Criteria for generalized macroscopic and mesoscopic quantum coherence

Transcription

1 Crtera for generalzed macroscopc and mesoscopc quantum coherence E. G. Cavalcant 1, and M. D. Red 1 Centre for Quantum Dynamcs, Grffth Unversty, Brsbane, Australa ARC Centre of Ecellence for Quantum-Atom Optcs, The Unversty of Queensland, Brsbane, Australa Receved 9 August 7; revsed manuscrpt receved 6 May 8; publshed 18 June 8 We consder macroscopc, mesoscopc, and S-scopc quantum superpostons of egenstates of an observable and develop some sgnatures for ther estence. We defne the etent, or sze S of a superposton, wth respect to an observable ˆ, as beng the range of outcomes of ˆ predcted by that superposton. Such superpostons are referred to as generalzed S-scopc superpostons to dstngush them from the etreme superpostons that superpose only the two states that have a dfference S n ther predcton for the observable. We also consder generalzed S-scopc superpostons of coherent states. We eplore the constrants that are placed on the statstcs f we suppose a system to be descrbed by mtures of superpostons that are restrcted n sze. In ths way we arrve at epermental crtera that are suffcent to deduce the estence of a generalzed S-scopc superposton. The sgnatures developed are useful where one s able to demonstrate a degree of squeezng. We also dscuss how the sgnatures enable a new type of Ensten-Podolsky-Rosen gedanken eperment. DOI: 1.113/PhysRevA I. INTRODUCTION Snce Schrödnger s semnal essay of , n whch he ntroduced hs famous cat parado, there has been a great deal of nterest and debate on the subject of the estence of a superposton of two macroscopcally dstngushable states. Ths ssue s closely related to the so-called measurement problem. Some attempts to solve ths problem, such as that of Ghrard, Rmn, Weber, and Pearle 3, ntroduce modfed dynamcs that cause a collapse of the wave functon, effectvely lmtng the sze of allowed superpostons. It thus becomes relevant to determne whether a superposton of states wth a certan level of dstngushablty can est epermentally 4. Evdence 5,6 for quantum superpostons of two dstngushable states has been put forward for a range of dfferent physcal systems ncludng superconductng quantum nterference devces, trapped ons, optcal photons, and photons n mcrowave hgh-q cavtes. Sgnatures for the sze of superpostons have been dscussed by Leggett 7 and, more recently, by Korsbakken et al. 8. Theoretcal work suggests that the generaton of a superposton of two truly macroscopcally dstnct states wll be greatly hndered by decoherence 9,1. Recently 11, we suggested to broaden the concept of detecton of macroscopc superpostons, by focusng on sgnatures that confrm, for some epermental nstance, a falure of mcroscopc or mesoscopc superpostons to predct the measured statstcs. Ths approach s applcable to a broader range of epermental stuatons based on macroscopc systems, where there would be a macroscopc range of outcomes for some observable, but not necessarly just two that are macroscopcally dstnct. Recent work by Marquardt et al. 1 reports epermental applcaton of ths approach. The paradgmatc eample 5,6,13,14 of a macroscopc superposton nvolves two states + and, macroscopcally dstnct n the sense that the respectve outcomes of a measurement ˆ fall nto regons of outcome doman, denoted + and, that are macroscopcally dfferent. We argue n Ref. 11 that a superposton of type PACS number s : 3.65.Ta, 4.5.Xa, b + + +, that nvolves a range of states but wth only some pars n ths case + and macroscopcally dstnct must also be consdered a type of macroscopc superposton we call these generalzed macroscopc superpostons, n the sense that t dsplays a nonzero off-dagonal densty matr element + connectng two macroscopcally dstnct states, and hence cannot be constructed from mcroscopc superpostons of the bass states of ˆ. Such superpostons are predcted to be generated n certan key macroscopc eperments, that have confrmed contnuous-varable 19 9 squeezng and entanglement, spn squeezng, and entanglement of atomc ensembles 3, and entanglement and volatons of Bell nequaltes for dscrete measurements on multphoton systems In ths paper, we epand on our prevous work 11 and derve new crtera for the detecton of the generalzed macroscopc or S-scopc superpostons usng contnuous varable measurements. These crtera confrm that a macroscopc system cannot be descrbed as any mture of only mcroscopc or s-scopc, where s S quantum superpostons of egenstates of ˆ. We show how to apply the crtera to detect generalzed S-scopc superpostons n squeezed and entangled states that are of epermental nterest. The generalzed macroscopc superpostons stll hold nterest from the pont of vew of Schrödnger s dscusson 1 of the apparent ncompatblty of quantum mechancs wth macroscopc realsm. Ths s so because such superpostons cannot be represented as a mture of states whch gve outcomes for ˆ that always correspond to one or other or nether of the macroscopcally dstnct regons + and. The quantum mechancal paradoes assocated wth the generalzed macroscopc superposton 1 have been dscussed n prevous papers 11,15,16,34,35. The crtera derved n ths paper take the form of nequaltes. Ther dervaton utlzes the uncertanty prncple and the assumpton of certan types of mtures. In ths respect they are smlar to crtera for nseparablty that have been derved by Duan et al. 36 and Hofmann and Takeuch /8/77 6 / The Amercan Physcal Socety

2 E. G. CAVALCANTI AND M. D. REID 37. Rather than testng for falure of separable states, however, they test for falure of a phase space macroscopc separablty, where t s assumed that a system s always n a mture never a superposton of macroscopcally separated states. We wll n ths paper note that one can be more general n the dervaton of the nequaltes, adoptng the approach of Leggett and Garg 13 to defne a macroscopc realty wthout reference to any quantum concepts. One may consder a whole class of theores, whch we refer to as the mnmum uncertanty theores MUTs and to whch quantum mechancs belongs, for whch the uncertanty relatons hold and the nequaltes therefore follow, based on ths macroscopc realty. The epermental confrmaton of volaton of these nequaltes wll then lead to demonstraton of a new type of Ensten-Podolsky-Rosen argument or parado 38, n whch the nconsstency of a type of macroscopc S-scopc realty wth the completeness of quantum mechancs s revealed 11,34. A drect analogy ests wth the orgnal EPR argument, whch s a demonstraton of the ncompatblty of local realsm wth the