Probability Constraints on the Classical/Quantum Divide

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1 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide 6 Pobbility Constints on the Clssicl/Quntum Divide Subhsh K STRACT This ppe consides the poblem of distinguishing between clssicl nd quntum domins in mcoscopic phenomen using tests bsed on pobbility nd it pesents condition on the tios of the outcomes being the (P s ) to being diffeent (P n ). Given thee events, P s /P n fo the clssicl cse whee thee e no 3 wy coiidees is one hlf whees fo the mximlly entngled quntum stte it is one thid. Fo non mximlly entngled objects ( ) we find tht so long s < 5.83, we cn septe them fom clssicl objects using pobbility test. Fo mximlly entngled pticles ( = ), we popose tht the vlue of 5/ be used fo P s /P n to septe clssicl nd quntum sttes when no othe infomtion is vilble nd mesuements e noisy. Key Wods: infomtion, Bell inequlity, quntum theoy, entnglement, pobbility constints NeuoQuntology 3; : 6-66 Intoduction Tht signl be consideed quntum is genelly detemined by diffeent chcteistics such s supeposition, entnglement, nd complementity. Infomtion ssocited with eithe clssicl o quntum vibles is mesued in eltion to the expeimentl ngement nd it is fution of the pobbilities of the outcomes ssocited with the expeiment. In the poblem of clcultion of infomtion, one should fist detemine whethe the vible being mesued Coesponding utho: Subhsh K Addess: Deptment of Compute Sciee, OSU, Stillwte, OK 778, USA Phone: e-mil subhsh.@ostte.edu Relevnt conflicts of inteest/finil disclosues: The uthos decle tht the esech ws conducted in the bsee of ny commecil o finil eltionships tht could be constued s potentil conflict of inteest. Received: 3 June 3; Revised: July 3; Accepted: August 3 eissn is clssicl o quntum, nd this my be difficult if the mesuement constints peclude detemintion of supeposition. If the vible is quntum, one must detemine if it is pue o mixed nd this would equie pio testing nd ssumption of sttionity of the pocess with espect to time. One must lso define the fmewo within which the question of infomtion is being sed sie, unde cetin conditions, n unnown pue stte ssocited with zeo von Neumnn entopy -- cn convey infomtion (K, 7). Thee e mny physicl pocesses whee the ntue of the vibles is well estblished both by theoy nd expeiment. But wht bout new situtions, outside of physics in mcoscopic systems, whee thee isn t consensus tht the pocesses hve quntum mechnicl bsis? Quntum biology is one such e (Collini ; Bll, ; Lmbet, 3).

2 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide Resons hve lso been dved fo consideing quntum models of the bin (K, 995; K, 996; K, ; Hmeoff nd Penose, 3; Feemn nd Vitiello, 6) nd if such futioning is tue thee ought to be evidee in fvo of coiidees coss spce nd time (K, 9). In mny situtions, such s coening detemintion of entnglement, the diffeentition between clssicl nd quntum effects is estimted by checing if the Bell inequlity is violted (Bell, 96; De Redt ; Fey, ). In his oiginl ppe, Bell showed tht unde conditions of independee clssicl ndom vibles A, B, C will stisfy P P C ) P ( B, C ) + () whee P ( X, Y ) is the pobbility tht the pi of ndom vibles X, Y hve some identicl popety. The Bell inequlity is consistey constint on futions of two ndom vibles nd it cn be descibed in mny othe wys (nd we will use slightly diffeent fom in ou discussion). Bell showed tht simil mesuement of entngled quntum vibles cn led to violtion of the inequlity nd, theefoe, such violtion cn seve s divide between clssicl nd quntum vibles. But Bell inequlity is lso violted in mny clssicl situtions whee long-nge coeltions pesist (Biggs nd Eisfeld, ; Mille, ). Sttistics my lso be used to distinguish between clssicl nd quntum systems (K, 3). But this pplicbility will be limited to situtions whee the quntum pocess is in stte of theml equilibium. In closed quntum system, thee my be lge numbe of eigensttes. When this system is opened to the envionment, mny of these sttes couple to the envionmentl sttes nd undego decoheee. Due to decoheee emeges envionment-induced supeselection tht leds to the suvivl of the pointe sttes (Zue, 3) tht emin coelted with the univese even though they e quntum mechnicl. The pointe sttes obey clssicl sttistics. The clssicl/quntum divide