Radboud Unverstet Njmegen Do Hdden Varables Improve Quantum Mechancs? Bachelor Thess Author: Denns Hendrkx Begeleder: Prof. dr. Klaas Landsman Abstract Snce the dawn of quantum mechancs physcst have contemplated f a hdden varable theory would be able to mprove quantum theory. The man goal of ths paper s to look at the artcle The completeness of quantum mechancs for predctng measurement outcomes (0) by Colbeck and Renner. We try to examne the methods used, and the proofs gven n ths paper. Through ths, we try to make an evaluaton of the strength of the result obtaned n ths artcle. Our concluson s that one has to make addtonal assumptons about the hdden varable theory, n order to complete the proof as gven n the artcle. 3 January 04
Contents Introducton 3 Probabltes and quantum mechancs 6 3 Varatonal dstance and correlaton measure 8 4 Compatablty of theores 5 Free parameters 6 Bpartte setup 3 7 Man theorem 6 7. No-sgnalng....................................... 6 7. Clam for the entangled state............................. 6 7.3 Orderng coeffcents of states............................. 0 7.4 Embezzlement ([5], [6])................................. 7.5 Generalzaton..................................... 5 7.5. Measurement devce.............................. 5 7.5. Embezzlement on measurement devce.................... 6 7.6 Proof of Theorem................................... 30 8 Conclusons and Dscusson 35 9 Bblography 37
Introducton Untl the twenteth century, physcs n prncple provded the exact descrpton of nature. Precse knowledge of all varables that concerned a problem, gave evoluton of that physcal process n space and tme. Thnk, for example, of Newtonan mechancs. Ths prncple of determnsm changed dramatcally when quantum mechancs was dscovered. Introduced as a method to predct the outcomes of measurements on mcroscopc systems such as partcles, quantum mechancs seems to clash wth the noton of determnsm. Quantum mechancs descrbes only the probablty of obtanng a certan measurement outcome. Ths yelds a stark contrast wth classcal physcs. Another mportant dfference between classcal physcs and quantum mechancs s the nfluence the observer seems to have n the latter. In most orthodox nterpretatons of quantum mechancs, performng a measurements causes a collapse of the state of the system. Roughly speakng, the system suddenly changes from a superposton of possble measurement outcomes, to the partcular measurement outcome actually obtaned. The brth of quantum mechancs occurred at the begnnng of the twenteth century. The probablstc nature of quantum mechancs caused a great rft between the physcsts of that tme. A large part of the communty saw the future of physcs n quantum mechancs, wllng to accept the loss of determnsm. One of the more famous defendants of quantum mechancs was Nels Bohr. The other part, smaller, of the communty agreed wth the powerful predctons quantum mechancs provde, but thought quantum mechancs was just an ntermedary theory: t had to be ncomplete. One of the most famous people on ths sde of the argument was Albert Ensten. Bohr and Ensten are well known for ther (ntally oral) dscusson concernng the nature and problems of quantum mechancs. Ths dscusson between Bohr and Ensten eventually led Ensten, Podolsky and Rosen to publsh a famous paper enttled: Can Quantum Mechancal Descrpton of Physcal Realty Be Consdered Complete? ([9]). Ths paper contans one of the most elegant and clear ways to llustrate what seems so skewed about quantum mechancs. Through ths, the paper also gves a motvaton why t should be reasonable and nterestng to look at the dea of a hdden varable. Let s consder an observer measurng the polarzaton of a photon usng a polarzed pece of glass. He can measure the polarzaton of the photon ether along a 0 degree axs, or a 90 degree axs. If, when measurng at 0 degrees, a photon passes through the glass, the measurement result s that ths photon s polarzed along the 0 degree axs. If t does not pass through, the photon s polarzed along the 90 degree axs. Nowadays, the EPR thought experment s usually reformulated n terms of the measurng of the polarzaton of an entangled par of photons. Ths measurement takes place n a confguraton wth two observers, Alce and Bob. Of the photon par, one reaches Alce, whlst to other reaches Bob. Alce and Bob agree to measure the polarzaton of these photons ether along a 0 degree axs, or a 90 degree axs. Quantum mechancs can descrbe a specal photon par (an entangled par) that ether s measured by both observers to have a polarzaton of 0 degrees (wth a probablty of of ths occurrng), or a polarzaton of 90 degrees. The photon par before measurement s a superposton of these two measurement outcomes. The surprsng results occur when Alce and Bob are extremely far away from each other. If Alce measures her photon to have a polarzaton of 0 degrees, she nstantly knows Bob s photon wll also have ths polarzaton. Behold: when Alce has measured her photon as havng a polarzaton of 0 degrees, Bob wll ndeed always measure hs photon to have a polarzaton of 0 degrees as well. It does not matter how far away Alce s. Ths means that Bob s photon 3
somehow nstantaneously knows the measurement result of Alce s photon. Keepng n mnd the the theory of relatvty assumes the speed of lght to be an absolute lmt, we are stuck wth a problem. But there seems to be a soluton: a hdden varable. We can for example postulate that, upon creaton of the photon par, both photons have a polarzaton along the same axs (ether the 0 degree or 90 degree axs). Alce and Bob just aren t aware of whch axs t s (t s hdden from them), and nether s quantum mechancs. If the two possble polarzatons each have a probablty of of occurrng, t gves rse to the same behavor, but there s no more need for nstantaneous nteracton between the photons. We can say the nformaton, of whch polarzaton the photons have, s determned by a varable, that s not accessble by quantum mechancs. A hdden varable. Ths hdden varable theory seemed to be a vable way to solve the frcton between classcal physcs and quantum mechancs. In 964 John Steward Bell publshed a paper named On the Ensten-Podolsky-Rosen Paradox ([8]). In ths paper he very elegantly proved that the assumpton of a hdden varable that reduces to both determnsm and localty leads to a contradcton. Generally speakng, he proved that every theory havng the prevously mentoned propertes of determnsm and localty, must satsfy a so called Bell nequalty. Quantum mechancs does not satsfy ths nequalty. The fact that the nequalty s not satsfed has been verfed expermentally on numerous occasons. Smlar results such as the Conway-Kochen Free Wll Theorem and everythng else concernng hdden varables has been the subject of numerous publshed papers. Qute recently, a new clam was made n ths terrtory by Roger Colbeck and Renato Renner. In two papers No extenson of quantum theory can have mproved predctve power ([], Nature, 0 Aug. 0) and The completeness of quantum theory for predctng measurement outcomes ([3], arxv, 0 Aug. 0) the authors make the clam that a theory usng hdden varables cannot gve better predctons than quantum mechancs, provded the hdden varable theory satsfes a number of propertes (notably a free choce assumpton) that seem weaker than the assmpton n ether Bell s theorem of the Free Wll Theorem etc. Hence, ths result would generalze prevous results lke those of Bell. Both papers have attracted attenton, most dscusson revolvng around the assumpton of free choce. The frst paper No extenson of quantum theory can have mproved predctve power ([]) contans lttle detal on the actual proof. The second paper The completeness of quantum theory for predctng measurement outcomes contans more techncal detals on how to prove the actual clam, but especally the crux of the proof s stll largely left to the reader. My goal for ths Bachelor thess s to gve the proof of the man clam n a complete and clear way. The man focus of ths paper wll be on the thrd paper publshed on ths subject by Colbeck and Renner ([3]). We have opted to keep both the assumptons made, and the structure of the proof, qute smlar to ths paper. In ths thess we wll look at one hdden varable wth a fnte number of possble values. Ths hdden varable descrbes some property of the partcle or system we are measurng. To arrve at the fnal result, we must frst specfy what s meant by probabltes gven by a physcal theory. Ths analyss s found n the frst secton. After that, we must make clear when a hgher theory s consdered compatble wth quantum mechancs, and when t gves better predctons. We can then proceed by ntroducng quanttes wth whch we can prove the result for a very specfc measurement and state. Fnally we wll expand ths to every state and measurement to obtan the fnal result mentoned n the artcle [3] by Colbeck and Renner. The result, may suggest that the topc of hdden varables s now closed once and for all: 4