completeness of quantum mechancs II. GENERALIZED S-SCOPIC COHERENCE We ntroduce n ths secton the concept of a generalzed S-scopc coherence 11, whch we defne n terms of falure of certan types of mtures. In the net secton, we lnk ths concept to that of the generalzed S-scopc superpostons 1. We consder a system whch s n a statstcal mture of two component states. For eample, f one attrbutes probabltes 1 and to underlyng quantum states 1 and, respectvely where denotes a quantum densty operator, then the state of the system wll be descrbed as a mture, whch n quantum mechancs s represented as = Ths can be nterpreted as the state s ether 1 wth probablty 1, or wth probablty. The probablty for an outcome of any measurable physcal quantty ˆ can be wrtten, for a mture of the type, as P = 1 P 1 + P, 3 where P =1, s the probablty dstrbuton of n the state. More generally, n any physcal theory, the specfcaton of a state where here s just a symbol to denote the state, but not necessarly a densty matr fully specfes the probabltes of outcomes of all eperments that can be performed on the system. If we then have wth probablty 1 a state 1 whch predcts for each observable ˆ a probablty dstrbuton P 1 and wth probablty a second state whch predcts P, then the probablty dstrbuton for any observable ˆ gven such mture s of the form 3. The concept of coherence can now be ntroduced. Defnton 1. The state of a physcal system dsplays coherence between two outcomes 1 and of an observable ˆ f and only f the state of the system cannot be consdered ρ 1 1 FIG. 1. Color onlne Probablty dstrbuton for outcomes of measurement ˆ. If 1 and are macroscopcally separated, then we mght epect the system to be descrbed as the mture, where 1 encompasses outcomes, and encompasses outcomes 1. Ths means an absence of generalzed macroscopc coherence, as defned n Sec. II. a statstcal mture of some underlyng states 1 and, where 1 assgns probablty zero for and assgns probablty zero for 1. Ths defnton s ndependent of quantum mechancs. Wthn quantum mechancs t mples that the quantum densty matr representng the system cannot be decomposed n the form. Thus, for eample, = where = 1 / does not dsplay coherence between 1 and because t can be rewrtten to satsfy Eq.. The defnton wll allow a state to be sad to have coherence between 1 and f and only f there s no possble ensemble decomposton of that state whch allows an nterpretaton as a mture, so that the system cannot be regarded as beng n one or other of the states that can generate at most one of 1 or. We net defne the concept of generalzed S-scopc coherence. Defnton. We say that the state dsplays generalzed S -scopc coherence f and only f there est 1 and wth 1 S we take 1, such that dsplays coherence between some outcomes 1 and. Ths coherence wll be sad to be macroscopc when S s macroscopc. If there s no generalzed S-scopc coherence, then the system can be descrbed as a mture where now states 1 and assgn nonzero probablty only for and 1, respectvely. Ths stuaton s depcted n Fg. 1. An mportant clarfcaton s needed at ths pont. It s clearly a vague matter to determne when S s macroscopc. What s mportant s that we are able to push the boundares of epermental demonstratons of S-scopc coherence to larger values of S. We wll keep the smpler termnology, but the reader mght want to understand macroscopc as S-scopc throughout the tet. Generalzed macroscopc coherence amounts to a loss of what we wll call a generalzed macroscopc realty. The smpler form of macroscopc realty that nvolves only two states macroscopcally dstnct has been dscussed etensvely by Leggett 13,14. Ths smpler case would be applcable to the stuaton of Fg. 1 f there were zero probablty for result n the ntermedate regon 1. Macroscopc realty n ths smpler stuaton means that the system must be n one or other of two macroscopcally dstnct states 1 and that predct outcomes n regons 1 and, respectvely. The term macroscopc realty s used 13 because the def- S ρ 618-

3 CRITERIA FOR GENERALIZED MACROSCOPIC AND... nton precludes that the system can be n a superposton of two macroscopcally dstnct states, pror to measurement. Generalzed macroscopc realty apples to the broader stuaton, where probabltes for outcomes 1 are not zero, and means that where we have two macroscopcally separated outcomes 1 and, the system can be nterpreted as beng n one or other of two states 1 and, that can predct at most one of 1 or. Agan, the term macroscopc realty s used, because ths defnton precludes that the system s a superposton of two states that can gve macroscopcally separated outcomes 1 and, respectvely. We note that Leggett and Garg 13 defne a macroscopc realty n whch they do not restrct to quantum states 1 and, but allow for a more general class of theores where 1 and can be hdden varable states of the type consdered by Bell 4. Such states are not restrcted by the uncertanty relaton that would apply to each quantum state, and hence the assumpton of macroscopc realty as appled to these theores would not lead to the nequaltes we derve n ths paper. Ths pont wll be dscussed n Sec. IV, but the reader should note that the defnton of S-scopc coherence wthn quantum mechancs means that 1 and are quantum states. III. GENERALIZED MACROSCOPIC AND S-SCOPIC QUANTUM SUPERPOSITIONS We now lnk the defnton of generalzed macroscopc coherence to the defnton of generalzed macroscopc superposton states 11. Generally we can epress as a mture of pure states. Thus =, 4 where we can epand each n terms of a bass set such as the egenstates of ˆ: = c. Theorem A. The estence of coherence between outcomes 1 and of an observable ˆ s equvalent, wthn quantum mechancs, to the estence of a nonzero offdagonal element n the densty matr,.e., 1. Proof. The proof s gven n Append A. Theorem B. In quantum mechancs, there ests coherence between outcomes 1 and of an observable ˆ f and only f n any decomposton 4 of the densty matr, there s a nonzero contrbuton from a superposton state of the type S = c c + c 5 1, wth c 1,c. Proof. If each cannot be wrtten n the specfc form 5, then each s ether of form 1 or, so that we can wrte as the mture. Hence the estence of coherence, whch mples cannot be wrtten as Eq., mples the superposton must always est n Eq. 4. The converse s also true: f the superposton ests n any decomposton, then there ests an rreducble term n the decomposton that assgns nonzero probabltes to both 1 and, and therefore the densty matr cannot be wrtten as Eq.. We say that a generalzed S-scopc superposton of states 1 and ests when any decomposton 