is centl notion of the Copenhgen Intepettion (CI) of quntum mechnics. The Mny Wolds Intepettion (MWI) tes the wvefution to be the pimy elity nd ssigns wvefution to the univese itself. In MWI thee is no collpse of the wvefution nd the obsevtion is consequee of decoheee bought bout by the envionment. If CI is n eissn inside-out view of the univese whee the elity is constucted out of the peceptions of the expeimente, MWI is n outside-in view in which the mthemticl fution of the univese is the pimy elity. In ou view CI is bette thn MWI in ddessing the question of fee-will which it does though the quntum Zeno effect if consciousness is seen to hve univesl bsis. In ppliction of quntum theoy to beliefs (which sidesteps the question whethe the bin must be descibed by quntum model), questions A, B, A B, nd B A e pesented in sequee to ech membe of goup of people nd the esponses veged (Khenniov, ). If it is shown tht P(A P( P(B A) P(A), tht would be evidee in suppot of non-clssicl bsis to such pobbility. In entngled sttes such s P() =.9 nd P() =.. Such stte fution my be ceted by pssing the stte though some ppopite unity opeto. Whethe such opetos cn be feely implemented is quite nothe mtte fo it ppes tht the poblem of gte design to deive specific stte futions is not esy (K, ). Six esons hve been dved fo quntum ppoch to mentl events: mentl sttes e indefinite, judgments cete the thn meely ecod, they distub ech othe, nd they do not lwys obey clssicl logic, they do not obey the piiple of unicity, nd cognitive phenomen my not be decomposble. We te nothe loo t wht is possible to infe fom expeimentl obsevtions elted to the ntue of the system o the signls. Given piwise dt fo vibles, Boole showed wht constints hd to be stisfied fo the piwise pobbilities of set of thee evens to be consistent (Boole, 86). In this sense, Boole s wo pefigues Bell s inequlities (Bell, 96) lthough the focus of Bell ws to find setting tht would clely point to the diffeees between clssicl nd quntum situtions. An entngled quntum stte such s ( ) hs the fom long ll mesuement bses. Thus fo ny ndom bses b, b ( ) b b b b nd ech of the qubits is in mixed stte. It is this fetue tht mes the mesuements long diffeent bses

3 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide on the quntum stte diffeent fom the mesuements of clssicl sttes. The conditions fo the pobbility constints e well-defined both fo clssicl objects nd quntum sttes nd we stess tht the use of the tios of to non events will be moe elible in the pesee of noise thn to conside just the bsolute vlue of events. Fo non-mximlly entngled objects ( ) we find tht so long s < 5.83, we cn septe them fom clssicl objects bsed on computtion of nd not events coss ppopitely chosen bses. Pobbility Constints It is quite cle tht constint on n vibles will be pojected s sevel constints on subsets of these vibles. Theefoe, if we e only given the constints on the subsets of the vibles, it cnnot be sid if they wee deived fom the fution on ll the vibles. Conside (with Boole) thee events A, B, nd C. Let P(A = ; P() = s; nd P(A = t. If Venn digm is dwn nd we wite P( ; P( ; P( ; P( Then ; s; t () s t s t t s These e of the fom P( A P( A P( ) which my be ewitten s: 6 (3) P( A P( A P( ) () This is fom simil to tht of the oiginl Bell inequlity (). Now we conside moe genel sitution whee thee e thee ndom vibles A, B, nd C ech one of which tes two vlues, sy nd. Le the pobbilities of the eight possible outcomes be defined s below: Tble. Pobbility of outcomes fo events. A B C Pobbility b c d e f g h In el dt, the set my not be complete due to noise nd othe expeimentl eos. In such cse, if the dt tht doesn t fit is ignoed, the pobbility vlues in the ight column will not dd up to one. Clely, P (,) b; P (,) c d; P (,) e f ; P (,) g h P (,) e; P (,) b f ; P (,) c g; P (,) d h eissn Figue. Venn digm fo thee events A, B, C. A stightfowd computtion shows tht the following constints need to be stisfied fo the dt to be consistent: P (,) c; P (,) b d; P (,) AC AC AC e g; P (,) f h AC Sie these e mutully dependent conditions, they should stisfy conditions such s: P (,) P (,) P (,) P (,) (5) AC AC P (,) P (,) P (,) P (,) (6) P (,) P (,) P (,) P (,) (7) AC AC tht expess pi () in two diffeent wys. Liewise, thee would be futhe set of conditions tht expess pi () in two diffeent wys.