hdden varables do not mprove quantum mechancal predctons. However, upon closer nspecton one mght crtcze the methods employed by Colbeck and Renner. Notably, the somewhat dubous noton of free choce, and the way n whch measurements are generalzed. The result certanly s a step n the rght drecton, but not as strong and straghtforward as prevous results such as Bell s Theorem. 5
Probabltes and quantum mechancs In ths paper we manly make use of fve random varables, called A N, B N, X, Y and Z (N N). We assume there s a probablty space Ω wth a probablty measure µ such that: X, Y : Ω {0, }, () A N : Ω A N = {0,..., N }, () B N : Ω B N = {, 3,... N }, (3) Z : Ω Z. (4) We do not fx the value set Z of Z. The only restrcton s for t to be fnte. The probablty of general dscrete random varable X havng a value x s gven by P (X = x) = µ(x (x)). (5) A condtonal dstrbuton functon for two random varables X and Z s gven by: P (X = x Z = z) = P (X = x, Z = z), (6) P (Z = z) defned whenever P (Z = z) > 0. We wll ntroduce some short-hand notaton for our dstrbutons. If we want to consder the probablty dstrbuton of X as a functon (of x) we wll wrte P X. For the condtonal probablty we wrte P X Z ( z). The condtonal probablty s defned for z Z such that P Z (z) 0. For the whole collecton of probabltes defned ths way we wrte P X Z. If we consder two random varables X and Y, for the jont probablty we ntroduce the followng notaton: P (X Y ) := P X,Y (x, y). (7) x,y x y In our applcaton the random varable X descrbes a measurement by observer A (called Alce) and takes possble values {0, }. Smlarly, the random varable Y descrbes measurements by observer B (called Bob). The probabltes of X or Y takng one of the possble values 0 and, then smply descrbe the probablty that a measurement of X or Y has the result 0 or, respectvely. The random varables A N and B N descrbe the possble settngs of the measurement by observers A and B; A N takes values a n A N, B N takes values b n B N. An mportant thng to consder s what we mean by P X and P Y (or P X,Y ) wthout consderng the values of A N and B N. The observers (A and B) carry out the experment. For A, measurement of X gves results whch the observer can see. The settng of A N can ether be hdden from the observer, gvng the dstrbuton P X, or t can be accessed by the observer, gvng the dstrbuton P X AN. In case the settng of A N s hdden, n our probablstc settng the results of P X are effectvely averaged over the values a that A N can take. The same process apples to observer B. Now that we have establshed our notaton, we can consder the queston how we obtan such probabltes. The man goal of ths paper s to compare probabltes gven by a hypothetcal theory T havng an extra varable Z, wth the probabltes gven by quantum theory. There s no need to specfy how the probabltes for theory T are obtaned, as actual measurements only concern the probabltes t produces. The probabltes quantum theory produce have so far been confrmed by experments, so t s natural to assume T s compatble wth quantum mechancs (ths wll be explaned n greater detal below), n the sense that averagng over the extra varable Z gves the same predctons as quantum mechancs. From now on, f we consder a probablty 6
P that explctly uses the hdden varable Z, we assume the probablty to be derved usng the theory T. A (pure) quantum-mechancal state ψ s a unt vector n some Hlbert space H. It descrbes the state of a quantum-mechancal system. An observer can preform measurements on ths system. Quantum mechancs gves probabltes for possble outcomes of such measurements for every possble state ψ. Therefore, when we are lookng at the probabltes of a specfc ψ, we wrte P ψ both for the quantum mechancal predcton and the predctons gven by our theory T. What s actually beng measured n a state ψ s some observable O, whch corresponds wth a hermtan operator O on H. In the scope of ths thess, the operator O has a dscrete spectrum of egenvalues {λ }. We have dm(h) < typcally. Quantum mechancs postulates the followng propertes: The outcome of a measurement of some observable O s one of the egenvalues λ of O. Let P λ be a projecton that projects on the egenspace of O spanned by the egenvectors wth egenvalue λ. The probablty of measurng λ n a state φ H s gven by (the Born rule): P φ O (λ) = φ, P λ(φ). (8) The Born rule s a postulate. At present t does not seem possble to derve the rule wthout ntroducng other, often questonable assumptons. 7
3 Varatonal dstance and correlaton measure We now defne two mportant concepts whch wll be used to prove the man theorem. Frst we construct a metrc on the space of probabltes wth that have the same value set. Defnton. Let X : Ω X and Y : Ω X be two random varables. For the correspondng probablty dstrbutons P X and P Y the varatonal dstance between P X and P Y s gven by D(P X, P Y ) = P X (x) P Y (x). (9) x X As the probabltes are functons n L ths metrc s actually the same as (half) the canoncal metrc d (.,.) on L. The fact that the varatonal dstance as defned above s a metrc, as well as other mportant propertes, s summarzed n the followng lemma: Lemma. The varatonal dstance D(.,.) has the followng propertes:. D(.,.) s a metrc on the space of probablty dstrbutons P X on X.. For all probablty dstrbutons P X and P Y : 0 D(P X, P Y ). 3. Suppose we have random varables X, X : Ω X and Y, Y : Ω Y. For the jont probablty dstrbutons P X,Y and P X,Y D(P X, P X ) D(P X,Y, P X,Y ). 4. D(.,.) s convex: let {α } I be a fnte set satsfyng I : α 0 and α =. Let {P X } I and {P Y )} I be sets of probablty dstrbutons. Then we have: I D( I α P X, I α P Y α D(P X, P Y ). I 5. D(P X, P Y ) P (X Y ). Proof. : It s clear that D(P X, P Y ) 0, as t s a sum over postve terms. Suppose D(P X, P Y ) = 0. We know that x : P X (x) P Y (x) 0 = x : P X (x) P Y (x) = 0. Ths means P X = P Y. Suppose P X = P Y, then x : P X (x) P Y (x) = 0 = D(P X, P Y ) = 0. 8
: As x As P X (x) P Y (x) = P Y (x) P X (x) we have D(P X, P Y ) = D(P Y, P X ). Suppose we have probablty dstrbutons P X, P X and P X. We see that: D(P X, P X ) = P X (x) P X (x) + P X (x) P X (x) x P X (x) P X (x) + P X (x) P X (x). Ths means D(P X, P X ) D(P X, P X ) + D(P X, P X ). P X (x) = and x x P Y (x) = we can see that: x 3: D(P X, P Y ) P X (x) + P Y (x) =. x D(P X, P X ) = P X (x) P X (x) x = P X,Y (x, y) P X,Y (x, y) x x y P X,Y (x, y) P X,Y (x, y). y D(P X,Y, P X,Y ). 4: ( D α P X, ) α P Y = α (P X (x) P Y (y)) x α P X (x) P Y (y) α D (P X, P Y ). x 5: See lemma 6 n the Colbeck and Renner artcle ([3]). We now ntroduce the so-called correlaton measure. Defnton. If P X,Y AN,B N s a collecton of condtonal probabltes, n the context of equatons () -(4), we defne, for N N, the correlaton measure I N as: I N (P X,Y AN,B N ) = P (X = Y A N = 0, B N = N ) + P (X Y a, b). (0) a A N,b B N a b = 9
We wll extensvely use ths correlaton measure on a specal state n the space C C, namely the maxmally entangled state defned as follows. Defnton 3. Let H A = C and H B = C be two Hlbert spaces (of dmenson ). The maxmally entangled state ψ 0 H A H B s defned by ψ 0 = (( ) ( ) ( ) ( )) 0 0 +. () 0 0 In a more general sense, f we defne a orthonormal bass on C by choosng e and e, we can wrte ψ 0 = (e e + e e ). 0
4 Compatablty of theores We wll be comparng the probabltes gven by quantum mechancs wth those of a theory T that has access to a hdden varable Z. All experments so far have shown us that measurements agree wth the predctons obtaned by quantum theory. So n order to stll explan the measurement results, our hdden varable theory T should be compatble wth quantum mechancs. By ths we mean that, f we calculate measurement outcomes wth our theory T wthout havng access to the extra parameter Z (effectvely summng over all possble values of Z), we should obtan the same predctons as gven by quantum theory. Defnton 4. Suppose we have random varables X, A N and Z. For measurements on a quantum state ψ, our hdden varable theory s compatble wth quantum mechancs f a, x : P ψ X A N (x a) = z Z P ψ X,Z A N (x, z a). () Here P ψ X,Z A N are the probabltes produced by our theory T, whereas P ψ X A N s the probablty obtaned from quantum mechancs. In order to compare the predctons gven by quantum mechancs wth those gven by the hdden varable theory, we have to specfy what t means for our theory T to be more nformatve. Defnton 5. Suppose we have random varables x, A N and Z. Let the hdden varable theory T be compatble wth quantum mechancs. For measurement on a quantum state ψ quantum mechancs s as least as nformatve as the hdden varable theory T f x, a, z(for whch P Z AN (z a) > 0): P ψ X A N (x a) = P ψ X A N,Z(x a, z). (3) In other words, knowng z adds nothng to the predctons done on a measurement on ψ. From now on we always assume our hdden varable theory T to be compatble wth quantum mechancs. Ths means we do not have to worry about dstngushng probabltes gven by both theores wthout usng the hdden varable Z, as these probabltes are the same.