4 must contan a nonzero probablty for a superposton 5, where 1 and are separated by at least S. Throughout ths paper, we defne the sze of the generalzed superposton = c k k 6 k where k are egenstates of ˆ and each c k to be the range of ts predcton for ˆ, ths range beng the mamum value of k j where k and j are any two components of the superposton 6 so c k,c j. From the above dscussons t follows that wthn quantum mechancs, the estence of generalzed S-scopc coherence between 1 and here 1 =S mples the estence of a generalzed S-scopc superposton of type 5, whch can be wrtten as = c + c + c + +, where the quantum state assgns some nonzero probablty only to outcomes smaller than or equal to 1, the quantum state + assgns some nonzero probablty only to outcomes larger than or equal to, and the state assgns nonzero probabltes only to ntermedate values satsfyng 1. Where S s macroscopc, epresson 7 depcts a generalzed macroscopc superposton state. In ths case then, only the states and + are necessarly macroscopcally dstnct. We regan the tradtonal etreme macroscopc quantum state c +c + + when c =. IV. MINIMUM UNCERTAINTY THEORIES We now follow a procedure smlar to that used to derve crtera useful for the confrmaton of nseparablty 36. The underlyng states 1 and comprsng the mture are themselves quantum states, and so each wll satsfy the quantum uncertanty relatons wth respect to complementary observables. Ths and the assumpton of Eq. wll mply a set of constrants, whch take the form of nequaltes. The volaton of any one of these s enough to confrm the observaton of a generalzed macroscopc coherence that s, of a generalzed macroscopc superposton of type 7. Whle our specfc am s to develop crtera for quantum macroscopc superpostons, we present the dervatons n as general a form as possble to make the pont that epermental volaton of the nequaltes would mply not only a generalzed macroscopc coherence n quantum theory, but a falure of the assumpton 3 n all theores whch place the system n a probablstc mture of two states, whch we desgnate by 1 and, and for whch the approprate uncertanty relaton holds for each of the states. In ths sense, our approach s smlar to that of Bell 4, ecept that the assumpton used here of mnmum uncertantes for outcomes of measurements would be regarded as more restrctve than the local hdden varable theory assumpton on whch Bell s theorem s based. We make ths pont more specfc by defnng a whole class of theores, whch we refer to as the MUT, that embody the assumpton that any state wthn the theory wll predct the same uncertanty relaton for the varances of two ncom

4 E. G. CAVALCANTI AND M. D. REID ρ L 1 +1 ρr Ref. 11 for completeness and also ntroduce more crtera of ths type. S/ S/ FIG.. Color onlne Probablty dstrbuton for a measurement ˆ. We bn results to gve three dstnct regons of outcome:, 1,+1. patble observables ˆ and pˆ as s predcted by quantum mechancs. Ths s a pror not an unreasonable thng to postulate for a theory that may dffer from quantum mechancs n the macroscopc regme but agree wth all the observatons n the well-studed mcroscopc regme. In ths paper we wll focus on pars of observables, such as poston and momentum, for whch the uncertanty bound s a real number, whch wth the use of scalng and choce of unts wll be set to 1, so we can wrte an uncertanty relaton assumed by all MUTs as p 1, 8 where and p are the varances of and p, respectvely. Ths s Hesenberg s uncertanty relaton, and quantum mechancs s clearly a member of MUT. Other quantum uncertanty relatons that wll be specfcally used n ths paper nclude + p, 9 whch follows for the same choce of unts as that of Eq. 8 and has been useful n dervaton of nseparablty crtera 36. V. SIGNATURES FOR GENERALIZED S-SCOPIC SUPERPOSITIONS: BINNED DOMAIN In ths secton we wll derve nequaltes that follow f there are no s-scopc superpostons where s S, so that volaton of these nequaltes mples estence of an S-scopc superposton or coherence, as defned n Secs. II and III. The approach s smlar to that often used to detect entangled states. Separablty mples nequaltes such as those derved by Duan et al. 36, and ther volaton thus mples estence of entanglement. Ths approach has been used to epermentally confrm entanglement, as descrbed n Ref., among others. An epermental descrpton of the approach we use here has been outlned by Marquardt et al. 1. We consder two types of crtera for the detecton of a generalzed macroscopc superposton or coherence. The frst, of the type consdered n Ref. 11, wll be consdered n ths secton and uses bnned outcomes to demonstrate a generalzed S-scopc superposton of states + and that predct outcomes n specfed regons denoted +1 and 1 respectvely Fg., where these regons are separated by a mnmum dstance S. We epand on some earler results of S A. Sngle system Consder a system A and a macroscopc measurement ˆ on A, the outcomes of whch are spread over a macroscopc range. We partton the doman of outcomes for ths measurement nto three regons, labeled l = 1,,1 for the regons S/, S/ S/, S/, respectvely. The probabltes for outcomes to fall n those regons are denoted,, and +, respectvely Fg.. If there s no generalzed S-scopc coherence then there s no coherence between outcomes n l=1 and l= 1, and the state of system A can be wrtten as m = L L + R R, 1 where L predcts outcomes n the regon S/, R predcts outcomes n the regon S/, and L and R are ther respectve probabltes. The assumpton of ths mture 1 mples P y = L P L y + R P R y. 11 Here y s the outcome of some measurement that can be performed on the system, and P R/L y s the probablty for a result y when the system s specfed as beng n state R/L. Where the measurement performed s ˆ, soy=, there s the constrant on Eq. 11 so that P R = for S/ and P L = for S/. Now consder an observable pˆ wth outcomes p ncompatble wth ˆ, such that the varances are constraned by the uncertanty relaton p 1. Our goal s to derve nequaltes from just two assumptons: frst, that ˆ and pˆ are ncompatble observables of quantum mechancs or of a mnmum uncertanty theory, so the uncertanty relaton holds for both R/L ; and, second, that there s no generalzed S-scopc coherence. Volaton of these nequaltes wll mply that one of these assumptons s false. Wthn quantum mechancs, for whch the frst assumpton s necessarly true, that would mply the estence of a generalzed macroscopc superposton of type 7 wth outcomes 1 and separated by at least S. If the quantum state s of form 1 or f the theory satsfes Eq. 11, then p L L p + R R p, 1 where p, L p, and R p are the varances of p n the states m, L, and R, respectvely. Ths follows smply from the fact the varance of a mture cannot be less than the average varance of ts component states. Specfcally, f a probablty dstrbuton for a varable z s of the form P z = N =1 P z, then z= N =1 z+ 1 z z. We can now, usng Eq. 1 and the Cauchy-Schwarz nequalty, derve a bound for a partcular functon of varances that wll apply f the system s descrbable as the mture Eq

