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## Large violation of Bell inequalities with low entanglement

Venue: | Comm. Math. Phys |

Citations: | 12 - 3 self |

### Citations

1154 |
Can the Quantum Mechanical Description of Physical Reality be considered Complete', Phys
- Einstein, Podolsky, et al.
- 1935
(Show Context)
Citation Context ... nonlocality dates back to the famous work of Einstein, Podolsky and Rosen (EPR) in 1935. They presented an argument which questioned the validity of quantum mechanics as a complete theory of Nature (=-=[28]-=-). However, it took almost 30 years to understand that the apparently dilemma presented in [28] could be formulated in terms of assumptions which naturally lead to a refutable prediction ([75]). Bell ... |

985 |
On the Einstein Podolsky Rosen paradox
- Bell
- 1964
(Show Context)
Citation Context ...hidden variable model implies some inequalities on the set of probabilities, since then called Bell inequalities, which are violated by certain quantum probabilities produced with an entangled state (=-=[6]-=-). For a long time after this, entanglement and violation of Bell inequalities were thought to be parts of the same concept. This changed in the late 1980s with a number of surprising results (see [74... |

443 |
The Volume of Convex Bodies and Banach Space Geometry
- Pisier
- 1989
(Show Context)
Citation Context ... E‖G : `n2 → `n2 (`n∞)‖ ≤ C1 √ n log n. In particular, E‖G : `n2 → `n1 (`n∞)‖ ≤ C1n √ log n. 2High probability means here that the probability tends to 1 exponentially fast as m→∞ (see Theorem 4.7 in =-=[60]-=-). LARGE VIOLATION OF BELL INEQUALITIES WITH LOW ENTANGLEMENT 15 Proof. Chevet’s inequality implies that E‖G‖ ≤ w2((ei)i; `n2 ) E‖ n∑ i,j=1 gi,jei ⊗ ej‖`n2 (`n∞) + ω2((ei ⊗ ej)i,j; `n2 (`n∞)) E‖ n∑ i=... |

362 | A parallel repetition theorem
- Raz
- 1998
(Show Context)
Citation Context ... √ n log n . Here, we use to denote inequality up to a universal constant independent of n ∈ N. The first unbounded violation of Bell inequalities dates back to the Raz parallel repetition theorem (=-=[66]-=-). Indeed, applying this result to the repetition of the magic square LARGE VIOLATION OF BELL INEQUALITIES WITH LOW ENTANGLEMENT 5 game (or any pseudo-telepathy game ([9])), one can deduce the existen... |

359 |
Operator spaces
- Effros, Ruan
- 2000
(Show Context)
Citation Context ...plexity Theory. Throughout the section, we will give some optimal results for this norm. 2. Basic tools 2.1. Operator spaces. We will recall some basic facts from operator spaces theory. We recommend =-=[27]-=- and [58] for further information and more detailed definitions. We will denote by Mn (resp Mm,n) the space of complex n× n (resp m× n) matrices. The theory of operator spaces came to life through the... |

317 | On the power of unique 2-prover 1-round games
- Khot
- 2002
(Show Context)
Citation Context ...n be stated in terms of these kinds of games. In particular, they are extremely useful to study problems of hardness of approximation. One example of this is the so called unique game conjecture (see =-=[42]-=-, [43]), which has become one of the crucial problems in Complexity Theory since it implies hardness of approximation results for several important problems (MaxCut, Multicut and Sparsest Cut, Vertex ... |

242 |
Completely bounded maps and operator algebras, Cambridge
- Paulsen
- 2002
(Show Context)
Citation Context ...letely contractions in the definition of ‖a‖|ψ〉. Let u : Rn ∩ Cn → Mk be a complete contraction. According to the Wittstock extension theorem (due independently to Haagerup, Paulsen and Wittstock see =-=[56]-=-) and the definition of Rn ∩ Cn ⊂ Rn ⊕ Cn, we can extend u : Rn ∩ Cn → Mk to Rn ⊕ Cn and find a decomposition u = uc + ur, where uc : Cn → Mk and ur : Rn → Mk are complete contractions. Therefore it i... |