5 Free parameters A theory (T ) that descrbes our (hypothetcal) measurements uses certan parameters. Common sense would dctate some of these parameters can be chosen freely. We expect a theory to predct outcomes for every ntal condton. For a physcally relevant theory we demand that the parameter A N s not correlated to other parameters, except to those parameters n the causal future of A N. If ths s the case we can consder A N to be a free parameter. Frst we wll have to defne an sem-order on the parameters of our theory T. A sem-order has all the propertes of a normal order, but s not necessarly ant-symmetrc. Defnton 6. Let V be a set. A sem-order on V s a bnary relaton that s both reflexve and transtve. In other words: v V : v v, (4) v, w, x V : If v w and w x v x. (5) If Γ s the set of all parameters of our theory T, we can defne a causal order on ths set. Defnton 7. Let Γ be the set of all parameters of our theory T. A causal order s a sem-order on Γ. From now on we say X les n the causal future of A N f A N X. The use of the the symbol ndcates the central dea of orderng the parameters: we would lke the causal order to be compatble wth relatvty. For example, the result of a measurement (X) should be free n relaton to the parameters that descrbe the settng of the experment (A N ), but the settng could possbly affect the measurement result. So: (X A N ) but A N X. To be compatble wth relatvty we demand that n the causal order we only have R S f S les n the future lght-cone of R. Wth ths n mnd, we defne the causal order as follows: Defnton 8. Let Γ be the set of all parameters wth the causal order. We say A N Γ s a free parameter f: X Γ wth (A N X) : P X,AN = P AN P X. (6)
6 Bpartte setup In ths paper we consder a specfc measurement setup n quantum mechancs. As mentoned before there are two observers A (Alce) and B (Bob). The possble quantum states for each observer ndvdually are descrbed by the Hlbert spaces H A = C for A, and H B = C for B. The space H A H B descrbes the states n the composte system that contans both A and B. As mentoned before, the random varables X and Y descrbe the measurement results for observers A and B, respectvely (wth possble results n {0, }). The random varables A N and B N descrbe the possble settng of the experment. A N has N possble values a n A N = {0,,..., N }, whlst B N has N possble values b n B N = {, 3,..., N }. We refer to ths stuaton as a bpartte setup. For a bpartte setup we want the causal order to have the followng propertes: A N X, B N Y, (7) (A N Z), (B N Z), (8) (A N Y ), (B N X). (9) If ether A or B preforms a measurement, the resultng quantum state n the total system H A H B wll be a projecton of our ntal state onto a certan subspace, namely for a gven vector v H we can defne a projecton onto the subspace spanned by ths vector. Ths projecton P v : H C v s gven by x H : P v (x) = x, v v, () where, s the nner product of H (taken to be lnear n the second varable). We can also project onto a subspace spanned by multple vectors. If Y s a subspace of H spanned by the vectors {ν,..., ν n } we wrte for the projecton of x onto the subspace Y P roj Y = (0) n P ν (x). () Suppose H A and H B are two Hlbert spaces. If P ν s a projecton on H A and P ω s a projecton on H B (so ν H A, ω H B ), we can construct a projecton P ν P ω : H A H B (C ν) (C ω) by takng: = P ν P ω,j α β j = (P ν (α ) P ω (β j )). (3),j These projectons on H A H B are used to fnd the resultng state after some measurement by A and B. In our bpartte setup we preform a measurement usng the followng vectors n C : ( cos(e a x ) e a x = π ( a N + x), Ea x = ); sn(e a x ) ( ) fx b = π ( b N + y), F y b cos(f b y = ). sn(fy b) For a fxed value of a, the measurement of A can have two possble outcomes X = 0 and X = whch are descrbed by projectng onto E0 a and E a, respectvely. The same goes for B. Ths means we have a probablty dstrbuton for the measurement n our bpartte setup on ψ H A H B gven by ) P ψ X,Y A N,B N (x, y a, b) = ψ, (P E P ax F ψ. by 3
An mportant property of ths bpartte setup n combnaton wth the maxmally entangled state and the correlaton measure s gven n the followng lemma. Lemma. Accordng to quantum mechancs, the correlaton measure of the maxmally entangled state ψ 0 n a bpartte setup wth fxed N N s equal to: I N (P ψ0 X,Y A N,B N ) = N sn ( π 4N ) π 8N. (4) Proof. Frst, we calculate P ψ0 X,Y A N,B N (x, y a, b). ψ 0, P E a x P F b y (ψ 0 ) = ( [ 0 ) ( ) + 0 ( ) 0 ( ) 0 ], [cos(e a x)ex a cos(fy)f b y b + sn(e a x)ex a sn(fy)f b y b ] ( ) ( ) = (, cos(e a 0 0 x)ex a cos(fy)f b y b ( ) ( ) + (, sn(e a 0 0 x)ex a sn(fy)f b y b ( ) ( ) 0 0 + (, cos(e a x)ex a cos(fy)f b y b ( ) ( ) 0 0 + (, sn(e a x)ex a sn(fy)f b y b = (cos (e a x) cos (f b y) + cos(e a x) cos(f b y) sn(e a x) sn(f b y) + sn (e a x) sn (f b y)) = (cos(ea x) cos(f b y) + sn(e a x) sn(f b y)) = cos (e a x f b y) = cos ( π (a b N + x y)). (5) 4
Now we wll calculate I N : I N (P ψ0 X,Y A N,B N ) = ψ 0, P E 0 0 P F N (ψ 0 ) + ψ 0, P 0 E 0 P F N (ψ 0 ) + ψ 0, P E a 0 P F b (ψ 0 ) + ψ 0, P E a P F b 0 (ψ 0 ) a b = = cos ( N 4N π) + (cos ( a b 4N π π) + cos ( a b 4N π + π)) a b = = sn ( 4N π) + a b = sn ( a b 4N π) = sn ( 4N π) + N sn ( 4N π) sn ( 4N π) = N sn ( 4N π) N( π 4N ) = π 8N. (6) 5
7 Man theorem Our ultmate goal s to prove that the assumptons of free varables and compatblty wth quantum mechancs lead to the concluson that the hdden varable Z does not gve mproved predctons compared wth quantum mechancs. In order to buld towards ths goal, we wll frst derve a drect consequence of our free varables. It turns out that free-varables force our hgher theory T to be no-sgnalng. The concept wll be explaned n the next subsecton. Subsequently we prove our clam for a very specfc state and measurement. Ths s the second subsecton. Then, n order to generalze, we need a process called embezzlement. Ths process s descrbed n the thrd subsecton. Fnally, the last subsecton wll state the clam, and use all the prevous work to prove the man clam of Colbeck and Renner:.e. that hdden varables do not mprove quantum mechancs. 7. No-sgnalng Lemma 3. The causal order satsfyng the condtons n equatons (7), (8) and (9) mply: P X,Z AN,B N = P X,Z AN and P Y,Z AN,B N = P Y,Z BN. We call a theory wth ths property a no-sgnalng theory. Proof. P X,Z AN,B N (x, z, a, b) = P X,Z,A N,B N (x, z, a, b) P AN,B N (a, b) The same holds for P Y,Z AN,B N. = P B N (b)p X,Z,AN (x, z, a) P AN (a)p BN (b) = P X,Z AN (x, z a). (As B N s free w.r.t by the causal order: eq (7), (8), (9)) 7. Clam for the entangled state Durng ths whole subsecton, we wll be workng n the bpartte setup (usng the measurement descrbed above). Ths means we know the value sets of X, Y, A N, B N and Z(see equaton () to (4)). The state for whch we calculate all the probabltes and correlatons below s ψ 0, the maxmally entangled state. Lemma 4. Suppose the dstrbuton P ψ0 X,Y A N,B N s a no-sgnalng theory. We ntroduce the unform dstrbuton defned by: P X (0) = ; P X () =. We then have the followng nequalty: a, b : D(P ψ0 X A N,B N,Z ( a, b, z), P X Z I N (P ψ0 X,Y A N,B N ), (7) where, Z s the average over z Z, where the z are dstrbuted accordng to P ψ0 Z A N,B N ( a, b). 6
