《The Universe and Quantum Computer 2022-11-2 232314 2.pdf》由会员分享，可在线阅读，更多相关《The Universe and Quantum Computer 2022-11-2 232314 2.pdf（15页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、arXiv:1009.6229v1 quant-ph 30 Sep 2010THE UNIVERSE ANDTHE QUANTUM COMPUTERStan GudderDepartment of MathematicsUniversity of DenverDenver,Colorado 80208sguddermath.du.eduAbstractIt is first pointed out that there is a common mathematical modelfor the universe and the quantum computer.The former is ca
2、lled thehistories approach to quantum mechanics and the latter is called mea-surement based quantum computation.Although a rigorous concretemodel for the universe has not been completed,a quantum measureand integration theory has been developed which may be useful for fu-ture progress.In this work w
3、e show that the quantum integral is theunique functional satisfying certain basic physical and mathematicalprinciples.Since the set of paths(or trajectories)for a quantum com-puter is finite,this theory is easier to treat and more developed.Weobserve that the sum of the quantum measures of the paths
4、 is unityand the total interference vanishes.Thus,constructive interference isalways balanced by an equal amount of destructive interference.Asan example we consider a simplified two-slit experiment1IntroductionBoth the universe and a measurement based quantum computer can be mod-eled as followsW U1
5、 M1 U2 M2 Un Mn(1.1)1In this process,W is the initial state given by a density operator,U1is aquantum gate(or propagator from time 0 to t1)given by a unitary operator,M1is a quantum event given by a projection operator P1(A1),U2is a quan-tum gate(or propagator from time t1to t2)given by a unitary op
6、erator,M2is a quantum event given by a projection operator P2(A2),.In the universe model,A1is one of various possible configurations andP1(A1)is the quantum event that the universe is in configuration A1at timet1.In the quantum computer model A1is a set of possible outcomes for ameasurement and P1(A
7、1)is the quantum event that one of the outcomes inA1occurs.In this model,the measurements can be adaptive in the sensethat a choice of measurements may depend on the results of previous mea-surements.The universe model is referred to as the histories approach toquantum mechanics 2,3,7,8,10,11,12,13.
8、A school of researchers believethat this approach is the most promising way to unify quantum mechanicsand gravitation.The measurement based quantum computer model is equiv-alent to the quantum circuit computer model but various researchers believeit has superior properties both in theory and for pra
9、ctical implementation1,4,9.It has been suggested that the universe is itself a gigantic quantumcomputer.This paper shows that there may be relevance to this statement.The mathematical background for these models consists of a fixed com-plex Hilbert space H on which the operators W,UiPi(Aj)act.In acc
10、ordancewith quantum principles,we assume that Piis a projection-valued measure,i=1,2,.In the universe model,H is infinite dimensional,t is contin-uous and the sets Aican be infinite.At the present time,this theory isnot complete and is not mathematically rigorous 3,8,12.Nevertheless,aquantum measure
11、 and integration theory has been developed to treat thisapproach 5,6,11,12,13.In this article we show that the quantum integraldefined in 6 is the unique functional that satisfies certain basic physical andmathematical principles.Since the universe is too vast and complicated for us to tackle in det
12、ailnow,we move on to the study of quantum computers.From another view-point,we are treating toy universes that are described by finite dimensionalHilbert spaces and finite sets.This experience may give us the power andconfidence to tackle the real universe later.In this work we make some obser-vatio
