Octob er 2020

Lecture Notes for 8.225 / STS.042, “Ph ysics in the 20th Cen tury”:

Bell’s Inequalit y and Quan tum En tanglemen t

Da vid Kaiser

Center for The or etic al Physics, MIT

In tro duction

These notes discuss the famous inequalit y deriv ed b y ph ysicist John S. Bell in 1964, regarding correlations of the outcomes of measuremen ts conduc ted on t w o or more par ticles. The first section in tro duces Bell’s inequalit y and the concep t of “lo cal realism.” The next section illustrates ho w systems that are prepared in a quan tum-en tangled state are predicted to violate Bell’s inequalit y , rev ealing stronger correlations than lo cal realism w oul d allo w. The app endix pro vides a deriv ation of Bell’s inequalit y .

Reading these notes is optional ; the notes are mean t to fill in some of the gaps in v arious deriv ations that w e will no t co v er during our class session.

Bell’s Inequalit y and “Lo cal Realism”

In 1964, theoretical ph ysicist John S. Bell published a short article in the first v o lume of an out-of-the-w a y ph ysics journal, Physics Physique Fizika . The journal folded just a few y ears later, b ut Bell’s arti cle w en t on to b ecome one of the most influen tial ph ysics pap ers of the t w en tieth cen tury . 1 In his article, Bell scrutinized the famous critque of quan tum mec han ics b y Alb ert Einstein , Boris P o dolsky , and Nathan Rosen from 1935, whic h is usually referred

1 John S. Bell, “On the Einstein-P o dolsky-Rosen parado x,” Physics Physiqu e Fizika 1 (1964): 195-200. Man y of Bell’s related articles and es sa ys on th e topic are a v ailable in B ell, Sp e akable and Unsp e akable in Quantum Me chanics (New Y ork: Cam bridge Univ ersit y Press, 1990). On the history of Bell’s inequalit y , see esp. Oliv al F reire, Jr., “Philosoph y en ters the optics lab or atory: Be ll’ s theorem and its first exp erimen tal tests (1965–1982),” Studies in H i sto r y and Philosophy of M o dern Physics 37 (2006): 577-616; Louisa Gilder, The A ge of Entanglement: When Quantum Physics was R eb orn (New Y ork: Knopf, 2008); and Da vid Kaiser, How the Hippies Save d Physics: Scienc e, Counter cultur e, and the Quantum R evival (New Y ork: W. W. Norton, 2011). Rece n t, ac cessible discussions include the graphic no v el b y T an y a Bub and Jeffrey Bub, T otal ly R andom: Why Nob o dy Understands Qu an tu m Me chanics (Princeton: Princeton Univ ersit y Press, 2018), and G eorge Greenstein, Quantum Str angeness: Wr estling with Bel l’s The or em and the Ultimate Natur e of R e ality (Cam bridge: MIT Press, 2019).

Figure 1: Sc hematic illustration of the original EPR though t exp e r im en t. A source σ emits a pair of particles th at tra v el in opp osite directions. A t e ac h d e tector, a ph ysicist selects a measuremen t to b e p erformed b y adjusting the detector settings ( a, b ); eac h detector then yields a measuremen t outcome ( A, B ).

to b y the authors’ initials: EPR. 2

In their article, the EPR authors describ ed a though t exp erimen t lik e the one sho wn in Fig. 1. A source σ emits pairs of particles tha t tra v el in opp osite directions. The particle tra v eling to the left encoun ters a device at w hic h a ph ysicist could select some prop ert y of the particle to measure, b y adjusting the detector setting a . Once the t yp e of measur emen t to p erform has b een selected and the particle has arriv ed at the detector, the detector completes a measurem en t of the selected prop ert y and yields a measuremen t outcome A . The parti cle tra v eling to the righ t encoun ters an iden tical devic e, at whic h a second ph ysicist could select v ario us t yp es of measure men ts to p erform b y adjustin g the detector setting b . Her device then yields the measuremen t outcome B .