197 |
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
- Werner
- 1989
(Show Context)
Citation Context ...[6]). For a long time after this, entanglement and violation of Bell inequalities were thought to be parts of the same concept. This changed in the late 1980s with a number of surprising results (see =-=[74]-=-, [64], [29]) which showed that, although entanglement is necessary for the violation of Bell inequalities, the converse is not true. On the other hand, up to our knowledge, violation of Bell inequali... |

172 | The unique games conjecture, integrability gap for cut problems and embeddings of negative type metrics into `1
- Khot, Vishnoi
- 2005
(Show Context)
Citation Context ...t seem to be much better that 10−5. In [41] the authors made a great improvement of the previous results. Via a highly non trivial construction of Khot and Visnoi in the context of Complexity Theory (=-=[43]-=-), the main result in [41] shows the existence of a quantum probability distribution P with n outputs and 2 n n inputs, which verifies that ν(P ) n 154 1. The prize in that estimate is a large numbe... |

155 |
Non-commutative vector valued Lp-spaces and completely p-summing maps
- Pisier
- 1998
(Show Context)
Citation Context ...K∞)→ `N1 (`K∞)⊗min `N1 (`K∞) . This implies ‖ id⊗ id : `N1 (`K∞)⊗ `N1 (`K∞)→ `N1 (`K∞)⊗min `N1 (`K∞)‖ K. To prove the upper bound as a function of the number of inputs N , recall (see for instance =-=[59]-=-) that ‖ id⊗ id : `N∞(`K∞)→ `N1 (`K∞)‖cb ≤ N. Then, the factorization id⊗ id : `N1 (`K∞)⊗ `N1 (`K∞)→ `N∞(`K∞)⊗min `N1 (`K∞)→ `N1 (`K∞)⊗min `N1 (`K∞) implies ‖ id⊗ id : `N1 (`K∞)⊗ `N1 (`K∞)→ `N1 (`K∞... |

137 |
Factorization of linear operators and geometry of banach spaces
- Pisier
- 1986
(Show Context)
Citation Context ...· · · , k and i = 1, .., n. Now, according to the little Grothendieck theorem, the 2-summing norm of Ti is bounded by K‖Ti‖, and hence Ti = uiDσi factors through a diagonal map Dσi(ej) = σi(j)ej (see =-=[61]-=-) with ‖ui‖ ≤ 1 and ‖σi‖2 ≤ K‖Ti‖. Let aj ∈Mk. Then we have ‖ ∑ j |σi(j)|2a∗jaj‖Mk ≤ ∑ j |σi(j)|2 sup j ‖aj‖2 . Hence we have ‖Dσi : `n∞ → Cn‖cb ≤ ‖σi‖2. On the other hand, it is well known (see [58])... |

134 |
Tensor Norms and Operator Ideals
- Defant, Floret
- 1993
(Show Context)
Citation Context ... = Mn⊗minE holds for every operator space E. The tensor norm min in the category of operator spaces will play the role of the so called norm in the classical theory of tensor norms in Banach spaces =-=[23]-=-. In particular min is injective, in the sense that if E ⊂ X and F ⊂ Y completely isometric (isomorphic), then E ⊗min F ⊂ X ⊗min Y holds completely isometrically (isomorphically). The analogue of the ... |

119 | Consequences and limits of nonlocal strategies
- Cleve, Høyer, et al.
- 2004
(Show Context)
Citation Context ... National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS it enriches the theory of multipartite interactive proof systems (=-=[7, 19, 18, 32, 24, 41, 39]-=-), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games ([41, 40]), etc. Bell inequalities and the... |

