aa r X i v : . [ qu a n t - ph ] J un More efficient Bell inequalities for Werner states
T. V´ertesi ∗ Institute of Nuclear Research of the Hungarian Academy of SciencesH-4001 Debrecen, P.O. Box 51, Hungary (Dated: October 26, 2018)In this paper we study the nonlocal properties of two-qubit Werner states parameterized by thevisibility parameter 0 ≤ p ≤
1. New family of Bell inequalities are constructed which prove the two-qubit Werner states to be nonlocal for the parameter range 0 . < p ≤
1. This is slightly widerthan the range 0 . < p ≤
1, corresponding to the violation of the Clauser-Horne-Shimony-Holt(CHSH) inequality. This answers a question posed by Gisin in the positive, i.e., there exist Bellinequalities which are more efficient than the CHSH inequality in the sense that they are violatedby a wider range of two-qubit Werner states.
PACS numbers: 03.65.Ud, 03.67.-a
I. INTRODUCTION
Quantum Mechanics is inherently nonlocal, clearlydemonstrated by the fact that measurements on quantumstates may violate the so-called Bell inequalities [1, 2].This has been verified experimentally as well, up to sometechnical loopholes [3]. On the other hand, when a quan-tum state cannot be prepared using only local operationsand classical communication, it possesses quantum cor-relations and we say that the state is entangled. It wasWerner who asked firstly what the relation is betweenquantum nonlocality and quantum correlations [4]. It isactually known that any pure entangled state of two ormore subsystems may violate a generalized Bell inequal-ity [5, 6], thus here nonlocality and entanglement coin-cide. For mixed states, however, the relation betweenentanglement and nonlocality is much complicated. In1989 Werner [4] constructed a family of bipartite mixedstates (became known as Werner states), which, while be-ing entangled, yield outcomes that admit a local hiddenvariables (LHV) model. This conclusively proved thatentanglement and nonlocality are different resources.However, if we want to describe quantitatively the dif-ference, the picture turns out to be quite subtle even inthe case of two-qubit Werner states, which are mixturesof the singlet | ψ − i = ( | i − | i ) / √ ρ Wp = p | ψ − ih ψ − | + (1 − p ) / . (1)Werner showed [4] that these states are separable if andonly if p ≤ /
3. With respect to the locality properties,on one hand, Werner states admit a LHV model for allmeasurements for p ≤ /
12 [7] and admit a LHV modelfor projective measurements for p ≤ . p > / √
2, in which case LHV model clearly cannotbe constructed. It is not known whether Werner states ∗ Electronic address: [email protected] admit an LHV model in the region 0 . < p ≤ / √ p where the state ceases to be non-local, designated p Wc , is particularly relevant from theviewpoint of experiments since this value specifies theamount of noise the singlet tolerates before losing its non-local properties. This issue was addressed by Gisin sometime ago [9] (see also [10]), who posed the question tofind Bell inequalities which are more efficient than theCHSH one for Werner states. In this paper we intend togive a definite answer to this question by providing Bellinequalities which can be violated slightly stronger thanthe CHSH one, resulting in the bound p Wc ≤ . p Wc ≤ / √ ∼ . p Wc forWerner states could not be decreased even on this way[12]. There is also an interesting line of research, whichexplores the parameter region of Bell violation for Wernerstates by restricting the class of possible LHV models[13, 14]. Actually, Ref. [14] could achieve violation ofcertain Bell inequalities, assuming the above limitationsfor p ≥ /
3, i.e., for the entire range of the nonsepara-bility region. An other way of generalization to obtainthe range of locality is the extension of the Werner statesto e.g., more parties [15] or higher dimensions [16, 17].However, let us mention, that a gap also remained inthese cases between the best known local model [16, 17]and the proven nonlocality threshold [18, 19].The outline of the present work is as follows. In Sec. IIwe briefly summarize the relation between Bell inequal-ities for two-qubit Werner states and Grothendieck con-stant of order 3, denoted by K G (3). In Sec. III a familyof Bell inequalities is constructed and in Sec. IV with theaid of these inequalities a lower bound, bigger than √ K G (3), implying that Werner states (1) with p < / √ K G (3) and for higherorders ( K G ( d ) with d = 4 , K G (4) > √
2, and in Sec. VII the relevanceproperty of the constructed family of Bell inequalitiesis demonstrated. Sec. VIII summarizes and poses someopen questions.
