PParameter Dependence and Bell nonlocality
Moji Ghadimi
Centre for Quantum Dynamics, Griffith University, Nathan, QLD, Australia
Bell’s theorem asserts that no modelthat satisfies all of the plausible physi-cal assumptions of outcome independence(OI), measurement independence (MI)and Parameter Independence (PI) can re-produce quantum mechanics. Here I findthe optimum model that saturates CHSHinequality for the case that outcome inde-pendence and measurement independencehold but parameter dependence is allowed.The symbolic optimizations to find the sat-urating models are performed using Ana-lytical Optimizer v1.0.
Bell inequalities demonstrate that any modelof some quantum mechanical experiments per-formed in space-like separated regions cannot sat-isfy all of the physically plausible properties ofoutcome independence (OI), measurement inde-pendence (MI) and Parameter Independence (PI)at the same time [1, 2]. The violation of theseinequalities by certain quantum correlations, im-plies that at least one such property must be re-laxed by any model of these correlations. Find-ing the minimum amount of relaxation neededto achieve some violation of these inequalities ishelpful for construction of fundamental theoriesof nature and also for quantifying resources inquantum computing and communication.Minimum relaxation requirement of some ofthese assumptions to simulate quantum mechan-ics has been investigated in the literature [3–8].For example, it is known that to simulate a sin-glet state 1 bit of randomness generation or out-come dependence, 1 bit of signaling or commu-nication (parameter dependence), or 1/15 of onebit of correlation between the underlying variableand the measurement settings (measurement de-pendence) is needed [8].In this paper I investigate the upper bound
Moji Ghadimi: [email protected] for violation of the Clauser-Horne-Shimony-Holt(CHSH) inequality for any given relaxation of pa-rameter independence. I write the right handside of the CHSH inequality as a function of jointoutcome probabilities and then optimise it un-der constraint of the measure of parameter de-pendence corresponding to the maximum possi-ble change in a probability in one region due toa measurement made in a distant region. To findthe optimum value of this function subject tothe constraints mentioned, I use Analytical Opti-mizer v1.0 (see Appendix C).I find a relationship between the sturatingbound of the CHSC inequality and a measureof parameter dependence for the case of one-wayand two-way dependence. The result answers thequestion posed in [8] and proves that the one waycommunication model of Pawlowski et al. [8, 9] isoptimal.
The CHSH scenario considers an experiment withtwo settings and two outcomes for measurementsperformed by distant experimenters, Alice andBob say, in space-like separated regions. For somefixed preparation procedure, a set of statisticalcorrelations, p ( a, b | x, y ) , can be assigned to thisexperiment where the pair ( a, b ) labels the pos-sible outcomes and the pair ( x, y ) labels possibleexperiment settings. Any underlying model ofthese correlations introduces an underlying vari-able λ on which the correlations depend, which istypically interpreted as representing informationabout the preparation procedure. From Bayestheorem one has the identity p ( a, b | x, y ) = Z dλp ( a, b | x, y, λ ) p ( λ | x, y ) , (1) with integration replaced by summation overany discrete ranges of λ . A given underlyingmodel specifies the type of information encoded a r X i v : . [ qu a n t - ph ] F e b y λ , and the underlying probability densities p ( a, b | x, y, λ ) and p ( λ | x, y ) . Definitions and quan-titative measures of relaxation for each of param-eter independence, outcome independence andmeasurement independence are introduced in [8].Based on those definitions MI can be simply de-fined as p ( λ | x, y ) = p ( λ ) that is probabilitiesof preparations are independent of measurementsettings. OI maybe defined as the property thatmeasurement outcomes are uncorrelated giventhe knowledge of the underlying variable. Out-come dependence then may be defined as maxi-mum variational distance between an underlyingjoint distribution and the product of its marginals[8] OD := sup x,y,λ X a,b | p ( a, b | x, y, λ ) − p ( a | x, y, λ ) p ( b | x, y, λ ) | (2) where ≤ OD ≤ and OD = 0 if and only if out-come independence is satisfied. Here I do not re-lax outcome independence assumption thereforeI only use OD = 0 as a constraint.The property that we intend to relax is parame-ter independence. Parameter independence holdswhen the underlying marginal distribution asso-ciated with one outcome is independent of theother parameter. A measure of parameter de-pendence can be maximum possible change in anunderlying marginal probability for one party, asthe consequence of changing the measurement pa-rameter of the other party. Following [8] and [10]we can define parameter dependence ( PD ) fromAlice to Bob as PD A → B := sup x,x ,y,b,λ | p ( b | x, y, λ ) − p ( b | x , y, λ ) | , (3) similarly for the other direction we have PD B → A := sup x,y,y ,b,λ | p ( a | x, y, λ ) − p ( a | x, y , λ ) | . (4) The two way parameter dependence allowed canbe defined as
