NNon-Local Boxes for Networks
Jean-Daniel Bancal and Nicolas Gisin , Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland SIT, Geneva, Switzerland (Dated: February 9, 2021)Nonlocal boxes are conceptual tools that capture the essence of the phenomenon of quantumnon-locality, central to modern quantum theory and quantum technologies. We introduce networknonlocal boxes tailored for quantum networks under the natural assumption that these networksconnect independent sources and do not allow signaling. Hence, these boxes satisfy the No-Signalingand Independence (NSI) principle. For the case of boxes without inputs, connecting pairs of sourcesand producing binary outputs, we prove that there is an essentially unique network nonlocal boxwith local random outputs and maximal 2-box correlations: E = √ − I. INTRODUCTION
Non-locality is a key feature of quantum physics andone of the major discovery - arguably the major discovery- of last century physics. Modern quantum technologypromises, in addition to quantum computers, quantumnetworks that will connect these quantum processors andoffer proven confidentiality of all communications. It isthus natural and timely to study quantum non-localityin networks, a field that has been burgeoning for about adecade under the names of bilocality and n -locality, for2 and n independent sources, respectively [1–21]. Let usstress that in networks the characteristic feature of quan-tum physics, namely entanglement, enters twice. First,entanglement of the subsystems emitted by the sources.Second, entanglement produced by the joint measure-ments that connect independent subsystems emitted bydifferent sources. This second form of entanglement ismuch less studied and understood than the first one [22].In this work we go beyond quantum physics and study n -locality for arbitrary boxes only limited by no-signalingand independence, a principle we name NSI [23]. This isin the spirit of the non-local boxes introduced by Popescuand Rochlich - the so-called PR-boxes [24] - as a concep-tual tool to study Bell non-locality. Here, however, wego beyond Bell non-locality, our boxes connect indepen-dent sources and have no inputs. A priori, such net-work non-local boxes could have any number of outputsand connect any number of sources, with sources con-nected to any number of boxes. In this paper we restrictourselves to boxes connecting two sources, producing bi-nary outputs and sources connected to two boxes, seeFig 1. We name such boxes binary boxes. Similarly tothe PR-boxes the aim is the set the limits of non-localityin networks imposed by the NSI principle and to offerconceptual tools to study this new form of non-locality.Assuming that locally each box’s output is totally ran-dom and that the outputs of two connected boxes aremaximally correlated, we find that there is a unique net-work binary box (unique up to flipping all outputs). FIG. 1:
Various networks. Boxes are indicated with a squarethat outputs x j . Sources are marks as ∗ .a) One box in a line. b) One box in a loop. c) n boxes in aline. d) n boxes in a loop. II. BINARY NETWORK NON-LOCAL BOXES
First, consider two independent sources, sending outsubsystems in two opposite directions, coupled by onebox as illustrated in Fig. 1.a. Each box ouputs a bit x j = ±
1, which we often label merely ± . We assumethat this bit is random, hence its expectation value iszero: E = (cid:104) x j (cid:105) = 0. Figure 1.b illustrates the case of abox connected to the two parts of a single source. Herealso we assume E o = (cid:104) x j (cid:105) = 0.Next, let us add more independent sources and boxes.Only two fully connected configurations are possible. Ei-ther all sources and boxes are on a line, Fig. 1c, or theyform a loop, Fig. 1d. It is important to realize that noother configurations is possible. Hence, proving the exis-tence of a binary network nonlocal box compatible withthese two configurations suffices to prove that the box isconceptually possible.In both configurations, the probability of the outputs x j can conveniently be expressed in terms of all correla-tors. Assuming all sources and boxes are identical, thecorrelators between k adjacent boxes are all equal, de-noted E k = (cid:104) x m +1 · x m +2 · ... · x m + k (cid:105) . In case of a poly- a r X i v : . [ qu a n t - ph ] F e b gon configuration with n boxes at the vertices, i.e. a loop,there is one additional correlator: E on = (cid:104) x · x · ... · x n (cid:105) ,where the subscript “o” indicates that this correlator cor-responds to a closed loop.Let us consider some examples. First, the joint proba-bility of outputs for 3 boxes on a line reads: p ( x , x , x ) = 12 (cid:0) x + x + x ) E