13、ns that may be useful in developing the structure for a general theory.For example,we show that the sum of the quantum measure of the paths isunity and that the total interference vanishes.Thus,constructive interfer-ence is always balanced by an equal amount of destructive interference.We2hope that
14、this article encourages an interchange of ideas between the twogroups working on measurement based quantum computation and the histo-ries approach to quantum gravity.It is fascinating to contemplate that thevery large and the very small may be two aspects of the same mathematicalstructure.2Quantum M
15、easures and IntegralsOne of the main studies of the universe is the field of quantum gravity andcosmology.In this field an important role is played by the histories approachto quantum mechanics 2,3,7,8,13.Let be the set of paths(or historiesor trajectories)for a physical system.We assume that there
16、is a natural-algebra A of subsets of corresponding to the physical events of the systemand that A for all .In this way,(,A)becomes a measurablespace.A crucial tool in this theory is a decoherence functional D:AA Cwhere D(A,B)roughly represents the interference amplitude between eventsA and B.Example
17、s of decoherence functionals for the finite case are givenin Section 3.In the infinite case,the form of the decoherence functionalis not completely clear 3.However,it is still assumed that D exists andthat future research will bear this out.It is postulated that D satisfies thefollowing conditions.(
18、1)D(A,B)=D(B,A)for all A,B A.(2)D(A B,C)=D(A,C)+D(B,C)for all A,B,C A withA B=.(3)D(A,A)0 for all A A.(4)|D(A,B)|2 D(A,A)D(B,B)for all A,B A.(5)If Ai A with A1 A2,then limD(A,A)=D(Ai,Ai)and if Bi A with B1 B2,then limD(Bi,Bi)=D(Bi,Bi).If D is the decoherence functional,then(A)=D(A,A)is interpretedas
19、 the“propensity”that the event A occurs.We refrain from calling(A)3the probability of A because does not have the usual additivity and mono-tonicity properties of a probability.For example,if A B=,then(A B)=D(A B,A B)=(A)+(B)+2ReD(A,B)Thus,the interference term ReD(A,B)prevents the additivity of.Eve
20、nthough is not additive in the usual sense,it does satisfy the more generalgrade-2 additivity condition(ABC)=(AB)+(AC)+(BC)(A)(B)(C)(2.1)for any mutually disjoint A,B,C A.Moreover,by Condition(5),satisfiesthe following continuity conditions.A1 A2 lim(Ai)=(Ai)(2.2)B1 B2 lim(Bi)=(Bi)(2.3)A grade-2 add
21、itive map:A R+satisfying(2.2)and(2.3)is called aq-measure 5,6,11.A q-measure of the form(A)=D(A,A)also satisfiesthe following regularity conditions.(A)=0 (A B)=(B)for all B A with A B=(2.4)(A B)=0 with A B=(A)=(B)(2.5)A q-measure space is a triple(,A,)where,A)is a measurable spaceand:A R+is a q-meas
22、ure 5,6.For a q-measure space(,A,)if A,B A we define the(A,B)inter-ference term IA,BbyIA,B=(A B)(A)(B)(A B)Of course,is a measure if and only if IA,B=0 for all A,B A and IA,Bdescribes the amount that deviates from being a measure on A and B.Since any q-measure satisfies()=0,if A B=thenIA,B=(A B)(A)(
23、B)(2.6)Lemma 2.1.Let(,A,)be a q-measure space with Ai A mutually dis-joint,i=1,.,n.We then have nI=1Ai!=nXi=1(Ai)+nXij=1IAi,Aj(2.7)ni=1Ai!ni=2Ai!=(A1)+nXi=2IA1,Ai(2.8)4Proof.Applying Theorem 2.2(b)5 and(2.6)we have ni=1Ai!=nXij=1(Ai Aj)(n 2)nXi=1(Ai)=nXij=1hIAi,Aj+(Ai)+(Aj)i(n 2)nXi=1(Ai)=nXi=1(Ai)+
24、nXi)d(2.9)where d is Lebesgue measure on R 6.If f:R is measurable,wecan write f in a canonical way as f=f+fwhere f+0,f 0 aremeasurable and f+f=0.We then define the quantum integralZfd=Zf+d Zfd(2.10)as long as the two terms in(2.5)are not both.If is an ordinary measure(that is,is additive),thenRfd is
25、 the usual Legesgue integral 6.Thequantum integral need not be linear or monotone.That is,R(f+g)d 6=Rfd+Rgd andRfd 6Rgd whenever f g,in general.However,the quantum integral is homogeneous in the sense thatRfd=Rfd,for all R.IfR|f|d we say that f is integrable and we denote byL1(,)the set of integrabl