4 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide tue: Let us now define P(,) P(,) P, P (,) P (,) P, AC AC nd P (,) P (,) P ( B,. It follows tht the following esult is A B + P C ) + P ( B, C ) = h (8) Theoem. Conside tht smple spce is completely descibed by the biny outputs ssocited with events A, B, nd C. Then P B ) + P C ) + P ( B, C ) = P B, whee P B, P (,,) P (,,). C C Theoem implies the genel inequlity: A B + P C ) + P ( B, C ) (9) It is obvious tht the stisfction of the inequlity (9) is necessy but not sufficient fo the dt to be consistent. Conside the pobbility vlues given below fo thee ndom vibles A, B, C: P (,).3; P (,).; P (,).3; P (,). P (,).5; P (,).35; P (,).5; P (,).5 P (,).; P (,).; P (,) AC AC AC.; P (,).3 AC () These vlues violte the inequlities (5), (6), nd (7). The dt does howeve stisfy the inequlity (9) sie the left hnd side fo this cse is >. Now conside specific cse of thee independent vibles ssocited with n expeiment whee ech vible tes the vlues o. Let the pobbilities of A, B, nd C being nd be given by the tble below: A B C p() s t p() s t Conside the poduct: s ( )( s) st ( s)( t) t ( )( t) () To find its minimum we diffeentite this expession with espect to nd put tht equl 63 to zeo. We find tht s t. Substituting bc in () we find the minimum to be t t whose lest vlue is. Its mximum vlue is obtined when, s, t e ech equl to which educes eqution () to 3. The left hnd side of () is nothing but P B ) + P C ) + B C nd, theefoe, we cn wite: A B + P C ) + P ( B, () Let P P B ) + P C ) + sc B C, nd P Pnot B ) + Pnot C ) + Pnot( B, whee Pnot mens tht the outputs A nd B e diffeent, nd the c in the subscipt fo Psc nd P efes to clssicl fo wht is being consideed is clssicl pobbility. A stightfowd computtion shows tht P P B ) + P C ) + not B C not not = b c d e f g P B, (3) By using Theoem on (3), it follows tht P sc P 3 () We cn summize the esult on the tio of the pobbilities of getting output to not output by the next esult: Theoem. Psc Ps B, P [ P B, ] s In the specil cse whee P B, C ) =, s (5) Psc P (6) Coeltion Acoss two Diffeent Bses We wish to detemine the coeltion coss two bses fo the genel quntum cse of entngled pticles ( ) (7) oiginlly descibed with espect to the A set of bses:,. (8) eissn

5 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide Let the second set of bses be b b, (9) Substituting, we see tht BB ( ( ) { bb ( ) bb ( ) ( ) bb bb } () Fo the mximlly entngled csed of =, ( b ) BB b bb. To conside the coeltion coss the two bses, we conside epesenting the pojection of one set of bses on nothe. We cn expess the stte s: = ( ) () The pobbilities P nd P not will simply be the modulus sque vlues of the mplitudes of nd on the one hnd, nd nd on the othe. Ming the equied computtion, we obtin: P = nd Pnot B ) = () Theoem 3. Fo the non-mximlly entngled pticles ( ) nd the two fmes defined by (8) nd (9), the pobbilities of output to dissimil output is given by P P _ quntum not _ quntum ( A, B ) 6 (3) It is significnt tht the esult is independent of the vlue of. This mens tht we will get the esult even if the entnglement is zeo nd we e deling with pue stte. The expession (3) gives the pobbilities fo simil nd dissimil outputs fo =. This is the cse which coesponds to mesuement bses b tht e othogonl to i. As is iesed, we otte b i wy fom. The vlue = coesponds to -5, =3 to -5, nd = to -6, nd so on. A stightfowd clcultion shows tht fo the stte b c d coeltion coss the bses BB of (9) leds to the following P : b ( ) c ( ) d () Loclity vesus Nonloclity The context fo the Bell inequlities is the EPR expeiment in which two obseves me mesuements on pi of entngled objects. Obseves A (Alice) nd B (Bob) me thei obsevtions using filtes whose oienttions e chosen independently immeditely befoe thei obsevtion nd, theefoe, these oienttions cnnot be consideed to be elted. Fo mximlly entngled pi of objects ( ), if the mesuement oienttions chosen by Alice nd Bob hppen to be the, the esults of the filtes will be identicl. Sie the pobbility of ech of the two choices being o is equl, this implies tht somehow the two objects emined coupled even though they could be so f pt tht signl fom one could not ech the othe in the time tht the decision to use specific mesuement oienttion ws mde. In n ctul expeiment, the mesuement oienttions will be ndomly distibuted. Nevetheless, the Bell inequlities descibe the sitution if the ttibutes of the two objects e defined in dve of the mesuement nd thee e no existing coeltions between the septed objects. A eissn

6 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide puely quntum desciption, on the othe hnd, violtes Bell inequlities. An essentil ingedient of the gument elted to the expeiment is tht the ctions of the souce C of the entngled objects cnnot possibly depend on the lte decisions elted to mesuement oienttions mde by both A nd B. One my sy tht fee will ws behind these decisions. Pentheticlly, it should be noted tht the mximlly entngled pi of objects e not in supeposition sttes but e the mixtues. The mesuement by Alice does not cuse collpse; it is choice between two outcomes ssocited with equl pobbility. The violtion of Bell inequlities hs been genelly viewed s the best poof of nonloclity. But ll expeiments tht hve been conducted until now hve loopholes nd theefoe they cnnot be consideed s definitive tests. The two piipl loopholes e those of loclity nd detection (Bunne et l, 3; Chistensen et l 3). The loclity loophole ddesses the possibility tht locl elistic theoy might ely on some type of signl sent fom one entngled pticle to its ptne. The detection loophole ddesses the fct tht when mesued with low-quntumefficiey detectos, s is cuently the cse, expeimentl esults cn be explined by locl elistic theoy ( t Hooft, 9; Vevoot, 3). The inequlity (9) is violted by mesuements on cetin quntum sttes. Thus if the stte fution is the non-mximlly entngled stte ( ), we cn te the thee mesuement bses to be the, 9 ; -6, 3 (=) nd -3, 6 (= nd flipping the fmes) to mximize the totl pobbility by picing unifom distibution ove the phse spce (s in Mccone, ). These fmes e witten s: eissn 33 55, ; (5) 3, b 3 b ; (6) 3 3 c, c (7) We hve ledy seen in eqution () tht = Pob ( + ) = / which P is ¼. Similly, P = ¼. The fution my be deived fom 65 (7) by substituting the invese tnsfomtions fo (6) nd (7): = ( 3) { bc 3 ( ) bc Theefoe, B, P ( = 3 bc 3 ( ) bc } (8) ( 3) (3 ) { } ( ) 6 6 8( ) (9) We find tht in the quntum exmple consideed bove, P sq= P (A, + P (A, P (B, = (3) 8( ) Eqution (3) violtes the Bell-type inequlity (9), tht is it is less thn, unless It is equl to when 6, which hs solutions 3 8 o 5.83 nd.7 which e the solution sie they e inveses of ech othe. Futhemoe, P P not + P not + nq P not ( B, = (3) 8( ) When =, P ( B, = ¼ nd so in ech of these cses, the not pobbility is ¾. Summizing, we cn wite: Theoem. Fo the stte ( ), fo P sq nd Pnq defined s in (7) nd (8), P sq < fo < We hve the futhe esult: Psq 3 3 P 5 5 (3) nq The pobbility of obtining the esults in the quntum cse (3) is lowe thn fo the clssicl cse (5) fo < Colusions This ppe consides the poblem of distinguishing between clssicl nd quntum

