Abstract
The key feature in correlations established by multiparty quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with nosignaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a ClauserHorneShimonyHolt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest PopescuRohrlich set of causal stochastic processes (aka a PR box) in the nosignaling polytope, related by polyhedral duality to Bell’s correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction \(\sqrt{2}\,1\approx 0.414\) of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
Introduction
The interest in quantum correlations as an informationprocessing resource^{1,2,3} puts under consideration the problem of finding the experimental setup that maximizes the strength of nonlocality that is achievable from a given multiparty quantum state being shared in a Bell scenario. But first, how to size up the strength of the nonlocality that might be present in quantum correlations^{4}?
In a bipartite (finite ab initio) Bell scenario involving two dichotomic measurements per party (a ClauserHorneShimonyHolt (CHSH) setup^{5} for short) quantum correlations may be stronger than just shared classical information, implying that an instant form of causal (noncommunicating) nonlocality between the two party locations is involved. Thus, if no communication between the parties is supported, what is the physical resource that is responsible for nonlocality in quantum bipartite setups? Entanglement is necessary but it is certainly not enough.
The measure of a physical setup as a resource is determined by the largest operational outcome that, in principle, the physical setup is able to provide. Traditionally, the interest was to distinguish classical from quantum correlations in Bell scenarios and it was then natural to quantify nonlocality as the maximum violation of a Bell inequality allowed by a given quantum state. Nowadays, operational interests in quantum information theory might decide about the unit to quantify nonlocality resources. The current views on Bell nonlocality are reviewed in^{6} and the resourcetheory point of view in^{4}. Currently, the PRbox is a choice^{7} to measure the amount of nonlocality that is supplied by bipartite setups sharing qubit pairs.
The PRbox is an elementary (but hypothetical) building block in probability theory that combines causality and nonlocality in one unit. It was first advanced by Popescu and Rohrlich (PR)^{8} and then was proved elementary in the nosignaling formalism^{6}. The PR box has become a new informationtheoretic entity with the clear meaning of a nonreducible (atomic) form of causal (noncommunicating) nonlocality. For instance, an unlimited number of PR boxes (together with a fixed amount of classical communication) allows two parties to give a simple solution to any communication complexity problem^{9}.
Our theoretical framework for nonlocality in quantum correlations in finite setups is the nosignaling formalism. This choice leaves off the discussion any form of noncausal nonlocality: classical communication does not have a place in the formalism. Any finite causalcorrelational setup—as the CHSH setup—is characterized in the nosignaling formalism by a point in the polytope of nosignaling stochastic matrices (or nosignaling boxes). The nosignaling polytope for the CHSH setup is cut by the CHSH inequalities producing the CHSH facets. The vertices the cuts leave outside are the nonlocal PR boxes. The skeleton of the CHSH polytope was computed in^{10} and the adjacency of PR boxes in the skeleton is given in Table A2 of the Appendix. Table A2 shows that every PR box is the apex of an 8simplex with a CHSH facet as its base.
Although the nonlocality produced by the PRbox per se is strongerthanquantum, it was shown in^{11} that it can be diminished (assisted with shared randomness, without introducing mixtures) to reproduce (without communication) the quantum correlations attainable by the singlet state in an operationally free (or continuous) binary bipartite setup. In Section 5 we demonstrate that one PR box is enough too to reproduce the quantum correlations in any CHSH setup sharing qubit pairs in any partially entangled pure state.
Thus, the PR box demonstrates by itself that, as an informationtheoretic operational unit, is capable to distinguish quantum from classical correlations in binary bipartite setups. Here we prove that in every CHSH setup the PR box is a physically meaningful ruler to measure the average amount of nonlocality it yields.
For CHSH setups, our questioning at the beginning is rephrased in the nosignaling formalism as, how much of a PR box is realizable by a given quantum entangled qubit pair? Our way to answer this question starts in Section 2 where the quantum probabilities of coincidences in a CHSH setup—sharing a qubit pair in an arbitrary pure state—are given the form of a nosignaling stochastic matrix (that we call a Q box), which puts the CHSH setup as a point in the polytope of nosignaling boxes. Next in Section 3, we introduce a novel measure of the CHSH box Q as a resource of bipartite nonlocality.