1+ ( x x + x x ) E + x x x E + x x E (cid:1) (1)= 12 (cid:0) x x + x x ) E + x x x E (cid:1) (2)where the first term guarantees normalization and thelast equality follows from the assumption E = 0. Next,for 4 boxes in a square we have: p o ( x x x x ) = 12 (cid:0) x x + x x + x x + x x ) E + ( x x x + x x x + x x x + x x x ) E + x x x x E o (cid:1) (3)and so on.The correlators corresponding to disconnected setsfactorize, because of the assumed independence of thesources, as in the last term in (1). It is important torealize that for hexagons and 5 boxes in a line and largernetworks, independence implies some non-linear correla-tors. For example, in a line with 5 boxes the last corre-lator equals the square of the 2-box correlator: p ( x ...x ) = 12 (cid:0) (cid:88) k =1 x k x k +1 E + (cid:88) k =1 x k x k +1 x k +2 E + (cid:88) k =1 x k x k +1 x k +2 x k +3 E + (cid:89) k =1 x k E + x x x x E (cid:1) (4)Obviously, the correlators have to be such that p ( x j ) ≥ { x j } . In [23], using 6 sources and6 boxes in an hexagonal loop, we proved the followingbound on the 2-box correlator: E ≤ √ − ≈ . E = 0 and E = √ − p ( (cid:126)x )and p o ( (cid:126)x ), where (cid:126)x = ( x ..x n ), are non-negative. Butfirst, in the next section, we consider 3 boxes in a lineand prove that the value of the corresponding correlator E is unique (up to its sign). III. 3-BOX CORRELATOR
Consider 3 boxes in a line as in eq. (2). We have p (+ − +) = (cid:0) − E − E (cid:1) ≥
0. Hence, E = √ − E ≤ − E = 3 − √
2. In the appendix weprove that lines of length 7 and longer do actually imposeequality: E = ± (3 − √
2) (6)Henceforth we assume the positive sign for E . We’llsee that this choice implies that all correlators are non-negative. Note that by flipping all outputs, all odd cor-relators, in particular E , merely change sign.The relation (6), inserted in (2), has the following im-portant consequence: p (+ − +) = 0. Accordingly, thereare never 3 adjacent boxes with outputs + − +: p n ( x ..x j − + − + x j +3 ..x n ) = p (+ − +) · (7) p ( x ..x x − x j +3 ..x n | x j = + , x j +1 = − , x j +2 = +) = 0This in turn implies that whenever two connected boxesoutput + − , then the next box necessarily outputs − .This is key in proving existence and uniqueness of ournetwork non-local box, as we show in the two next sec-tions. IV. RECURSION FORMULA FOR THE E k Let us define q n ( (cid:126)x ) = 2 n p n ( (cid:126)x ), so that the normaliza-tion factors drop out. We still have the non-negativityconditions q ( (cid:126)x ) ≥ (cid:126)x . The case of n + 1 boxes ina line can be reduced to n boxes as follows: q n +1 ( x ..x n +1 ) = q n ( x ..x n ) + (Π n +1 j =1 x j ) E n +1 + n − (cid:88) k =0 (Π n +1 j = k +2 x j ) q k E n − k (8)= q ( x ..x n )+ (Π n +1 j =1 x j )( E n +1 + Q n ) (9)where Q n = (cid:80) n − k =0 (Π k +1 j =1 x j ) q k E n − k , q k = q k ( x ..x k ) andwe used x j = 1 for all j . The first term on the left of (9)takes into account the first n boxes, the second term all n + 1 boxes and the last term all cases that involve thelast box but not all boxes.Considering successively the cases (Π n +1 j =1 x j ) = ± q n ( x ..x n ) − Q n ≥ E n +1 ≥ − q n ( x ..x n ) − Q n (10)Apply this to the string of alternating outputs q n (+ − + − + − ... ), the result of the previous section implies q k = 0for all k ≥
3. Hence, from (9) and (10) one obtains therecursion formula: E n +1 = − E n + E n − + (1 − E ) E n − (11)This leads to the following closed form for all correla-tors of boxes in a line: E n = 1 µ (cid:2)(cid:0) µ − (cid:1) n − − (cid:0) − µ + 1 (cid:1) n − (cid:3) (12)where µ = (cid:112) √ ≈ . E n > n ≥
