~y ) | + 12 s cd Z Ω dc × Ω dc | X ( ~x − ~y ) | dµ ( ~x ) dµ ( ~y ) + 12 s ab Z Ω ba × Ω ba | X t ( ~x − ~y ) | dµ ( ~x ) dµ ( ~y ) (cid:21) , (10)where X is defined by (8). X t stands for the transposition of X , and s αβ = R Ω βα dµ ( ~x ). Themaximization on the right side of the inequality goes over all the integral area Ω ba × Ω dc with0 ≤ a < b ≤ π and 0 ≤ c < d ≤ π .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 localunitary equivalent to ρ . Thus to compare the difference of the maximal violation for ρ andthat for ρ ′ , it is sufficient to just consider the difference between σ and ρ ′ .Without loss of generality, we setΣ A = x
00 1 ! and Σ B = y
00 1 ! (11)with x, y ≥
0. According to the definition of δ k and η l in (7), one computes that δ = − x
00 1 ! , δ = xx ! and δ = ix − ix ! ; (12) η = − y
00 1 ! , η = yy ! and η = iy − iy ! . (13)Let σ = ! . Set ~δ = ( δ , δ , δ ), ~η = ( η , η , η ), and ~σ = ( σ , σ , σ , σ ). We have ~δ = C~σ and ~η = D~σ , where C = (1 − x ) (1 + x ) 0 00 0 x
00 0 0 x and D = (1 − y ) (1 + y ) 0 00 0 y
00 0 0 y respectively . (14)4hen one has x kl = ( CW D † ) , where W is a 4 × w αβ = tr [ σσ α ⊗ σ β ].Let ˜ O A = O A ! and ˜ O B = O B ! where O A and O B are 3 × ~r and ~s be three dimensional vectors with entries r i = tr [ ρσ ⊗ σ i ]and s j = tr [ ρσ j ⊗ σ ] respectively. And let ˜ T = ~r~s T ! . One can further show that X = CW D † = C ˜ O A ˜ T ˜ O † B D † , (15)and N = x + y + + 4 x − y + ( O A ~s ) + 4 x + y − ( O B ~r ) + 4 x − y − ( O A T O tB ) , (16)where x + = (1 + x ), x − = (1 − x ), y + = (1 + y ) and y − = (1 − y ). Numerically, onecan parameterize O A and O B and then search for the maximization in theorem 1. For thelower bound in theorem 2, we refer to [20]. Corollary:
For two-qubit Werner state [27] ρ w = p | ψ − ih ψ − | + (1 − p ) I , with | ψ − i =( | i − | i ) / √
2, one computes T = − p − p
00 0 − p . Then by using the symmetricproperty of the state, (15) and (16), together with theorem 1, we havemax B CHSH |hB
CHSH i ρ ′ | = 2 p τ ′ + τ ′ , (17)where τ ′ and τ ′ are the two largest eigenvalues of the matrix X † X/N with X given by x kl = tr [ ρ w δ k ⊗ η l ] , k, l = 1 , , . (18) Applications
In the following we discuss the applications of local filtering. First we show that astate which does not violate the CHSH and the V´ e rtesi’s inequalities could violate theseinequalities after local filtering. Consider the following density matrix for two-qubit systems: ̺ = 14 ( I ⊗ I + rσ ⊗ I − p X i σ i ⊗ σ i ) , (19)where − . ≤ p ≤ . ̺ . By using the positive partialtransposition criteria one has that ̺ is separable for − . ≤ p ≤ . r = 0 .
3. It is direct to verify that both the CHSH inequality and V´ e rtesi’sinequalities fail to detect the non-locality for the whole region − . ≤ p ≤ .
7. Afterfiltering, non-locality can be detected for 0 . ≤ p ≤ . . ≤ p ≤ . p = 0 . r = 0 . ̺ is 1 .
994 without local filteringand 1 . on - localitydetected by Vertesi'sinequality after Local filteringNon - localitydetected by CHSHafter Local filteringSeparable - - Figure 1: For r = 0 .