For any finite causally correlated multiparty setup, we propose to measure the amount of nonlocality it yields by the proximity (accepting the usual L_{2} distance for probability space) of its nosignaling stochastic matrix Q to the nearest nonlocal vertex of the nosignaling polytope. The nonlocal box to be sized is located in the nosignaling polytope by the convex combination with the least number of extreme (elementary) boxes, including in the combination only the closest nonlocal vertex (just one PR box in the case of CHSH setups), the vertex that is going to fix the amount and nature of nonlocality. A similar way of weighing the content of nonlocality in Bell scenarios is proposed in^{12} as a linear program in the L_{1} metric, aka the classical trace distance. On the other hand, we should note that characterizing nonlocality in continuous setups is a subject that lies beyond polyhedral convex geometry^{13}. Models of quantum nonlocality in unrestricted continuous setups have been considered in^{11,14,15,16,17}.
For finite setups, convex geometry gives a statistical meaning and an operational interpretation to our measure of nonlocality. We illustrate by example these two subjects in Sections 4 and 5. But first, in Section 3 we find that the CHSH quantum matrix Q—as a point in the nosignaling polytope–is located within one of the PR nonlocal 8simplices. We treat the case of CHSH setups lying in the neighborhood of the prototypical PR box^{8} (with stochastic matrix denoted by R). Other geometries are treated similarly.
A point inside a simplex has a unique representation in barycentric coordinates. We expand Q in Section 3 as the unique convex combination of 8 local vertices and the PR box R. The PR coefficient in the barycentric expansion of Q is denoted by N_{ Q } ∈ [0, 1]. Geometrically, N_{ Q } is a measure of its proximity to R (with the L_{2} distance). Statistically, N_{ Q } is the probability for box Q to behave nonlocally, as the PR box R does. Operationally, the convex combination of elementary (extreme) nosignaling boxes is interpreted as a set of stochastic processes. Such a full and definite physical significance is what we want of a measure of the causal nonlocality being supplied by Q in the average. The PR box appears thus as the informationtheoretic choice for the unit of causal nonlocality. The only nosignaling box P with N_{ P } = 1 is the PR box R.
Our Lemma 3.1 states that the average strength of nonlocality N_{ Q } is the probability to find the CHSH quantum box Q behaving as the PR box does: N_{ Q } is the fraction of times the nonlocal box Q realizes a PR box in the average. The complementary fraction of times the quantum box Q behaves locally. This statistical description was first discussed and termed a 2species model by (with due respect) EPRbis in^{14}. We illustrate this point with an example in Section 5.
In Section 2 we have discovered that the Bell correlation functions are the linear functionals that place PR boxes and CHSH facets in duality^{18}. For the prototypical PR box the Bell correlation linear functional on Q is denoted [Q]. Then, the two corresponding CHSH inequalities are [Q] ≤ 2, and for the PR box [R] = 4, which is the largest Bell correlation for a nosignaling box. A consequence of polyhedral duality we find in Section 3 is the relation between the average measure of causal nonlocality and Bell correlation: \({N}_{Q}=\frac{1}{2}[Q]1\).
In Section 4 we find the optimal geometry (the set of setup vectors) that maximizes the average strength of bipartite nonlocality that is realizable in a CHSH setup from two qubits in an arbitrary state ψ. The optimization results constitute our Lemma 4.1. The maximal average nonlocality \({N}_{{Q}_{\psi }}\) that is available from state ψ, in terms of Wootters’ concurrence W_{ ψ }, is given by \({N}_{{Q}_{\psi }}=\sqrt{1+{W}_{\psi }}1\le \sqrt{2}1\approx 0.414\), which is an increasing function of concurrence, proving that \({N}_{{Q}_{\psi }}\) is an entanglement monotone. Optimization shows us that maximally nonlocal CHSH boxes are restricted to lie in the nosignaling polytope within a 4dimensional simplex (denoted Δ^{4}) with apex the PR box R. We show the locus of maximally nonlocal CHSH boxes as a function of entanglement, projected into the five 3D facets of Δ^{4}.