In the con text of thi s though t exp erimen t, the EPR authors articulated t w o criteria that they argued an y reasonabl e ph ysical theory should meet:

• “Realit y crit erion”:“If, without in an y w a y disturbing a system, w e can predict with certain t y (i.e., with probabilit y equ al to unit y) the v alue of a ph ysical quan tit y , then there exists an elemen t of realit y corresp onding to this ph ysical quan tit y .”

• “Lo calit y”: “Since at the time of measuremen t the t w o systems [that is, the pair of particles emitted from source σ ] no longe r in tera ct, no real c hange can tak e pl ace in the secon d system in consequence of an ything that ma y b e done to the first system.” 3

The com bination of these cri teria is often referred to as “ lo c al r e alism ”: that ph ysical systems, including elemen tary particles, p ossess definite v alues for v arious prop erties on their o wn, prior to and indep enden t of our c hoice of what to measure, and that no influence can tra v el across space arbitrar ily quic kly .

2 Alb ert Einstein, Boris P o dolsky , and Nathan Rosen, “Can quan tum mec hanical description of realit y b e considered complete?,” Physic al R eview 47 (1935): 777-780.

3 Einstein, P o dolsky , and Rose n , “Can quan tum mec hanical description,” on pp. 777, 779.

The EPR authors argued that since quan tum mec hanics is incompatible with lo cal re - alism, quan tum mec hanics could not b e a complete or final theory of nature. In his 1964 analysis, Bell agreed that qua n tum mec hanics is inco mpatible with lo cal realism; but he also emphasized that lo cal realism itself w as an assumption , not (as y et) a demonstrated fact. No matter ho w reasonable lo cal re alism ma y app ear — judged either b y common exp eri - ence or b y its easy fit with other w ell-established ph ysical theories, including Ein s t ein’s o wn relativit y — the question remained whether natur e b eha v ed in accordance w ith lo cal realism. Bell prop osed a new t yp e of exp erimen t that coul d test whether the b eha vior of ph ysical systems at the atomic sca le w as consisten t with lo cal realism. His first step w as to suggest that the t w o detectors in Fig. 1 p erform measuremen ts whose outcomes could only ev er tak e one of t w o v al ues (so-called “dic hotomic observ ables”), suc h as the spin of a n electron or the p olarization of a photon. F or example, if one w ere m easuring an electron’s s p in along z ˆ , the only p ossible measuremen t outcomes w ould b e spin up or spin do w n along z ˆ . If one measured the electron’s spin in units of ~ / 2, then these t w o outcomes w ould corr esp ond to +1 (spin up) and − 1 (spin do wn). Sim ilarly for photon p olarization: o nce one orien ted the p olarizing filter along some direction, a photon could only ev er b e measured with p olarization parallel (+1) or p erp endicular ( − 1) to the c hosen orien tation. F o r measuremen ts of dic hotomic observ ables, the measuremen t outcom es can alw a ys b e represen ted as simply A = ± 1 and

B = ± 1.

Next, Bell con tin ued, the ph ysicists at eac h detec tor could adju s t the detector settings at their device to select among v a rious measuremen ts to p erform. T he ph ysicist at the left detector could c ho ose to measure an electr on’s s p in along z ˆ or along some direction inclined b y an angle θ from z ˆ , and sim ilarly for the ph ysicist at the righ t detec tor. In other w ords, the ph ysicist on the left could c ho ose to p erform a measuremen t either with detector setting a or a 0 ; and the ph ysicist on the righ t could c ho ose among detector settings b or b 0 .