116 |
Probabilistic methods in the geometry of Banach spaces
- Pisier
- 1986
(Show Context)
Citation Context ...Moreover, ‖M̃‖`n1 (`n+1∞ )⊗`n1 (`n+1∞ ) = ‖M‖`n1 (`n∞)⊗`n1 (`n∞) ≤ K 2 1 log n. follows immediately from the injectivity of the -norm for the first equality and from Lemma 3.5. Indeed, we refer to =-=[63]-=- for the simple fact that E‖ ∑ x,a,k kx,aek ⊗ (ex ⊗ ea)‖`n2⊗`n1 (`n∞) ≤ √ pi 2 E‖ ∑ x,a,k gkx,aek ⊗ (ex ⊗ ea)‖`n2⊗`n2 (`n∞) . Let E : `n2 → `n1 (`n∞) denotes the map E(ek) = kx,aex ⊗ ea. Then our ... |

111 |
Tensor products of operator spaces
- Blecher, Paulsen
- 1991
(Show Context)
Citation Context .... 46 M. JUNGE AND C. PALAZUELOS Proof. The first inequality follows from the fact that for any pair of Banach spaces X e Y , the map id : min(X)⊗hmin(Y )→ X ⊗γ Y defines an isometry (see for instance =-=[8]-=-). Here, given a Banach space Z we denote by min(Z) the operator space structure endowed by any embedding Z ↪→ C(K). To see the second inequality, consider an element z such that ‖z‖h ≤ 1. Now, accord... |

108 |
The operator Hilbert space OH, complex interpolation and tensor norms
- Pisier
- 1996
(Show Context)
Citation Context ...}. In this work we will also use Pisier’s operator space OH as a technical tool. We refer to its definition as complex interpolation space OH = (R,C) 1 2 and further properties to [58, Chapter 7] and =-=[62]-=-. The operator space `n1 carries a natural operator space structure as the dual of ` n ∞, i.e. `n1 = (` n ∞) ∗. Note that for any operator space X the natural operator space structure on `1(X) ⊂ (c0 ⊗... |

74 | Parallel repetition: simplifications and the no-signaling case
- Holenstein
- 2006
(Show Context)
Citation Context ...at for every n we have quantum probability distributions P with n inputs, n outputs and dimension n such that ν(P ) nx. However, regarding the sharpest estimates on the parallel repetition theorem (=-=[30]-=-, [65], [67]), the best known value for the previous x doesn’t seem to be much better that 10−5. In [41] the authors made a great improvement of the previous results. Via a highly non trivial construc... |

47 | Near-optimal algorithms for unique games
- Charikar, Makarychev, et al.
- 2006
(Show Context)
Citation Context ...a ... ... ... ... nx,a n x,a 1 x,a · · · 1 for a = 1, · · · , n , 1−∑na=1Eax for a = n+ 1 for x = 1, · · · , n. 1Actually, one can obtain n 1 24 up to terms of lower order via a claim in (=-=[17]-=-, pag. 3). 6 M. JUNGE AND C. PALAZUELOS c) States : Let (αi) n+1 i=1 be a decreasing and positive sequence and |ϕα〉 = n+1∑ i=1 αi|ii〉. Theorem 1.2. There exist universal constants C and K such that fo... |

43 |
Device-independent security of quantum cryptography against collective attacks
- Aćın, Brunner, et al.
- 2007
(Show Context)
Citation Context ... theoretical interest, Bell inequalities have found applications in many areas of QIT: quantum cryptography, where it opens the possibility of getting unconditionally secure quantum key distribution (=-=[1, 4, 51, 50]-=-), complexity theory, where The authors are partially supported by National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS ... |

42 | Parallel repetition in projection games and a concentration bound
- Rao
- 2008
(Show Context)
Citation Context ... every n we have quantum probability distributions P with n inputs, n outputs and dimension n such that ν(P ) nx. However, regarding the sharpest estimates on the parallel repetition theorem ([30], =-=[65]-=-, [67]), the best known value for the previous x doesn’t seem to be much better that 10−5. In [41] the authors made a great improvement of the previous results. Via a highly non trivial construction o... |