Proof. We wll omt the ψ 0 n the probabltes, and wrte a 0 = 0, b 0 = N. I N (P X,Y AN,B N,Z(,,, z)) = P (X = Y A N = a 0, B N = b 0, Z = z) + P (X Y A N = a, B N = b, Z = z) a A N,b B N a b = Lemma + D( P X AN,B N,Z(x a 0, b 0, z), P Y AN,B N (y a 0, b 0, z)) D(P X AN,B N,Z(x a, b, z), P Y AN,B N,Z(y a, b, z)) a A N,b B N a b = Lemma 3 = D( P X AN,Z(x a 0, z), P Y BN,Z(y b 0, z)) + D(P X AN,Z(x a, z), P Y BN,Z(y b, z)). a A N,b B N a b = As D(, ) s a metrc, we have, usng the trangle nequalty, D( P X AN,Z( a 0, z), P X AN,Z( a 0, z)) D( P X AN,Z( a 0, z), P Y BN,Z( b 0, z)) + D(P X AN,Z( a 0, z), P Y BN,Z( b 0, z)). Now for the second term D(P X AN,Z( a 0, z), P Y BN,Z( b 0, z)), by usng the trangle nequalty multple tmes can get an expresson whch only contans a s and b s whch have a dstance of one. For a 0 = 0, b 0 = N we have: D(P X AN,Z( a 0, z), P Y BN,Z( b 0, z)) D(P X AN,Z( a 0, z), P Y BN,Z(, z)) whch gves us... + D(P X AN,Z(, z), P Y BN,Z( b 0, z)) D(P X AN,Z( a 0, z), P Y BN,Z(, z)) + D(P X AN,Z(, z), P Y BN,Z(, z)) + D(P X AN,Z(, z), P Y BN,Z( b 0, z)) a A N,b B N a b = D(P X AN,Z(x a, z), P Y BN,Z(y b, z)). I N (P X,Y AN,B N,Z(,,, z) D( P X AN,Z( a 0, z), P X AN,Z( a 0, z)) (8) = P X AN,Z(x a 0, z) P X AN,Z(x a 0, z) x = ( P X A N,Z(x a 0, z) ) x = D(P X AN,B N,Z( a 0, b 0, z), P X )). (9) Note that the probablty P X AN,B N,Z( a 0, b 0, z) for ψ 0 only depends on the dstance of a and b modulus N. For a 0 and b 0 above the dstance (modulo N ) s. Ths means we can 7
replace a 0 and b 0 for any a and b wth dstance. Eventually we only use a 0 (equaton 8), so equaton 9 hold for all a and b. We now average both sdes of nequalty (9) over all z Z. Takng the average on the left hand sde (I N (P X,Y A,B,Z (,,, z)) wll complete the proof of the lemma. Note that as A N and B N are free varables, we have for all z, a, b: P Z AN,B N (z a, b) = P Z,A N,B N (z, a, b) P AN,B N (a, b) = P Z,A N,B N (z, a, b) P AN (a)p BN (b) = P Z(z)P AN (a)p BN (b) P AN (a)p BN (b) = P Z (z). (as (A N B N )) (as (A N Z), (A N B N ) and (B N Z), (B N A N )) Usng the fact that for all z one has P Z AN,B N (z a, b) = P Z (z), we can average I N over z: I N (P X,Y AN,B N,Z(,,, z) Z : = z = z P Z AN,B N (z a, b)i N (P X,Y AN,B N,Z(,,, z) P Z (z)i N (P X,Y AN,B N,Z(,,, z) = z + P Z AN,B N (z a 0, b 0 )P (X = Y A N = a 0, B N = b 0, Z = z) a A N,b B N a b = Ths s the expresson gven n the lemma. P Z AN,B N (z a, b)p (X Y A N = a, B N = b, Z = z) z = P (X = Y A N = a 0, B N = b 0 ) + P (X Y A N = a, B N = b) a A N,b B N a b = = I N (P X,Y AN,B N ). Now we average over z the rght hand sde of equaton (9): D(PX AN,B N,Z( a, b, z), P X Z := P Z AN,B N (z a 0, b 0 )D(P X AN,B N,Z( a, b, z), P X ) z = P Z AN,B N (z a 0, b 0 )( P X AN,B N,Z(x a, b, z) P X (x) ) z x = P X AN,B N,Z(x a, b, z)p Z AN,B N (z a 0, b 0 ) x,z P X (x)p Z AN,B N (z a 0, b 0 ) = D(P X,Z AN,B N (, a, b), P X P Z AN,B N ( a 0, b 0 )). (30) Now we wll use Lemma and Lemma 4 (whch we have just proven) on P ψ0 X,Y A N,B N. 8
Lemma 5. x, a, z (such that P ψ0 Z A N (z a) > 0): P ψ0 ψ0 X A N,Z(x a, z) = PX A N (x a). (3) Proof. It s easy to see that P X (x) = P ψ0 X A N,B N (x a, b). In combnaton wth the average (30) we just calculated, and the prevous lemma (4), ths gves the nequalty D(P ψ0 X,Z A N,B N (, a, b), P ψ0 X A N,B N ( a, b)p ψ0 Z A N,B N ( a 0, b 0 )) I N(P X,Y AN,B N ). The dea s to take the lmt N. By dong ths we ncrease the range of possble values taken by A N and B N. The quantum-mechancal probabltes P ψ0 are gven by projectng onto the vector E a x = ( ) cos(e a x ) sn(e a x ) a = ak we project on the same vector. So X A N,B N (where e a x = π ( a N + x)). If we choose N = kn(k N) and P ψ0 X A N,B N (x a, b) = P ψ0 X A kn,b kn (x ak, bk). Increasng N does not change the probablty of P ψ0 X A N,B N ( a, b) as long as we scale a and b accordngly. However, for P X,Z AN,B N (x, z a, b) we cannot drectly conclude that scalng gves the same probabltes. A condton we have to place on our theory T s that for the same physcal measurement, the probabltes gven by T stay the same. Thus f we scale a and b as above, we are measurng the state ψ 0 along the same angles, therefore physcally gvng the same measurement. From ths we can conclude: The last term P ψ0 Z A N,B N Now t follows that: P X,Z AN,B N (x, z a, b) = P X,Z AkN,B kn (x, z ak, bk). can easly be scaled as: P ψ0 Z A N,B N = P ψ0 Z = P ψ0 Z A kn,b kn. D(P ψ0 X,Z A N,B N (, a, b), P ψ0 X A N,B N ( a, b)p ψ0 Z A N,B N ( a 0, b 0 )) = D(P ψ0 X,Z A kn,b kn (, ak, bk), P ψ0 X A kn,b kn ( ak, bk)p ψ0 Z A kn,b kn ( a 0 k, b 0 k)) I N (P X,Y AkN,B kn ) π 6kN. Takng the lmt k now forces the metrc to zero, makng the dstrbutons equal, whch n turn gves: P ψ0 X,Z A N,B N (, a, b) = P ψ0 X A N,B N ( a, b)p ψ0 Z A N,B N ( a 0, b 0 ); = P ψ0 ψ0 X A N,Z(x a, z) = PX A N (x a) (f P ψ0 Z A N (z a) 0). 9
7.3 Orderng coeffcents of states Before we contnue to buld a constructon to explot the results we have so far, we need to ntroduce some notaton and facts that are needed n the next sectons. Defnton 9. Let φ = n = α e be a state n Hlbert space H of dmenson m. We defne the sequence {α r } (where r n) to be the sequence of coeffcents α ordered n descendng order. To be clear, f for example α 5 s the largest coeffcent, we defne = 5, such that α = α 5. We use ths dea of rearrangng coeffcent n descendng order to compare dfferent states. Defnton 0. Suppose we have two states φ = n = α e and ψ = n j= β je j n Hlbert space H. We wrte: φ ψ f k ( α r ) r= k ( β r ) for all k {,..., n} It turns out maxmally entangled states are mnmal n ths sense. r= Lemma 6. Suppose we have a Hlbert space H of dmenson m. Let ψ m = m = m e be the maxmally entangled state of rank m and let φ = m = α e be another state n H. We have φ ψ m. Proof. We wll use nducton on the number of elements we sum over. Frst we clam α m. Suppose α < m. Ths mples α r < m for all r, as α s the largest coeffcent. But then we have n n > α r = α =. r= So ths s a contradcton wth φ havng norm. Suppose now that we know for all l < k : l r= α r l = m. We agan prove by contradcton, so suppose k r= α r < k = m. Ths means ( α k < k ) k m α r m r= r= r= = α r < for all r > k m m k m α r < α r + m m m α r < m = r= = k+ Ths s a contradcton wth φ havng norm, so φ ψ m. Ths dea can also be expanded to states whch conssts of tensor products. Corollary. Suppose we have state φ = m = α e and the maxmally entangled state ψ m = m = m e of rank m n Hlbert space H of dmenson m. Let ψ = n j= β j be another state n Hlbert space H of dmenson n. Then φ ψ ψ m ψ = 0