Because the outcomes of measuremen ts at eac h detector could only ev er b e A = ± 1 and B = ± 1 — no matter what the selection of detector settings ( a, b ) had b een on a giv en exp erimen tal run — Bell suggested that ph ysicists consider the c orr elation function :

E ( a, b ) = h A ( a ) B ( b ) i , (1)

where the angular brac k ets d enote a v eraging o v er man y exp erimen tal runs in whic h the particles w ere sub jected to measuremen ts with the pair of detecto r sett ings ( a, b ). Giv en A ( a ) = ± 1 and B ( b ) = ± 1, the pro duct A ( a ) B ( b ) can only ev er b e ± 1 on an y giv en exp erimen tal run. Up on a v e raging o v er man y runs in whic h the detector settings w ere ( a, b ), the correlation function w ould therefore satisfy − 1 ≤ E ( a, b ) ≤ 1. On e ma y then consider a

c ombination of these correlatio n functions, as one v aries the detector settings at eac h side:

S ≡ E ( a, b ) + E ( a 0 , b ) − E ( a, b 0 ) + E ( a 0 , b 0 ) . (2)

This form of the parameter S is often called the Bell-CHSH parameter: it w as deriv ed b y ph ysicists John Clauser, Mic hael Horne, Abner Shimon y , and Ric hard Holt so on after Bell’s original deriv ation. 4

The correlation functions E ( a, b ) (and hence the Bell-CHSH paramet er S ) can b e mea - sured exp erimen tally b y conducting measuremen ts on man y pairs of particles, reco rding the measuremen t outco mes A and B for a giv en pair of detector settings ( a, b ), and rep eating the exercise for the other com binations of detector settings, suc h as ( a, b 0 ), ( a 0 , b ), and so on. One can also mak e pr e dictions for the v alues of E ( a, b ) (and hence S ) using v arious ph ysi - cal theories, and co mpare tho s e prediction s with exp erimen tal observ ations. In general, the theoretical prediction for the v alu e of the co rrelation function E ( a, b ) can b e written as

E ( a, b ) =

A , X B = ± 1

A B p ( A, B | a, b ) , (3)

where p ( A, B | a, b ) is an expression for the probabilit y that the measuremen t outcomes on a giv en exp erimen tal run will b e ( A, B ) when the detecto r settings are selected to b e ( a, b ). F or example, one ma y use a particular theoretical mo del to calculate p (+1 , +1 | a, b 0 ), whic h is the probabilit y to find A = +1 and B = +1 when the detector settings are set to a and b 0 . The question then comes do wn to ho w v arious ph ysical theorie s mak e predictions f or the correlation functions E ( a, b ). Bell and the CHSH authors demonstrated that any ph ysical theory that satisfies lo cal realism w ould mak e predictions for E ( a, b ), and hence for S , in

whic h S w as sub jec t to an ine quality :

| S | ≤ 2 . (4)

I review Bell’s deriv ation of this inequalit y in the App endix. The main t ak e-a w a y m essage is that quan tum mec hanics is not compatible with lo cal realism, and it predicts t hat mea - suremen ts on certain systems could b e mor e str ongly c orr elate d than lo cal-realist the ories w ould allo w. That is, quan tum mec hanics predicts that p h ysicists should measur e violations of Bell’s inequalit y , finding | S | > 2 under certain conditions, whereas an y lo c al-realist theory should ob ey Eq. (4).

4 John F. Clauser, M ic hael A. Horne, Abner Shimon y , an d R i c hard A. Holt, “Prop osed exp erimen t to test lo cal hidden v ariable theories,” Physic al R eview L etters 23 (1969): 880-884. Their deriv ation follo w ed closely along Bell’s original argumen t in Bell, “On the Einstein-P o dolsky-Rosen parado x.”

Quan tum En tanglemen t and Bell’s Inequalit y

Quan tum-Mec hanical Description of P olarized Photons

F or definiteness, let us consider the case in whic h the source σ emits pairs of photons, a nd the ph ysicists at eac h detect or c ho ose to measure t heir photon’s linear p olarization. Once a ph ysic ist fixes the orien tati on of the p olarizing filter at her detector station (along some direction in space describ ed b y a unit v ector n ˆ ), she can register either horizon tally p olarized photons (in state | H i ) or v ertically p olarized photons (in state | V i ). A horizon tally p olarized photon registers as measuremen t outcome +1 (analogous to spin u p for an electron), and a v ertically p olarized photon registers as − 1 (aki n to spin do wn). These states ha v e no o v er lap; they are orthogonal to eac h other:

h V | H i = 0 , h H | V i = 0 , (5)

where h ψ | is the “Hermitian conjugate” of the state | ψ i . 5 The states are also normalized suc h that

h H | H i = 1 , h V | V i = 1 . (6)