40 |
Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs 38
- Tomczak-Jaegermann
- 1989
(Show Context)
Citation Context ...the literature. Remark 4.2. (Random variables) Although we have proved Theorem 1.1 (via Theorem 3.1) using gaussian variables, it is well known that the same estimations work for Bernoulli variables (=-=[69]-=-) and random unitaries ([49], [33]) (in this last case one has to normalize by a factor √ n). Thus Theorem 1.2 can be stated using Bernoulli variables (kx,a)x,a,k (as it is stated), gaussian variable... |

39 | A counterexample to strong parallel repetition
- Raz
- 2008
(Show Context)
Citation Context ... n we have quantum probability distributions P with n inputs, n outputs and dimension n such that ν(P ) nx. However, regarding the sharpest estimates on the parallel repetition theorem ([30], [65], =-=[67]-=-), the best known value for the previous x doesn’t seem to be much better that 10−5. In [41] the authors made a great improvement of the previous results. Via a highly non trivial construction of Khot... |

37 |
A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
- Navascués, Pironio, et al.
- 2008
(Show Context)
Citation Context ...lities (related to the study of deciding whether a given probability distribution belongs to the quantum set Q), has captured the interest of many researchers in QIT (see e.g. [46], [24], [72], [54], =-=[55]-=-). On the other hand, since any two-prover one-round game can be seen as a Bell inequality, Game Theory and, in general Computer Science, can be considered as a very important source of results about ... |

35 | The quantum moment problem and bounds on entangled multi-prover games - Doherty, Liang, et al. - 2008 |

34 |
Unbounded violation of tripartite Bell inequalities
- Pérez-García, Wolf, et al.
- 2009
(Show Context)
Citation Context ...the theory of multipartite interactive proof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space (=-=[11, 13, 57, 71, 73]-=-), entangled games ([41, 40]), etc. Bell inequalities and their connection to quantum entanglement have remained quite mysterious despite the recent research on this topic. In the few last years, the ... |

32 |
Bounding the set of quantum correlations
- Navascués, Pironio, et al.
(Show Context)
Citation Context ...inequalities (related to the study of deciding whether a given probability distribution belongs to the quantum set Q), has captured the interest of many researchers in QIT (see e.g. [46], [24], [72], =-=[54]-=-, [55]). On the other hand, since any two-prover one-round game can be seen as a Bell inequality, Game Theory and, in general Computer Science, can be considered as a very important source of results ... |

31 |
Bilinear forms on exact operator spaces and
- Junge, Pisier
- 1995
(Show Context)
Citation Context ... interested in identifying Bell inequalities where the dimension free version (2.7) LV|ψ〉(M) = sup k LV|ψk〉(M) remains bounded. In fact similar objects have been studied in operator space theory (see =-=[38]-=-). Given two operator spaces X and Y and a ∈ X ⊗ Y , we may define a modified min-norm ‖a‖ψ−min = sup{〈ψk|(u⊗ v)(a)|ψk〉}, where the sup runs over all k ∈ N and all completely contractions u : X → Mk a... |

28 | Bell’s inequalities and density matrices. Revealing ‘hidden’ nonlocality
- Popescu
- 1995
(Show Context)
Citation Context ...For a long time after this, entanglement and violation of Bell inequalities were thought to be parts of the same concept. This changed in the late 1980s with a number of surprising results (see [74], =-=[64]-=-, [29]) which showed that, although entanglement is necessary for the violation of Bell inequalities, the converse is not true. On the other hand, up to our knowledge, violation of Bell inequalities i... |

26 | Perfect parallel repetition theorem for quantum XOR proof systems
- Cleve, Slofstra, et al.
- 2007
(Show Context)
Citation Context ... of approximation of the quantum value (commonly called entangled value) of games ([39],[31]), the unique game conjecture in the quantum context ([41]) or the parallel repetition theorem ([30], [41], =-=[20]-=-, [40]). The standard way to tackle these kinds of problems is to show that the entangled value (resp. non signally value) of the considered games can be approximated by some semidefinite programming ... |