Proof. The coeffcents of φ ψ are of the form α β j, those of ψ m ψ are m β j. We know from the prevous lemma that φ ψ m. Lookng at the sum of the k largest m β j squared, we know there are k α, such that the sum over these α squared s greater than k m (as φ ψ m). Ths means the sum over the coeffcents whch conssts of the b j of the k largest coeffcents of ψ m ψ pared wth these α s larger than the sum over the k largest m β j squared 7.4 Embezzlement ([5], [6]) To generalze the prevous results to the man theorem we are tryng to prove, we are gong to make use of a constructon called the embezzlng state. These embezzlng states are vectors n the space n C C n. For C we take the bass vectors = ( ( 0) and 0 = 0 ). A tensor product over n such vectors may, for example, have the form 0... 0 0. Any natural number such that n can be wrtten n base : b () = k n n + k n n +... + k + k 0 (k j {0, }). For ths decomposton n base two we wrte = (k n... k 0 ). We now defne a bass of n C as Defnton. For n N we decompose any 0 n n base two. If b () = (k n... k 0 ) we take: n C e + := k n... k 0. (3) The {e } =,... n are parwse orthonormal and therefore form a bass of n C n. Usng ths bass we now defne a specal collecton of vectors µ n. Defnton. Take HÃ = n C C n and HÃ = H B. For all n N we have an embezzlng state µ n defned as: HÃ H B µ n := n e HÃ Cn j j e H B j, (33) where C n = j=n j= j (to normalze ths µ n). We look at H A = m C = C m and H B = H A. Suppose, for an arbtrary m, we are gven a bpartte state n H A H B = C m C m, wrtten n ts Schmdt decomposton as φ = m α θ H A θ H B, where θ H A θ H B H A H B. The α are greater than 0. = The stuaton for ths embezzlement protocol s very smlar to the bpartte setup. We have observers A and B. Ths tme, the states belongng to A are n the Hlbert spaces H A and HÃ. Those of B n spaces H B and H B. The dea behnd these embezzlng states s to transform the ntal state e H A e H B µ n to a vector arbtrarly close to φ µ n, but dong t wthout any communcaton between observers A and B. Ths means we can only use untary transformatons on H A and HÃ for A, and untary transformaton on H B and H B for B. Such transformatons are called local untary transformatons. The ablty to extract an arbtrary φ from the ntal state wll enable us to apply our prevous results for the maxmally entangled states on more general measurements. Defnton 3. Take HÃ = H B = n C C n and H A = H B = m C C m. For µ n HÃ H B the embezzlng state, and φ H A H B an arbtrary state (e. unt vector), the vector φ µ n = j, j= γ,j ((θ H A θ H B )) (e HÃ j e H B j ),
(n H A H B HÃ H B) gves us coeffcents {γ,j },j. Takng the n largest of those and orderng them as a descendng sequence we obtan a sequence γ r,j r such that γ,j γ,j... γ m,j. For ths sequence and the gven unt vector φ we defne the embezzlement of n φ as E(φ) n,m := n r= Cn r= r ((θ H A r θ H B Ths E(φ) n,m s also a vector n H A H B HÃ H B. r ) (e HÃ j r e H B j r )). (34) Ths embezzlement of φ s mportant because we are able to transform µ n to E(φ) n,m usng only local untary transformatons. Lemma 7. Let {e H A... ẽ H B } be an orthonormal bass of m C C m m. Wth e H A e H B we can consder the state (e H A e H B ) (µ n ) = Cn n j= (e H A j e H B ) (e HÃ j e H B j ), n H A H B HÃ H B. There exst local sometres U n,m;φ (for A) on H A HÃ and V n,m;φ on H B H B (for B) such that these transform (e H A e H B ) (µ n ) to E(φ) n,m. Proof. We wll prove ths lemma for U n,m;ψ on H A HÃ. The proof for V n,m;ψ s exactly the same, only ths tme takng place on H B H B (whch s effectvely the same space as H A HÃ). The space H A HÃ s somorphc to the space C n+m. We have two orthonormal bases on ths space. Bass one s e H A are part of ths bass. The second base s θ H A =,..., m. The vectors θ H A r e HÃ j wth j =,... n and =,..., m. The vectors e H A e HÃ e HÃ j (θ H A H A ) wth j =,... n and e HÃ j r are part of ths bass. A bass transform between two orthogonal bass s untary. Therefore, we take U n,m;φ to be the bass transformaton that maps bass vector e H A e H B j to C n (θ H A j e HÃ j j ). We let V n,m;φ be the bass transform that maps e H B e H B j to C n (θ H B j e H B j j ). We want to apply U n,m;φ and V n,m;φ to (e H A e H B ) µ n. Note that U n,m;φ V n,m;φ s a map on H A HÃ H B H B. So to apply ths map we have to permute the order of the Hlbert spaces. Let T be such a permutaton,.e. T : H A H B HÃ H B H A HÃ H B H B. We can then transform (e H A e H B ) µ n to E(φ) n,m wth ad of the untary transformaton T (U n,m;φ V n,m;φ )T. Now that we know how to create such a state E(φ) n,m from (e H A e H B ) µ n, we complete our embezzlement procedure by showng that f we take n to be large enough, E(φ) n,m s arbtrarly close to µ n φ. Lemma 8. Take a fxed H A H B φ = m and a fxed ɛ (0 < ɛ < ). If n m ɛ = α θ H A θ H B (where θ H A θ H B H A H B ) the followng nequalty holds: φ µ n, E(φ) n,m ɛ. (35)
Proof. We wll frst fully expand the state φ µ n, after whch we wll gve estmates for the nner product φ µ n, E(φ) n,m. The proof gven here s smlar to the proof provded by [5]. We know φ µ n = ( m = α θ H A θ H B = α (θ H A Cn j j, ) θ H B Remember that E(φ) n,m := r= n Cn r= e j and θ are orthonormal we can conclude: φ µ n, E(φ) n,m α = (θ H A Cn j = C n = n r=,j,r j, α jr e HÃ j α r. jr C n rcn θ H B, e HÃ j r e H B j ) (e HÃ j n Cn j= ) (e HÃ j j e HÃ j r ((θ H A r θ H B r ) (e HÃ j r e H B j ), e H B j e H B j ) (j =... n, =... m ). n r= Cn r= e H B j r )). Usng the fact that the r ((θ H A, e H B j r θ H A, θ H A r θ H B, θ H B r r θ H B r ) (e HÃ j r e H B j r )) (j, r =... n ; =... m ) So we have a sum over the n largest coeffcents of φ µ n (whch are by defnton the n coeffcents of E(φ) n,m ) tmes the frst n coeffcents of µ n. We wll now verfy that n fact, the j-th coeffcent of E(φ) n,m s smaller than the correspondng coeffcent of µ n. Ths observaton and the proof come from [5]. For a fxed t and, we defne N t α to be the number of coeffcents jcn that are strcly greater than α tcn. Sayng jcn s strcly greater than tcn, s equvalent to takng j s such that j < a t. From ths t follows N t < α t. As φ s a vector of length one, we have m = α =. Ths gves us m = N t = t. Ths s an upper bound on the number of coeffcents of E(φ) n,m that are strctly bgger than tcn. As the coeffcents of E(φ) n,m are n descendng order, ths means for k n we have: γ k,j k < Cnk. Ths means φ µ n, E(φ) n,m C n n r= α r. j r We have φ µ n ψ m µ n by Corollary. Here ψ m s the maxmally entangled state of rank m n the bass θ θ. Ths means the sum over the n largest coeffcents of φ µ n squared s greater than or equal too the frst n coeffcents of ψ m µ n. We know ψ m µ n = m n Cnj. m = j= 3
Ths gves us: n C n r= α r n m m j r j= = = n m j= C n j = C n m C n. C n j m The C n can be estmated usng the natural logarthm. n = s the Remann-sum for n + (where we evaluate on the left of every nterval). Ths mples: C n = n = n + ln( n ). x dx Note that n+ = n = = n+ and that ln(n + ) ln(n) = ln( + n ). We know that n ln( + n ) whch, for every n, mples: x dx C n ln( n ) C n+ ln( n+ ) 0. (36) Wrte: α := Cn m ln(n m ) ln() and β := Cn ln(n ) ln(). As α > β (because of equaton (36)) we know αn β(n m) n(n m) + αn (n m)n + (n m)β n(n m + α) (n m)(n + β) n m + α n m n + β n ln( n m ) + C n m ln( n m ) ln( n ) + C n ln( n ) n m n C n m C n + m n. So we have proven: φ µ n, E(φ) n,m ɛ. (37) 4