If a ph ysicist w ere to r otate the p olarizing filter at her detector so that it p oin ted along a new direction n ˆ 0 that made an angle θ ≡ n ˆ · n ˆ 0 with resp ect to the original orien tation, then a differ ent set of states | H ˜ i and | V ˜ i w ould register as either horizon tally or v ertical ly p olarized at her detector. These states share some o v erlap with the original states | H i and

| V i . In fact, the new stat es are simply rotations b y the angle θ from the origina l states:

| H ˜ i = R ( θ ) | H i

| V ˜ i | V i

(7)

= cos θ sin θ

| H i ,

— sin θ cos θ | V i

5 If w e r e p re sen t the state ψ as a column v ector in its abstract v ector space, then ψ is a ro w v ector in that space, giv en b y the conjugate transp ose. F or e xampl e :

| ψ i = x − → h ψ | = ( x ∗ , y ∗ ) ,

where the comp onen ts x and y are n um b ers (whic h could b e complex). The inner pro d uc t b et w een t w o states | ψ 1 i and | ψ 2 i can then b e ev aluated:

inner pro duct of | ψ 1 i = m with | ψ 2 i = s is giv en b y h ψ 2 | ψ 1 i = ( s ∗ , t ∗ ) m = ms ∗ + nt ∗ .

If | ψ 1 i = | H i and | ψ 2 i = | V i , represen ting photon s tat e s with horizon tal and v ertical p olarization (in a particular basis), resp ectiv ely , then m = 1, n = 0, s = 0, and t = 1, and w e see that h V | H i = h H | V i = 0. More lo osely , w e can think of the inner pro duct h ψ 2 | ψ 1 i as the analogy of a dot pro duct among ordinary v ectors, k eeping in mind that the comp onen ts of the quan tum states | ψ i i can, in general, b e complex n um b ers.

or

| H ˜ i = | H i cos θ + | V i sin θ ,

| V ˜ i = − | H i sin θ + | V i cos θ .

(8)

In Eq. (7), R ( θ ) is the usual rotatio n matrix in t w o spatial dimensions. 6 When measured along the orien tation n ˆ 0 , these states yield measuremen t outcomes +1 (for stat e | H ˜ i ) and − 1 (for state | V ˜ i ). Using Eqs. (5) and (6), it is easy to confirm that the pair of p olarization states in the rotated basis are orthogona l to eac h other and also prop erly normalized: h V ˜ | H ˜ i = h H ˜ | V ˜ i = 0, and h H ˜ | H ˜ i = h V ˜ | V ˜ i = 1.

Maximally En tangled States

W e can arrange for the source σ to emit pairs of photons in a maximally en tangled state, suc h as

1

| Ψ i = √ 2 | H i A | V i B − | V i A | H i B

. (9)

The subscript “ A ” refers to the state of the particle tra v eling to w ard the left detector, and “ B ” refers to the particle tra v eling to w ard the righ t dete ctor. Eq. (9 ) re presen ts an “en tangled” state b ecause it cannot b e factorize d : there is no w a y to describ e the b eha vior of particle A without also referring to some asp ect of the b eha v ior of particle B , and vice v ersa. The state | Ψ i is “maximally” en tangled b ecause the states within the sup erp osition are w eigh ted with co effi cien ts of equal magnitude (in this case, 1 / 2).

When a pair of photons is prepared in the sta te | Ψ i and eac h photon is measu red b y a p olarizer orien ted in the d irection n ˆ , the measuremen t outcomes should alw a ys b e opp osite to eac h other: A = +1 at the left detector together with B = − 1 at the righ t detector, or A = − 1 with B = +1. The s e t w o outcomes for the pair of measu remen ts ( A, B ) should o ccur with equal probabilit y , since the co efficien ts in fron t of eac h term on the righ t side of Eq. (9) are of equal magn itude.