25 |
From bell’s theorem to secure quantum key distribution
- Aćın, Gisin, et al.
(Show Context)
Citation Context ... theoretical interest, Bell inequalities have found applications in many areas of QIT: quantum cryptography, where it opens the possibility of getting unconditionally secure quantum key distribution (=-=[1, 4, 51, 50]-=-), complexity theory, where The authors are partially supported by National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS ... |

23 | The uniqueness theorem for entanglement measures
- Donald, Horodecki, et al.
(Show Context)
Citation Context .... Indeed, for bipartite pure states there exists a universal measure of entanglement, the so called entropy of entanglement : E(|ψ〉) = S((|ψ〉〈ψ|)A), where S denotes the usual von Neumann entropy (see =-=[25]-=-). It is easy to see that E(|ψ〉) ≥ 0 for every state |ψ〉 and that the maximally entangled state in dimension n is |ψn〉 = 1√ n n∑ i=1 |ii〉, verifying E(|ψn〉) = log2(n). For a given bipartite pure state... |

21 |
Banach spaces with a unique unconditional basis up to permutation
- Bourgain, Casazza, et al.
- 1985
(Show Context)
Citation Context ...every Bell inequality M with n inputs and n outputs we have ωop(M) supP∈L |〈M,P 〉| n. LARGE VIOLATION OF BELL INEQUALITIES WITH LOW ENTANGLEMENT 45 Furthermore, as a consequence of a deep result in =-=[14]-=-, we can prove a much sharper result about the previous optimality when we consider the rank of the operator M . Indeed, we will show Theorem 7.5. Let’s denote, for each n ∈ N, An = sup { A : ωop(M) s... |

21 |
Oracularization and twoprover one-round interactive proofs against nonlocal strategies
- Ito, Kobayashi, et al.
- 2009
(Show Context)
Citation Context ...m physics or under the assumption of the non-signally condition. Some examples of this are the study of hardness of approximation of the quantum value (commonly called entangled value) of games ([39],=-=[31]-=-), the unique game conjecture in the quantum context ([41]) or the parallel repetition theorem ([30], [41], [20], [40]). The standard way to tackle these kinds of problems is to show that the entangle... |

20 |
Factorization Theory for Spaces of Operators,’’ Habilitationsschift, Universita t
- Junge
- 1996
(Show Context)
Citation Context ...m variables) Although we have proved Theorem 1.1 (via Theorem 3.1) using gaussian variables, it is well known that the same estimations work for Bernoulli variables ([69]) and random unitaries ([49], =-=[33]-=-) (in this last case one has to normalize by a factor √ n). Thus Theorem 1.2 can be stated using Bernoulli variables (kx,a)x,a,k (as it is stated), gaussian variables (gkx,a)x,a,k and random unitarie... |

19 |
Background level and counter efficiencies required for a loophole-free Einstein–Podolsky–Rosen experiment, Phys
- Eberhard
- 1993
(Show Context)
Citation Context ...nonlocality where the maximally entangled state has been shown not to be the most nonlocal one. We can find some of these anomalies ([53]) in the study of Bell inequalities ([2]), detection loophole (=-=[26]-=-), extractable secrete key ([68]), K-L distance ([3]), etc. Here we will show that there are Bell inequalities which avoid violation of the maximally entangled state in high dimension. The examples ar... |

18 |
Connes’ embedding problem and Tsirelson’s problem
- Junge, Navascues, et al.
- 2011
(Show Context)
Citation Context ...atural operator space structure as dual space of NSG(N,K)∗. Remark 6.5. Duality in the category of operator system is in general a tricky point and we will disregard this problem here. Remark 6.6. In =-=[34]-=- the authors show that the map ι : NSG(N,K)∗ → ?Ni=1`K∞ defined by ι(ex,a) = pix(ea), where ea is the a-th canonical vector in ` N ∞ and pix : ` N ∞ ↪→ ?Ni=1` K ∞ is the canonical embedding of ` N ∞ i... |