5 CRITERIA FOR GENERALIZED MACROSCOPIC AND... L L + R R p p =L,R =L,R p 1. =L,R 13 The left-hand sde s not drectly measurable, snce t nvolves varances of ˆ n two states whch have overlappng ranges of outcomes. We must derve an upper bound for L/R n terms of measurable quanttes. For ths we partton the probablty dstrbuton P R accordng to the outcome domans l =,1, nto normalzed probablty dstrbutons P R P R S/ and P + P R S/ : P R = R P R + R+ P Here R+ = S/ P R d= + and R = S/ P R d. It follows that R = R R + R+ + + R R+ + R, where + + and R R are the averages varances of P + and P R, respectvely. Usng the bounds R / + +, R S /4, R+ 1, and + R + +S/, we derve R + + S/ S/ + + and, by smlar reasonng, 15 L + S/ + S/ Here and are the mean and varance of the measurable P, whch, snce the only contrbutons to the regons + and are from P R and P L respectvely, are equal to the normalzed + and parts of P, so that P + = P S/ and P = P S/. We substtute Eq. 15 n Eq. 13, and use + + R and + L to derve the fnal result whch s epressed n the followng theorem. Theorem 1. The assumpton of no generalzed S-scopc coherence between outcomes n regons +1 and 1 of Fg. or, equvalently, of no generalzed S-scopc superpostons nvolvng two states and + predctng outcomes for ˆ n the respectve regons +1 and 1 wll mply the uncertanty relatons ave + p 1 17 and ave + p, 18 where we defne ave = and + +S/ + S/ +S² / Thus, the volaton of ether one of these nequaltes mples the estence of a generalzed S-scopc quantum superposton, and n ths case the superposton nvolves states + and predctng outcomes for ˆ n regons +1 and 1, of Fg., respectvely. As llustrated n Fg., the and are the varance and mean of P, the normalzed dstrbuton over the doman l= 1. s the total probablty for a result n the doman l= 1, whle = The measurement of the probablty dstrbutons for ˆ and pˆ are all that s requred to determne whether volaton of the nequalty 17 or 18 occurs. Where ˆ and pˆ correspond to optcal feld quadratures, such dstrbutons have been measured, for eample, by Smthey et al. 43. Proof. The assumpton of no such generalzed S-scopc superposton mples Eq. 1. We have proved that Eq. 17 follows. To prove Eq. 18, we start from Eq. 1 and the uncertanty relaton 9, and derve a bound that wll apply f the system s descrbable as Eq. 1 : L L + R R + p =L,R + =L,R p =L,R + p. Usng Eqs. 15 and 16 and + + R and + L we get the fnal result. B. Bpartte systems One can derve smlar crtera where we have a system comprsed of two subsystems A and B. In ths case, a reduced varance may be found n a combnaton of observables from both subsystems. A common eample s where there s a correlaton between the two postons X A and X B of subsystems A and B, respectvely, and also between the two momenta P A and P B. Such correlaton was dscussed by Ensten, Podolsky, and Rosen 38 and s called EPR correlaton. If a suffcently strong correlaton ests, t s possble that both the poston dfference X A X B and the momenta sum P A + P B wll have zero varance. Where we have two subsystems that may demonstrate EPR correlaton, we may construct a number of useful complementary measurements that may reveal generalzed macroscopc superpostons. The smplest stuaton s where we agan consder superpostons wth respect to the observable X A of system A. Complementary observables nclude observables of the type P = P A gp B, 19 where g s an arbtrary constant and P B s an observable of system B. We denote the outcomes of measurements X A, P A, P B, P by the lower case symbols A, p A, p B, p, respectvely. The Hesenberg uncertanty relaton s A nf,l p A = A p 1. We have ntroduced nf,l p A = p so that a connecton s made wth notaton used prevously n the contet of demonstraton of the EPR parado 44,41. More generally 39,41, we defne an nference varance nf p A = P p B p A p B, 1 p B whch s the average condtonal varance for P A at A gven a measurement of P B at B. The p A p B are the varances of the condtonal probablty dstrbutons P p A p B. We note that nf,l p A s the lnear regresson estmate of nf p A, but that we have nf p A = nf,l p A for the case of Gaussan states 41. The uncertanty relaton A nf p A 1 and also p A nf A 1, holds true for all quantum states 35, so that we can nterchange nf p A wth nf,l p A n the proofs and theorems below. Theorem. Where we have a system comprsed of sub

6 E. G. CAVALCANTI AND M. D. REID systems A and B, the absence of generalzed S-scopc superpostons wth respect to the measurement X A mples ave A + B + pa + p B 1 7 ave A + nf p A 1. 3 ave A,, and are defned as for Theorem 1 for the dstrbuton P A. nf p A s defned by Eq. 1 and nvolves measurements performed on both systems A and B. The nequalty 3 also holds replacng nf p A wth nf,l p A whch s defned by Eq.. Thus volaton of Eq. 3 mples the estence of the generalzed S-scopc superposton, nvolvng states predctng outcomes for X A n regons +1 and 1. Proof. The proof follows n dentcal fashon to that of Theorem 1, ecept n ths case the L and R of Eq. 1 are states of the composte system, and there s no constrant on these ecept that the doman for outcomes of X A s restrcted as specfed n the defnton of R/L. The epanson 4 for the densty matr as a mture s = r r r r where now r =,j c,j A j B, j B beng egenstates of an observable of system B that form a bass set for states of B. The generalzed superposton 5 thus becomes n ths bpartte case r = c 1 1 A u 1 B + c A u B + c j A j B, 1, 4 where u 1 and u are pure states for system B. Ifweassume no generalzed S-scopc superposton, then can be wrtten wthout contrbuton from a state of form 4 and we can wrte as Eq. 1. The constrant 1 mples P p = I=R,L I P I p where P R L p s the probablty dstrbuton of p for state R/L. Thus Eq. 1 also holds for p replacng p, as do all the results nvolvng the varances of A. Also, Eq. 1 holds for nf p A see Append B. Thus we prove Theorem by followng Eqs In order to volate the nequalty 3, we would look to mnmze nf p A, or nf,l p A = p. For the optmal EPR states, P A + P B has zero varance, and one would choose for P the case of g= 1, so that p = p A + p B, where p B s the result of