W e can unpac k those statemen ts a bit more explicitly , whic h should mak e it easier to understand ho w things c hange once t he ph ysicists rotate their p olarizers t o directions other than n ˆ . When b oth p olarizers rem ain aligned along n ˆ (that is, a and b are eac h r otated b y 0 ◦ from the d irection n ˆ ), w e ma y calculate p ( A, B | a, b ) → p ( A, B | 0 ◦ , 0 ◦ ) for v arious outcomes A = ± 1 and B = ± 1. In general in q uan tum theory , the probabilit y for a giv en outcome is giv en b y the absolute square of the corresp onding amplitude , A . The amplitude, in turn, is calculated from the overlap (or inner pro duct) b et w een the incoming qu an tum state | Ψ i and

6 W e are assuming that the p olarizing filters are orien ted at some direction within the t w o-dimensional plane that is p erp endicular to the photons’ d irec t ion of tra v e l ; hence w e ma y use this form for R ( θ ).

the state corresp onding t o the relev an t measuremen t outcome. F o r example, w e ma y consider measuremen ts along n ˆ that yield o utcomes A = +1 and B = +1. The state corresp onding to this measuremen t outcome is | H i A | H i B . Using the expression f or the incoming state | Ψ i in Eq. (9), w e ma y calculate the amplitude as

A (+1 , +1 | 0 ◦ , 0 ◦ ) = B h H | A h H | | Ψ i

1

= √ 2 B h H | A h H | | H i A | V i B − | V i A | H i B

1

(10)

= √ 2 B h H | V i B A h H | H i A − B h H | H i B A h H | V i A

1

= √ 2 0 × 1 − 1 × 0

= 0 .

In mo ving from the second line to t he third, w e ha v e ma de use of the fact that the states p er - taining to particles A and B b elong to d istinct v ector spaces, so w e alw a ys ha v e B h ψ 2 | ψ 1 i A = 0 and A h ψ 1 | ψ 2 i B = 0 for any states | ψ 1 i A and | ψ 2 i B . In m o ving from the third to the fourth line, w e ha v e made use of th e ortho gonality of the states | H i and | V i , as in Eq. (5): for eac h particle, A h H | V i A = A h V | H i A = B h H | V i B = B h V | H i B = 0. W e ha v e also u s e d the fact that these states are normalize d , as in Eq. (6). Th us w e see that for a pair of particles prepared in the state | Ψ i of Eq. (9), the amplitude to measure A = +1 and B = +1 along the orien tation n ˆ vanishes . Therefore w e find the probabilit y

p (+1 , +1 | 0 ◦ , 0 ◦ ) =

2

A (+1 , +1 | 0 , 0 ) = 0 . (11)

According to quan tum mec hanics, there shoul d b e zer o probabilit y of measuring A = +1 and B = +1 along n ˆ for pairs of particles prepared in state | Ψ i . A similar calculation yields p ( − 1 , − 1 | 0 ◦ , 0 ◦ ) = 0 for particles prepared in state | Ψ i .

On the other hand, w e find a nonzero probabilit y to m easure A = +1 and B = − 1:

A (+1 , − 1 | 0 ◦ , 0 ◦ ) = B h V | A h H | | Ψ i

1

= √ 2 B h V | A h H | | H i A | V i B − | V i A | H i B

1 h

| V i

h H | H i

−

h V | H i

h H | V i

(12)

= √ 2 B V B A

A B B A A

1

= √ 2

1

= √ 2 .