17 | Unbounded violations of bipartite Bell inequalities via operator space theory
- Junge, Palazuelos, et al.
(Show Context)
Citation Context ...om different areas of mathematics has started to clarify the situation. This includes the previous works of the authors, which are based on operator space techniques. Indeed, in the consecutive works =-=[35]-=- and [57], the authors have shown the operator space theory as a natural framework for the study of Bell inequalities (see also [36]). Using this connection the authors proved in [35] the existence of... |

16 | Entanglement-resistant two-prover interactive proof systems and non-adaptive PIR. Quantum Information and Computation
- Cleve, Gavinsky, et al.
- 2009
(Show Context)
Citation Context ... National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS it enriches the theory of multipartite interactive proof systems (=-=[7, 19, 18, 32, 24, 41, 39]-=-), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games ([41, 40]), etc. Bell inequalities and the... |

15 | Rounding parallel repetitions of unique games
- Barak, Hardt, et al.
- 2008
(Show Context)
Citation Context ... of the considered games can be approximated by some semidefinite programming (SDP) relaxation with some good properties. In this sense, the following relaxation has been shown to be very useful (see =-=[5]-=- and [41] for details). For every M consider the following optimization problem (OP), which maximizes over complex vectors {uax}nx,a=1, {vby}ny,b=1 and z: OP 7.1. ωop(M) := max {∣∣ n∑ x,y,a,b=1 Ma,bx,... |

15 |
Hidden quantum nonlocality revealed by local filters
- Gisin
- 1996
(Show Context)
Citation Context ...long time after this, entanglement and violation of Bell inequalities were thought to be parts of the same concept. This changed in the late 1980s with a number of surprising results (see [74], [64], =-=[29]-=-) which showed that, although entanglement is necessary for the violation of Bell inequalities, the converse is not true. On the other hand, up to our knowledge, violation of Bell inequalities is the ... |

15 | The unique games conjecture with entangled provers is false
- Kempe, Regev, et al.
- 2007
(Show Context)
Citation Context ... National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS it enriches the theory of multipartite interactive proof systems (=-=[7, 19, 18, 32, 24, 41, 39]-=-), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games ([41, 40]), etc. Bell inequalities and the... |

14 | inequalities for arbitrarily high-dimensional systems - Collins, Gisin, et al. |

14 | The communication complexity of nonsignaling distributions
- Degorre, Kaplan, et al.
(Show Context)
Citation Context ...enote the set of non-signalling probability distributions by C. We must point out that the elements in C were initially called behaviors (see [70]). However, following the more recent literature (see =-=[22]-=- and [35]), we will not use that terminology. b) LHV (Local Hidden Variable) if P (a, b|x, y) = ∫ Ω Pω(a|x)Qω(b|y)dP(ω) 4 M. JUNGE AND C. PALAZUELOS for every x, y, a, b, where (Ω,Σ,P) is a probabilit... |

13 |
Quantum nonlocality in two three-level systems
- Aćın, Durt, et al.
(Show Context)
Citation Context ...in the context of quantum nonlocality where the maximally entangled state has been shown not to be the most nonlocal one. We can find some of these anomalies ([53]) in the study of Bell inequalities (=-=[2]-=-), detection loophole ([26]), extractable secrete key ([68]), K-L distance ([3]), etc. Here we will show that there are Bell inequalities which avoid violation of the maximally entangled state in high... |

13 |
Introduction to Operator Spaces
- Pisier
- 2003
(Show Context)
Citation Context ...heory. Throughout the section, we will give some optimal results for this norm. 2. Basic tools 2.1. Operator spaces. We will recall some basic facts from operator spaces theory. We recommend [27] and =-=[58]-=- for further information and more detailed definitions. We will denote by Mn (resp Mm,n) the space of complex n× n (resp m× n) matrices. The theory of operator spaces came to life through the work of ... |

12 | Tsirelson bounds for generalized ClauserHorne-Shimony-Holt inequalities
- Wehner
(Show Context)
Citation Context ... Bell inequalities (related to the study of deciding whether a given probability distribution belongs to the quantum set Q), has captured the interest of many researchers in QIT (see e.g. [46], [24], =-=[72]-=-, [54], [55]). On the other hand, since any two-prover one-round game can be seen as a Bell inequality, Game Theory and, in general Computer Science, can be considered as a very important source of re... |