PD := max { PD A → B , PD B → A } . (5) Here ≤ PD ≤ and PD = 0 if and only if pa-rameter independence is satisfied. This is a mea-sure of maximum possible change in an underly-ing marginal probability of one experimenter, as the result of changing the measurement setting ofthe other observer. A CHSH type inequality foran underlying model having value of parameterdependence PD can be written as h XY i + h XY i + h X Y i − h X Y i ≤ B (PD)(6) where h XY i denotes the average product of themeasurement outcomes for setting X for Aliceand Y for Bob. It is well-known that B (PD = 0) (assuming also measurement independence andoutcome independence) is 2 at maximum and Bcan reach √ in certain quantum mechanical ex-periments. Here I find the maximum possible violation ofCHSH inequality assuming outcome indepen-dence (
OD = 0 in Equ. 2) and measurementindependence ( MI ) but allowing parameter de-pendence (0 ≤ P D ≤ . MI corresponds to p ( λ | x, y ) = p ( λ ) in Equ.1. I consider both one-way and two-way parameter dependence.For the case of two-valued measurements, I de-note the possible outcomes by a, b = ± and pos-sible measurements by x or x and y or y . If wedefine c = p ( a = +1 , b = +1) , m = p ( a = +1) and n = p ( b = +1) , for joint measurement set-ting ( x, y ) , the corresponding joint measurementdistribution can be written in the form p ( a = +1 , b = +1 | x, y, λ ) = c ( λ ) p ( a = +1 , b = − | x, y, λ ) = m ( λ ) − c ( λ ) p ( a = − , b = +1 | x, y, λ ) = n ( λ ) − c ( λ ) p ( a = − , b = − | x, y, λ ) =1 + c ( λ ) − m ( λ ) − n ( λ ) . (7) Similarly we use subscript 2 to define c ( λ ) , m ( λ ) and n ( λ ) for joint measurement setting ( x, y ) , 3 for ( x , y ) and 4 for ( x , y ) .If we define left hand side of equation 6 to be E λ = h XY i λ + h XY i λ + h X Y i λ − h X Y i λ (8) sing Equ. 7, for any given λ , we have h XY i λ = 1 + 4 c − m − n . (9) And similarly we can use labels 2, 3 and 4 for h XY i λ , h X Y i λ and h X Y i λ respectively. UsingEqu. 9, Equ. 8 becomes E λ = (1 + 4 c − m − n )+(1 + 4 c − m − n )+(1 + 4 c − m − n ) − (1 + 4 c − m − n ) . (10) To find the upper bound, we need to maximize E λ with three constraints below:1. Since all the probabilities in Equ. 7 needsto be between 0 and 1 and the sum of thoseneeds to be 1 we have ≤ m i ≤ ≤ n i ≤ { , m i + n i − } ≤ c i ≤ min { m i , n i } i = 1 , , , . (11)
2. Outcome independence constraint: | c i − m i n i | = 0 i = 1 , , , . (12) With this c i can be eliminated and the thirdconstraint above is automatically satisfied.3. Parameter dependence constraint: This caneither be the one-way parameter dependenceconstraint (allowing parameter dependence → ) | m − m | , | m − m | ≤ PD | n − n | , | n − n | = 0 (13) or the two-way parameter dependence con-straint | m − m | , | m − m | ≤ PD | n − n | , | n − n | ≤ PD (14)
The symbolic optimization of this equation wasdone using Analytical Optimizer v1.0 (see Ap-pendix C). The results are summarised in the fol-lowing sections.
Outcomes + , + + , − − , + − , − Measurements x, y x, y − PD 0 PD x , y x , y Table 1: A saturating model for one-way parameter de-pendence. x and x are measurement choices for oneobserver and + and − signs on the left side of each pairshows that observer’s outcome. The scond observer haschoices y and y and signs on the right of the pair ofsigns, show this observers outcome. C H S C I n e qu a lit y B ound ( B ) CHSC Quantum Bound T w o - w a y P D CHSH Bound for one-way PDCHSH Bound for two-way PD O n e - w a y P D Figure 1: The bounds for CHSC inequality for differentvalues of one-way and two-way parameter dependence(PD). The horizontal line shows the quantum (Tsirelson)bound for the CHSH inequality. Vertical lines show thevalue of PD required to simulate quantum mechanicsfor one-way and two-way parameter dependence. For the case of one-way parameter dependencethe maximum value that can be found for B (PD) is (see Appendix A) B ow (PD) = 2PD + 2 f or ≤ PD ≤ . (15) This means that to simulate quantum mechanicsone-way parameter dependence of at least