In Section 5 the barycentric expansion of the CHSH box—supporting the measure of nonlocality \({N}_{{Q}_{\psi }}\)—is interpreted as a 2species stochastic causal simulator of the quantum correlations produced by the CHSH box. The simulator in Section 5 has the least of elementary nosignaling boxes and does not involve any form of classical communication between the parties, the PR box is the only source of nonlocality. Data produced with a computer implementation of the simulator agrees with the quantum mechanical formulas.
The NoSignaling Polytope of CHSH Scenarios
The players in a CHSH scenario (Alice and Bob) are identified by the labels i = 1 and i = 2, respectively. They share in the setup a pair of qubits (prepared in the quantum state ψ〉) and have access each to a local binary input x_{ i } ∈ {0, 1} allowing them to choose one out of two setup vectors, \({\overrightarrow{a}}_{{x}_{1}}=({a}_{x}({x}_{1}),{a}_{y}({x}_{1}),{a}_{z}({x}_{1}))\) for Alice (i = 1) and \({\overrightarrow{b}}_{{x}_{2}}=({b}_{x}({x}_{2}),{b}_{y}({x}_{2}),{b}_{z}({x}_{2}))\) for Bob (i = 2), to align the quantization axes of the local observables \({\sigma }_{{\overrightarrow{a}}_{{x}_{1}}}={\overrightarrow{a}}_{{x}_{1}}\cdot \overrightarrow{\sigma \,}\) and \({\sigma }_{{\overrightarrow{b}}_{{x}_{2}}}={\overrightarrow{b}}_{{x}_{2}}\cdot \overrightarrow{\sigma \,}\). The local outcomes may result in one of the values \({(1)}^{{w}_{i}}\), with w_{ i } ∈ {0, 1}. The CHSH setup has sixteen configurations (w_{1}w_{2}; x_{1}x_{2}) with probability
where the \(T({w}_{i}{x}_{i})={\textstyle \tfrac{1}{2}}(1+{(1)}^{{w}_{i}}{\sigma }_{{\overrightarrow{v}}_{{x}_{i}}})\) are orthogonal projections.
The quantum probabilities (1) satisfy the nosignaling property (or the impossibility to use correlations to exchange information between the parties^{6})
Then, Alice’s marginal probabilities are independent of Bob’s input choice and conversely. Besides, \(\sum _{{w}_{1}{w}_{2}}P({w}_{1}{w}_{2}{x}_{1}{x}_{2})=1\) and P(w_{1}w_{2}x_{1}x_{2}) ≥ 0.
The foregoing equations (nosignaling and normalization) reduce the sixteen probabilities to eight independent variables and the inequalities (the nonnegativity conditions) define halfspaces in 16Euclidean space. The region in 16space where all conditions are satisfied is the 8dimensional polytope of the 4 × 4 nosignaling stochastic matrices (or boxes), with entries P(w_{1}w_{2}x_{1}x_{2}): the binary word x = x_{1}x_{2} is the matrix row index and the binary word w = w_{1}w_{2} the column index. The probabilities in (1) are the entries of the quantum box Q supplied by state ψ in the CHSH setup.
The stochastic matrices that are the vertices of the CHSH nosignaling polytope were computed with the enumeration algorithm^{10}. The matrix entries P(w_{1}w_{2}x_{1}x_{2}) of the vertex matrices are presented in Table A1 of the Appendix. All the local vertices in Table A1 (that we call Bboxes) are parameterized in the single formula
with αaβb ∈ {0, 1}^{4}. Bboxes factor out into local 2 × 2 deterministic transition matrices. Any mixture of product matrices is a local box.
