Necessary and sufficient criterion for extremal quantum correlations in the simplest Bell scenario
aa r X i v : . [ qu a n t - ph ] M a y Necessary and sufficient criterion for extremal quantum correlationsin the simplest Bell scenario
Satoshi Ishizaka
Graduate School of Integrated Arts and Sciences, Hiroshima University,1-7-1 Kagamiyama, Higashi-Hiroshima 739-8521, Japan (Dated: September 12, 2018)In the study of quantum nonlocality, one obstacle is that the analytical criterion for identifyingthe boundaries between quantum and postquantum correlations has not yet been given, even inthe simplest Bell scenario. We propose a plausible, analytical, necessary and sufficient conditionensuring that a nonlocal quantum correlation in the simplest scenario is an extremal boundarypoint. Our extremality condition amounts to certifying an information-theoretical quantity; theprobability of guessing a measurement outcome of a distant party optimized using any quantuminstrument. We show that this quantity can be upper and lower bounded from any correlation in adevice-independent way, and we use numerical calculations to confirm that coincidence of the upperand lower bounds appears to be necessary and sufficient for the extremality.
Since Einstein, Podolsky and Rosen proposed a para-dox [1] in 1935, quantum nonlocality has been a centraltopic in fundamental science. In 1964, Bell showed thatthe nonlocal correlations predicted by quantum mechan-ics are inconsistent with local realism [2]. The nonlo-cal correlations do not contradict the no-signaling prin-ciple, but it was later found that the strength of quantumcorrelations is more restricted than that allowed by theno-signaling principle [3, 4]. Since then, many effortshave been made to determine the fundamental princi-ples limiting quantum nonlocality [5–8]. In these studies,however, one serious obstacle is that the analytical crite-rion for identifying the boundaries between quantum andpostquantum correlations has not yet been given, even inthe simplest Bell scenario.In the simplest Bell scenario, where two remote par-ties, Alice and Bob, each perform two binary measure-ments on a shared quantum state, Tsirelson showed thatthe Bell inequality of the Clauser-Horne-Shimony-Holt(CHSH) type [9] is violated up to 2 √ x ( y ) andobtains an outcome a = ± b = ± | ψ i . The properties of anonlocal correlation are described by a set of conditionalprobabilities p = { p ( ab | xy ) } . The set p specifies a pointin the probability space, whereas a Bell inequality, whichhas the form P abxy V abxy p ( ab | xy ) ≤ c , specifies a hyper-plane in the probability space. The left-hand side of theBell inequality is called the Bell expression. Let A x ( B y )be the observable of Alice’s (Bob’s) projective measure-ments, which satisfy A x = B x = I . Due to the no-signalingcondition such that the marginal p ( a | x ) and p ( b | y ) doesnot depend on y and x , respectively, all Bell expressioncan be recast, without loss of generality, in the form B = X x V Ax h A x i + X y V By h B y i + X xy V xy h A x B y i , (1)where h· · · i is the abbreviation of h ψ | · · · | ψ i . The set oflocal correlations is tightly enclosed by facet Bell inequal-ities, all of which are the CHSH type, together with thepositivity constraints p ( ab | xy ) ≥
0. As a result, C CHSH = max |h A B i + h A B i + h A B i−h A B i| (2)exceeds 2 if and only if the correlation is nonlocal, wherethe maximization is taken over the four positions of theminus sign in the CHSH expression. A local correlationis an extremal point of the quantum set if and only if itis a deterministic correlation. In this paper, therefore,we exclusively consider extremal correlations made froma nonlocal quantum correlation.Let us then recall the bound on nonlocality in termsof the guessing probability [19], i.e., X xy s x u xy ( − xy h A x B y i ≤ (cid:2) X x s x ( D Bx ) (cid:3) / . (3)Any quantum realization must satisfy this inequality forany real s x and u xy such that u u = u u and P xy u xy = 1. The quantity D Bx describes the guessingprobability; Bob’s optimal probability of guessing Alice’soutcome a is (1+ D Bx ) / ∀ s x , u xy ) is (cid:12)(cid:12)(cid:12) ˜ C ˜ C − ˜ C ˜ C (cid:12)(cid:12)(cid:12) ≤ (1 − ˜ C ) / (1 − ˜ C ) / +(1 − ˜ C ) / (1 − ˜ C ) / (4)for both ˜ C xy = h A x B y i /D Bx and ˜ C xy = h A x B y i /D Ay [19].When ˜ C xy = h A x B y i , Eq. (4) reproduces the TLM in-equality, and the saturation is necessary and sufficientfor the extremality of nonlocal quantum correlations inthe case of unbiased marginals ( h A x i = h B y i = 0) [18, 20].Therefore, Eq. (4) is said to be the scaled TLM inequal-ity, as the correlation function h A x B y i is scaled by D Bx and D Ay . As preliminarily mentioned in [19], every ex-tremal correlation including the case of biased marginalsappears to saturate the scaled TLM inequality, whosenumerical evidence is explicitly shown later. However, itwas also shown that the saturation alone is insufficientfor identifying the extremality.To search for a complete set of conditions, let us focuson the fact that, for a given {h A x B y i , h A x i , h B y i} , theupper bounds of D Bx and D Ay can also be determinedirrespective of the details of the realizations. This can bedone by using the method based on the NPA hierarchy[13, 14] as follows: Let us consider the states of | A x i ≡ A x | ψ i and | B y i ≡ B y | ψ i . In addition, | X i ≡ X | ψ i isintroduced to obtain the bound of D Bx , where X is anyHermitian operator on Bob’s side that satisfies h X | X i =1. The Gram matrix Γ of the states {| ψ i , | A x i , | B y i , | X i} has the formΓ = h A i h A i h B i h B i γ γ h A B i h A B i γ h A B i h A B i γ γ γ γ (5) where only the upper triangular part is shown. Since D Bx = max h ψ | X | ψ i =1 h ψ | A x X | ψ i = max h X | X i =1 h A x | X i , (6)the upper bound of D B ( D B ) is obtained by maximizing γ ( γ ) under the constraint that the real symmetricmatrix Γ is positive semidefinite. Here, the maximiza-tions of γ and γ are done separately. However, thismethod, corresponding to the lowest level of the NPAhierarchy, does not work well, as | X i = | A x i is always asolution and the maximum cannot be less than 1. Letus then move on to the next 1 + AB level of the NPAhierarchy. Namely, let us further introduce A x B y | ψ i and A x X | ψ i , and construct the 12 ×
12 Gram matrix(with constraints between the matrix elements). Then,the bounds less than 1 can be obtained (mainly numeri-cally, though). The upper bounds of D Ay are obtained inthe same way. Throughout this paper, these bounds arecalled device-independent upper bounds.Now, consider a realization such that D Bx and D Ay co-incide with the device-independent upper bounds, andfurther saturates Eq. (3) for an appropriate choice of s x and u xy , hence saturating Eq. (4). Such a correlationhas a significant property: D Bx and D Ay are unique irre-spective of the realizations, as they are tightly boundedfrom above and below. Namely, they can be certifiedfrom {h A x B y i , h A x i , h B y i} , as of the certification of, e.g.,randomness [21]. Note that, even in this time, B y itselfgenerally does not coincide with an optimal operator forguessing the outcome of A x , and vice versa (see AppendixC in [19]). This certifiability of D Bx and D Ay , despite thatthey depend on the state and the measurements, mayimplicitly imply that the realization is unique up to localisometry, i.e., the realization can be self-tested [22], asin the case of unbiased marginals where every nonlocalboundary correlation self-tests the maximally entangledstate [20]. Therefore, such a correlation is a good candi-date of an extremal correlation, as a correlation must beextremal if it is self-testable [18]. Moreover, if this insightis true, the certifiability of D Bx and D Ay ensures that thedevice-independent bounds are attained by a two-qubitrealization, as every extremal correlation in the simplestBell scenario has a two-qubit realization [3, 23].In two-qubit realizations, where projective measure-ments of rank 1 are performed on a two-qubit entangledpure state | ψ i = cos χ | i + sin χ | i , since the guessingprobability is given by D Bx = tr | ρ B | x − ρ B − | x | [24], with ρ Ba | x being Bob’s local state conditioned on a , and similarly for D Ay , we have (see Appendix A for details)( D Bx ) = h A x i + sin χ, ( D Ay ) = h B y i + sin χ. (7)It is then found that, for a given {h A x B y i , h A x i , h B y