II. BELL INEQUALITIES LINKED TOGROTHENDIECK CONSTANTS
Define the expression I = | m X i,j =1 M ij a i b j | , (2)where M is any m × m matrix with real entries and a , . . . , a m , b , . . . , b m ∈ {− , +1 } . Now let us define I d = | m X i,j =1 M ij ~a i · ~b j | , (3)where the unit vectors ~a , . . . , ~a m ,~b , . . . ,~b m are in R d and ~a · ~b is the dot product of ~a and ~b . Grothendieckconstant plays a prominent role in the theorem of linearoperators on Banach spaces [20]. Grothendieck constantof order d , designated K G ( d ), for any integer d ≥
2, canbe defined as [21] I d ≤ K G ( d ) max a i ,b j = ± I (4)for all unit vectors ~a , . . . , ~a m ,~b , . . . ,~b m in R d and forall m × m matrix M . The constant K G ( d ) is taken to bethe smallest possible one.Now let us discuss briefly the connection with Bell in-equalities. In the Bell scenario we consider two parties,Alice and Bob, each chooses from m ± { A , . . . , A m } and { B , . . . , B m } . Thejoint correlation of Alice and Bob’s measurement out-comes, designated α i and β j respectively, is given by h α i β j i = Tr ( A i ⊗ B j ρ ), where ρ denotes the density ma-trix of the bipartite state. A correlation Bell inequalitycan be written as m X i,j =1 M ij h α i β j i ≤ L, (5)where L signifies the bound which can be achieved bylocal models and M is a m × m matrix with real coef-ficients defining a Bell inequality. The local bound canalways be achieved by a deterministic local model, i.e.,for all real numbers a i , b j = ± a i ,b j m X i,j =1 M ij a i b j = L. (6) In this way the expression I defined by (2) is linked toa correlation Bell inequality with matrix M and localbound max a i ,b j = ± I = L .On the other hand, for the singlet state ρ = | ψ − ih ψ − | we have h α i β j i ψ − = h ψ − | A i ⊗ B j | ψ − i = − ~a i · ~b j , wherethe observables A = ~a~σ and B = ~b~σ corresponding toAlice and Bob’s projective measurements are specifiedby the unit vectors ~a and ~b in R . Then substitut-ing into (5) one obtains the expression I in (3). Fur-thermore, Tsirelson [22] proved that correlations whichare dot products of unit vectors ~a,~b ∈ R d can alwaysbe realized by performing projective measurements onmaximally entangled states in some higher dimensionalHilbert spaces. Thus the value max I d can always beachieved by means of quantum mechanics.Since joint correlations vanish for the maximally mixedstate, it follows that the critical point at which Wernerstates in (1) cease to violate any Bell inequality is p Wc =1 /K G (3). This key correspondence has been establishedin Ref. [8]. Though, the exact value of K G (3) is notknown, but known bounds establish that 0 . ≤ p Wc ≤ . K G (3) ≥ . . ≤ p Wc ≤ . K G (5) ≥ / . > √
2. AlsoToner has shown that K G (4) > √ K G (3) isbigger than √ p Wc < / √ III. CONSTRUCTING FAMILY OF BELLINEQUALITIES
For Bell diagonal states, such as for Werner states,under projective measurements Alice and Bob’s localmarginals (defined by h α i i = Tr ( A i ⊗ ρ ) for Alice andlikewise for Bob) are zero, thus it is sufficient to dealwith generic correlation Bell inequalities defined by (5) toobtain maximal Bell violation for Werner states. More-over, in this respect, the tight correlation Bell inequali-ties, which can be considered as facets of the correlationpolytope [24], specified by the number of two-outcomemeasurements m on each side, are the most efficient ones.For m = 2 one obtains as the only nontrivial correlationinequality the CHSH one [25]. For m > m = 4 all the correlation inequal-ities have been computed [26], and the two inequivalentinequalities obtained are in fact less efficient than theCHSH one for Werner states. However, the complexityof the computation exponentially grows with m (in fact,this is an NP-complete problem [24]), therefore there isno hope to completely characterize all the facets of thecorrelation polytope for any given m . Thus in general oneneeds to look for alternative methods. For instance