11 |
Operator space theory: A natural framework for Bell inequalities
- Junge, Palazuelos, et al.
(Show Context)
Citation Context ...f violation of Bell inequalities and the NSG space 6.1. Geometric interpretation of violation of Bell inequalities. We have seen in Section 1 that for certain applications in QIT (see [22], [35], and =-=[36]-=-) the value supP∈Q ν(P ) is very interesting. In this section we want to provide a geometric interpretation of this value in terms of convex sets of probabilities. We begin recalling some basic notion... |

11 | No strong parallel repetition with entangled and non-signaling provers
- Kempe, Regev
- 2010
(Show Context)
Citation Context ...roof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games (=-=[41, 40]-=-), etc. Bell inequalities and their connection to quantum entanglement have remained quite mysterious despite the recent research on this topic. In the few last years, the application of techniques fr... |

9 |
Mixed-norm inequalities and operator space Lp embedding theory
- Junge, Parcet
(Show Context)
Citation Context ...a and sup l ‖Fl‖ ≤ ‖F‖∞ . Since the underlying measure space is a probability space, we obtain the assertion in the case q = ∞. For 1 ≤ q ≤ ∞, this follows from a complex interpolation argument as in =-=[37]-=-. Just note that in operator space jargon, the Lemma is an immediate consequence of the fact that id : `nq → `n∞ and id : Lq(Ω;X)→ L1(Ω;X) are complete contractions. Although the techniques to prove... |

9 |
A lower bound on the dimension of a quantum system given measured data, Phys. Rev. A78
- Wehner, Christandl, et al.
- 2008
(Show Context)
Citation Context ...the theory of multipartite interactive proof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space (=-=[11, 13, 57, 71, 73]-=-), entangled games ([41, 40]), etc. Bell inequalities and their connection to quantum entanglement have remained quite mysterious despite the recent research on this topic. In the few last years, the ... |

8 | Nonlocality, closing the detection loophole, and communication complexity - Massar - 2002 |

7 |
Optimal Bell tests do not require maximally entangled
- Aćın, Gill, et al.
- 2005
(Show Context)
Citation Context ...een shown not to be the most nonlocal one. We can find some of these anomalies ([53]) in the study of Bell inequalities ([2]), detection loophole ([26]), extractable secrete key ([68]), K-L distance (=-=[3]-=-), etc. Here we will show that there are Bell inequalities which avoid violation of the maximally entangled state in high dimension. The examples are closely related to Theorem 1.2, but we need more i... |

7 | Detection Loophole in Asymmetric Bell Experiments - Brunner, Gisin, et al. |

7 |
Security of key distribution from causality constraints
- Masanes, Winter, et al.
- 2006
(Show Context)
Citation Context ... theoretical interest, Bell inequalities have found applications in many areas of QIT: quantum cryptography, where it opens the possibility of getting unconditionally secure quantum key distribution (=-=[1, 4, 51, 50]-=-), complexity theory, where The authors are partially supported by National Science Foundation grant DMS-0901457. 1 ar X iv :1 00 7. 30 43 v2s[ qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS ... |

6 |
Multiplayer XOR games and quantum communication complexity with clique-wise entanglement. Manuscript at http://arxiv.org/abs/0911.4007
- Briet, Buhrman, et al.
- 2009
(Show Context)
Citation Context ...aximally entangled state is a poor candidate to get large violations. A similar statement holds in the context of tripartite correlations, see [57] and the recent generalization to diagonal states in =-=[10]-=-. However, in [16] the authors showed the existence of a Bell inequality (constructed with 2n inputs and n outputs) with positive coefficients for which the maximally entangled state in dimension n gi... |