1 × 1 − 0 × 0

So then w e find

p (+1 , − 1 | 0 ◦ , 0 ◦ ) = A (+1 , − 1 | 0 ◦ , 0 ◦ ) 2 = 1 . (13)

The s a me series of steps yields p ( − 1 , +1 | 0 ◦ , 0 ◦ ) = 1 / 2. Therefore w e see that when b oth p olarizers are orien ted along the original direction n ˆ , quan tum mec hanics predicts that ph ysicists should find the results A = +1 , B = − 1 half the time (probabilit y = 1 / 2), and A = − 1 , B = +1 half the time. Note that th is calculation do es not enable us to predict whic h sp ecific result will arise on a given exp erimen tal run. When using quan tu m mec hanics, w e are only able to calculate the pr ob ability to find a sp ecific outcome on an y giv en run.

Measuremen t Correlations with Rotated P olarizers

The real genius of John Bell’s w ork com es from allo wing eac h ph ysicist to rotate the p olarizing filter at her d etector station b y some angle a w a y from the original direction n ˆ , and then considering ho w the correlations in measur emen t outcomes v ary . Su pp ose that the p h ysicist at th e left, whose detector station p erforms measuremen ts on p article A , c ho oses to rotate her p olarizing filter b y som e angle θ a a w a y f rom the original direction n ˆ , while the ph ysicist at the righ t, who measures particle B , rotates her p olariz ing filter b y some other angle θ b a w a y from n ˆ . The angle b et w een the t w o p olarizing filters is then giv en b y θ ab = θ a − θ b .

Using Eq. (8), the states that co rresp ond to measuremen t ou tcomes A = ± 1 once the ph ysicist at the left detector has rotated her p olarizing filter b y angle θ a ma y b e written

| H ˇ i A = | H i A cos θ a + | V i A sin θ a ,

| V ˇ i

= − | H i sin θ

+ | V i cos θ ,

(14)

while the states that corresp ond to measuremen t outcomes B = ± 1 when the p olarizing filter at the righ t detector has b een rotated b y angle θ b ma y b e written

| H ˜ i B = | H i B cos θ b + | V i B sin θ b ,

| V ˜ i B

= − | H i B sin θ b

+ | V i B cos θ b .

(15)

Then w e ma y c alculate v arious ampli tudes, suc h as

A (+1 , +1 | θ a , θ b ) = B h H ˜ | A h H ˇ | | Ψ i

= B h H | cos θ b + B h V | sin θ b A h H | cos θ a + A h V | sin θ a | Ψ i

(16)

= B h H | A h H | cos θ b cos θ a + B h H | A h V | cos θ b sin θ a

+ B h V | A h H | sin θ b cos θ a + B h V | A h V | sin θ b sin θ a | Ψ i .

Giv en the form of | Ψ i in Eq. (9) an d the orhogonalit y of | H i and | V i states, the on ly non v a - nishing con tributions in Eq. (16) come from terms prop ortional to B h V | A h H | and B h H | A h V | ,

so w e then find

1

A (+1 , +1 | θ a , θ b ) = √ 2

— B h H | H i B A h V | V i A cos θ b sin θ a

+ B h V | V i B A h H | H i A sin θ b cos θ a

1

= √ 2 −

1

cos θ b sin θ a + sin θ b cos θ a

(17)

= √ 2 sin( θ b − θ a )

1

= − √ 2 sin θ ab ,

where the p en ultimate step mak es use of the trigonometric angle-addition form ula, and the final step fo llo ws up on using θ ab = θ a − θ b and sin( − θ ab ) = − sin θ ab . F ollo wing the same series of steps, w e ma y calculate the other amplitudes:

˜ ˇ 1

A ( − 1 , − 1 | θ a , θ b ) = B h V | A h V | | Ψ i = − √ 2 sin θ ab ,

˜ ˇ 1

A (+1 , − 1 | θ a , θ b ) = B h V | A h H | | Ψ i = √ 2 cos θ ab , (18)

˜ ˇ 1

A ( − 1 , +1 | θ a , θ b ) = B h H | A h V | | Ψ i = − √ 2 cos θ ab .

T aking the absolute squares, w e then find the pro babilities for these v arious outcomes:

p (+1 , +1 | θ

, θ ) = p ( − 1 , − 1 | θ , θ ) = 1 sin 2 θ ,

a b a b 2 ab

1

(19)

p (+1 , − 1 | θ a , θ b ) = p ( − 1 , +1 | θ a , θ b ) = cos 2 θ ab .