The eight vertices of the nosignaling polytope in Table A1 that are stochastic matrices that cannot be factorized are the (nonlocal) PR boxes. We denote them by R_{ ijk }, with ijk ∈ {0, 1}^{3}, and have entries parameterized in the following form,
Every PR box in the polytope connects through edges with eight of the Bboxes, conforming an 8simplex. The adjacency of PR boxes in the polytope is given in Table A2. The prototypical PR box R_{000} and its eight neighbors are highlighted with an asterisk in Table A1. We denote by \({ {\mathcal B} }_{\ast }\) the set of Bboxes with an asterisk in Table A1.
Every pair of antipodal PR boxes defines in the nosignaling polytope the axis σ_{ ij } = 2(R_{ij0}−R_{ij1}) of nonlocality. Each axis (a pair of PR boxes) is dual to a pair of CHSH facets through the inner product (σ_{ ij }, Q), where (A, B) = tr(A^{T}B) for real matrices. This fact is confirmed using the matrices R_{ ijk } in (4) and the definition 〈x_{1}x_{2}〉 = P(00x_{1}x_{2}) + P(11x_{1}x_{2}) − P(01x_{1}x_{2}) − P(10x_{1}x_{2}). Then, for every binary word ij there is the CHSH inequality (σ_{ ij }, Q) ≤ 2 for any local nosignaling box Q,
The prototypical axis of nonlocality of our interest is σ_{00} (or just σ), for which the scalar product
is the usual Bell correlation [Q] of box Q. Notice that the Bell correlation [⋅] is just the linear functional that puts the corresponding CHSH facets in duality with the axis of nonlocality σ. The PR box R_{000} (or just R) has [R] = 4, the largest Bell correlation for a nosignaling box in the polytope. Since [B] = 2 for every box \(B\,\in \,{ {\mathcal B} }_{\ast }\) and \({ {\mathcal B} }_{\ast }=8\), then the affine span of \({ {\mathcal B} }_{\ast }\) is the hyperplane in 8space that supports the CHSH facet that is the dual of R.
The Average Causal Nonlocality of a CHSH Quantum Box
We’ll look for the maximally nonlocal CHSH setup for the very general quantum state
The content of entanglement in ψ〉 is measured by its concurrence \({W}_{\psi }={\sin }^{2}2\beta \).
The probability entries of the quantum box Q that is supplied by the CHSH setup with the general state ψ in (6), are calculated with (1), resulting in
where α = α_{2} − α_{1}.
For nosignaling boxes in the vicinity of the PR box R it is convenient to choose vertex R as the origin and adopt the set of edges \( {\mathcal B} =\{\beta =BR:B\,\in \,{ {\mathcal B} }_{\ast }\}\) (see the adjacency Table A2 in the Appendix) as a vector basis. Let \({ {\mathcal B} }^{r}\) be the reciprocal to basis \( {\mathcal B} \). By convexity we have that cone \(( {\mathcal B} )\) covers the polytope. Thus matrix Q relative to R is expanded in the \( {\mathcal B} \) basis as \(QR={\sum }_{\beta }\beta {C}_{\beta }\), with coefficients C_{ β } = (β^{r}, Q − R) ≥ 0. The expansion provides the barycentric representation
where we have used the Bell correlations [B] = 2 and [R] = 4 to get that all coefficients add up to one,
Recall that the Bell correlation [⋅] used in (8) and in (9) is the linear functional that puts the PR box and the CHSH facet in duality.
By definition, the expansion (8) of any nosignaling box has C_{ β } ≥ 0 for all \(\beta \,\in \, {\mathcal B} \), while the PR coefficient \({N}_{Q}:=\frac{1}{2}[Q]1\) is positive only when box Q is in the interior of the 8simplex with R as its apex, since then [Q] > 2. Our definition of the average strength of causal nonlocality of box Q is the coefficient of the PR box R in the barycentric representation (8),
and N_{ Q } = 0 otherwise. The connection of the average strength of causal nonlocality with the Bell correlation of box Q established in (10) is a consequence of polyhedral duality. It does not suffer of any ambiguity given the uniqueness of the barycentric expansion (8). Besides, only the PR box has the largest amount of nonlocality, N_{ R } = 1. The results are summarized in the following.
