Zeno Toffano
email: zeno.toffano@centralesupelec.fr

Cassandre Chatard

Kaicheng Ding

Alice Portalier

Gregoire Vilde

Grégoire Vildé

Logical Families of Nonlocal Boxes

Bell Inequality for the 256 combinations of logical functions in the two-party correlation logical boxes

The well known nonlocal PR Box [1] correlates outputs (a, b) to inputs (x, y) in a two-party correlation by means of the logical constraint equation a ⊕ b = x ∧ y . It violaties the CHSH Bell Inequality (BI) [2] maximmaly : giving the number 4 beyond the quantum limit 2 √ 2, this last being obtained for a maximally nonlocal entangled quantum state (Bell state). The PR box is also no-signaling.

Here we consider families of boxes with all possible combinations of logical functions f correlated by the equation : fo(a, b) = fi(x, y). The indexes o (output) and i (input) correspond to different Boolean functions and the varaibles (x, y, a, b) {0, 1} are Boolean. There are 16 different Boolean functions for 2 binary variables (n = 2), these are ordered with increasing binary number in the truth table : f0 has the truth table (0, 0, 0, 0), f1 is NOR (1, 0, 0, 0) and so on... For example OR (∨) is f14 (0, 1, 1, 1) and in the PR Box, XOR (exclusive or :⊕) is fo=6(a, b) (0, 1, 1, 0) and AND (∧) is fi=8(x, y) (0, 0, 0, 1).

We define the joint mean value for the possible outcomes of the box as a function of the marginal probabilities :

C (i,o) x,y = a,b [P (a, b | fo(a, b) = fi(x, y)) • A • B] [3]. The measurement outcomes A = 2a -1 (Alice) and B = 2b -1 (Bob)
give the values ±1. The Bell parameter considering the four input possibilities is :

S (i,o) = C (i,o) 00 + C (i,o) 01 + C (i,o) 10 -C (i,o)
11 . If S > 2 we have nonlocality. There are actually four options by changing the place of the "-" in the expression of S (i,o) . We calculated every case and kept the value of S with the maximal absolute value. We also tested no-signaling by the condition :

P (a | x, y) = P (a | x) and P (b | x, y) = P (b | y).
For n = 2, there are 16 × 16 = 256 equations. A family of 16 equations giving maximal violation of the BI (|S| = 4) and no-signaling, corresponds to the following combinations of logical functions : output o : {6, 9} and input i : {1, 2, 4, 8} (8 cases) and also o : {6, 9} and i : {7, 11, 13, 14} (8 cases). The PR Box is part of this family (o = 6 , i = 8). Another family of 32 equations giving a BI violation |S| = 10 3 ≈ 3.33 > 2 √ 2 > 2, exceeding the quantum limit, but now signaling, corresponds to : o : {7, 11, 13, 14} and i : {1, 2, 4, 8}, and the converse o : {1, 2, 4, 8} and i : {7, 11, 13, 14}. An example of this kind of nonlocal logical box is :

a ∨ b = x ∧ y (o = 14 , i = 8).
For three-party correlations, n = 3, we have 256 logical functions. The expression of the joint outcome mean value C (i,o) x,y,z is similar as before. We use the Svetlichny BI [4], which is written as follows : In this work we show that in the case of two-party correlations, we have two families of nonlocal boxes : one with 16 boxes, comprising the PR box, and the other one, to our knowledge unknown up to now, giving 32 nonlocal boxes beyond the quantum limit but signaling. For three-party correlations the number increases and the picture is much more complex.

M (i,o) = C (i,o) 000 + C (i,o) 001 + C (i,o) 010 + C (i,o) 100 -C (i,o) 011 -C (i,o) 101 -C (i,o) 110 -C (i,o) 111 , if M >

Bell Inequality tests on logical boxes for two and three-party correlations

The well known nonlocal PR box [1] correlates outputs ሺܽ, ܾሻ to inputs ሺ,ݔ ݕሻ in a two-party correlation by means of the logical constraint correlation equation ܽ ⊕ ܾ ൌ ݔ ∧ .ݕ This box violates the CHSH Bell Inequality (BI) [2] maximally: giving the number 4 beyond the quantum limit 2 2 (Tsirelson bound) obtained for a Bell state which is a maximally nonlocal entangled quantum state, this box is also no-signaling.

Here we consider families of boxes with all possible combinations of logical functions correlated by the equation:

݂ ሺܽ, ܾሻ ൌ ݂ ሺ,ݔ ݕሻ (1)
݂ ሺܽ, ܾሻ (output) and ݂ ሺ,ݔ ݕሻ (input) stand for the 16 possible different Boolean functions (see the list below) and ሺ,ݔ ,ݕ ܽ, ܾሻ ∈ 0,1 are the Boolean variables of these functions.