i} ,the entanglement of | ψ i specified by sin χ is determinedas a consistent solution of four quadratic equations to be S ± xy ≡ h J xy ± q J xy − K xy i ,J xy ≡ h A x B y i − h A x i − h B y i + 1 , K xy ≡ h A x B y i − h A x ih B y i . (8)For each x and y , one of the two solutions S ± xy agrees withsin χ . Since D Bx and D Ay are increasing functions ofsin χ as in Eq. (7), we immediately obtain the followinganalytical upper bounds in two-qubit realizations:( D Bx ) ≤ h A x i + S + xy and ( D Ay ) ≤ h B y i + S + xy . (9)These hold for every x and y . Note that the simultane-ous saturation of these eight inequalities requires thatsin χ = S + xy for every x and y , while cases such assin χ = S +00 = S +01 = S − = S +11 frequently occur ingeneral two-qubit realizations. We have compared Eq.(9) with the corresponding device-independent bound ob-tained numerically (by the random tests as used in Fig.2 below). The results indicate that, for two-qubit realiza-tions saturating both Eqs. (4) and (9) , the two boundsagree with each other within numerical accuracy, as ex-pected. Moreover, it is found that any correlation, whose(non two-qubit) realization saturates both Eqs. (4) and(9), and fulfills one more condition Y xy [(1 − S + xy ) h A x B y i − h A x ih B y i ] ≥ , (10)always has a two-qubit realization (see Appendix A).Note that Eq. (10) is merely redundant, when two-qubitrealizations only are considered.Therefore, the necessary and sufficient condition wepropose for the extremality is the simultaneous satura-tion of the two inequalities given by Eq. (4) and the eightinequalities given by Eq. (9), and fulfillment of Eq. (10).To check the validity, it suffices to investigate two-qubitrealizations, because of the existence of a two-qubit re-alization due to [3, 23] and Eq. (10), and the certifiabil-ity of D Bx and D Ay already confirmed numerically. Wehave performed numerical calculations to check the ne-cessity of the proposed extremal condition as follows: Fora randomly constructed Bell expression Eq. (1), wherewithout loss of generality all coefficients are randomlyselected from [ − , −0.2 −0.1 022.8 RHSLHS 〈 A x B y 〉 C C H S H ∏ xy sin (φ Bx −θ By ), 〈 A x B y 〉 / D Bx 〈 A x B y 〉 / D Ay ∏ xy sin (φ Ay −θ Ax ) FIG. 1: The relation between the RHS and LHS of Eq. (4)for 20,000 randomly chosen nonlocal realizations ( C CHSH > h A x B y i are not scaled (black dots) and are scaled by D Bx or D Ay (red dots). The fraction of nonlocal realizationsamong all realizations is roughly 0.6%. In the inset, both( Q xy sin( φ Bx − θ By ) , C CHSH ) and ( Q xy sin( φ Ay − θ Ax ) , C CHSH ) areplotted for the same realizations (black dots). The bluedots show the results, where V is randomly selected from[ − . , . V Ax , V By are from [ − . , . − , functions are not scaled by either D Ax or D By , the LHSis less than or equal to the RHS, as shown by the blackdots, which is an expected behavior of the (non-scaled)TLM inequality. However, as shown by the red dots, theequality holds when the correlation functions are scaledby D Ax or D By , hence suggesting that the saturation ofEq. (4) is indeed necessary for the extremality. Notethat the saturation is equivalent to the fulfillment of (seeAppendix A) Y xy sin( φ Bx − θ By ) ≤ Y xy sin( φ Ay − θ Ax ) ≤ , (11)which is numerically more feasible for verifying the sat-uration of Eq. (4) than checking Eq. (4) itself. Here, φ Bx is the angle of the Bloch vector of ρ Bx , θ By is the angleof the measurement basis of B y , and similarly for Alice.The inset in Fig. 1 shows the distributions of C CHSH , Q xy sin( φ Bx − θ By ), and Q xy sin( φ Ay − θ Ax ) for realizationschosen randomly. The results indicate that Eq. (11) issatisfied for all extremal correlations, which strengthensthe results of the main body of Fig. 1.We have also performed similar numerical calculationsand confirmed that ( D Bx ) −h A x i and ( D Ay ) −h B y i arecloser to S + xy than S − xy for every x and y and for all ex-tremal correlations chosen randomly. Namely, the nu-merical results suggest that the saturation of Eq. (9) isalso necessary for the extremality.Let us then investigate the sufficiency. We present the ∆λ max C C H S H ∆ I p λ max N PA λ=1 FIG. 2: For each of 40,000 