Lemma 3.1.
The nosignaling box Q that is provided by a CHSH setup with the quantum state (6) is the mixture
of a local box L and the PR box R. The average strength of causal nonlocality of box Q is \({N}_{Q}=\frac{1}{2}[Q]1\), of box L is N_{ L } = 0 and N_{ R } = 1.
Proof
: Normalize the C_{ β } coefficients in (8) to \({c}_{\beta }={C}_{\beta }/\sum {C}_{\beta }\). Then, using \(\sum {C}_{\beta }=2\,\frac{1}{2}[Q]\) from (9), we obtain from (8) that \(Q=(2{\textstyle \tfrac{1}{2}}[Q])\sum {c}_{\beta }B+({\textstyle \tfrac{1}{2}}[Q]1)\,R\), where \(L=\sum {c}_{\beta }B\) and \(\sum {c}_{\beta }=1\). That N_{ L } = 0 follows from \([L]=\sum {c}_{\beta }[B]=2\) and definition (10).□
Following the procedure leading to (8) on our quantum nosignaling box Q, with probability entries in (7), we get the convex combination
where the coefficients C_{ β } are easily identified and the stochastic matrices B_{ αaβb } involved in (12) (elements of \({ {\mathcal B} }_{\ast }\)) are highlighted with an asterisk in Table A1 (see the parameterized form in (3) too). The pair correlations in the CHSH setup are
The average strength of causal nonlocality, N_{ Q }, of the CHSH box Q provided by state ψ in (6), is to be computed by substituting the pair correlations (13) in the Bell correlation (5) and then using definition (10). The result for N_{ Q } is an involved function of the setup vectors and the angle of entanglement β.
Maximally Nonlocal CHSH Setups
The stochastic matrix Q of the correlational setup that is produced when the quantum state ψ in (6)—containing an arbitrary amount of entanglement—is being shared in a CHSH setup is given in (12). The average measure of causal nonlocality of the quantum box Q, N_{ Q } introduced in Lemma 3.1, is computed using (10), (5) and (13). Thus, we are in position to tackle the problem of finding the optimal setup vectors \({\overrightarrow{a}}_{opt}({x}_{1})\) and \({\overrightarrow{b}}_{opt}({x}_{2})\) that maximize the average strength of nonlocality that is realizable by state ψ.
The optimal setup vectors and the corresponding measure of nonlocality and optimal local box L (denoted \(\overline{L}\) when optimal) are given in the following.
Lemma 4.1.
The CHSH quantum no signaling box Q in (12) showing the maximum strength of nonlocality is
where \({V}_{\psi }:\,\,=\sqrt{1+{W}_{\psi }}\) is the Bell violation factor; \({N}_{{Q}_{\psi }}={V}_{\psi }1\) is the maximal strength of nonlocality for ψ; and optimal box \(\bar{L}\) is the local box L in Lemma 3.1, evaluated at the optimal setup vectors
where \(\hat{r\,}:\,\,=\,\cos \,(\theta \alpha )\,\hat{x\,}+\,\sin \,(\theta \alpha )\hat{y\,}\) is a unit vector and θ ∈ [0, 2π] is any angle.