measurement of P B at B. Ths case gves nf p A =. More generally for quantum states that are not the deal case of EPR, our choce of p becomes so as to optmze the volaton of Eq. 3 and wll depend on the quantum state consdered. Ths wll be eplaned further n Sec. VIII. A second approach s to use as the macroscopc measurement a lnear combnaton of observables from both systems A and B, so, for eample, we mght have ˆ = X A +X B / and pˆ = P A + P B /. Relevant uncertanty relatons nclude based on X A, P A = whch gves A p A =1 and A + B p A + p B A + B + p A + p B 4, 5 6 and from these we can derve crtera for generalzed S-scopc coherence and superpostons. Theorem 3. The followng nequaltes f volated wll mply estence of generalzed S-scopc superpostons and ave A + B + pa + p B. 8 We wrte n terms of the normalzed quadratures so that, followng Eq. 5, A + B 1 would mply squeezng of the varance below the quantum nose level. The quanttes ave,, and are defned as for Theorem 1, but we note that P n ths case s the dstrbuton for ˆ = X A +X B /. S now refers to the sze of the superposton of X A +X B /. Proof. In ths case the R/L of Eq. 1 are defned as specfed orgnally n Eq. 1 but where s now defned as the outcome of the measurement ˆ = X A +X B /. The falure of the form 1 for s equvalent to the estence of a generalzed superposton of type 4 where now refers to egenstates of X A +X B. Thus the egenstates are of the general form = j c j j A j B. The mture 1 mples Eq. 1 where now p refers to the outcome of pˆ = P A + P B, and wll mply a smlar nequalty for ˆ. Applcaton of uncertanty relaton 5 for the products can be used n Eq. 13, and the proof of theorem follows as n Eqs of theorem 1. The second result follows by applyng the procedure for proof of Eq. 18 but usng the sum uncertanty relaton 6. VI. SIGNATURES OF NONLOCATABLE GENERALIZED S-SCOPIC SUPERPOSITIONS A second set of crtera wll be developed, to demonstrate that a generalzed S-scopc superposton ests, so that two states comprsng the superposton predct respectve outcomes separated by at least sze S, but n ths case there s the dsadvantage that no nformaton s obtaned regardng the regons n whch these outcomes le. Ths lack of nformaton s compensated by a far smpler form of the nequaltes and ncreased senstvty of the crtera. For pure states, a measurement of squeezng p mples a state that when wrtten n terms of the egenstates of s a superposton such that 1/ p. Wth ncreasng squeezng, the etent S of the superposton ncreases. To develop a smple relatonshp between S and p for mtures, we assume that there s no such generalzed coherence between any outcomes of ˆ separated by a dstance larger than S. Ths approach gves a smple connecton between the mnmum sze of a superposton descrbng the system and the degree of squeezng that s measured for ths system. The drawback s the loss of drect nformaton about the locaton n phase space for eample of the superposton. We thus refer to these superpostons as nonlocatable. A. Sngle systems We consder the outcome doman of a macroscopc observable ˆ as llustrated n Fg. 3, and address the queston of whether ths dstrbuton could be predcted from mcro-

7 CRITERIA FOR GENERALIZED MACROSCOPIC AND....5 scopc, or s-scopc s S, superpostons of egenstates of ˆ alone. The assumpton of no generalzed S-scopc coherence between any two outcomes of the doman for ˆ or, equvalently, the assumpton of no generalzed S-scopc superpostons, wth respect to egenstates of ˆ, means that the state can be wrtten n the form S = S. 9 Here each S s the densty operator for a pure quantum state that s not such a generalzed S-scopc superposton, so that S has a range of possble outcomes for ˆ separated by less than S. Hence S = S S where S = c k k 3 k but the mamum separaton of any two states k, k, nvolved n the superposton that s wth c k,c k s less than S, so k k S. Assumpton 9 wll mply a constrant on the measurable statstcs, namely, that there s a mnmum level of uncertanty n the predcton for the complementary observable pˆ. The varances of each S must be bounded by It s also true that S FIG. 3. Color onlne We consder an arbtrary probablty dstrbuton for a measurement ˆ that gves a macroscopc range of outcomes. S S p S p. 3 Now the Hesenberg uncertanty relaton apples to each S the nequalty also apples to the MUT s dscussed n Sec. IV so for the ncompatble observables ˆ and pˆ S S p Thus a lower bound on the varance of p follows: p S p, 34 1 S 4 S. We thus arrve at the followng theorem. Theorem 4. The assumpton of no generalzed S-scopc S S =1 S= 4 6 FIG. 4. Color onlne P for a coherent state : = p =1. coherence n ˆ wll mply the followng nequalty for the varance of outcomes of the complementary observable pˆ p S. 35 The man result of ths secton follows from Theorem 4 and s that the observaton of a squeezng p n pˆ such that p /S wll mply the estence of an S-scopc superposton c + c +S + S +, namely, of a superposton of egenstates of ˆ, that gve predctons for ˆ wth a range of at least S. The parameter S gves a mnmum etent of quantum ndetermnacy wth respect to the observable ˆ. Here c and c +S represent nonzero probablty ampltudes. In fact, usng our crteron 36 squeezng n p p 1 wll rule out any epanson of the system densty operator n terms of superpostons of wth S Fg. 4. Thus onset of squeezng s evdence of the onset of quantum superpostons of sze S, the sze S= correspondng to the vacuum nose level. Ths nose level may be taken as a level of reference n determnng the relatve sze of the superposton. The epermental observaton 9 of squeezng levels of p.4 confrms superpostons of sze at least S=5. B. Bpartte systems For composte systems comprsed of two subsystems A and B upon whch measurements X A, P A, X B, P B can be performed, the approach of the prevous secton leads to the followng theorems. Theorem 5a. The assumpton of no generalzed S-scopc coherence wth respect to X A mples nf p A S. 38 nf p A s defned as n Eq. 1. The result also holds on replacng nf p wth nf,l p as defned n Eq.. Theorem 5b. The assumpton of no generalzed S-scopc coherence wth respect to ˆ = X A +X B / mples pa + p B S