randomly constructed realizations, λ max is obtained using the NPA hierarchy, and (∆ , λ max ) isplotted. The inset shows the distribution of C CHSH and ∆,which indicates that C CHSH ≥ numerical evidence that a correlation generated by a two-qubit realization, which saturates both Eqs. (4) and (9),is always located at a quantum boundary. In the calcu-lations, we randomly construct a realization by selecting θ Ax , θ By , and χ uniformly. The realization is discarded ifit does not satisfy Eq. (11). Otherwise, it is kept, and∆ = max xy {h A x i + S + xy − ( D Bx ) , h B y i + S + xy − ( D Ay ) } is calculated. The realization constructed in this way sat-urates both Eqs. (4) and (9) only when ∆ = 0. Letting p be the correlation generated by the realization, we theninvestigate the quantum realizability of q = λ p + (1 − λ ) I ,where I is the completely random correlation given by h A x B y i = h A x i = h B y i = 0. Concretely, we obtain themaximum possible value of λ , denoted by λ max , usingthe 1+ AB level of the NPA hierarchy method for each ofthe realizations constructed randomly (including the caseof ∆ = 0). Since λ max obtained via the NPA method isan upper bound such that q is quantum realizable but p is known to be quantum realizable, λ max = 1 means that p is located at a quantum boundary (see the schematicpicture in Fig. 2). Figure 2 shows the results of the calcu-lations, which indicate that λ max = 1 always holds when∆ = 0. We have also confirmed that all data points with λ max = 1 for ∆ > p ( ab | xy ) = 0]. Note that the device-independent upper bounds of D Bx and D Ay are typicallymonotonically decreasing in λ , while the lower bounds inEq. (3) is monotonically increasing [19]. These mono-tonicities also suggest that p , where the two boundsmeet, must be located at a quantum boundary.Unfortunately, however, the above calculations cannotexclude the possibility that p is located at a non-extremalboundary. In the first place, does there exist any two-qubit realization that can generate such a non-extremal β( D xy ) ( D Bx ) , ( D A y ) QM IPR P L T ( D Bx ) , ( D A y ) FIG. 3: In the correlation space α PR + β P L +(1 − α − β ) I , thequantum boundary is the red straight line as proved in [18].Here, PR is the postquantum correlation produced by thePopescu-Rohrlich box [4], P L is a local deterministic correla-tion with h A x i = h B y i = 1, and T is a correlation attaining theTsirelson bound. Along this boundary, ( D Bxy ) ≡ h A x i + S + xy and ( D Axy ) ≡ h B y i + S + xy are plotted. Since D Bx = D Ay = 1 atboth edges of the boundary, the correct device-independentupper bounds are equal to 1 over the entirety of the boundary. (and nonlocal) boundary correlation? This alone is an in-triguing but difficult problem as discussed in [26, 27]. Inthe case of a correlation whose two-qubit realization sat-urates both Eqs. (4) and (9), however, the certifiability of D ≡ ( D B , D B , D A , D A ) (confirmed numerically) stronglyconstrains the possibility of being a non-extremal bound-ary correlation. For a correlation written by two ex-tremal correlations as λ p +(1 − λ ) p , a realization with p λ D ( p )+(1 − λ ) D ( p ) necessarily exists [19], but itmust coincide with D ( λ p +(1 − λ ) p ) so that D is unique.This coincidence is quite unlikely due to the nonlinearcharacteristics of the bounds Eq. (9), unless the boundsare constant over the entirety of the boundary. For ex-ample, Fig. 3 plots the bounds along the non-extremalboundary illustrated in the figure, where the nonlinearityindeed prevents the coincidence. Figure 3 also indicatesthat the disagreement between S +11 and S +00 = S +01 = S +10 prevents the simultaneous saturation of Eq. (9). Notethat any two-qubit realization does not exist on the mid-dle of this boundary as proved in [26]. In the case of localcorrelations, however, there exist non-extremal bound-aries such that the bounds Eq. (9) are constant. Forexample, the local non-extremal boundary correlations h A x i = h B y i = 0 , h A B y i = 1 , h A B y i = λ, (12)saturate both Eqs. (4) and (9), where D = (1 , , ,
1) [18].However, such a non-extremal boundary (i.e., where thebounds Eq. (9) become constant) is also unlikely, exceptfor D = (1 , , , Conjecture 1.