7.5 Generalzaton Suppose we have a general measurement quantum-system S. We are measurng some observable M wth correspondng operator M. The state of the system pror to measurement s called ψ, whch s a unt vector n Hlbert space H S. We descrbe the measurement as a projectve measurement wth K possble outcomes. Ths means M = K λ P roj λ. Here, λ are the K egenvalues of ths operator. Each P roj λ projects onto the egenspace of H S belongng to the egenvalue λ. Wth ths measurement we have the assocated probabltes P ψ M (λ ) = ψ, P roj λ ψ. We wrte p = P ψ M (λ ). Rght after the measurement wth result λ, the state of the system s gven by ρ := P roj λ ψ p. Note that the ρ are a set of K orthonormal vectors n H S. Ths s what we consder a general measurement. Wth ths general measurement we can state the man clam. Theorem. Let M be a general measurement on ψ H S wth K possble outcomes. Frstly we assume our hdden varable theory T s compatble wth quantum mechancs (as defned n secton 4). Secondly, we demand that for the specfc measurement on ψ 0 n the bpartte setup, the varables A N and B N are free, w.r.t the causal order defned n equatons (7), (8) and (9). Then quantum mechancs s at least as nformatve as T,.e., for any λ σ(m) and z Z we have P ψ M (λ ) = P ψ M Z (λ z). We have the general measurement S on ψ (wth ψ n Hlbert space H S ). We wll descrbe ths measurement n two alternatve ways. Frst, n a system consstng of the combnaton of S wth a system called the measurement devce D (specfed by yet another Hlbert space H D ). Secondly, we wll descrbe the measurement n an even larger space. On ths last space we wll use embezzlement together wth our prevous results to extract the clam. Usng ths embezzlement, we wll fnally try to reduce the stuaton to the orgnal measurement on ψ. 7.5. Measurement devce We now descrbe the measurement M on the state ψ n an alternatve way. Consder a Hlbert space H D of dmenson K. Choose an orthonormal bass gven by e H D,... e H D. Now defne H S H D φ := K = = N p ρ e H D. (38) Ths, together wth the projectons S P H e D, gves rse to probabltes: P φ M (λ ) = φ, S P H e D φ = p ρ, ρ e H D = ψ, P roj λ ψ = P ψ M (λ )., e H D 5
After the measurement wth result λ, the state collapses to ( S P H e D )φ φ, ( S P e H D )φ = p ρ e H D p = ρ e H D. Ths means that after measurement on H S H D wth result λ the H S -part of φ collapses to ρ just as t would when measurng ψ on H S wth result λ. 7.5. Embezzlement on measurement devce As a last step we consder φ as defned above n a larger Hlbert space, on whch we use embezzlement. Ths space s H := H S H D H S H D H S H D. Lookng back to the prevous subsecton on embezzlement, the space H S and H D correspond to HÃ and H B, whereas H S and H D correspond to H A and H B. For the szes of these spaces, take H S and H D to be n C. We wll take H S and H D to be r C. The exact choce for n and r wll be determned below, as t s dependent on how close of an approxmaton we are tryng to acheve va embezzlement. We can chose a vector n H S H D to embezzle. We choose a maxmally entangled state of rank m, whch s ψ m = m m = eh S e H D (n H S H D ). Note that due to the fact that H S and H D have dmenson r, we can have ψ up to ψ r. Corollary. For a fxed ɛ (0 < ɛ < ) and n N: r n there exst local sometrcs U n,r;ψ m and V n,r;ψ m such that T (U n,r;ψ m V n,r;ψ m)t : (e H S e H D ) µ n E(ψ m ) n,r, wth: ψ m µ n, E(ψ m ) n,r ɛ. Here T agan permutes the order of the Hlbert spaces to let U n,r;ψ m V n,r;ψ m map (e H S e H D ) µ n. Proof. It s evdent that ths corollary s just embezzlement appled to the specfc state ψ m. The valdty of the nequalty s guaranteed by lemma 8. Wth transformatons U n,r;ψm and V n,r;ψm, we construct the followng map on H: Û ˆV = K K = j= P ρ P e H D j T (U n,r;ψ m V n,r;ψ m j )T. (39) Ths s a untary map we can apply to H Φ := φ (e H S e H D ) µ n. For the -th poston n the sum, we embezzle to extract a maxmally entangled state of rank m. We do ths n order to embezzle the Φ to a state that looks very much lke a maxmally entangled state. Ths opens up ways to apply the prevous results n ths more general settng. The next lemma descrbes whch vectors we end up wth after embezzlement va Û ˆV. Lemma 9. Applyng the operator Û ˆV on Φ gves, for approprate choces for m, r and n, a state arbtrarly close to H Ψ := ( N m r = j= ρ e H D e H S j e H D j ) µ n n the sense that for every 0 > ɛ > we have (Û ˆV )Φ, Ψ ɛ. 6
Proof. We want to use the trangle nequalty. For ths we need to establsh a connecton between the norm and the nequalty assumed n the lemma. The norm s gven by ψ = ψ, ψ. Suppose for a real number ɛ (0 < ɛ < ) and two vectors ν and ω of norm we have ν ω ɛ. An easy calculaton gves us: ν ω = ν ω, ν ω ν ω, ν ω ɛ ν, ν ν, ω ω, ν + ω, ω ɛ Thus we can conclude that the followng are equvalent: ɛ ν, ω + ω, ν ɛ Re(ν, ω) ɛ ν, ω. ν, ω ɛ ν ω ɛ. Frst we look at the what applyng the operator Û ˆV to Φ does: (Û ˆV )(Φ) = ( P ρ P H e D T U n,r;ψ m V n,r;ψ m j T )(( j,j k = pk ρ, ρ k ρ e H D j, e H D k e H D j,j,k pk ρ k e H D k (e H S e H D ) µ n (T (U n,r;ψ m V n,r;ψ m j )T ((e H S e H D ) µ n ) K = p ρ e H D (T (U n,r;ψ m V n,r;ψ m )T ((e H S e H D ) µ n ) = =: ˆΦ. Now, when we embezzle by applyng the operator U V we obtan somethng close to: p ρ e H D ( m m j= eh S j e H D j ) µ n, whch we wll call Φ emb H. Φ emb := = K ( = p ρ e H D ( p m m j= m e H S m j= j e H D ) µ n ρ e H D e H S j e H D j ) µ n. j We would lke the state Φ emb to be close to a state of ths form: H Ψ := ( r K m ρ e H D e H S e H D j ) µ n. = j= Not surprsngly, ths can be accomplshed to arbtrary precson by choosng the approprate m and n. Each m descrbes the rank of entanglement we acheve by embezzlement n the th 7
poston of the sum. The n was a parameter n the embezzlement to control the precson. We want p m to be close to, choosng the m r such that K = m = r. Ths means we have to choose the natural number m close to r p. We do ths by takng m = p r for =... K and for m K = r (m + m +... + m K ). Ths satsfes the condton that the sum of the m s equal to r. Note that m K s always well defned, as m + m... + m K (p +... + p K ) r < r. We know the dstance between m and p r s bounded. For =... K we know m r p. For the K-th m we have m K p K r K. Therefore, r p m r p m m r p r m K r m K r. Lookng at Φ emb Ψ, we can now easly see that ( ) K Φ emb Ψ K. (40) Wth these estmates, we are able to complete the proof of the lemma. Let ɛ be a real number such that 0 < ɛ <. Choose an r such that Φ emb Ψ ɛ (see equaton (40)). Then we pck n r ɛ, and the m as defned above. Lookng at ˆΦ and Φ emb and usng lemma 8 we can 4 conclude they are close as well: K ˆΦ, Φemb = p T U n,m;ψ m V n,m j;ψ mt (eh S e H D µ n ), ( = K p ( 4 ɛ) = 4 ɛ. r m m j= e H S j e H D j ) µ n Ths mples ˆΦ Φ emb ɛ, whch mples ˆΦ Ψ ˆΦ Φemb + Φ emb Ψ ɛ. Ths, mples: ˆΦ, Ψ ɛ. Ths concludes the proof of the lemma. Let us quckly summarze what we have done untl now. We have ntroduced a untary operator Û ˆV on H whch transforms Φ to Ψ up to arbtrary precson. Takng a closer look at 8
Ψ = ( N m r = j= ρ e H D e H S j e H D j ) µ n, we see t conssts of a sum of r orthogonal vectors n H. Ths s very much lke the result of takng the tensor product of the maxmally entangled state of order (prevously wrtten as ψ 0 ) r tmes. Ths allows us to use our prevous results concernng measurements on such a ψ 0. We now try to make the dea that Ψ looks lke a tensor product of maxmally entangled states more explct. Remember that H S = H D = r C. Let us look at the tensor product of r such maxmally entangled states ψ 0. To make sure we are stll dstngushng the Hlbert spaces belongng to S and those belongng to D correctly, we wll wrte H s = H d = C. Wrte: ψ 0 = ( ) e H s e H d + e H s e H d. We look at the tensor product of r such entangled states, whch gves (H s H d )... (H s H d ) r = ( ) e H s e H d + e H s e H d = j {,} j=,...,r (e H s e H d )... (e H s r e H d r ). Now ntroduce an operator T σ. Ths T σ changes the order of all Hlbert spaces n queston such that t groups those belongng to S and those belongng to D,.e. T σ ((H s H d )... (H s H d )) = (H s... H s ) (H d... H d ). Ths operator s an sometry. Now applyng T σ to the tensor product of maxmally entangled states gves the vector we wll call ψ0 r : 0 : = T σ ψ r = j {,} j=,...,r = r j {,} j=,...,r (e H s e H d )... (e H s r e H d r ) (e H s... e H s r ) (e H d... e H d r = e H S e H D. The last step gong from (e H s... e H s r ) (e H d... e H d r ) to e H S e H D constructon of the bass of r C ntroduced n Defnton. r ) s exactly the We want to mplement our ψ0 r n H, so we defne the vector: H Ψ ent = ρ e H D ψ0 r µ n. Both Ψ and Ψ ent are vectors n H consstng of r orthonormal vectors. Ths means we can easly transform Ψ nto Ψ ent usng local untary transformatons (just lke U and V n the embezzlement procedure). Let S be a untary operator on H S H S that transforms a vector ρ e H S j to ρ e H S k. W s a untary operator on H D H D that transforms e H D e H D j Agan usng the operator T to change the order of the Hlbert-spaces we have ( T (S W )T ) Ψ = Ψ emb. to e H D e H D k. 9