3, both the CHSH inequality and V´ertesi’s inequality fail to detectthe non-locality of ̺ for the whole parameter region of p . After local filtering, non-localityis detected for 0 . ≤ p ≤ . . ≤ p ≤ . ̺ is computed to be less than one,implying the non-locality can not be detected by the lower bound for MVVI derived in [20]without local filtering. However, by taking x = y = 1 . , a = c = 0 . , b = d = 1 . . ̺ = 14 ( I ⊗ I + pσ ⊗ I + p X i σ i ⊗ σ i ) . (20)According to the positivity of a density matrix, we have − . ≤ p ≤ . ̺ is entangled for − . ≤ p ≤− . ̺ admits LHV models for − . ≤ p ≤ − . ̺ as a convex combination of singlet and separable states, ̺ = q | ψ − ih ψ − | + (1 − q )[ 12 ( I − q − q σ ) ⊗ I , (21)where | ψ − ih ψ − | = ( I ⊗ I − P i =1 σ i ⊗ σ i ) and q = − p . According to [16], with a visibilityof q = , the correlations of measurement outcomes produced by measuring the observables A = −→ a · −→ σ and B = −→ b · −→ σ on the singlet state can be simulated by an LHV model in which6he hidden variable −→ λ s ∈ S is biased distributed with probability density ρ ( −→ λ s |−→ a ) = |−→ a · −→ λ s | π . (22)With probability 0 < q ≤ , Alice and Bob can share the biased distributed variableresource and output a = − sgn ( −→ a · −→ λ s ) and b = sgn ( −→ b · −→ λ s ), respectively. With probability1 − q , Alice outputs a = ± p ( a |−→ a ) = tr [ ( I − q − q σ z ) I ±−→ a ·−→ λ s ], and Boboutputs ± p ( b |−→ b ) = . Then we can simulate the correlations producedby measuring obesrvables A and B on ̺ , p ( a, b |−→ a , −→ b , ̺ ) = tr ( I + a −→ a −→ σ ⊗ I + b −→ b −→ σ ρ ) = 1 − qab −→ a · −→ b − aa q , (23)which can be given by the following LHV model, p ( a, b |−→ a , −→ b , ̺ ) = q Z S p ( a |−→ a , −→ λ s ) p ( b |−→ b · −→ λ s ) ρ ( −→ λ s ) d −→ λ s + (1 − q ) p ( a |−→ a ) p ( b |−→ b )= q Z Ω a,b |−→ a · −→ λ s | π d −→ λ s + (1 − q ) p ( a |−→ a ) p ( b |−→ b ) , (24)where Ω a,b = {−→ λ s | − sgn ( −→ a · −→ λ s ) = a } ∩ {−→ λ s | b = sgn ( −→ b · −→ λ s ) } . Explicitly, p (1 , |−→ a , −→ b , −→ λ s ) = q Z Ω , |−→ a · −→ λ s | π d −→ λ s + 1 − q tr [ 12 ( I − q − q σ z ) I + −→ a · −→ λ s ,p (1 , − |−→ a , −→ b , −→ λ s ) = q Z Ω , − |−→ a · −→ λ s | π d −→ λ s + 1 − q tr [ 12 ( I − q − q σ z ) I + −→ a · −→ λ s ,p ( − , |−→ a , −→ b , −→ λ s ) = q Z Ω − , |−→ a · −→ λ s | π d −→ λ s + 1 − q tr [ 12 ( I − q − q σ z ) I − −→ a · −→ λ s ,p ( − , − |−→ a , −→ b , −→ λ s ) = q Z Ω − , − |−→ a · −→ λ s | π d −→ λ s + 1 − q tr [ 12 ( I − q − q σ z ) I − −→ a · −→ λ s , where Ω , = {−→ λ s |−→ a ·−→ λ < }∩{−→ λ s |−→ b ·−→ λ ≥ } , Ω , − = {−→ λ s |−→ a ·−→ λ < }∩{−→ λ s |−→ b ·−→ λ < } ,Ω − , = {−→ λ s |−→ a · −→ λ ≥ } ∩ {−→ λ s |−→ b · −→ λ ≥ } , Ω − , − = {−→ λ s |−→ a · −→ λ ≥ } ∩ {−→ λ s |−→ b · −→ λ < } .Therefore the state ̺ admits LHV model for − . ≤ p ≤ − . − . ≤ p ≤ − . Remark:
In [17] Horodeckis have presented the connection between the maximal viola-tion of the CHSH inequality and the optimal quantum teleportation fidelity: F max ≥
12 (1 + 112 max B CHSH |hB
CHSH i ρ | ) (25)which means that any two-qubit quantum state violating the CHSH inequality is useful forteleportation and vice versa. Ac´ i n et al. have derived the relation between the maximal7 - - - - - - - - - - - - - H H p L L Figure 2: The MVCI of ̺ (dashed line) v.s. the MVCI after Local filtering (solid line). f ( p )stands for the MVCI. Note that the classical bound of the CHSH inequality is 2.violation of the CHSH inequality and the Holevo quantity between Eve and Bob in device-independent Quantum key distribution(QKD) [18]: χ ( B : E ) ≤ h ( 1 + p (max B CHSH |hB
CHSH i ρ | / −
12 ) , (26)where h is the binary entropy. From our theorem, max B CHSH |hB