In nonlocal quantum correlations, acorrelation is extremal if and only if it fulfills Eq. (10)and the realization saturates Eqs. (4) and (9).
Note that, in the case of unbiased marginals, the sat-uration of Eq. (9) implies D Bx = D Ay = 1, and Eq. (4)is reduced to the TLM inequality, as it should be. Ourconjectured criterion also correctly identifies the analyt-ical examples of extremal correlations in [28, 29], andeven the non-exposed extremal correlation of the Hardypoint [18, 30]. Moreover, in the case of local correlations,the inset of Fig. 2 suggests that the criterion ensures theboundaryness ( C CHSH = 2), but not necessarily extremal-ity due to the correlation of, e.g., Eq. (12). Note furtherthat Fig. 2 also suggests the following.
Conjecture 2.
The AB level of the NPA hierarchy(i.e., almost quantumness [31]) is sufficiently strong totightly bound every extremal correlation. It is immediately noticed that we can eliminate D Bx and D Ay by combining Eqs. (4) and (9). The resultant set ofinequalities must suffice for identifying the extremalityby virtue of the certifiability of D Bx and D Ay . However,separating the extremal condition into Eqs. (4) and (9)will be advantageous in searching for fundamental prin-ciples that limit nonlocal correlations, as the principlesleading to Eqs. (4) and (9) will be independent of eachother. For example, the information causality (IC) prin-ciple [32] successfully explains the Tsirelson bound andeven some curved quantum boundaries [33], and it wasexpected that the IC principle could explain every quan-tum boundary. As noted in [19] (see also Appendix B),however, the IC principle cannot explain extremal bound-aries generated from a partially entangled state, as it can-not explain the saturation of Eq. (4) unless D Bx = D Ay = 1,where Eq. (9) plays no role, i.e., the IC principle is unre-lated to Eq. (9). On the other hand, the cryptographicprinciple possibly explains the saturation of Eq. (4) when D Bx , D Ay < χ = 1), Bob can perfectly guess Alice’s outcomefor both x = 0 , D Bx = 1, and the uncertainty be-tween A and A vanishes [1]. The guessing probabilitydecreases as the entanglement decreases, and for an un-entangled state (sin χ = 0), the guessing probability issolely determined by the uncertainty ∆ A x = p −h A x i as D Bx = |h A x i| (see [29, 36, 37] for a slightly different linkbetween nonlocality and uncertainty). Since correlationswith biased marginals are generated from a partially en-tangled state, it is natural that the amount of entangle-ment is involved in the extremality condition. Hence, afundamental principle that leads to Eq. (9) must be theone that more or less explains the entanglement boundin an information-theoretical way.The plausible analytical condition that limits thestrength of extremal quantum correlations in the simplestBell scenario was determined. We hope that this analyt-ical condition will result in a new fundamental principlebehind quantum mechanics to be found.The author would like to thank the anonymous ref-erees for pointing out Eq. (12) and for suggestions on improving the presentation of this paper. This work wassupported by JSPS KAKENHI Grant No. 17K05579. Appendix A: Two-qubit realization