26、e functions.If is a measure on A,then it is well known that the Lebesgue integralf 7Rfd from L1(,)to R is the unique linear functional satisfyingRAd=(A)for all A A where Ais the characteristic function of Aand if fi,f L1(,)with 0 f1 f2,limfi=f,then limRfid=Rfd.We now show that the quantum integral i
27、s also the unique functionalsatisfying certain basic principles.5Theorem 2.2.If(,A,)is a q-measure space,then F(f)=Rfd is theunique functional F:L1(,)R satisfying the following conditions.(i)If 0 ,0 and A B=,thenF A+(+)B=(A)+(+)(B)+u(,)IA,Bfor some function u:(,)R2:0 R.(ii)If f,g,h L1(,)are nonnegat
28、ive and have mutually disjoint support,thenF(f+g+h)=F(f+g)+F(f+h)+F(g+h)F(f)F(g)F(h)(iii)If fi,f L1(,)with 0 f1 f2,limfi=f,then limF(fi)=F(f).(iv)F(f)=F(f+)F(f).Before we present the proof of Theorem 2.2,let us interpret the fourconditions.Letting =1,=in(i)we obtain F(A)=(A)whichshows that F is an e
29、xtension of.Letting =0 in(i)we obtainF(A+B)=(A)+(B)+u(,)IA,B(2.11)which is an extension of(2.6)for.This also shows that F is linear for thesimple function A+Bexcept for an interference term.With 6=0,wecan write(i)asF(A+B)+B=F(A+B)+F(B)which essentially states that Bdoes not interfere with A+B.Condi-
30、tion(ii)is grade-2 additivity for F which is natural to expect for a functionalextension of.Condition(iii)is a generalization of the continuity property(2.2).Finally,Condition(iv)is the natural way to extend F from nonnega-tive functions to arbitrary real-valued functions in L1(,).Proof.(of Theorem
31、2.2).It is shown in 6 that f 7Rfd satisfies(i)(iv)where(iii)is called the q-dominated monotone convergence theorem.Conversely,suppose F satisfies(i)(iv).Applying(2.11)with B=we havethat F(A)=(A)for all A A.If IA,B=0 for all A,B A,then is a measure and both F(f)andRfd are the Lebesgue integral of f.I
32、fIA,B6=0 for A,B A with A B=we have by(2.11)and(i)that(A)+(+)(B)+u(,+)IA,B=F A+(+)B=(A)+(+)(B)+u(,)IA,B6Hence,u(,+)=u(,)and it follows that u(,)does not depend on.Letting v()=u(,)for all 0 we have thatF(A+B)=(A)+(B)+v()IA,BIf IA,B6=0,then letting =we obtain for A B=that(A)+(B)+IA,B=(A B)=F(AB)=F(A+B
33、)=(A)+(B)+v()IA,BHence,v()=for every 0.We conclude thatF(A+B)=(A)+(B)+IA,B(2.12)It follows from(ii)and induction that if f1,.,fnhave mutually disjointsupport,thenF?Xfi?=nXij=1F(fi+fj)(n 2)nXi=1F(fi)(2.13)If 0 1 nand Ai A are mutually disjoint,it follows from(2.12),(2.13)thatF nXi=1iAi!=nXij=1F(iAi+j
34、Aj)(n 1)nXi=1i(Ai)=nXij=1hi(Ai)+j(Aj+iIAi,Aji(n 2)nXi=1i(Ai)=nXi=1i(Ai)+nXij=1iIAi,AjIt is shown in 5 thatZ nXi=1iAi!d=1 ni=1Ai!ni=2Ai!#+n1(An1 An)(An)+n(An)7By(2.8)and similar expressions for the other terms,we conclude thatZ nXi=1iAi!d=F nXi=1iAi!Hence,F(f)=Rfd for every nonnegative simple functio
35、n.It follows from(iii)and the q-dominated monotone convergence theorem that F(f)=Rfdfor every nonnegative f L1(,).By(iv),F(f)=Rfd for every f L1(,).3Quantum ComputersThis section considers measurement based quantum computers 1,4,9.Thetheory is simpler than that in Section 2 because the Hilbert space
36、 H is finitedimensional and the sample space is finite.As discussed in Section 1,wehave n measurements given by projection-valued measures P1,.,Pn.LetOibe the set of possible outcomes for measurements Pi,i=1,.,n.ThenOiis a finite set with cardinality|Oi|=mi,i=1,.,n,where mi dimH.Using the notation P
37、i(a)=Pi(a),since Piis a projection-valued measure,we have thatXPi(a):a Oi=I(3.1)i=1,.,n,where I is the identity operator.Also,if A,B OiwithA B=it follows that Pi(A)Pi(B)=0.For Ai Oi,i=1,.,n,wecall A1 Ana homogeneous event(or course-grained history)and forai Oiwe calla1 an=(a1,.,an)a path(or trajecto