8 E. G. CAVALCANTI AND M. D. REID Proof. The proofs follow as for Theorem 4, but usng the uncertanty relatons and 5 n Eq. 34 nstead of Eq. 33. The observaton of squeezng such that Eq. 38 s volated wll mply the estence of an S-scopc superposton c A u 1 B + c +S + S A u B +, 4 namely, of a superposton of egenstates A that gve predctons for X A separated by at least S. Smlarly, the observaton of two-mode squeezng such that Eq. 39 s volated wll mply estence of an S-scopc superposton of egenstates of the normalzed poston sum X A +X B /. VII. CRITERIA FOR GENERALIZED S-SCOPIC COHERENT STATE SUPERPOSITIONS The crtera developed n the prevous secton may be used to rule out that a system s descrbable as a mture of coherent states, or certan superpostons of them. If a system can be represented as a mture of coherent states the densty operator for the quantum state wll be epressble as = P d, 41 whch s, snce P s postve for a mture, the Glauber- Sudarshan P representaton 45. The quadratures ˆ and pˆ are defned as =a+a and p= a a /, so that = p=1 for ths mnmum uncertanty state, where here a, a are the standard boson creaton and annhlaton operators, so that a =. Provng falure of mtures of these coherent states would be a frst requrement n a search for macroscopc superpostons, snce such mtures epand the system densty operator n terms of states wth equal yet mnmum uncertanty n each of and p, that therefore do not allow sgnfcant macroscopc superpostons n ether. The coherent states form a bass for the Hlbert space of such bosonc felds, and any quantum densty operator can thus be epanded as a mture of coherent states or ther superpostons. It s known 46 that systems ehbtng squeezng p 1 cannot be represented by the Glauber- Sudarshan representaton, and hence onset of squeezng mples the estence of some superposton of coherent states. A net step s to rule out mtures of s -scopc superpostons of coherent states. To defne what we mean by ths, we consder superpostons s = c, 4 where for any, j such that c,c j, we have j s for all, j s s a postve number. We note that for a coherent state, =. Thus the separaton of the states wth respect to ˆ s defned as S =s. The separaton of the two coherent states and where s real n terms of corresponds to S =4 =s, as llustrated n Fg. 5. We net ask whether the densty operator for the system can be descrbed n terms of the s -scopc coherent superpostons, so that. S=4α+ S α =4α FIG. 5. Color onlne a P for a superposton of coherent states 1/ e /4 +e /4 here the scale s such that =1 for the coherent state. = r r s r s, 43 r where each r s s of the form 4. Each r s predcts a varance n whch has an upper lmt gven by that of the superposton 1/ e /4 s / +e /4 s /. Ths state predcts a probablty dstrbuton P = 1 P G, where P G = 1 ep s 44 Fg. 5, whch corresponds to a varance = =1+s =1+S /4. Ths means each r s s constraned to allow only 1+s, whch mples for each r s a lower bound on the varance p so that p 1/ 1/ 1+s. Thus usng the result for a mture 43, we get that f ndeed Eq. 43 can descrbe the system, the varance n p s constraned to satsfy p 1/ 1+s. Thus observaton of squeezng p 1, so that the nequalty p 1/ 1+s 45 s volated, wll allow deducton of superpostons of coherent states wth separaton at least s. Ths separaton corresponds to a separaton of S =s n between the two correspondng Gaussan dstrbutons Fg. 5, on the scale where =1 s the varance predcted by each coherent state. We note that measured values of squeezng p.4 9 would mply s.. Ths confrms the estence of a superposton of type S = c = c + + c +, 46 where a separaton of at least s = j =. occurs between two coherent states comprsng the superposton, so that we may wrte =1.1. Note we have defned reference aes n phase space selected so that the as s the lne connectng the two most separated states and j so that j = and the p as cuts bsects ths lne. Equaton 46 can be compared wth epermental reports 6 of generaton of etreme coherent superpostons of type 1/ e /4 +e /4, where =.79, mplyng =.89. The correspondng generalzed s -scopc superposton 46 as confrmed by the squeezng measurement n

9 CRITERIA FOR GENERALIZED MACROSCOPIC AND... < ρ > < ρ > FIG. 6. Plot of for a coherent state, where =.5. volves at least the two etreme states wth =1., but could nclude other coherent states wth 1.1. VIII. PREDICTIONS OF PARTICULAR QUANTUM STATES We wll now consder epermental tests of the nequaltes derved above. An mportant pont s that the crtera presented are suffcent to prove the estence of generalzed macroscopc superpostons, but there are many macroscopc superpostons whch do not satsfy the above crtera. Nevertheless there are some systems of current epermental nterest whch do allow for volaton of the nequaltes. We analyze such cases below, notng that the volaton would be predcted wthout the epermenter needng to make assumptons about the partcular state nvolved. A. Coherent states The wave functon for the coherent state s 1 = 1/4ep Ths gves the epanson n the contnuous bass set, the egenstates of ˆ. Thus for the coherent state = c = d. 48 The probablty dstrbuton for s the Gaussan Fg. 4 P = 1 = 1/ep 49 we take to be real centered at and wth varance =1. The coherent state possesses nonzero off-dagonal elements where s large and thus strctly speakng can be regarded as a generalzed macroscopc superposton. However, as and devate from, the matr elements decay rapdly, and the off-dagonal elements decay rapdly wth ncreasng separaton: 1 = 1/ep In effect then, the off-dagonal elements become zero for sgnfcant separatons 1 Fg. 6. We can epect that the detecton of the macroscopc aspects of ths superposton FIG. 7. Plot of for the superposton state 51, where =.5. wll be dffcult. Snce p=1, t follows that we can use the crteron 35 to prove coherence between outcomes of separated by at most S= Fg. 4, whch corresponds to the separaton S=. B. Superpostons of coherent states The superposton of two coherent states 47 = 1/ e /4 + e /4, 51 where s real and large s an eample of a macroscopc superposton state. The wave functon n the poston bass s = e /4 e /4 e + e. 1/4 We consder the two complementary observables ˆ and pˆ, and note that the probablty dstrbuton P for ˆ dsplays two Gaussan peaks centered on = Fg. 5 : P = 1 P G, where P G =ep / /. Each Gaussan has varance =1. The macroscopc nature of the superposton s reflected n the sgnfcant magntude of the off-dagonal elements, where = and =, correspondng to =4. In fact = e + /4 cosh cosh 5 as plotted n Fg. 7 and whch for these values of and becomes 1 e 8. Wth sgnfcant off-dagonal elements connectng macroscopcally dfferent values of, ths superpo- 1/ ston s a good eample of a generalzed macroscopc superposton 7. Nonetheless we show that the smple lnear crtera 35 and 17 derved from Eq. 4 are not suffcently senstve to detect the etent of the macroscopc coherence of ths superposton state 51, even though the state 51 cannot be wrtten n the form 1. We pont out that t may be possble to derve further nonlnear constrants from Eq. 1 to arrve at more senstve crtera. To nvestgate what can be nferred from crtera 35, we note that ˆ s the macroscopc observable. The complementary observable pˆ has dstrbuton P p =ep p / 1 +sn p / whch ehbts frnges and has varance p=1 4 ep 4 Fg. 8. There s a mamum squeezng of p.63 at =.5. However, the squeezng dmnshes as ncreases, so the crteron becomes less ef