Here, the details of two-qubit realizations, where pro-jective measurements of rank 1 are performed on a two-qubit entangled state | ψ i , are described. By applyingappropriate local unitary transformations, Alice’s andBob’s observables are written as A x = cos θ Ax σ + sin θ Ax σ , B y = cos θ By σ + sin θ By σ , (A1)where ( σ , σ , σ ) are the Pauli matrices. Since any Bellexpression is then maximized when ρ = | ψ ih ψ | is realsymmetric, | ψ i can be expressed by further rotating thelocal bases as | ψ i = cos χ | i + sin χ | i (0 < χ ≤ π/ . (A2)Under this parameterization, we have h A x B y i = sin θ Ax sin θ By + cos θ Ax cos θ By sin 2 χ, (A3) h A x i = sin θ Ax cos 2 χ, (A4) h B y i = sin θ By cos 2 χ. (A5)Moreover, define the angles φ Bx and φ Ay astr A A x | ψ ih ψ | = h A x i I + D Bx φ Bx σ + sin φ Bx σ ) , tr B B y | ψ ih ψ | = h B y i I + D Ay φ Ay σ + sin φ Ay σ ) . (A6)It is found that D Bx = q sin θ Ax + cos θ Ax sin χ = tr | ρ B | x − ρ B − | x | ,D Ay = q sin θ By + cos θ By sin χ = tr | ρ A | y − ρ A − | y | , (A7)where ρ Ba | x = tr A I + aA x | ψ ih ψ | and ρ Ab | y = tr B I + bB y | ψ ih ψ | .Let us then determine the entanglement specified bysin χ for a given {h A x B y i , h A x i , h B y i} . Eliminating θ Ax and θ By from Eqs. (A3)–(A5), we have h A x B y i = h A x ih B y i cos χ ± sin 2 χ s − h A x i cos χ s − h B y i cos χ , and thus we have the quadratic equation for cos χ , i.e.,cos χ + ( J xy −
2) cos χ + K xy − J xy + 1 = 0 . (A8)Since this must hold for every x and y , there are fourquadratic equations in total. Two solutions of eachquadratic equation are given by sin χ = S ± xy .Let us see that, when a (non two-qubit) realization ofa correlation simultaneously saturates the scaled TLMinequalities, ( D Bx ) ≤ h A x i + S + xy , ( D Ay ) ≤ h B y i + S + xy ,and further fulfills Q xy [(1 − S + xy ) h A x B y i−h A x ih B y i ] ≥ S + xy = 1, since D Bx , D Ay ≤
1, it is found that D Bx = D Ay = 1 inthe original realization, and the existence of a two-qubitrealization is obvious from the TLM criterion. When0 < S + xy <
1, a two-qubit realization can be constructed asfollows: first determine χ from sin χ = S + xy , and nextdetermine sin θ Ax and sin θ By from Eq. (A4) and Eq. (A5),respectively. This two-qubit realization can reproduce h A x B y i of the original realization by adjusting the signsof cos θ Ax and cos θ By , as S + xy is a solution of Eq. (A8), and Q xy [ h A x B y i− h A x ih B y i cos χ ] ≥
0. Moreover, since D Bx and D Ay of the two-qubit realization are the same as those of theoriginal realization, the two-qubit realization saturatesthe scaled TLM inequality, if the original realization does.Note thatsin θ Ax = D Bx sin φ Bx , cos θ Ax = D Bx cos φ Bx / sin 2 χ, sin θ By = D Ay sin φ Ay , cos θ By = D Ay cos φ Ay / sin 2 χ, (A9)hence, h A x B y i D Bx = cos( φ Bx − θ By ) , h A x B y i D Ay = cos( φ Ay − θ Ax ) . (A10)When ˜ C xy = cos( φ x − θ y ) ≡ cos δ xy , by noticing that˜ C ˜ C − ˜ C ˜ C = cos δ cos δ − cos δ cos δ = cos( δ − δ ) − cos( δ − δ ) − sin δ sin δ + sin δ sin δ = − sin δ sin δ + sin δ sin δ , it is not difficult to see that the necessary and sufficientcondition for the saturation of the scaled TLM inequalityis given by sin δ sin δ sin δ sin δ ≤ Appendix B: Insufficiency of IC principle