2. Obviously, flipping all outputs would changethe sign of E n for all odd n ’s. Table I lists the 12 firstvalues of E on , remarkably they are all of the form n + m √ n and m integers.It remains to prove that these correlators guarantee q n ( (cid:126)x ) ≥ (cid:126)x .First, notice that by inserting the recusion formula (11)into (9) one gets, after some algebra: q n ( (cid:126) n ) = ( √ n − (13)where (cid:126) n = (+1 , +1 , ..., +1), with n +1’s.Second, using q n ( x ..x j − + − + x j +3 ..x n ) = 0 for all2 ≤ j ≤ n − x = +1. Then, eitherall outputs are +1, i.e. (cid:126)x = (cid:126) n in which case q n ( (cid:126) n ) = √ n +1 >
0, or the first output − j , i.e. x k = +1 for all k < j and x j = −
1. In sucha case, the box at position j + 1 can be removed, asanyway the output − j + 1 box isremoved, then the independence assumption implies thatthe probability factorizes: p n ( x x ...x n ) = p j (+ + ... ++ − ) · p n − j − ( x j +2 ...x n ).Eventually, all probabilities are products of the fol-lowing terms: p n ( (cid:126) n ), p n ( (cid:126) n − − ), p n ( − (cid:126) n − ) and p n ( − (cid:126) n − − ), where − (cid:126) n represents an output stringstarting with an output − n outputs +1.Remains to prove that these 4 terms are all non-negativefor all n . Relation (13) proves the positivity of the firstterm and will be heavily used to prove the positivity ofthe 3 other term: p n +1 ( (cid:126) n − ) = p n +1 ( − (cid:126) n ) = p n ( (cid:126) n ) − p n +1 ( (cid:126) n +)= 1 √ n +1 (cid:0) − √ (cid:1) > p n +2 ( − (cid:126) n − ) = p n ( (cid:126) n ) − p n +2 ( − (cid:126) n +) − p n +2 (+ (cid:126) n − ) − p n +2 (+ (cid:126) n +) (15)= 1 √ n +1 (cid:0) − √ − √ − (cid:1) (16) > n boxes in a line are fixed by the assumptions E = 0and E = √ − E ≥
0) and they guarantee thatall probabilities p ( (cid:126)x ) ≥
0, for all output strings (cid:126)x . Inthe next section we prove that these correlators are alsocompatible with n boxes in any polygon configuration. V. POLYGONS
The smallest closed loop has a single box fed by thetwo links produces by a single source, see Fig. 1a. Inthis case we assume, similarly to a single box in a line, E o = 0, where the upper index o indicates a closed loop.Accordingly, single boxes always produce fully randomoutputs. The second shortest loop has 2 boxes and 2sources. Since E o is not limited by the NSI principle, weassume it takes the largest possible value : E o = 1.Let’s now consider polygons with n + 1 vertices andedges, for n ≥
2. Using a similar technique as for the linewe have: q on +1 ( x ..x n +1 ) = q n ( x ..x n ) + Π n +1 j =1 x j · E on +1 + n (cid:88) k =1 E k · k (cid:88) (cid:96) =1 (cid:0) Π n + (cid:96)j = n +1 − k + (cid:96) x j (cid:1) · q n − − k ( x n − − k + (cid:96) ..x n + (cid:96) +2 ) (17)where all indices of x j ’s are assumed modulo n + 1 (e.g. x n + (cid:96) +2 = x (cid:96) +1 ) and we set q − = q = 1.Using q (+ − +) = 0 we deduce that for all n ≥ q n +1 ( (cid:126) n , − ) = 0. Let’s apply this to the above formula: q on +1 ( (cid:126) n , − ) = q n ( (cid:126) n ) − E on +1 + n (cid:88) k =1 E k · k (cid:88) (cid:96) =1 x .. ˆ x (cid:96) .. ˆ x (cid:96) + n − k ..x n +1 · q n − − k ( x (cid:96) ..x (cid:96) + n − k − ) (18)= √ n − − E on +1 − n − (cid:88) k =1 E k · k · √ n − k − − ( n − E n − − nE n = 0 (19)Consequently: E on +1 = √ n − − n − (cid:88) k =1 E k · k · √ n − k − − n · E n − ( n − · E n − (20)Using eq. (12), with the same µ = (cid:112) √
2, one gets,for all n ≥ E on = (cid:0) µ − (cid:1) n + (cid:0) − µ + 1 (cid:1) n (21)Note that E on > n . Table I lists the 12 first valuesof E on , remarkably they are all of the form n + m √
2, with n and m integers, as we found for the correlators in a line. At first sight, one may fear that combining the two end sourcesof the 2-box configuration in a line into a single source, thuschanging the line into a loop configuration, leads to signaling,since E (cid:54) = E o . However, the change in the source takes time toinfluence the boxes, the time of flight of the subsystems emittedby the sources. Hence, E (cid:54) = E o does not lead to signaling. Remains to prove that q on +1 ( (cid:126)x ) ≥ (cid:126)x . First,assume that not all outputs are equal. Then, there are 2adjancent boxes with outputs +1 −
1. Since p (+ − +) =0, we can remove the next box as it necessarily outputs −