Proof
: Maximization of the average causal nonlocality N_{ Q } in (10) is equivalent to maximization of the Bell correlation [Q]. The Bell correlation [Q] as a function of the setup vectors is obtained by replacing the pair correlations (13) in (5). We require the maximum value of [Q] for setup vectors subjected to lie on the unit sphere. The method of Lagrange multipliers yields the optimal unit \({\overrightarrow{a}}_{opt}(0)=(0,\,0,\,{a}_{z})\) with \({a}_{z}=\pm 1\), \({\overrightarrow{a}}_{opt}(1)\,=\,(\cos \,\theta ,\,\sin \,\theta ,\,0)\) with \(\theta \in \mathrm{[0,}\,2\pi ]\) an arbitrary angle, and \({\overrightarrow{b}}_{opt}({x}_{2})=\lambda \,((1{)}^{{x}_{2}}\,\sin \,2\beta \,\overrightarrow{r\,}{a}_{z}\hat{z\,})(1+{\sin }^{2}2\beta {)}^{1/2}\), with λ = ±1. Putting the optimal vectors back into [Q] yields the extreme values \([Q]=2\lambda \sqrt{1+{\sin }^{2}2\beta }\). For the maximum value of [Q] let λ = 1. We are free to choose a_{ z } = 1.□
The stochastic matrix Q_{ ψ } in Lemma 4.1 is the maximally nonlocal CHSH box for the given state ψ. The result and main interest in Lemma 4.1 is not the upper bound but the location of the optimal box in the nosignaling polytope, equation (14). The upper bound is well known. It has been obtained in the classic reference^{19} and by the spectral decomposition of Bell operators^{20,21}. An extra bonus of Lemma 4.1 is the geometry of the optimal CHSH setup for ψ.
The nonlocality \({N}_{{Q}_{\psi }}\) attainable in the optimal CHSH setup for state ψ as a function of the angle of entanglement β is shown in Fig. 1. The corresponding Bell correlation as given by (14) is [Q_{ ψ }] = 2V_{ ψ }, that proves \({V}_{\psi }=\sqrt{1+{W}_{\psi }}\le \sqrt{2}\) is the Bell violation factor. The maximal average strength of causal nonlocality \({N}_{{Q}_{\psi }}=\sqrt{2}1\approx 0.4142\) is attained by a maximally entangled state ψ with concurrence W_{ ψ } = 1 when shared in its optimal CHSH setup (15).
The local box \(\bar{L}\) in the mixture (14) that reproduces the maximally nonlocal Q_{ ψ }, is obtained by substituting the optimal setup vectors (15) in the local part of the expansion (12) of Q. The result is the convex combination
which is a local box lying inside a 3simplex with vertices
Written in factorized form makes evident the local character of the bboxes. The Bell correlation of the bboxes is \([{b}_{{s}_{1}{s}_{2}}]=2\), which is the largest value for a local box.
The mixing coefficients \({p}_{{s}_{1}{s}_{2}}\) in (16) are obtained by substituting the optimal setup vectors (15) in the c_{ β } coefficients of the expansion (12) of Q,
and \({\sum }_{{s}_{1}{s}_{2}}{p}_{{s}_{1}{s}_{2}}=1\). Thus, \([\bar{L}]=2\) and \({N}_{\bar{L}}=0\).
The maximally nonlocal boxes Q_{ ψ } are a mixture of the PR box R and a local box \(\bar{L}\) (the mixture of local boxes in (16)). As a function of the angle of entanglement β of state ψ, the map \(\beta \mapsto {Q}_{\psi }\) generates the locus of maximally nonlocal boxes in the space of stochastic matrices.
The locus of maximally nonlocal boxes lies within the 4simplex with apex the PR box R and base the 3simplex with vertices the local boxes \({b}_{{s}_{1}{s}_{2}}\) in (17)–(20) (let us denote it Δ^{4}). The projections of the locus of Q_{ ψ } boxes onto each of the five facets (3simplices) of Δ^{4} are shown as the blue curves in Fig. 2. The arrows along the locus indicate the direction of increasing β. The end points of the locus of Q_{ ψ } boxes are the product boxes b_{00} (when β = 0, V_{ ψ } = 1; see (21)) and b_{01} (when β = π/2, V_{ ψ } = 1; see (21)), with stochastic matrices in (17) and (18). The box with the strongest causal nonlocality (\(\sqrt{2}1\)) is marked with a red dot, which is reached when β = π/4. The green dot is the barycenter of the facet. All boxes Q in the 3simplex with vertices {b_{ ij }} are local, with the biggest Bell correlation [Q] = 2 for a local box. In this facet of local boxes the barycenter and the projection of the box with the strongest nonlocality coincide.