We briefly noted in [19] that the information causality(IC) principle is insufficient for the full identification ofthe quantum boundaries for bipartite settings, no matterwhat protocol is considered. However, the paper wascriticized because the explanation was considered unclearor the point was completely misunderstood. Here, weexplain the point in more detail.Let us recall the derivation of the IC principle. In thegeneral setting of communication, where Alice is given abit string ~x = ( x , x , · · · ) and sends ~m to Bob as a mes-sage, the information about ~x obtainable by Bob is char-acterized by the mutual information I ( ~x : ~mρ B ), where ρ B is the state of Bob’s half of the no-signaling resources.Using the no-signaling condition and the information-theoretical relations respected by quantum mechanics, itwas shown in [32] that I ( ~x : ~mρ B ) ≤ H ( ~m ) − H ( ~m | ~xρ B ) ≤ H ( ~m ) . (B1) u u a a u a v b v b v b Alice Bob m → x → classical message ••• ρ B premeasurement local state FIG. 4: A protocol may connect the quantum boxes in acomplicated way; regardless, let us denote the outputs of theboxes as ~a = ( a , a , · · · , a n ). To achieve the maximum limitset by the IC principle, the protocol must satisfy H ( ~m | ~xρ B ) =0, i.e., Bob must be able to completely determine ~m from ~x and his local state ρ B . The message ~m is constructed from ~a and ~x , but ~a is ambiguous for Bob. Can Alice constructan unambiguous message ~m by using the ambiguous output ~a from the quantum boxes? Since the entropy H ( ~m ) cannot exceed the number ofbits in ~m , the IC principle is derived. Here, we considerthe case where the number of message bits is finite suchthat H ( ~m ) is finite.The point is that the term H ( ~m | ~xρ B ) in Eq. (B1) isinevitably nonzero in some cases; hence, the saturationof Eq. (B1) is impossible. Namely, the IC principle hasomitted the nonnegligible term in its derivation.Suppose that Alice and Bob share n identical “quan-tum boxes”, each of which accepts inputs ( u, v ) and pro-duces outputs ( a, b ) according to the conditional proba-bilities p ( ab | uv ), where the simplest Bell scenario is con-sidered (see Fig. 4). A protocol may connect the inputsand outputs of the n boxes in a complicated way, but letus denote Alice’s outcomes as ~a = ( a , a , · · · , a n ), where a i is the outcome of the i -th quantum box. Now, con-sider a correlation located at an extremal boundary andshowing D Bu <
1. This means that Bob’s local states (ofa single box) for different values of a become nonorthog-onal; thus, he cannot completely determine a . Since D Bu is generally upper bounded in a device-independent way,his ambiguity about a is inevitable, irrespective of thedetails of the realization of the quantum box. In thisway, each a i has ambiguity for Bob. This ambiguity isso strong that he cannot determine ~a even if he knows ~x (and even if he knows all of Alice’s inputs ~u to the boxes),i.e., H ( ~a | ~xρ B ) >
0. Since ~m is constructed from ~x and ~a , it is clear from Eq. (B1) that any protocol whose ~m contains the information of ~a and H ( ~m | ~xρ B ) > I ( ~x : ~mρ B ) = H ( ~m ).Note that no redundant coding technique can reduceBob’s ambiguity about ~a , as the ambiguity originatesfrom ρ B , which is not under Alice’s control. For exam-ple, ~a~a~a has exactly the same ambiguity as ~a for Bob.If Alice postselects the boxes with the same output a tomultiply Bob’s local state such as ρ Ba | u ⊗ ρ Ba | u ⊗ ρ Ba | u ⊗· · · , shecan reduce Bob’s ambiguity, but such postselection is notallowed. Although Alice can control the value of a viathe input u to some degree, a is nevertheless determinedin a probabilistic way by p ( a | u ), and it is impossible tocompletely eliminate Bob’s ambiguity about a .More concretely, let us consider the quantum box thatis realized by a pure partially entangled state and show-ing D B , D B <
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