QM13: The Observability of Counterfactuals The Elitzur-Vaidman Bomb Test, Ref.[1] Last Update: 13/3/11

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1 . Coutrfactuals QM3: Th Obsrvablty of Coutrfactuals Th Eltzur-Vadma Bomb Tst, Rf.[] Last Updat: 3/3/ Suppos somthg could hav happd, but actually dd ot happ. I classcal physcs th fact that a vt could hav happd but dd t ca mak o dffrc to ay futur outcom. Oly thos thgs whch actually happ ca fluc th futur voluto of th world. But quatum mchacs t s othrws. Th pottal for a vt to happ ca fluc futur outcoms v f th vt dos ot happ. Somthg that could happ but actually dos ot s calld as coutrfactual. I quatum mchacs coutrfactuals ar obsrvabl thy hav masurabl cosqucs. Th Eltzur-Vadma bomb tst provds a strkg llustrato of ths. But frst w must rvs how trfromtrs work spcfcally Mach-Zhdr trfromtrs. Fgur Th Mach-Zhdr Itrfromtr RB = Rfrc Bam; SB = Sampl Bam Th ky to th bhavour of th trfromtr s th phas chag suffrd by th rflctd ad trasmttd wavs at th mrrors. Thr s pottal cofuso ovr ths sc th phas chag dpds upo th typ of obct usd as a mrror. Hr w shall ust stat th ruls for th phas chags for two typs of mrror. Th ruls ar drvd from frst prcpls th Appdx.. Rflcto/Trasmsso Phas Chag Ruls. Slvrd or Half-Slvrd Mrrors Ths ar smply covtoal mrrors whch cosst of a flat pc of glass oto o sd of whch has b dpostd a th layr of slvr, or smlar rflctv mtal. Th trms slvrd ad half-slvrd rfr to dffrt thckss of mtal flm, th frst bg suffct to prvt ay trasmsso of lght through th mrror, whras th lattr s calbratd to allow about as much trasmsso as rflcto. Th phas shft causd by such mrrors dffrs accordg to whthr th comg bam s cdt o th slvrd surfac (th frot of th mrror) or th glass surfac (th back of th

2 mrror). Not that mrrors domstc us usually hav th back (th glass) facg forwards, ad th frot s grally covrd wth somthg opaqu to protct th slvr flm. Th ruls for th phas shfts ar, [] A bam rflctd from th frot (slvrd) surfac udrgos a phas chag of 80 o (.. a factor of ); [] A bam rflctd from th back surfac udrgos o phas chag du to rflcto but a phas chag of du to ts passag (twc) through th glass, fa whr ka ad whr th rfractv dx of th glass c ad a th thckss of th glass; [3] A bam trasmttd through a (half-slvrd) mrror udrgos a phas chag of. All ths phas chags ar wth rspct to what th phas would b at th sam plac f propagato had volvd passag through ar alo (.., wth rspct to th phas k r of a propagatg wav). Th phas chag ruls ar th sam for slvrd ad half-slvrd mrrors. I th aalyss blow th phas chag wll b assumd th sam for all four mrrors. Ths mpls that th mrrors all hav th sam thckss to optcal prcso,.., to a accuracy much lss tha th wavlgth of lght. I practc ths s mprobabl ad som adustmt (calbrato) of th trfromtr wll b rqurd pror to us to compsat for ths practcal lmtato.. Pla Plats Ay traspart plat wth a rfractv dx,, gratr tha ar wll caus a mxtur of trasmsso ad rflcto. Ths rsults from th rqurmt to satsfy th rlvat boudary codtos at both surfacs of th plat, rathr tha bg a rsult of rflcto at ust o surfac (as for a mrror). Cosqutly th phas chags ar dffrt. Th rul s smply, Th rflctd wav acqurs a phas factor of compard to th trasmttd wav. Th trasmttd wav actually pcks up a phas factor rlatv to th cdt wav, but th abov rul s all that s dd to aalys th trfromtr. 3. Th Mach-Zhdr Itrfromtr No Sampl Itally w aalys what sgals w xpct at th dtctors ad (Fgur ) f thr s o sampl, or aythg ls put th way of th bams (.g., ay masurg dvc). Frstly lt s assum mrrors ad half-slvrd mrrors ar usd, th ortato show Fgur. Th phas factors acqurd by th rfrc bam ad th sampl bam at dtctor ar, Dtctor, Mrrors: RB: SB: Th thr trms ach xprsso abov rfr to th rflcto or trasmsso at ach mrror squc. Th phas factors for th rfrc ad sampl bams ar qual ad oppost, so cosqutly thy dstructvly trfr. Th prdcto, bor