Gisinexplored special form of families of tight correlation in-equalities, the so-called D -inequalities in Ref. [10]. Aviset al. [26] applied triangular elimination to the list ofknown facet inequalities of the cut polytope to constructmany new tight correlation inequalities. Alternatively,one can construct (possibly not tight) correlation inequal-ities which however can be easily generalized to arbitrarynumber of settings, such as in the cases [27, 28, 29, 30].In the present work we have chosen this latter directionby modifying the correlation inequalities Z n introducedin [30].Let us specify the form of M in (2) through the follow-ing formula, I n A ,n B = n A X i =1 n B X j =1 a i b j + X ≤ i 1. However,in this particular case one can exploit the symmetry withrespect to change of indices within the sets { a i } n A i =1 and { b i } n B i =1 . Thus one needs to check altogether n A n B caseswhere +1 occurs 1 ≤ k ≤ n A times in the set { a i } n A i =1 and +1 occurs 1 ≤ l ≤ n B times in the set { b } n B i =1 (therest being − k, l pair we have max I n A ,n B =max { ( n A − k )( n B − l ) + 2( n A − k ) k + 2( n B − l ) l } . Thisexpression is maximal by k − l = ⌊ ( n A − n B ) / ⌋ resultingin the local boundmax I n A ,n B =( n A + n B − / , for | n A − n B | oddmax I n A ,n B =( n A + n B ) / , for | n A − n B | even . (9)In this paper we focus on two particular cases n A = n B +1 and n A = n B , but first let us restrict our attentionto the latter, symmetric case n A = n B = n . The LHVbound gives max I n,n = n by inserting n A = n B = n in(9). On the other hand, the expression I dn,n , symmetric in the two parties, reads I dn,n = n X i n X j ~a i · ~b j + X ≤ i In fact, in the particular case I nn,n one can obtain theexact maximum, max I nn,n = 3 / − / (2 n ). This result(noticing that for large n the violation tends to 1 . 5) wouldindicate that there may be some hope to get a lowerbound K G (3) > √ 2. Below we show that the maximumabove can indeed be attained.First let us observe that in (12) only n vectors occur,thus we have max I mn,n = max I nn,n with m = n ( n + 1) / m is the number of measurement settings on eachside. Now we take the unit vectors ~a i in such a waythat ~a i · ~a j = 1 / i = j . This can be achieved, bynoting that the n × n Gram matrix G , defined by elements G ij = ~a i · ~a j is positive definite, and every positive definitematrix is a Gram matrix for some set of vectors. Thusit is enough to show that the Gram matrix G , definedas above ( G ij = 1 , ∀ i = j and G ij = 1 / , ∀ i = j ), ispositive definite. However, using the Sylvester criterion[32] one can establish that G defined above is positivedefinite iff det G > n . One may obtain byinduction the closed formula det A = ( a − b ) n − ( a + ( n − b ) for the determinant of any n × n matrix A havingin the diagonals the value a and in all off-diagonals thevalue b . By choosing particularly a = 1 and b = 1 / G > n , which proves ourassertion.Beside, all the elements in (12) can be obtained as theonly function of ~a i · ~a j , since | P ni =1 ~a i | = P ni,j =1 ~a i · ~a j and | ~a i − ~a j | = p − ~a i · ~a j . Thus by substitution weobtain I nn,n = n (3 n − / 2. Then it follows using (9), that I nn,n / max I n,n = 3 / − / (2 n ). It is also possible to verifythat this is in fact the maximum value. The verification,which is not detailed here, goes the same line as discussedby Wehner in [33] for the chained Bell inequality [27]through the dual solution of a semi-definite optimizationproblem [34]. Note that this optimization problem is justthe first step in the hierarchy introduced by Navascueset al. [35, 36].Now we wish to obtain a lower bound K G (3) ≥ I n,n / max I n,n bigger than √ K G (3). Since owing to (9), max I n,n = n , we are left with the calculation of I n,n which thoughmight be not maximal, but large enough to supply a goodlower bound for K G (3). This is achieved by substitutingin (12) in the place of ~a i explicit values on