38、ry or fine-grained history).Let be the set of all pathsand A=2the set of events.Then|=m1mnand|A|=2m1mn.In accordance with Section 1,we have an initial state W given by adensity operator on H and unitary operators Uion H describing quantumgates,i=1,.,n.For two paths =(a1,.,an)and=(b1,.,bn),thedecoher
39、ence functional D(,)is defined byD(,)(3.2)=tr WU1P1(a1)U2P2(a2)UnPn(an)Pn(bn)UnP2(b2)U2P1(b1)U18Notice that D(,)=0 if an6=bn;that is,the paths dont end at the samepoint.For A,B A,we extend the decoherence functional by bilinearity toget 3D(A,B)=XD(,):A,BThen D satisfies the usual properties(1)(5)(Se
40、ction 2)of a decoherencefunctional.It follows that(A)=D(A,A)is a q-measure on A.If W=|ih|is a pure state,we have()=D(,)=kPn(an)UnPn1(an1)Un1U2P1(a1)U1k2(3.3)As in Section 2,we define the(,)interference term byI,=(,)()()=2ReD(,)We then have that(A)=XD(,):,A=X():A+X2ReD(,):,A=X():A+X?I,:,A?(3.4)Althou
41、gh the next result is well known,we include the proof because it isparticularly simple in this case.Theorem 3.1.The q-measure satisfies the regularity conditions(2.4),(2.5).Proof.To prove(2.4),suppose that(A)=0 and A B=.ApplyingCondition(4)of Section 2 we conclude that D(A,B)=0.Hence,(A B)=XD(,):,A
42、B=(A)+(B)+2ReD(A,B)=(B)To prove(2.5)suppose that A B=and(A B)=0.Again,applyingCondition(4)we have that0=(A B)=(A)+(B)+2ReD(A,B)(A)+(B)2|D(A,B)|(A)+(B)2(A)12(B)12=h(A)12(B)12i2Hence,(A)12(B)12=0 so that(A)=(B).9Although the next result is elementary,it does not seem to be well known.This result shows
43、 that the sum of the quantum measure of the paths is unityand that the total interference vanishes.This indicates that at least oneof the outcomes occurs and that constructive interference is alwaysbalanced by an equal amount of destructive interference.Theorem 3.2.For the q-measure space(,A,)we hav
44、eX():=1,X?I,:,A?=0Proof.We prove this result for a pure state W=|ih|and the general resultfollows because any state is a convex combination of pure states.Applying(3.3)we haveX()=X?kPn(an)UnPn1(an1)Un1U2P1(a1)U1k2:(a1,.,an)?By(3.1)we have thatX()=X?kUnPn1(an1)Un1U2P1(a1)U1k2:(a2,.,an)O2 On?.=kUnUn1U
45、2U1k2=1Applying(3.2)we have that()=XD(,):,=1Hence,by(3.4)and what we just proved we obtain1=()=1+X?I,:,A?and the result follows.The next theorem is a straightforward application of(3.2)10Theorem 3.3.(a)If W=|ih|is a pure state and A1 Anis ahomogeneous event,then(A1 An)=kPn(An)UnPn1(An1)P1(A1)U1k2(b)
46、If Bi A are mutually disjoint,then(A1 An1(Bi)=Xi(A1 An1 Bi)The result of Theorem 3.3(b)is consistent with the fact that the lastmeasurement does not affect previous ones.We now make two observations.Since?I,?=2|ReD(,)|2()12()12we have the inequalitiesh()12()12i2(,)h()12+()12i2(3.5)Finally,it is not
47、hard to show that the quantum integral becomesZfd(3.6)=Xf()():+X?I,min(f(),f():,A?We close with an example of a simplified two-slit experiment.Supposewe have two measurements P1,P2that have two values a1,a2and b1,b2,re-spectively.We can assume that dimH=2.We interpret a1,a2as two slitsand b1,b2as tw
48、o detectors on a detection screen.Suppose we have an initialpure state W=|0ih0|.Letting =U10and U=U2for a homogeneoushistory A1 A2by Theorem 3.3(a)we have that(A1 A2)=kP2(A2)UP1(A1)k2(3.7)We have the four paths(ai,bj),i,j=1,2 and(3.7)gives(ai,bj)=kP2(bj)UP1(ai)k2,i,j=1,2We see directly or by Theorem
49、 3.2 that2Xi,j=1(ai,bj)=111so at least one of the paths occurs.The“probability”that detector b1registers is(ai,a2 b1)=kP2(b1)Uk2and similarly(ai,a2 b2)=kP2(b2)Uk2SincekP2(b1)Uk2+kP2(b2)Uk2=1one of the detectors registers.Letting =(a1,b1),=(a2,b1),the(,)interference term becomesI,=2ReD(,)=2RehP1(a2)U
50、P2(b1)UP1(a1),iIn general I,6=0 and hence,(,)6=()+().Thus,the twopaths,ending at detector b1interfere.Similarly,the two paths endingat detector b2interfere.The“probability”that the particle goes through slit a1is(a1 b1,b2)=kUP1(a1)k2=kP1(a1)k2and the“probability”that the particle goes through slit a