3 out by xprmt, s that o lght mrgs to dtctor. If th lasr brghtss s turd dow utl thr s oly o photo passg through th qupmt at ay tm, th sam cocluso appls: o photos ar rgstrd by dtctor. W must hop that w gt costructv trfrc to dtctor, or our photos wll hav go mssg! W do dd gt costructv trfrc to dtctor, thus, Dtctor, Mrrors: RB: SB: Now lt s xam what happs f pla plats wr usd stad of mrrors. I ths cas th phas rul s for rflcto rlatv to trasmsso. So cosdr th phas chags of th sampl bam wth rspct to th rfrc bam for xt to dtctor. At th frst mrror th SB pcks up a phas factor of wrt th RB, ad th sam s tru at th last (3 rd ) mrror, sc at both ths mrrors th SB s rflctd but th RB s trasmttd. As for th d mrror, both bams ar rflctd so thr s o rlatv phas chag. Cosqutly th rlatv phas chag btw th SB ad th RB at dtctor s, ad w aga coclud that thr s dstructv trfrc ad hc o photos at dtctor. Rpatg ths aalyss for dtctor th phas chag of th SB rlatv to th RB du to th frst two mrrors s dtcal,..,. Howvr at th thrd mrror t s ow th RB whch s rflctd ad th SB whch s trasmttd. Hc th RB acqurs a phas factor of wrt th SB, whch s quvalt to th SB acqurg a phas factor of wrt th RB. So ovrall th phas chag of th SB rlatv to th RB to dtctor s, ad so thr s costructv trfrc. All th photos tr dtctor, as for th mrrors. 4. Th Mach-Zhdr Itrfromtr Wth Masurmts To aalys th ffct of cludg a masurg dvc thr of th bam paths, frstly lt s stablsh som otato. Suppos th stat of a photo whch follows path RB to dtctor s wrtt RB. Th th stat of a photo followg path SB to dtctor s SB RB. Th prcdg aalyss has show that ad. Th stat at th dtctors s thus, Hc th flux of photos to th dtctors s proportoal to, RB RB () cos () thus gvg costructv trfrc, trfrc, 0, at dtctor., at dtctor ad dstructv How dos ths chag f a masurg dvc s srtd to thr bam path? A dvc to dtrm whch path th photo taks must hav two stats, o of whch corrspods to th photo was dtctd path RB ad th othr corrspodg to th photo was dtctd path SB. Ths stats of th masurg dvc wll b wrtt M : RB ad M : SB rspctvly. Now a prfct masurmt must b such that f th photo s masurd path RB,.., f th masurg dvc stat bcoms M : RB aftr masurmt, th th photo s dftly ot o path SB. Ths mas

4 th two masurmt stats must b orthogoal, M : SB M : RB 0. Howvr, t s possbl to vsag a poor masurmt whch mght dcat a probablty of th photo bg path RB, but wth som rsdual possblty of bg path SB. For such a mprfct masurmt w would hav M : SB M : RB 0. Th combd stat of th photo ad th masurg dvc, at thr dtctor, s thus, RB M : RB Th flux of photos to th dtctors s thus, RB M : SB (3) M : SB M : RB (4) Cosqutly, f w hav prformd a prfct masurmt whch dftly dtcts whch path th photo took, th, bcaus M : SB M : RB 0, w hav from (4) that th photo flux s uty,, to both dtctors. All trfrc has b lost. Ths stablshs qut grally that ay mas of dtctg whch path th photo taks aroud th trfromtr wll dstroy th trfrc ( th ss that both dtctors wll dtct qual umbrs of photos). Howvr, th occurrc or ot of trfrc s ot a all-or-othg affar. A mprfct masurmt, whch has 0 M : SB M : RB, wll stll lav rsdual trfrc ad hc mor photos to dtctor tha to dtctor. 5. Th Obsrvato of Coutrfactuals: Th Eltzur-Vadma Bomb Tst Evrythg s ow st up for ths lovly xampl. Imag that you hav maufacturd a larg umbr of bombs. Th bombs ar so sstv that th slghtst tracto, say wth a sgl photo, would mak th bomb xplod. Th troubl s that you kow that som bombs ar duds but you d to dtfy o whch s dftly ot a dud. How o arth ca you do ths? By th problm statmt, th bombs ar so sstv that th slghtst physcal tracto wth ay gv bomb wll mak t xplod. So you d to dtrm f a bomb s ot a dud wthout tractg wth t at all! Of cours you ca asly dtrm whch ar th duds by pokg all th bombs. Th os that do t xplod ar th duds. But ufortuatly that lavs you wth o lv bombs. I classcal physcs th problm s solubl. But quatum physcs, amazgly, t ca b solvd. A bomb s placd wth th RB bam path of th trfromtr such a way that a passg photo may or may ot tract wth t. For xampl, ths mght b accomplshd by attachg a rod to th mrror at th bottom rght such that thr was th tst gap btw th rod ad th bomb. If th mrror s fr to mov wh struck by a photo, th rod would th strk th bomb ad st f off. (Th mchasm s mpractcal, of cours, but th prcpl s what mattrs a truly practcal dvc s prfctly fasbl). Of cours, you ca t actually pck th bomb up ad mov t, or t would go off. So ths s rally shorthad for a Mach-Zhdr trfromtr s costructd aroud a gv bomb. You mght also b wodrg what us ths bombs could possbly b, sc you could vr mov thm to whr you mght wat to dstroy somthg. Hy, do t tak ths xampl so ltrally!