Corollary 3. For every 0 < ɛ < there s a Φ wth Φ, Φ ɛ such that (T W ST )(Û ˆV ) Φ = Ψ emb. Proof. From lemma 9 we know that there s an r such that (Û ˆV )Φ, Ψ ɛ. We know that T S W T s untary, so (T S W T )(Û ˆV )Φ, Ψ emb ɛ. For Φ we take [ Φ = (T W ST )(Û ˆV )] (Ψemb ). It follows that Φ wth Φ, Φ ɛ. There s one last thng we need to consder before we can fnally prove our clam. We have a several vectors that are close. It s qute ntutve that, when the vectors are close, projectve measurements on such vectors gve smlar results. Ths ntutve dea s made precse n the followng lemma. Lemma 0. Consder a general measurement of some observable M wth outcomes λ, wth λ σ(m). Let ν and ω be two states. If there s an 0 < ɛ < such that ν, ω ɛ, then for the probablty dstrbutons P φ M (λ ) and P ψ M (λ ) we have D(P ν M, P ω M ) ν, ω ɛ. Proof. The proof wll not be gven. See Chapter 9 n the book [4]. An mportant matter s how the hdden varable theory behaves for measurements on states that are close. We assume the hdden varable theory to be compatble wth quantum mechancs (as descrbed n secton 4), but ths alone s not enough to guarantee that the hdden varable theory also gven smlar probabltes on smlar states. To follow the proof as gven n the Colbeck and Renner artcle ([3]), we do need such a property n order to complete the proof. We wll say the theory T exhbts contnuous behavor f measurements on close states gve rse to a probablty dstrbutons that are close (n the varatonal dstance). Defnton 4. Consder a general measurement of some observable M wth outcomes λ. Let ν and ω be two states. We say the hdden varable theory T has contnuous behavour f the follow holds: for every ɛ > 0, there s a 0 < δ < such that f ν, ω δ we have ) D (PM Z ν ( z), P M Z ω ( z) ɛ 7.6 Proof of Theorem Now that we know how to reach, from a general measurement, somethng resemblng the rank two maxmally entangled state, we can ntroduce a measurement on Φ. Frst let us ntroduce some notatons. We wll name the operator, that transforms Φ to somethng close to Ψ emb, U. So U := T (W S)T (Û ˆV ). 30
Next, we would lke to make the projectons ntroduced n the bpartte setup r tmes on the H S H D part of Ψ emb, leavng all the other spaces unaltered. We defne ) ( r O r x, a; y, := b H S H D T σ P a E P b x F y = T σ H S H D. Agan we quckly recap. We start wth the state Φ. Usng the operator U (whch only uses local untary transformatons), we know U(Φ) s close to Ψ emb. Ths s equvalent to sayng there s a Φ close to Φ whch gets mapped to Ψ emb by U (by corollary 3). On ths Ψ emb we can apply O r x, a; y, to acheve the projectons on the maxmally entangled state. We now use the followng b operator to descrbe measurements on Φ: U O r x, a; y, b U. We frst transform to somethng resemblng a maxmally entangled state usng U, perform projectve measurements on ths maxmally entangled state (wth the operator O r x, a; y, ), after whch b we transform back usng U. Ths measurement s descrbed by random varables X, Y, AN and B N. The new random varable X conssts of r copes of X, n the sense that the possble values of X are vectors x of length r, wth x {0, }. In other words, x {0, } r. The same holds for Y. The varables A N and B N work the same way; they take values a (and b) n the r-fold Cartesan product of A N (and B N ) whch we wrte as A r N (and B r N ). Usng the operator U O r x, a; y, U as a projectve b measurement on Φ we obtan the followng probabltes: Φ, U O r x, a; y, b U(Φ) P Φ X, Y AN, B N ( x, y a, b) := = UΦ, O r x, a; y, b U(Φ). Ths same measurement on Φ, the state close to Φ gves (usng U( Φ) = Ψ emb ) P Φ X, Y AN, B ( x, y a, b) = U Φ, O r N x, a; y, U Φ b = Ψ emb, O r x, a; y, Ψ b emb ( r ) = ψ0 r σ P a E P b x F y Tσ ψ0 r = = r = r = = ψ 0, P a E P b x F (ψ 0 ) y P ψ0 X,Y A N,B N (x, y a, b ). 3
Let us sum ths over all possble y {0, } r P Φ X AN, B N ( x a, b) = y {0,} = = = y {0,} = r = r = P Φ X, Y AN, B N ( x, y a, b) r P ψ0 X,Y A N,B N (x, y a, b ) ( ) P ψ0 X,Y A N,B N (x, 0 a, b ) + P ψ0 X,Y A N,B N (x, a, b ) P ψ0 X A N,B N (x a, b ). (4) Note that, when we assume that A N and B N are free varables for the bpartte setup, the probabltes above are ndependent of b. Let us look more closely at P Φ X AN, for A,B N = a 0 N wth a 0 = (0,..., 0). Wrtng t n a slghtly dfferent way, yelds ( r ) P Φ X AN, B ( x a 0, b) = ψ r N 0, T σ P E 0 x Hd Tσ ψ0 r = = ψ 0, T σ ψ 0, r = r T σ = ( P E 0 x Hd ψ 0 ) [ r = ( ) ( ) ] P E 0 x Hd e H s e H d + P E 0 x Hd e H s e H d = ψ0 r, r eh S k e H D k (for a certan k {,..., r }) = Ψ emb, ρ e H D r e H S k e H D k µ n = Ψ, r ρ e H D e H S j e H D j µ n (for certan {,..., K}, j {,... m }). So the dstrbuton for X havng a value of x corresponds to Ψ, ρ r e H D e H S j e H D j µ n for certan and j (on the state Φ). These and j are unquely determned by the k because T (S W )T (e H S k e H D k ) = ρ e H D e H S j e H D j. The k s agan unquely determned by how we have constructed the bass on r C. To express ths connecton between the x and the and j to whch x corresponds, we wll wrte x j for the vector correspondng to ρ e H D e H S j e H D j µ n. Ths relaton also allows us to group the r vectors x j n K sets, such that every set contans all x j wth a fxed and j rangng from to m. Such a set we wrte as S = { x j j {,..., m }}. In order to keep the notaton short, we replace the random varable X wth S. The varable S takes value λ f X S. The choce for λ wll become clear f we look at the probablty for 3
X S for the orgnal state Φ. Lookng at the probabltes for Φ, we get P Φ S A N, B N (λ a 0, b) := P Φ( X S A N = a 0, B N = b) Ths also mples, by defnton of Φ, that P Φ S A N, B N (λ a 0, b) = x S P Φ X AN, B N ( x a 0, b) = Ψ, ρ m e H D r e H S j e H D j µ n. = x S P Φ X AN, B N ( x a 0, b) j= = Φ, [ p ρ e H D T U n, r ;ψ m V n, r ;ψ m T = φ, HS P H e D φ ( e H S e H D µ n )] = P ψ M (λ ) (4) = p. We are now able to prove Theorem. Proof. We are gong to use a proof by contradcton. For a general measurement M we assume the followng: The hdden varable theory T s compatble wth quantum mechancs For the measurement n the bpartte setup, the varables A N and B N are free w.r.t the causal order defned by equatons (7), (8) and (9). The theory T has contnuous behavor (per defnton 4) For a z Z we have D(P ψ M, P ψ M Z ( z)) > ɛ. As llustrated by equaton (4), descrbng the measurement M as applyng U O r x, a; y, b U to Φ (and summng over all x j S ) s equvalent to applyng M to ψ. As both dstrbutons descrbe the same measurement, we necessarly have and P Φ S A N, B N ( a 0, b) = P ψ M. (43) P Φ S A N, B ( a N,Z 0, b, z) = P ψ M Z ( z). (44) We have fnally establshed a connecton between the orgnal measurement on ψ, and the more complcated measurement (usng embezzlement) on Φ. We assume A N and B N to be free varables (n the bpartte setup). Ths, together wth lemma 5, means P ψ0 X A N,B N (x a, b) = P ψ0 X A N (x a) = P ψ0 X A N,Z (x a, z). 33