the follow-ing way. Since ~a i are unit vectors in R , the first twocomponents P i = ( x i , y i ) of ~a i , which can be consideredas points in the XY -plane, completely specify the vec-tor itself. Let n = 30 and distribute these points onthree co-centric circles centered at the origin (0 , 0) withradii ρ I = 22 / ρ II = 52 / 100 and ρ III = 77 / P = ( ρ I cos π/ , ρ I sin π/ P = (0 , ρ II ) and P = (0 , ρ III ). The other P i vectors are constructedfrom the above vectors by rotating them with angles π/ π/ 5, and π/ 8, respectively, such as to form regu-lar polygons with vertices 4, 10 and 16 (as it is shownin the figure). By inserting the explicit values of thecorresponding set { ~a i } i =1 into the expression to be max-imized in (12) one obtains I n,n /I n,n = 1 . 415 199 with n = 30. This implies that K G (3) is indeed bigger than √ . 414 2136 . . . . The specific values of ~a i have beenfound by performing optimization with respect to theradii of the three circles by choosing regular polygonswith various number of vertices. FIG. 1: The 30 points which are projection of the vectors ~a i on the XY -plane. They are equally distributed on threeco-centric circles with radii 22 / , / , / 100 centeredat the origin. The outer circle represents the grand circleprojected on the XY -plane, thus having radius 1. V. BETTER LOWER BOUNDS FORGROTHENDIECK CONSTANT OF ORDERS 3,4,5 In this section a general method is discussed to obtainlocal maximum on I dn,n for any n, d , which for many in-stances are presumably the global or close to the globalmaximum. Then, recalling K G ( d ) ≥ I dn,n / max I n,n andmax I n,n = n , this method yields lower bounds for K G ( d ). In particular we present results for d = 3 , , n = 100.Let us consider the following iteration scheme, whichis a simplified version of the see-saw iteration method,already used in the literature to solve optimization prob-lems in similar context entering many optimization pa-rameters [37, 38]. Note, that the matrix M of I n,n definedthrough (7) is symmetric, thus we may write I dn,n = m X i,j =1 M ij ~a i · ~b j = m X i =1 ~b i · m X j =1 M ij ~a j = m X i =1 ~a i · m X j =1 M ij ~b j , (13)with m = n ( n + 1) / ~a i ,~b j are unit vectors in R d . In this notation we contracted the double indices ij appearing in (10), so that { ~a i } mi =1 stands for the set( { ~a i } ni =1 , { ~a ij } ≤ i 5, that thevalue | P j M ij ~a j | in the denominator of the iteratedexpression was nonzero (actually, it was no less than10 − for all i in each case of d ). On the other hand,the iteration was performed with machine precision ∼ − in the Mathematica package, and we checkedthat | ~a i · ~a i − | < − for all 1 ≤ i ≤ m after the 1000iteration steps completed.For n = 100, we obtained the following numbers, I n,n / max I n,n = 1 . 417 241, I n,n / max I n,n = 1 . 445 207,and I n,n / max I n,n = 1 . 460 065. These numbers arelower bounds for the Grothendieck constants K G (3), K G (4) and K G (5), respectively. We mention that for n = 100 the dimension of the respective matrix M is n ( n + 1) / K G (5) presented so far in the literature K G (5) ≥ / . 428 571 . . . comes from the Fishburn-Reeds inequality[23]. Our result for K G (3) provides us with the betterlower bound p Wc ≤ . 705 596 for the critical value p Wc owing to the formula p Wc = 1 /K G (3). VI. MINIMAL NUMBER OF MEASUREMENTS One may also ask, what is the smallest number of set-tings on Alice and Bob’s side, where K G ( d ) can exceed √ d > 2. To the best of our knowledge, sofar it has been provided by the Fishburn-Reeds inequal-ities [23]. Their construction, giving K G (5) ≥ / . 