5 A actv bomb has thus b mad a masurg dvc rgardg whch of th paths, RB or SB, th photo taks. If t taks path RB, th bomb xplods. If th bomb dos ot xplod thr th photo took path SB or th bomb s a dud. Th vry sstvty of a actv bomb maks t a prfct masurg dvc. Now f th bomb s a dud, th th bomb dos ot costtut a masurg dvc. A dud bomb mght as wll ot b thr. So, wth a dud bomb, th photos wll always mrg to dtctor, vr to dtctor. But f th bomb s ot a dud, ad assumg t dos ot xplod, th th masurmt (of th photo path bg SB) dstroys th trfrc ad th photo could b dtctd thr dtctor or. But f t s dtctd dtctor th bomb caot b a dud! So w hav succssfully dtfd a bomb whch s dftly ot a dud but wthout t xplodg. Mraculous! Th fact that th bomb mght hav go off, but actually dd ot, s crucal to th bomb costtutg a masurg dvc ad hc to th dtfcato of th uxplodd bomb as ot bg dud. Th coutrfactual has had a obsrvabl cosquc, amly that th bomb s ow kow wth crtaty to b lv. Ths curous phomo, ad th bomb scaro dscrbd abov, was orgally dscrbd by Eltzur & Vadma, Rf.[]. Do ot thk that t s too thortcal to b dmostratd xprmtally. O th cotrary, ths was do almost as soo as th ffct was dscovrd, by Zlgr s group Va, Rf.[]. As dscrbd, th bomb tst s trrbly ffct. Of th lv bombs, half ar xplodd, ad of th rmag 50% oly half of ths rsult a photo at dtctor ad hc ar dtfd dftly as lv bombs. Thus, a sgl applcato of th bomb tstr dtfs ¼ of th lv bombs. But aothr ¼ stll rma uxplodd ad udtfd. Rug ths through th bomb tstr aga rsults a furthr ¼ of ths ¼ bg dtfd as lv. Hc, rpatd applcatos dtfy, ¼ + ¼ x ¼ + ¼ x ¼ x ¼ = / 3 of th lv bombs. But th rmag / 3 ar dstroyd trrbly ffct. Howvr, ths s avodabl. I prcpl vrtually all th lv bombs ca succssfully b dtfd, as show by Kwat t al, Rf.[]. 6. Rfrcs [] Eltzur A. C. ad Vadma L. (993). Quatum mchacal tracto-fr masurmts. Foud. Phys. 3, arxv:hp-th/ [] P. G. Kwat, H. Wfurtr, T. Hrzog, A. Zlgr, ad M. A. Kasvch (995). "Itracto-fr Masurmt". Phys. Rv. Ltt. 74 (4): A. Mrrors Appdx Drvato of th Phas Chag Ruls for Rflcto/Trasmsso Propagato through th glass of a mrror causs a phas chag of for a dstac kx x of travl, compard wth th phas chag of through ar. Th wav-umbrs k x

6 ar rlatd by k k, so th phas factor rlatv to ar propagato s as gv th abov txt. kx ka If a wav ar,, mts a glass surfac, whch th wav wll b C, th boudary codtos at th surfac ar that th wavfucto ad ts x-drvatv b kx cotuous. W must also accout for a rflctd wav, B. Hc w rqur, assumg th surfac s at x 0, Ths quatos ar radly solvd to gv, B C ad k Bk k C (A.) B ad Sc t s clar that B s ral ad gatv, corrspodg to a phas chag wth rspct to th cdt wav of 80 o, a phas factor of. Ths cofrms th phas chag rul [] of.,.., rflcto at th frot (slvrd) fac of a mrror causs a phas chag by a factor of. Now cosdr rflcto from th back fac. If a wav glass mts th ar boudary, th abov aalyss stll appls xcpt that th rols of k ad k ar rvrsd. Hc k k / ad hc s rplacd by / throughout. So th rflcto ad trasmsso coffcts ar ow rspctvly, B ad C C Th rflcto coffct s ow ral ad postv, ad so thr s o phas chag btw th rflctd ad cdt wavs, cosstt wth rul [].. QED. A. Pla Plat k x, (A.) (A.3) W must ow cosdr trasmsso through a ft thckss, a, of traspart kx kx matral. Th cdt plus rflctd wavs th rgo x 0 ar B. Wth k x k x th plat matral, 0 x a, th rght plus lft gog wavs ar E F. I th kx rgo x a th trasmttd wav s C. By applyg th boudary codtos (cotuty of th wavfucto ad ts x-drvatv) at both boudars x 0 ad x a, th coffcts B, E, F, C ar foud. I partcular th rato of th B ad C coffcts s, B C s k a ka (A.4) ka Now th factor of ust accouts for th fact that th phas of th rflctd wav has b rfrc to posto x 0 whras th trasmttd wav phas s rfrcd to x a. Rfrcg thm both to th sam pot, as rqurd, lavs th phas factor from (A.4) as ust, as clamd.. QED.

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