CHSH i ρ | can be enhanced byimplementing a proper local filtering operation from smaller to larger than 2, which makes ateleportation possible from impossible, or can be improved to obtain a better teleportationfidelity. The proper(optimal) local filtering operation can be selected by the optimizingprocess 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 localfiltering operations which makes a chance for attacking the protocol. Discussions
It is a fundamental problem in quantum theory to recognize and explore the non-localityof a quantum system. The Bell inequalities and their maximal violations supply powerfulability to detect and qualify the non-locality. Furthermore, the constructing and the com-putation of the maximal violation of a Bell inequality is in close relationship with quantumgames, minimal Hilbert space dimension and dimension witnesses, as well as quantum com-munications such as communication complexity, quantum cryptography, device-independentquantum key distribution etc. [6]. A proper local filtering operation can generate and en-hance the non-locality. We have investigated the behavior of the maximal violations of theCHSH inequality and the V´ e rtesi’s inequality under local filtering. We have presented ananalytical method for any two-qubit system to compute the maximal violation of the CHSHinequality and the lower bound of the maximal violation of V´ e rtesi’s inequality under local8ltering. We have shown by examples that there exist quantum states whose nonlocality canbe revealed by local filtering operations in terms of the V´ e rtesi’s inequality instead of theCHSH inequality. Methods
Proof of Theorem 1 and Theorem 2
The normalization factor N has the following form, N = tr [ U Σ A U † ⊗ V Σ B V † ρ ] = tr [Σ A ⊗ Σ B U † ⊗ V † ρU ⊗ V ]= tr [Σ A ⊗ Σ B ̺ ] , (27)where ̺ = U † ⊗ V † ρU ⊗ V . Since ρ and ̺ are local unitary equivalent, they must have thesame value of the maximal violation for CHSH inequality.We have that t ′ ij = tr [ ρ ′ σ i ⊗ σ j ] = 1 N tr [( F A ⊗ F B ) ρ ( F † A ⊗ F B ) † σ i ⊗ σ j ]= 1 N tr [ ρU Σ A U † σ i U Σ A U † ⊗ V Σ B V † σ j V Σ B V † ]= 1 N X kl tr [ U † ⊗ V † ρU ⊗ V Σ A O Aik σ k Σ A ⊗ Σ B O Bjl σ l Σ B ]= 1 N X kl O Aik O Bjl tr [ ̺ Σ A σ k Σ A ⊗ Σ B σ l Σ B ]= 1 N X kl O Aik O Bjl tr [ ̺δ k ⊗ η l ]= 1 N X kl O Aik x kl O Bjl = 1 N ( O A XO TB ) ij . (28)In deriving the fourth equality in (28) we have used the double cover relation betweenthe special unitary group SU (2) and the special orthogonal group SO (3): for any givenunitary operator U , U σ i U † = P j =1 O ij σ j , where the matrix O with entries O ij belongs to SO (3) [25, 26].Finally, one has that T ′ = 1 N O A XO † B , (29)and ( T ′ ) † T ′ = 1 N O B X † O † A O A XO † B = 1 N O B X † XO † B . (30)By noticing the orthogonality of the operator O B we have that the eigenvalues of ( T ′ ) † T ′ and X † X/N must be the same, which proves theorem 1.We can further obtain theorem 2 by substituting (29) into (5).9 eferences [1] Bell J.S. On the Einstein Podolsky Rosen Paradox. Physics , 195-200 (1964).[2] Brukner ˇC., ˙Zukowski M. & Zeilinger A. Quantum Communication Complexity Protocolwith Two Entangled Qutrits. Phys. Rev. Lett. , 197901 (2002).[3] Buhrman H., Cleve R., Massar S., & de Wolf R. Nonlocality and communication com-plexity. Rev. Mod. Phys. , 665 (2010).[4] Scarani V., & Gisin N. Quantum Communication between N Partners and Bell’s In-equalities. Phys. Rev. Lett. , 117901 (2001).[5] Ekert A.K. Phys. Rev. Lett.
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Author contributions
M. Li and H.H. Qin wrote the main manuscript text. All authors reviewed the manuscript.