10 E. G. CAVALCANTI AND M. D. REID P(p) p 1 p α 3 FIG. 8. a P p for a superposton 51 of two coherent states where =.5 and b the reduced varance p 1, versus. fectve as the separaton of states of the macroscopc superposton ncreases. The mamum separaton S that could be conclusvely nferred from ths crteron s S.5 at =.5. As dscussed n Sec. VII, the detecton of squeezng n p s enough to confrm the system s not that of the mture =1/ + 53 of the two coherent states. In fact, the squeezng rules out that the system s any mture of coherent states. We note though that snce the degree of squeezng p s small, our crtera s not senstve enough to rule out superpostons of macroscopcally separated coherent states. C. Squeezed states Consder the sngle-mode momentum squeezed state 48 = e r a² a. 54 Here s the vacuum state. For large values of r these states are generalzed macroscopc superpostons of the contnuous set of egenstates of ˆ =a+a, wth wave functon 4 1 = 1/4ep and assocated Gaussan probablty dstrbuton 55 1 P = 1/ep 56. The varance s =e r. As the squeeze parameter r ncreases, the probablty dstrbuton epands, so that eventually wth large enough r, can be regarded as a macroscopc observable. Ths behavor s shown n Fg. 9. The dstrbuton for p s also Gaussan but s squeezed, meanng that t has reduced varance: p 1. In fact, Eq. 54 s a mnmum uncertanty state, wth p=1/ =e r. Where squeezng s sgnfcant, the off-dagonal elements = where s large are sgnfcant over a large range of values Fg. 9. The crteron 17 for the bnned outcomes s volated for the deal squeezed state 54 for values of S up to.5. The < ρ > =.5 = (a) (b) FIG. 9. Color onlne a Probablty dstrbuton for a measurement X for a momentum-squeezed state. The varance ncreases wth squeezng n p, to gve a macroscopc range of outcomes, and for the mnmum uncertanty state 54 satsfes p=1. b The for a squeezed state 54 wth r=13.4 =3.67 whch predcts a a =.5. crteron can thus confrm macroscopc superpostons of states wth separaton of up to half the standard devaton of the probablty dstrbuton of, even as. Ths behavor has been reported n 11 and s shown n Fg. 1. Squeezed systems that are generated epermentally wll not be descrbable as the pure squeezed state 54. Ths pure state s a mnmum uncertanty state wth p=1. Typcally epermental data wll generate Gaussan probablty dstrbutons for both and p and wth squeezng p 1 np, but typcally p 1. The mamum value of S that can be proved n ths case of the Gaussan states reduces to as (a) S ma 1 (b) S S/ S/ S ma /.5 S= S= p FIG. 1. Color onlne Detecton of underlyng superpostons of sze S for the squeezed mnmum uncertanty state 54 by volaton of Eqs. 17 dashed lne of b and 35 full lne of b. S ma s the mamum S for whch the nequaltes are volated. Inset of b shows behavor of volaton of Eq. 17 for general Gaussansqueezed states. Inequalty 35 depends only on p. The sze of S ma relatve to P s llustrated n a. 1 p 618-1

11 CRITERIA FOR GENERALIZED MACROSCOPIC AND... p or nf p ncreases to 1.6. Ths s shown n Fg. 1. Analyss of recent epermental data for mpure states that allows a volaton of Eq. 17 has been reported by Marquardt et al. 1. The crteron 35, as gven by Theorem 4, s better able to detect the superpostons Fg. 1, partcularly where the uncertanty product gves p 1, though n ths case the superpostons are nonlocatable n phase space, so that we cannot conclude an outcome doman for the states nvolved n the superposton. Ths crteron depends only on the squeezng p n one quadrature and s not senstve to the product p. For deal squeezed states wth varance =, one can prove a superposton of sze S=, four tmes that obtaned from Eq. 17 Fg. 1. Epermental reports 9 of squeezng of orders p.4 confrms superpostons of sze at least S=5, whch s.5 tmes that defned by S =, whch corresponds to two standard devatons of the coherent state, for whch =1 Fg. 4. D. Two-mode squeezed states Net we consder the two-mode squeezed state 49 e r ab a b. 57 Here a,b are boson annhlaton operators for modes A and B, respectvely. The wave functon and dstrbuton P are as n Eqs. 55 and 56, but the varance n ˆ =X A s now gven by =cosh r. The ˆ =X A s thus a macroscopc observable. In the two-mode case, the squeezng s n a lnear combnaton P A + P B of the momenta P A and P B at A and B, rather than n the momentum pˆ = P A for A tself. The observable that s complementary to X A s of form P = P A gp B, where g s a constant, whch s Eq. 19 of Sec. V. We can select to evaluate one of the crtera 3, 38, and 39. Choosng as our macroscopc observable and our complementary one P A gp B, we calculate nf p A =1/ =1/cosh r 58 for the choce g= P A P B / P B = tanh r whch mnmzes nf p A 44. The applcaton of results to crteron 3 gves the result as n Fg. 1, to ndcate detecton of superpostons of sze S where S=.5 for the deal squeezed state 57, and the result shown n the nset of Fg. 1 f A nf p A 1. The predcton for the crteron of Theorem 3, to detect superpostons n the poston sum X A +X B by measurement of a narrowed varance n the momenta sum P A + P B, s also gven by the results of Fg. 1. Calculaton for the deal state 57 predcts pa +p B =e+r whch corresponds to that of the one-mode squeezed state. The predcton for the mamum value of S of Theorem 3 s therefore gven by the dashed curves of Fg. 1, and the nset. A better result s gven by Eq. 38, f we are not concerned wth the locaton of the superposton. Where we use =e r and A + B Eq. 38, the degree of reducton n nf p A determnes the sze of superposton S that may be nferred. By Theorem 5, measurement of nf p A allows nference of superpostons of egenstates of ˆ separated by at least S =/ nf p A. 59 Realstc states are not lkely to be pure squeezed states as gven by Eq. 57. Nonetheless the degree of squeezng ndcates a sze of superposton n X A, as gven by Theorem 5. Epermental values of nf p A.76 have been reported, to gve confrmaton of superpostons of sze S.3, whch s 1.1 tmes the level of S= that corresponds to two standard devatons A =1 of the vacuum state Fg. 4. More frequently, t s the practce to measure squeezng n the drect sum P A + P B of momenta. The macroscopc observable s then the poston sum X A +X B. The reports of