W e no w ha v e all the pieces w e need to calcul ate the quan tum-mec hanica l prediction for the correlation function E ( a, b ) as in Eq. (3), applied to a pair of particles prepared in the state

| Ψ i of Eq. (9). F or an angle θ ab b et w een the o rien tations of the p olarizers at the left an d righ t stations, the correlatio n function tak es the f orm

E ( a, b ) =

A , X B = ± 1

A B p ( A, B | a, b )

= (+1)(+1) p (+1 , +1 | θ a , θ b ) + (+1)( − 1) p (+1 , − 1 | θ a , θ b )

+ ( − 1)(+1) p ( − 1 , +1 | θ a , θ b ) + ( − 1)( − 1) p ( − 1 , − 1 | θ a , θ b ) (20)

= sin 2 θ ab − cos 2 θ ab

= − cos(2 θ ab ) ,

Figure 2 : In a test of the Bell-CHSH inequalit y , ph ysic ists can selec t v arious angles b et w een their detector s ettin gs ( a, a 0 ) and ( b, b 0 ). A common c hoice i s to se t θ ab = θ a 0 b = θ a 0 b 0 = θ , and then θ ab 0 = 3 θ .

where the last step follo ws up on using the usual double-angle form ula. As a quic k realit y c hec k, note that for the case in whic h the t w o p olarizers are eac h a ligned along n ˆ , so that θ ab = 0, the expression in Eq. (20) indicates that the t w o photons should alw a ys b e measured to ha v e opp osite p olarizations: either A = +1 and B = − 1, or A = − 1 and B = +1, exactly as w e w ould exp ect from the form of | Ψ i in Eq. (9).

Predictions for the Bell-CHSH P arameter

Our quan tum-mec hanical expression for E ( a, b ) in Eq. (20) holds for any v alue of θ ab . So w e ma y us it to calculate the quan tum-mec hanical prediction for the Bell-CHSH parameter S defined in Eq. (2):

S ≡ E ( a, b ) + E ( a 0 , b ) − E ( a, b 0 ) + E ( a 0 , b 0 )

= − cos(2 θ ab ) − cos(2 θ a 0 b ) + cos(2 θ ab 0 ) − cos(2 θ a 0 b 0 ) .

(21)

W e can con s id er a particular arrange men t among the detector settings ( a, a 0 ) and ( b, b 0 ), as in Fig. 2. In that case, w e set θ ab = θ a 0 b = θ a 0 b 0 = θ , and hence θ ab 0 = 3 θ . Fig . 3 sho ws ho w the v alue of − S v aries with the angle θ . 7 Note that for sev eral c hoices of the angle θ , the quan tum-mec hanical predict ion for the Bell-CHSH parameter S exc e e ds the Bell-CHSH inequalit y of Eq. (4): rather than | S | remaining restricted to | S | ≤ 2, there are scenar ios predicted in whic h measuremen ts on quan tum systems should b e mor e str ongly c orr elate d than the Bell-CHSH inequalit y w ould allo w, with | S | > 2. In particular, the predicted correlations are maximized for θ = 22 . 5 ◦ . F or that c hoice of relativ e orien tations among the

detector settings, w e ha v e cos(2 θ ) = cos(45 ◦ ) = 1 / √ 2 and cos(6 θ ) = cos(135 ◦ ) = − 1 / √ 2 ,

7 In Fig. 3 I plot S rather than S b ecause the Bell-CHSH in e q ualit y applies to S , and I find it easier to visualize the main p eaks as in Fig. 3. If w e had considered corr e lati o n s among measureme n ts on pai r s o f particles prepared in some other maximally en tangled state, suc h as Φ = H A H B + V A V B / 2, then our expression for S w ould ha v e had the opp osite o v erall sign from the expression in Eq. (21), though, ob viously , the same b eha vior of | S | .