PD = √ − ≈ . is required (see Fig. 1).One possible model to saturate this bound isshown in table 1 (see Appendix A). Note thatthese values are for one particular λ and if we onlyhave this one λ , the model allows signalling at theobserver level. It is trivial to prevent signalling bytaking equal mixture of this model and one withthe outcomes flipped [11]. This does not changethe value of PD . This argument applies to all themodels in this paper. The result proves that the utcomes + , + + , − Measurement x, y x, y PD(1-PD) PD − x , y x , y − , + − , − Measurements x, y x, y PD PD − PD x , y x , y Table 2: A saturating model for two-way parameter de-pendence, ≤ PD ≤ . x , x , y and y are measurementchoices and + and − are outcomes. The left side ofeach pair corresponds to one observer and the right sideto the other. one way communication model of Pawlowski etal. [8, 9] p ( a, b | x, y, λ ) = p ( a, b | x, y , λ ) = p ( a, b | x , y, λ )= δ aλ δ bλ p ( a, b | x , y , λ ) = [ p (1 − δ aλ ) + (1 − p ) δ aλ ] δ bλ , (16) is optimal. Here λ = ± and p ( λ ) = , and itis straightforward to calculate that for arbitrary p ∈ [0 , , PD = p and the violation of CHSHinequality is 2p (the same as the optimum foundhere). For the case of two-way parameter dependencethe optimum value for B is (see Appendix B) B tw (PD) = 4PD(1 − PD) + 2 f or ≤ PD ≤ f or < PD ≤ . (18) This shows that if two-way parameter depen-dence is allowed there will be an advantage for thecase ≤ PD ≤ comparing to one-way parame-ter dependence but for ≤ PD ≤ no advantagecan be gained (See Fig. 1). To simulate quantummechanics PD = (1 − q − √ / ≈ . is Outcomes + , + + , − − , + − , − Measurements x, y
PD 1-PD 0 0 x, y x , y x , y Table 3: A saturating model (outcome probabilities fordifferent measurement settings) for two-way parameterdependence, ≤ PD ≤ . required that is lower than what is needed forone-way parameter dependence (see Fig. 1).Two possible saturating models for two-way pa-rameter dependence for the cases of ≤ PD ≤ and ≤ PD ≤ are shown in tables 2 and 3respectively. Here I found the upper bound for a CHSH in-equality when outcome independence and mea-surement independence are satisfied but param-eter independence is not. I found the saturat-ing bound for both one-way and two-way pa-rameter dependence and compared the boundwith previously published models of communi-cation. For future work it is of interest to find B (OD , PD = 0 , MD = 0) and B (OD , PD , M =0) . One problem with B (OD , PD = 0 , MD =0) is that outcome dependence (Eq. 2) is aquadratic constraint and the current version ofAnalytical Optimizer is not designed to handlequadratic constraints. An additional problemwith B (OD , PD , MD = 0) is that the current ver-sion of the code does not support more than oneconstant in the constraints but here we have both OD and PD as free parameters. Acknowledgement
I thank Michael J. W. Hall, Mirko Lobino andTim Gould for several helpful discussions.
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Table 4: One possible set of optimal input values forone-way parameter dependence. [12] Moji Ghadimi. Analytical opti-mizer. https://github.com/moji131/Analytical-Optimizer , 2020.
A One-way Parameter DependenceBound
For the case of one-way parameter dependenceusing Equ. 12 and equlities in Equ. 13, we caneliminate c i , n and n . Therefore Equ. 10 be-comes E λ = (1 + 4 m n − m − n )+(1 + 4 m n − m − n )+(1 + 4 m n − m − n ) − (1 + 4 m n − m − n ) . (19) Expanding rest of the constraints we have m − m ≤ PD , m − m ≤ PD ,m − m ≤ PD , m − m ≤ PD0 ≤ m ≤ , ≤ m ≤ , ≤ m ≤ ≤ m ≤ , ≤ n ≤ , ≤ n ≤ . (20) To find the maximum value of E λ in Equ. 19subject to the constraints above I used the codeAnalytical Optimizer (see Appendix C). Thiscode finds the optimum to be and for ≤ PD ≤ .Values for one possible set of input parameterto achieve the presented maximum function valueis listed in table 4 which leads to the model shownin table 1. B Two-way Parameter DependenceBound
For the case of two-way parameter dependenceusing Equ. 12 we can eliminate c i and Equ. 10 m n1 1 1 12 PD(1-PD) (1-PD) PD3 1 1 14 0 1 0 Table 5: A set of input values that maximizes the boundfunction for the two-way parameter dependence for thecase of ≤ PD ≤ . c m n1 PD 1 PD2 1 1 13 0 0 04 0 0 1 Table 6: One example of optimal input values for pa-rameter dependence when ≤ PD ≤ . becomes E λ = (1 + 4 m n − m − n )+(1 + 4 m n − m − n )+(1 + 4 m n − m − n ) − (1 + 4 m n − m − n ) . (21) Expanding rest of the constraints we have m − m ≤ PD , m − m ≤ PD , m − m ≤ PD ,m − m ≤ PD , n − n ≤ PD , n − n ≤ PD ,n − n ≤ PD , n − n ≤ PD , ≤ m ≤ , ≤ m ≤ , ≤ m ≤ , ≤ m ≤ , ≤ n ≤ , ≤ n ≤ , ≤ n ≤ , ≤ n ≤ Analytical Optimizer (Appendix C) finds themaximum value of E λ in Equ. 21 to be − PD) + 2 for ≤ PD ≤ and for ≤ PD ≤ .For ≤ PD ≤ one possible saturating modelcan be achieved with values in table 5. Thecorresponding model is shown in table 2. For ≤ PD ≤ a saturating model can be madewith values in table 6 and the resulting model isshown in table 3. C Analytical Optimizer Code