Maximally Nonlocal Boxes as Stochastic Processes
The nosignaling boxes and the associated average measure of causal nonlocality (10) have, by definition, a direct interpretation as a set of stochastic processes that simulate quantum correlations without using classical forms of communication between the parties.
The random variable of a boxprocess is the binary word w_{1}w_{2} ∈ {0, 1}^{2}, and every row of the associated stochastic matrix–selected with the binary word x_{1}x_{2}–determines a mode of operation of the box, having its own distribution of probability over the possible random outcomes for word w_{1}w_{2}.
For instance, depending on the choice of the product x_{1} ⋅ x_{2}, the PR box R has two modes of operation (see the R_{ ijk } matrices in Table A1 or the parameterized form in (4)). A perfectly anticorrelated mode for x_{1} ⋅ x_{2} = 1 and a perfectly correlated mode for x_{1} ⋅ x_{2} = 0. The nonlocality of the PR box relies on the switching between those two correlational modes, without allowing the parties to communicate.
The maximally nonlocal boxes Q_{ ψ } are a mixture of the PR box R and the four \({b}_{{s}_{1}{s}_{2}}\) boxes in (17)–(20), combined in the mixture (16) as box \(\bar{L}\). As 4 × 4 stochastic matrices the \({b}_{{s}_{1}{s}_{2}}\) boxes are the following,
Every row in a bmatrix has at most two options with 1/2 probability each. Then, a single random bit r shared by the parties (with 1/2 probabilities for 0 and 1) is sufficient to reproduce the statistics of the output words of the \({b}_{{s}_{1}{s}_{2}}\) nosignaling boxes in (22) by some stochastic processes. Let us denote the output word of the \({b}_{{s}_{1}{s}_{2}}\) boxes by v_{1}v_{2}. The processes
asymptotically reproduce the probability entries in every \({b}_{{s}_{1}{s}_{2}}\) matrix. This is proved by comparison of the probabilities \({P}_{{s}_{1}{s}_{2}}({v}_{1}{v}_{2}{x}_{1}{x}_{2})\) that follow from (23) with the entries of the bmatrices in (22). When x_{1} = 0 or s_{1} = 0, v_{1} and v_{2} in (23) are independent variables and \({P}_{{s}_{1}{s}_{2}}({v}_{1}{v}_{2}{x}_{1}{x}_{2})={P}_{{s}_{1}{s}_{2}}({v}_{1}{x}_{1}){P}_{{s}_{1}{s}_{2}}({v}_{2}{x}_{2})\), with
where 〈r〉 = 1/2. When x_{1} = s_{1} = 1, then
The comparison has resulted in coincidence in every case.
The optimal local box \(\bar{L}\) in (16) is the mixture of the four \({b}_{{s}_{1}{s}_{2}}\) boxes, with probabilities \({p}_{{s}_{1}{s}_{2}}\) given in (21): the random bits s_{1} and s_{2} are not independent. Then, to reproduce the local box \(\overline{L}\) by the mixture of processes in (23), with probabilities \({p}_{{s}_{1}{s}_{2}}\), the binary word s_{1}s_{2} is in the stochastic process a piece of random information that is shared by the parties too.
The process reproducing the local box \(\overline{L}\) (16) is represented graphically in Fig. 3. The “inputoutput flow” in the diagram goes from top to bottom and the parties are far apart horizontally. The classical information that is shared by the parties consists of the three random bits r, s_{1} and s_{2}. The two bits in the word s_{1}s_{2} are not independently random.
The stochastic process that simulates the maximally nonlocal quantum box Q_{ ψ } derives from the mixture (14), of the local box \(\overline{L}\) and the PR box R, with mixing probability the strength of nonlocality \({N}_{{Q}_{\psi }}\). Then, sharing a single random bit n_{ ψ } is sufficient to produce the mixture of the process reproducing box \(\overline{L}\) in Fig. (3) with the PR box R. The combined process is shown in Fig. 4.