We know that P Φ X AN, B N ( x a, b) = r = P ψ0 X A N (x a ). The dstrbutons for the measurement on the maxmally entangled state are ndependent of Z (as proven n lemma 5). The dstrbuton for Φ s just a product of dstrbutons for ψ 0, so s ndependent of Z as well. Ths gves P Φ S A N, B N (λ a 0, b) = P Φ S A N, B N,Z (λ a 0, b, z). (45) Usng corollary 3, we can conclude that we can choose r (and the m assocated wth t) such that Φ, Φ ɛ 8. Usng lemma 0 ths mples ( D P Φ S A N, B ( a 0, b), P Φ N S A N, B ( a 0, ) b) N ɛ. Next, usng the constrant that the theory T has contnuous behavor, we know that for ɛ, there s a δ such that when Φ, Φ δ we know We can choose r, such that the maxmum of r and r : ( D P Φ S A N, B ( a N,Z 0, b, z), P Φ S A N, B ( a N,Z 0, ) b, z) ɛ. Φ, Φ δ. We want to use both nequaltes, so defne r to be r = max(r, r ). Combnng the above nequaltes, we are able to gve an upper bound on the varatonal dstance between P ψ M and P ψ M Z ( z). ) D (P ψm, P ψm Z ( z) ( D P ψ M, P Φ S A N, B ( a 0, ) ( b) + D P ψ N M Z ( D P ψ M, P Φ S A N, B ( a 0, ) ( b) + D P ψ N M Z ( D P Φ S A N, B ( a 0, b), P Φ N S A N, B ( a 0, b) N + D ɛ. ( P Φ S A N, B N,Z ( a 0, b, z), P Φ S A N, B N ( a 0, b) Φ ( z), P S A N, B ( a 0, ) b) N Φ ( z), P S A N, B ( a N,Z 0, ) b, z) ) We have used equatons ) (45), (43) and (44). Ths s a contradcton wth the assumpton that D (P ψm, P ψm Z ( z) > ɛ. As ths constructon s possble for every ɛ > 0, t follows that P ψ M = P ψ M Z ( z). ) 34
8 Conclusons and Dscusson The man goal has been acheved. We have constructed a proof for the clam of Colbeck and Renner that, under some natural assumptons, hdden varables do not mprove the predctons made by quantum mechancs. If some hdden varable theory has free choce (to deduce no-sgnalng) and t s compatble wth quantum mechancs, quantum mechancs s at least as nformatve as the hdden varable theory. The compatblty states, n more detal, that the predctons gven by quantum mechancs should be dentcal to those provded by a hdden varable theory that s averaged over the hdden varable. As quantum predctons seem sound, ths assumpton s non-negotable. The assumpton on free varables however, s more controversal. In the paper About possble extensons of quantum theory ([7]) by Ghrard and Romano, a dummy hgher theory s suggested whch does not conform to the noton of free choce. On the other hand, t does have the nosgnalng property and seems to be a legtmate hdden varable theory. Ths mples the free choce as descrbed by Colbeck and Renner s actually an assumpton stronger than no-sgnalng, and therefore somewhat redundant. Assumng just no-sgnalng, the proof can be completed n exactly the same way. It can even be slghtly smplfed by, descrbng the settngs A N and B N as fxed parameters n the form of angels, nstead of descrbng the settngs by random varables wth N possble values. Ths would make the proof of lemma 5 somewhat easer, as one does not have to use the lmt of N anymore. But n general, the man proof would reman the same. A far more pressng matter s contaned n the crux of the proof, namely the generalzaton from one state and one measurement to any state and any measurement. Ths s acheved by descrbng the orgnal general measurement (on ψ) as an alternatve measurement (on Φ) n a larger Hlbert Space. One can show that the predctons gven by both descrptons are the same. After measurement wth a certan result, n both cases we collapse to the same egenstate (n the S part of the system). The queston remans though, f ths agan mples the predctons usng the hdden varable theory T are the same. Intutvely, the predctons made by T should be the same f we have equvalent descrptons of the same measurement. As the hdden varable s assumed to be a property of the system, one mght expect that as long as we are descrbng the same experment, equvalent descrptons of ths measurement should gve rse to the same predctons, even usng the hdden varable theory. But t s not really clear that the measurement va embezzlement we perform n the larger Hlbert space s n fact an equvalent descrpton of the general measurement. But ths fact s crucal n tyng the general measurement and the specfc projectons on the maxmally entangled state together. Therefore, n our vew, the result s not complete wthout a specfcaton on when exactly the theory T should gve the same predctons for alternatve descrptons of a measurement. Secondly, n order to complete the proof, we have to assume an addtonal property for a hdden varable theory, that s not mentoned n the Colbeck and Renner artcle. Throughout the proof of the clam, we make use of states that are close. For a gven r, the probabltes obtaned for Φ are equal to a product of r probabltes obtaned by measurement on ψ 0 n the bpartte setup. But Φ and Φ are now only guaranteed to be close up to a bound determned by r. If we want them to be closer, we have to choose a larger r, whch leads to a dfferent Φ and operator U O r x, a; y, U. There s no reason to assume that the constructon for one partcular b r would yeld a Φ equal to Φ. For quantum mechancs, we know when the states are close, the probablty dstrbutons assocated wth a measurement on these states are close as well. 35
For T we have, a pror, no reason to assume the probablty dstrbutons for measurements on close states are close. In order to say somethng about the predctons gven by T for our measurement on Φ (whch s equvalent wth the general measurement on ψ), we use knowledge about the predctons of T on Φ. We have assumed the theory T to have an extra property we have called contnuous behavor, whch enables us to compare the probabltes gven by T on Φ and Φ, f Φ and Φ are close. Wthout ths assumpton, we cannot compare the predctons gven by T on Φ and Φ. In concluson, we have reached our goal. But along the way, we had to make some concessons n the form of extra assumptons on how the hdden varable theory T relates to quantum mechancs. Therefore, I thnk the result of Colbeck and Renner s not qute strong enough to exclude the exstence of a hdden varable theory whch assumes compatblty wth quantum mechancs and free choce n a bpartte setup. 36
9 Bblography References [] Roger Colbeck and Renato Renner, No extenson of quantum theory can have mproved predctve power. arxv: 005.573v3 09 aug 0. [] Roger Colbeck and Renato Renner, No extenson of quantum theory can have mproved predctve power. Nature communcatons, Artcle number: 4, 0 Aug 0. [3] Roger Colbeck and Renato Renner, The completeness of quantum theory for predctng measurement outcomes. arxv, 08.43 0 Aug 0. [4] Mchael A. Nelsen and Isaas L. Chuang, Quantum Computaton and Quantum Informaton: Chapter and Chapter 9. Cambrdge, 0th edton. [5] Wm van Dam and Patrck Hayden, Unversal entanglement transformatons wthout communcaton. Physcal Revew A 67, 06030(R), 003. [6] Mateus de Olvera Olvera, Embezzlement States are Unversal for Non-Logal Strateges. arxv: 009.077v 03 sep 00. [7] GanCarlo Ghrard and Raffaele Romano, About possble extenson of quantum theory. arxv 30.5040v Jan 03. [8] John S. Bell, On the Ensten Podolsky Rosen Paradox, Physcs Vol., No. 4, pp. 95-00, 964 [9] A. Ensten, B. Podolsky, N. Rosen, Can Quantum-Mechancal Descrpton of Physcal Realty Be Consdered Complete?, Phys. Rev 47, 777-780, 935 37