7 NeuoQuntology Decembe 3 Volume Issue Pge 6 66 K S., Pobbility constints on the clssicl/quntum divide domins in mcoscopic phenomen using tests bsed on pobbility nd it pesents condition on the tios of the outcomes being the (P s) to being diffeent (P n). Given thee events, P s/p n fo the clssicl cse whee thee e no thee-wy coiidees is one-hlf whees fo the mximlly-entngled quntum stte it is one-thid. In the pesee of noise, the middle point of 5/ could seve s the dividing line between these two sttes. Fo the nonmximlly entngled stte ( ), the thee-wy coiidees cn distinguish between quntum nd clssicl cses so long s < It is indeed tue tht the conditions outlined hee will lso be stisfied by mny egimes of clssicl wves (Fey, ). Clely the pplicbility of these pobbility constints is elevnt when we e consideing the pticle view of the phenomenon. Acnowledgements This esech ws suppoted in pt by the Ntionl Sciee Foundtion gnt CNS-768. I m lso thnful to the Chop Foundtion fo hospitlity t the Sges nd Scientists Symposium when this wo ws completed. Refeees Bll P. The dwn of quntum biology. Ntue ; 7: 7-7. Bell J. On the Einstein Podolsy Rosen pdox. Physics 96; (3): 95. Boole G. On the theoy of pobbilities. Philos Tns R Soc London 86; 5: 5-5. Biggs JS nd Eisfeld A. Equivlee of quntum nd clssicl coheee in electonic enegy tnsfe. Phys Rev E ; 83: Bunne N et l., Bell nonloclity. 3. Xiv: Chistensen BG et l., Detection-loophole-fee test of quntum nonloclity nd pplictions. 3. Xiv: Collini E et l., Coheently wied light-hvesting in photosynthetic mine lge t mbient tempetue. Ntue ; 63: De Redt H, Hess K, Michielsen K. Extended Boole-Bell inequlities pplicble to quntum theoy. J Comp Theo Nnosci ; 8: 39. Fey DK. Pobing Bell s inequlity with clssicl systems. Fluctution nd Noise Lettes ; 9: Feemn W nd Vitiello G. Nonline bin dynmics s mcoscopic mnifesttion of undelying mny-body dynmics. Physics of Life Reviews 6; 3: Hmeoff S nd Penose R. Conscious events s ochestted spce-time selections. NeuoQuntology 3; : -35. K S. Quntum neul computing. In Adves in Imging nd Electon Physics, vol. 9, P. Hwes (edito). Acdemic Pess, 59-33, 995. K S. The thee lnguges of the bin: quntum, eogniztionl, nd ssocitive. In Lening s Self- Ogniztion, K. Pibm nd J. King (editos). Lwee Elbum Assocites, Mhwh, NJ, 85-9, 996. K S. Active gents, intelligee, nd quntum computing. Infomtion Sciees ; 8: -7. K S. Quntum infomtion nd entopy. Intentionl Jounl of Theoeticl Physics 7; 6, K S. Anothe loo t quntum neul computing. 9. Xiv: K S. The poblem of testing quntum gte. Infocommunictions Jounl ; (): 8-. K S. The univese, quntum physics, nd consciousness. Jounl of Cosmology 9; 3: 5-5. K S. Biologicl memoies nd gents s quntum collectives. NeuoQuntology 3; (3): Khenniov AY. Ubiquitous Quntum Stuctue: Fom Psychology to Fine. Spinge, Belin,. Lmbet N et l., Quntum biology. Ntue Physics 3; 9: 8. Mccone A. A simple poof of Bell s inequlity.. Xiv:.5. Mille WH. Pespective: Quntum o clssicl coheee? J Chem Phys ; 36: 9. t Hooft G. Entngled quntum sttes in locl deteministic theoy. 9. Xiv: Vevoot L. Bell s theoem: Two neglected solutions. Foundtions of Physics 3; 3: Zue, W.H. Decoheee, einselection, nd the quntum oigins of the clssicl. Rev. Mod. Phys. 3; 75: eissn

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2. Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i