measured epermental values ndcate 3 pa +p B.4, whch accordng to Theorem 5 mples superpostons n X A +X B / of sze S 3., of order 1.6 tmes the standard vacuum state level. The slghtly better epermental result for the superpostons n the poston sum may be understood snce t has been shown by Bowen et al. that, for the Gaussan squeezed states, the measurement of nf p A s more senstve to loss than that of p A + p B. The nf p A s an asymmetrc measure that enables demonstraton of the EPR parado 39,44, a strong form of quantum nonlocalty 41,5. IX. CONCLUSION We have etended our prevous work 11 and derved crtera suffcent to detect generalzed macroscopc or k S-scopc superpostons k1 c k k of egenstates of an observable ˆ. For these superpostons, the mportant quantty s the value S of the etent of the superposton, whch s the range n predcton of the observable S s the mamum of j where c j,c. Ths quantty gves the etent of ndetermnacy n the quantum predcton for ˆ. In ths sense, there s a contrast wth the prototype macroscopc superposton of type c +c 1 1 that relates drectly to the essay of Schrödnger 1. Such a prototype superposton contans only the two states that have separaton S n ther outcomes for. Nonetheless, we have dscussed how the generalzed superposton s relevant to testng the deas of Schrödnger, n that such macroscopc superpostons are shown to be nconsstent wth the hypothess of a quantum system beng n at most one of two macroscopcally separated states. We have also defned the concept of a generalzed S-scopc coherence and the class of MUTs wthout drect reference to quantum mechancs. The former s ntroduced n Sec. IV as the assumpton 3 and s assocated to the falure of a generalzed assumpton of macroscopc realty. Ths assumpton s that the system s n at most one of two macroscopcally dstngushable states, but that these underlyng states are not specfed to be quantum states. The assumpton of MUTs s that these component states do at least satsfy the quantum uncertanty relatons. In the dervaton of the crtera of ths paper, only two assumptons are made: that the system does satsfy ths generalzed macroscopc S-scopc realty and that the theory s a MUT. These assumptons lead to nequaltes, whch, when volated, generate evdence that at least one of the assumptons must be ncorrect. We pont out that f, n the event of volaton of the nequaltes, we opt to conclude the falure of the MUT as

12 E. G. CAVALCANTI AND M. D. REID sumpton, then ths does not mply quantum mechancs to be ncorrect, but rather that t s ncomplete, n the sense that the component states can themselves not be quantum states. It can be sad then that volaton of the nequaltes of ths paper mples at least one of the assumptons of generalzed macroscopc S-scopc realty and the completeness of quantum mechancs s ncorrect. There s a smlarty wth the Ensten-Podolsky-Rosen argument 38. In the EPR argument, the assumpton of a form of realsm local realsm s shown to be nconsstent wth the completeness of quantum mechancs. Therefore, as a concluson of that argument, one s left to conclude that at least one of local realsm and the completeness of QM s ncorrect EPR opted for the frst and took ther argument as a demonstraton that quantum mechancs was ncomplete. Only after Bell 4 was t shown that ths was an ncorrect choce. Here, as n the EPR argument, the assumpton of a form of realsm macroscopc S-scopc realsm can only be made consstent wth the predctons of quantum mechancs f one allows a knd of theory n whch the underlyng states are not restrcted by the uncertanty relatons 11. ACKNOWLEDGMENTS We thank C. Marquardt, P. Drummond, H. Bachor, Y-C. Lang, N. Mencucc, N. Korolkova, A. Caldera, P. K. Lam, H. Wseman, A. Bradley, M. Olsen, F. De Martn, E. Gacabno, B. Whaley, G. Leuchs, C. Fabre, A. Leggett, L. Plmak, and others for nterestng dscussons. We are grateful for the support from the ARC Centre of Ecellence Program, the ARC Grant No. FF458313, and the Queensland State Government. AB = B, A1 where and B are orthonormal and,1. The superscrpt B denotes the states of the anclla and the absence of a superscrpt denotes the states of the system of nterest A. We decompose each pure state that appears n the Schmdt decomposton n the bass of egenstates of ˆ as = k c,k k. By assumpton 1 = and therefore 1 = c,1 c, =. We can epand AB as AB = 1 1 B + B + c,k k B, k, A where we defne the unnormalzed 1 B c,1 B and B c, B. The nner product of these two vectors s 1 B B = c,1 c,. But as shown above c,1 c, =, so 1 B and B are orthogonal. We can therefore defne an orthonormal bass wth the normalzed 1 B = 1 / B c,1 and B = / B c,, plus addtonal j B wth 3 j D, where D s the dmenson of subsystem B s Hlbert space. Takng the trace of AB = AB AB therefore yelds =Tr B AB = 1 B AB 1 B + B AB B + j B AB j B. j A3 Now referrng to epanson A, we see that 1 B AB 1 B = c,1 1 1 and B AB B = c,. We then defne 1 1 1, 1 c,1, =1 1, and 1 c, + j j B AB j B. Obvously 1 =, and by substtutng Eq. A nto we see that 1 1 =. Therefore can be decomposed as = wth 1 1 = 1 = as desred. APPENDIX A: PROOF OF THEOREM A We wll now prove the statement that coherence between 1 and s equvalent to a nonzero off-dagonal element 1 n the densty matr. As dscussed n Sec. II, wthn quantum mechancs the statement that there ests coherence between 1 and s equvalent to the statement that there s no decomposton of the densty matr of form where 1 and are densty matrces such that 1 1 = 1 =. Therefore Theorem A can be reformulated as sayng that 1 = f and only f such a decomposton does est. It s easy to prove the frst drecton of the equvalence: f 1,, 1, such that = and 1 1 = 1 =, then 1 =. To show ths, frst note that for any densty matr and,, f = then =, where =,. Snce by assumpton 1 1 = 1 =, then 1 = 1 =. The converse can also be proved. We use the facts that any can always be wrtten as the reduced densty matr of an enlarged pure state, where the system of nterest call t A s entangled wth an anclla B,.e., =Tr B AB AB ; and that any bpartte pure state can always be wrtten n the Schmdt decomposton 51 APPENDIX B We wsh to prove that f can be wrtten as m = L L + R R, then nf,m p A L nf,l p A + R nf,r p A, where nf,j p A = J p B J p A p B. p B The subscrpt J refers to the J from whch the probabltes are calculated. We have nf,m p A = P m p B m p A p B p B = p B p A P m p A,p B p A p A p B m p B p A I P I p A,p B p A p A p B I. I=R,L The nequalty follows because p A p B m s the mean of P p A p B for m, and the choce a= p P p p= p wll mnmze p P p p a. From ths the requred result follows