We define the joint mean value for the possible outcomes of the box as a function of the marginal probabilities [3]:

ܥ ௫,௬ ሺ,ሻ ൌ ∑ ܲሺܽ, ܾ|݂ ሺܽ, ܾሻ ൌ ݂ ሺ,ݔ ݕሻሻ ⋅ ܣሺܽሻ ⋅ ܤሺܾሻ , where ܣ ܽ ൌ 2ܽ െ 1 and ܤ ܾ ൌ 2ܾ െ 1 (2)
The measurement outcomes ሺ,ܣ ܤሻ , Alice and Bob, give the values േ1 . The Bell parameter considering the four input possibilities is:

ܵ ሺ,ሻ ൌ ܥ ሺ,ሻ ܥ ଵ ሺ,ሻ ܥ ଵ ሺ,ሻ െ ܥ ଵଵ , If ܵ 2 ( * ) we have nonlocality (3)
We also tested no-signaling by the condition:

ܲ ܽ ,ݔ ݕ ൌ ܲ ܽ ݔ and ܲ ܾ ,ݔ ݕ ൌ ܲ ܾ ݕ (4)
For two party correlations we have to consider 16 ൈ 16 ൌ 256 equations. We find two families of nonlocal boxes: one with 16 no-signaling boxes, comprising the PR box, giving maximal violation ( ܵ ൌ 4) and the other one, to our knowledge unknown up to now, giving 32 nonlocal boxes beyond the quantum limit but signaling with ܵ ൌ భబ య ൎ 3.33 2 2 2 which also violates the BI above the quantum limit.

We also tested three party correlations (256 ൈ 256 ൌ 65536 equations) using the Svetlichny Bell Inequality [4], which is written as follows:

ܯ ሺ,ሻ ൌ ܥ ሺ,ሻ ܥ ଵ ሺ,ሻ ܥ ଵ ሺ,ሻ ܥ ଵ ሺ,ሻ െ ܥ ଵଵ , െ ܥ ଵଵ , െ ܥ ଵଵ , െ ܥ ଵଵଵ , if ܯ 4 ( * ) we have nonlocality (5)
Non locality is obtained for six families corresponding to different values of ܯ 4. Among these three are also no-signaling.

( * ) There are actually four options to set S by changing the place of the " െ " sign in Eq.3. We have therefore chosen to calculate in each case the value of S and keep the one whose absolute value is maximum. Same procedure in Eq.5 for ܯ for the three-party correlations.

The PR box [1] corresponds to the AND function, ݂ ଼ ,ݔ ݕ , at the input and to the XOR function, ݂ ܽ, ܾ , at the output. Bell parameter is maximal ܵ ൌ 4.

It is part of the no-signaling family of 16 nonlocal boxes (orange cases).

The other family of 32 nonlocal boxes (green cases) is signaling and giving a BI violation ܵ ൌ భబ య ൎ 3.33 2 2 2, exceeding the quantum limit. An example of this kind of nonlocal logical box has the logical constraint ܽ ∨ ܾ ൌ ݔ ∧ ݕ .

Boolean function truth tables

We consider the 16 possible different 2-input Boolean functions

two-party correlation box logical equation

Logical function for p, q f n (p,q)

Truth table for 2 inputs (p,q) (1,1) (1,0) (0,1) (0,0)

Bell Inequality for the 65536 combinations of logical functions in the three-party correlation logical boxes

Calculation (computer) of the Svetlichny [4]

f15

  According to standard numeration, logical functions ሺ, ሻ are ordered with increasing binary number in the truth table.

  the PR box : a⊕ ⊕ ⊕ ⊕b = x∧ ∧ ∧ ∧y C xy = P(0,0|x,y) . (-1) . (-1) + P(0,1|x,y) . (-1) . (+1) + P(1,0|x,y) . (+1) . (-1) + P(1,1|x,y) . (+1) . (+1) (Eq 2) C 00 =C 01 =C 10 =1/2+0+0+1/2=+1 C 11 = 0-1/2-1/2+0=-1 S=C 00 +C 01 +C 10 -C 11 = +4 (Eq 3) in out

  MNL = 8 ; 6.4 ; 6 ; 5.33 ; 4.8 ; 4.57 with the corresponding number of cases Ncase(MNL) = 71 ; 560 ; 112 ; 840 ; 896 ; 560. The cases of non-locality and non-signaling are less and occur only for MNL+NS = 8 ; 6 ; 5.33.

4

we have non-locality. n = 3 corresponds to 256 × 256 = 65536 equations. Non-locality is obtained for six different values:

  parameter ܯ (Eq. 5) for 256 ൈ 256 ൌ 65536 equations. Non-locality ܯ 4 is obtained for six different values of ܯ .

		8	6.4	6	5.33	4.8	4.57
	Number of	71	560	112	840	896	560
	boxes
	Signaling :	NO	YES	NO	NO	YES	YES