The process in Fig. 4 reproduces the statistics of the outcomes for the binary word w_{1}w_{2} of the maximally nonlocal quantum box Q_{ ψ } when the probability that the random bit n_{ ψ } gets the value 1, equals the average strength of causal nonlocality \({N}_{{Q}_{\psi }}={V}_{\psi }1\le \sqrt{2}1\approx 0.414\).
We have tested the process in Fig. 4 by comparing the pair correlations 〈x_{1}x_{2}〉 resulting from a computer implementation of Fig. 4 with the corresponding quantum mechanical formulas, which are obtained by substituting the optimal setup vectors (15) in the pair correlations (13). The result consists of the following quantum mechanical formulas,
The pair correlations 〈x_{1}x_{2}〉 estimated numerically by the computer version of the process in Fig. 4 were compared with the formulas in (24). The comparison in Fig. 5 shows a very good agreement between the numerical results produced by the computer version of the stochastic process and the results of quantum mechanics.
Concluding remarks
The harmonious blend of causality and nonlocality in quantum correlations is an intriguing phenomenon. Entanglement simulation was intended to quantify the nonlocality of EPR pairs in terms of the amount of communication necessary to simulate the correlations produced in a CHSH setup. However, the approach disregards causality and is not appropriate to physically understand the phenomenon. Instead, the nosignaling formalism has introduced the methods and the elements (e.g., the PR box as a basic element that is causally nonlocal) to gain understanding about violations of CHSH inequalities, without recurring to any form of communication.
In search of how and when nonlocality operates, the model for the “continuous Bohm setup” by Cerf et al. in^{11} behaves nonlocally all the times through the PR box. In the model^{11} shared randomness disguises nonlocality and does not dilute it in a mixture of “local and non local qubits”. That it must be so for the singlet state in a continuous setup was proved by EPRbis in^{14}.
On the opposite sense, the nosignaling formalism proves that every finite causally correlated setup admits a 2species simulator^{14} that sometimes behaves locally and some others nonlocally. For “continuous” setups (those taking into account the whole set of possible measurements) the situation is not that final. The 2species problem in continuous setups has been considered in^{14,15,16,17}.
We have developed in full detail the 2species description of correlations in CHSH setups. We have shown how convex geometry (i) decomposes quantum correlations into local and nonlocal parts, disclosing the nonlocal content, and (ii) provides the local and nonlocal resources (e.g., the PR box) to build the most economical 2species simulators of quantum correlations, i.e., the simulators using the least number of elementary resources to compose the 2species mixture. Figure 4 is the most economic 2species simulator for a pair of qubits in a partially entangled state when shared in its optimal CHSH geometry.
The average measure of the amount of causal nonlocality in the class of 2species models is the fraction of nonlocal species in the ensemble^{14}. The average nonlocality provided by the simulator in Fig. 4 is \({N}_{{Q}_{\psi }}=\sqrt{1+{W}_{\psi }}1={V}_{\psi }1\), where W_{ ψ } is ψ’s concurrence and V_{ ψ } is the Bell violation factor. The quantum limit on the nonlocal content is \({N}_{{Q}_{\psi }}\le \sqrt{2}1\).
The methods of convex geometry we have applied for the CHSH setups have a general value. They equally apply to decompose quantum correlations into local and nonlocal parts and to identify the necessary elementary resources for mixed states in any finite multiparty scenario. A hard task we foresee is the construction of the skeleton of the degenerate nosignaling polytope^{22}. An algorithm to alleviate the degeneracy problem was provided in^{10}.
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Everything necessary to reproduce the results we are reporting is included in the manuscript. Anyway, readers who want more information about data or protocols should contact the corresponding author.
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Acknowledgements
J.M.M.M. was a CONACyT (México) fellow when this work was in progress. The APC were paid with funds from “Cátedra Marcos Moshinsky 2016’’.
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Urías, J., Méndez Martínez, J.M. Maximally nonlocal ClauserHorneShimonyHolt scenarios. Sci Rep 8, 7128 (2018). https://doi.org/10.1038/s41598018249703
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