6 |
A generalized Grothendieck inequality and entanglement
- Briët, Buhrman, et al.
- 2009
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Citation Context ...the theory of multipartite interactive proof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review [15]); Estimates for the dimension of the underlying Hilbert space (=-=[11, 13, 57, 71, 73]-=-), entangled games ([41, 40]), etc. Bell inequalities and their connection to quantum entanglement have remained quite mysterious despite the recent research on this topic. In the few last years, the ... |

5 |
Testing the Hilbert space dimension
- Brunner, Pironio, et al.
- 2008
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5 |
Entangled games are hard to approximate. arXiv:0704.2903v2 [quant-ph
- Kempe, Kobayashi, et al.
- 2007
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5 |
Universally-composable privacy amplification from causality constraints
- Masanes
- 2009
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5 | Bounding the dimension of bipartite quantum systems - Vertesi, Pal |

4 |
Quantummulti prover interactive proofs with communicating provers
- Ben-Or, Hassidim, et al.
- 2008
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4 | An anomaly of non-locality
- Methot, Scarani
- 2007
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Citation Context ...not new. Indeed, there are many examples in the context of quantum nonlocality where the maximally entangled state has been shown not to be the most nonlocal one. We can find some of these anomalies (=-=[53]-=-) in the study of Bell inequalities ([2]), detection loophole ([26]), extractable secrete key ([68]), K-L distance ([3]), etc. Here we will show that there are Bell inequalities which avoid violation ... |

3 |
Bounds on quantum correlations in Bell inequality experiments, Phys. Rev. A 75
- Liang, Doherty
- 2007
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Citation Context ...tum value of Bell inequalities (related to the study of deciding whether a given probability distribution belongs to the quantum set Q), has captured the interest of many researchers in QIT (see e.g. =-=[46]-=-, [24], [72], [54], [55]). On the other hand, since any two-prover one-round game can be seen as a Bell inequality, Game Theory and, in general Computer Science, can be considered as a very important ... |

2 |
Non-locality and Communication Complexity, to appear in Reviews of Modern Physics
- Buhrman, Cleve, et al.
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Citation Context ...qu an t-p h]s9sSe p 2 01 0 2 M. JUNGE AND C. PALAZUELOS it enriches the theory of multipartite interactive proof systems ([7, 19, 18, 32, 24, 41, 39]), communication complexity (see the recent review =-=[15]-=-); Estimates for the dimension of the underlying Hilbert space ([11, 13, 57, 71, 73]), entangled games ([41, 40]), etc. Bell inequalities and their connection to quantum entanglement have remained qui... |

2 |
Random Fourier series with applications to Armonic Analysis
- Marcus, Pisier
- 1981
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Citation Context ...(Random variables) Although we have proved Theorem 1.1 (via Theorem 3.1) using gaussian variables, it is well known that the same estimations work for Bernoulli variables ([69]) and random unitaries (=-=[49]-=-, [33]) (in this last case one has to normalize by a factor √ n). Thus Theorem 1.2 can be stated using Bernoulli variables (kx,a)x,a,k (as it is stated), gaussian variables (gkx,a)x,a,k and random un... |

2 |
Secrecy extraction from no-signalling correlations, Phys
- Scarani, Gisin, et al.
- 2006
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Citation Context ...entangled state has been shown not to be the most nonlocal one. We can find some of these anomalies ([53]) in the study of Bell inequalities ([2]), detection loophole ([26]), extractable secrete key (=-=[68]-=-), K-L distance ([3]), etc. Here we will show that there are Bell inequalities which avoid violation of the maximally entangled state in high dimension. The examples are closely related to Theorem 1.2... |

1 |
E-mail address: junge@math.uiuc.edu 54
- Comp
- 2001
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Citation Context ...Nature ([28]). However, it took almost 30 years to understand that the apparently dilemma presented in [28] could be formulated in terms of assumptions which naturally lead to a refutable prediction (=-=[75]-=-). Bell showed that the assumption of a local hidden variable model implies some inequalities on the set of probabilities, since then called Bell inequalities, which are violated by certain quantum pr... |