1. In this way one opens the loop and reduces it toa line for which we already proved non-negativity of allprobabilities. Next, assume all outputs are +1. Since E on ≥ n , p on ( (cid:126) n ) ≥
0. Finally, assume all outputsare -1: p on ( −→− n ) = 12 n (cid:0) n n − (cid:88) k =2 ( − k E k + ( − n E on (cid:1) (22)From (12) and (21) straightforward computations showsthat E n − E n +1 > E n − − E on >
0. Hence, p on ( −→− n ) ≥ n . VI. CONCLUSION
We proved that under the natural assumption E = E o = 0 there is a single binary network non-local box thatmaximizes the 2-box correlator E . The existence anduniqueness of such a box is highly non-trivial. Admit-tedly, the deep reason why such a box exists and - espe-cially - is unique is left open. Another question is whetherthis binary non-local box can be realized within quantumtheory and, if not, how close quantum can come?A natural challenge is whether a similar result holds also for boxes with more outputs, and/or boxes connect-ing more than 2 sources. We investigated numericallythe case of 4 outputs under the natural assumption ofoutput permutation symmetry, i.e. for all permutations π of 1,2,3,4 p n ( (cid:126)x ) = p n ( π ( (cid:126)x )) (same permutations for alloutputs). However, we found no sign of the uniquenessfor 4-output network boxes, though the question remainsopen and should be investigated in future work. Also,other network nonlocal boxes should be studied, in par-ticular configurations beyond lines and loops, like, e.g.star networks.Network non-locality is a new and fascinating form ofnon-locality. It combines non-locality and “entangling”joint measurements. It deserves to be analyzed withingquantum theory and, as we do here, from outside quan-tum under the very general NSI principle: no-signalingand independence. The presented binary non-local boxfor networks is a conceptual tool, similar in spirit to thePR- boxes introduced by Popescu and Rochlich [24]; suchconceptual tool are useful to understand and simulatequantum correlations, and for applications in the spiritof device independent quantum information processing.Among the many fascinating research to which thisone contributes are all questions on the limits of quan-tum non-locality (e.g. information causality [25]). Inparticular, macroscopic locality [27] and classical limitsof non-local boxes [28, 29] are especially interesting andshould be apply to network boxes. n E n √ − − √ √ − − √ − √ √ −
14 27 − √ √ −
31 10 − √ − √ √ − E on − √ √ − − √ √ − √ − − √ √ −
52 71 − √ √ −
45 44 − √ Table of the first 12 correlators in line, E n , and in aloop, E on . (The indicated values of E o and E o are the maximalvalues compatible with p o ( x x ) ≥ and p o ( x x x ) ≥ ,respectively.) Acknowledgment
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Appendix A: Proof of (6)
Let us consider the case where E = 0 and E = √ − E ≥ (3 − √ . (A1)Since we have anyway E ≤ (3 − √ , this implies that E = ± (3 − √
2) (with the sign being a symmetry of theproblem).
Theorem A.1.
In a line configuration with 7 couplers, when-ever E = 0 and E = √ − , the following inequality holds: E ≥ (3 − √ . (A2) Proof.
Let us identify 17 probabilities: P = P ( − − − − − − − ) (A3) P = P ( − − − + − − − ) (A4) P = P ( − − + − + − − ) (A5) P = P (+ − + − + − +) (A6) P = P (+ + − + − + +) (A7) P = P (+ + + − + + +) (A8) P = P (+ + + + + + +) (A9) P = P ( − + + − + − +) (A10) P = P ( − + − + − − +) (A11) Q = P ( − − − − + − +) (A12) Q = P ( − − − − + + +) (A13) Q = P ( − − − + − + +) (A14) Q = P ( − − − + + + +) (A15) Q = P ( − − + − + + +) (A16) Q = P ( − − + + − + − ) (A17) Q = P ( − + − + + + +) (A18) Q = P (+ − + − − + +) . (A19)We define three vectors in R : u = (8 , , , , , , , , , , , , , , , , , , , , , , , v = (4 , , , , , , , , , , , , , , , , , , , , , , , c = (cid:16) √ , √ , √ ,
109 + 2003 √ , √ , √ , √ , − √ , √ , √ , −
38 + 337 √ , √ ,
227 + 1805 √ , √ , √ , − √ , √ , √ , √ , √ ,
44 + 25 √ , √ , − √ , √ (cid:17) (A21) A lenghty but direct computation shows that2 · (cid:88) i =1 c i P u i Q v i = 12 √ −
17 + E . (A22)In other words, this expression does not involve any correla-tor with more than 3 parties. Since the probabilities and all components of c are positive, this expression also is positive,i.e. 12 √ −
17 + E ≥ ,,