428 571 . . . can be obtained by 20 measurement set-tings on each side. Now we choose n A = n B + 1 = 5 inexpression I n A ,n B of (7), giving the number of settings11 and 14 on Alice and Bob’s side, respectively. Thusthe matrix M in this particular instance has dimensions11 × 14. We show that this expression I , provides uswith an example where K G (4) > √ n A = 5 and n B = 4 into the for-mula (9) for odd | n A − n B | one obtains the value 20 forthe local bound. On the other hand, the maximum valuecorresponding to the vectorial case max I , , can be ob-tained by the mean of semidefinite techniques [33] as afirst step of the hierarchy in [35], where we used the Se-DuMi package [39] for Matlab by the explicit numericalcomputation. This algorithm solves both the primal and the dual optimization problem at the same time and thusyields bounds on the accuracy of the obtained solutionas well. Actually, we obtained the same optimal value28 . 390 139 for both cases. This yields the ratio 1 . 419 507for the violation of the Bell inequality I , ≤ 20 clearlybeating the √ VII. TIGHTNESS AND RELEVANCE OF BELLINEQUALITIES It would be interesting to know whether the familyof correlation inequalities defined by (7) is tight, i.e.,whether it is a facet or not of the local Bell polytope [40]consisting of local marginals as well. This can be doneby computing the dimension of the subspace spanned byall deterministic strategies saturating the inequality. Ifthis subspace is found to be a hyperplane with dimen-sion d = m A m B + m A + m B , then the inequality istight. Numerically, we treated the n A = n B + 1 and n A = n B = n cases in the expression I n A ,n B . Computa-tionally we found that in the former case the inequalityis tight up to n A = 4. On the other hand, the lattersymmetric inequality proved to be not tight, but by theaddition of some local terms a i , b j = ± I ′ n,n = n X i,j =1 a i b j + X ≤ i 4, as well.Let us discuss the concept of relevant Bell inequali-ties, whose definition we quote from [10], Sec. A.1: “Aninequality is relevant with respect to a given set of in-equalities if there is a quantum state violating it, but notviolating any of the inequalities in the set.” Collins andGisin [41] showed that the I inequality is relevant tothe famous CHSH inequality [2]. Interestingly, they alsofound that given I the CHSH inequality is no longerrelevant. Furthermore, Ito, Imai and Avis [26] have re-cently conjectured supported by numerical optimization,that there exist Bell inequalities relevant for the I in-equality for 3-level isotropic states. However, limiting theHilbert space dimension to a qubit pair, they did not finda Bell inequality which would be relevant with respect to I . Our new inequalities, I n,n and I ′ n,n with n = 100,however, are examples to this latter case, demonstratingthat in the parameter range 0 . 705 596 < p ≤ . I inequal-ities but violate I n,n or I ′ n,n for n = 100 (note that forthe Werner states the local marginals become identicallyzero, thus in this respect I n,n and I ′ n,n are equivalent).Moreover, one can demonstrate, that there is an inclu-sion relation, a notion introduced in [43], between I ′ n,n and I ′ n − ,n − , meaning that one can obtain the inequality I ′ n − ,n − by measuring the identity for some settings inthe inequality I ′ n,n (i.e., performing degenerate measure-ments). This implies that I ′ n,n for any n > I ′ , ≡ I . The proof is simple, actu-ally by setting a n = +1 , b n = +1 and a in = − , b in = − ≤ i < n in I ′ n,n one obtains I ′ n − ,n − , and then byinduction one arrives at I ′ , .Altogether, one can say that if one limits the Hilbertspace dimension to two qubits ([10], Sec. A.2) the I ′ n,n inequality for n → ∞ is the only relevant one with respectto all presently known Bell inequalities. VIII. SUMMARY We provided a new family of Bell inequalities whichproves that Werner states in (1) are nonlocal for theparameter range p > . p > . K G (3), is bigger than √ 2, in partic-ular, K G (3) ≥ . √ K G (3) stronger. The possibility for such inequali-ties is suggested by the fact that an upper bound for K G ≡ lim n →∞ K G ( n ) is 1 . I n,n for n → ∞ givesthe lower bound 1 . K G , which is even smaller thanthe lower bound 1.6770 for K G presented in Ref. 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