Information*Loss*and* Entanglement*in* Quantum*Theory* Jürg%Fröhlich% ETH%Zurich%&%IHES%

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1 Information*Loss*and* Entanglement*in* Quantum*Theory* Jürg%Fröhlich% ETH%Zurich%&%IHES%

2 Some-basic-claims-to-be-justi>ied,-afterwards)- 1. The%quantum'mechanical-state-the% wave%function )%does% not-have%an% ontological%status.%it%does%not-describe% what% is.%it%is%a%mathematical%device%enabling%us%to%make%bets% about%what%may%happen%in%the%future.% 2. In%the%presence%of%information%loss%and%entanglement%with% lost%degrees%of%freedom,%pure-states-typically%evolve%into% mixed-states-without%violation%of%any%basic%principles%of% quantum%theory),%and%it%is%this%fact%that%makes%a%rational% theory%of%measurements/observations%possible!%% 3. Without%fundamental%loss-of-information-and%entanglement%% of%observed%degrees%of%freedom%with%unobservable/ lost - degrees%of%freedom,%retrieval%of%information%about% quantum%systems%by%measurements/observations%would% actually%be%impossible.%%

3 Basic-claims,-ctd.- 4. No%information'%or%unitarity-paradoxes%in%quantum% theory,%even%if%spacertime%is%curved,%e.g.,%in%the% presence%of%black%holes)!%time%evolution%of%states%of% quantum%systems%exhibiting%information%loss%and% entanglement%with%unobservable%degrees%of%freedom% that,%for%example,%may%have%disappeared%in%a%heat% bath,%or%escaped%towards%intinity,%or%have%fallen% through%a%horizon%of%a%black%hole)%is%actually%never% unitary% %it%is% treerlike!% 5. Operator%algebras%including%type%III 1 %factors!)%and% other%sophisticated%mathematical%tools%have%been% invented%to%be%used-in%quantum%theory,%rather%than%to% be%ignored!%% %

4 CreditsandContents: Motivation from recent experiments, in particular the onesoftheharoche1raimondgroup;papersbybauer& Bernard,Maassen&Kümmerer,andothers;jointwork with my former PhD student Baptiste Schubnel;some joint efforts with Ballesteros, Faupin, Fraas, Pickl, Schilling, Schubnel); discussions with Detlev Buchholz andothers. 1. Introduction 2. InformationLoss 3. ProjectivevonNeumann)measurements 4. Conclusions

5 1.%Introduction% %Questions%to%be%Addressed% In our courses, we tend to describe quantumg mechanical systems as pairs of a Hilbert space,, and a propagator, Ut,s), describing timegevolution. Unfortunately,thesedataencodealmostnoinvariant structurebesides spectral properties of Ut,s)) and give the erroneous impression that quantum theory might be deterministic. Thus, among fundamental problemsofquantumtheoryare: Whatdowehavetoaddtotheusualformalism ofquantumtheoryinordertoarriveatamatheg maticalstructurethatthrough interpretation ) canbegivenphysicalmeaning,independentlyof observers?

6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%Questions,%ctd.% Where% does% intrinsic- randomness- in% quantum% theory% come% from,% given% the% deterministic% character% of% the% Schrödinger%equation?%In%which%way%does%it%differ%from% classical%randomness?% Do% we% understand% the% effective- dynamics- of% quantum% systems,% e.g.,% ones% in% contact% with% a% heat% bath% Quantum- Brownian- Motion ),! or% systems% under% repeated%measurements"exhibiting quantum-jumps )?% What% do% we% mean% by% an% isolated- system - in% quantum% mechanics,% and% why% is% this% an% important% notion?% How% does%one%prepare%a%system%in%a%prescribed-state?% %%%%%Answers%and%insight%come%from-understanding%roles%of% Information-Loss-&-Entanglement-

7 2.%Information%Loss% Asimple1mindeddeUinitionofquantum1mechanicalsystems: Anisolatedquantumsystem,S,ischaracterizedby followingchoices: i)u=unitarypropagator) ii)alist,,ofbounded,selfadjoint operatorsonrepresentingphysicalquantities/ potentialpropertiesofsthatcanbemeasured/ observedindirect projectivemeasurements ; Smustbeanisolatedsystemchosenlargeenoughfor quantitiesrepresentedby,tobemeasureable).

8 Information%Loss,%ctd.% ChooseTiducialtime,t 0,anddeTineHeisenbergpicture) tobetheoperatorrepresentingthepot.prop.corresp. toattimet;!listofops. Pot.properties,,measureable/observableattimes s tgenerateaw*1alg.,: 1) " 2) #InformationLoss!

9 DeTine Information%Loss,%ctd.% sothat isa*endom;nota*autominformationloss """"""""""""" entanglementwith lost degreesoffreedom! Itiseasytoconstructexamplesofgenerallynon1 autonomous)quantumsystemsexhibitinginformationloss, inthesenseofeq.2): 1) Independent probes,e j,j=1,2,3,,withe j being destroyedattimeτj,τdiscretetimestep) 2) Small systemstemporarilyinteractingwithquantized wavemedium,e.g.,photons,phonons,etc.)

10 %%3)Information-loss-in-theories-like-QED% %see%buchh.rrob.- Allops.in arelocalized in for t > t 0 worldlineofjf

11 3.%An%application:%Projective%von%Neumann)% measurement% Wewillnowdiscusshowphys.quantitiesaremeasuredprojectively.% Somefundamentalquestionstobeaddressed: 1) Whatismeantbya measurement of?aroundwhich timetdoesittakeplace?ameasurementofoughttoresult in havingavalue,i.e.,becomean empirical/objective property ofs at%some%time%t.%proj.-measnts.-vs.-indirect-kraus)-measnts.)- 2)%%Given%a%state%of%S,%does%QM-predict%which%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%will%be%measured%tirst;%what%does%qm-predict%about%the%% %%%%%%%%outcome%of%%measnt.%of%%%%%%?%in%which%way%is%qm-intrinsically% %%%%%%%%indeterministic?%why%does%a%measnt.%of%%%%%%%have%a-random outcome?%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "

12 Projective%measurements,%ctd.% Projective-measurements- We%have%to%clarify%what%may%be%meant%by%a%projectivemeasurement-of%a%potential%property%%%%%%%%%%%%%%%%%%%%and%what%the% role%of%information-loss-&-entanglement-in%measnts.%is:% % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%A4)%%%%%%%%%%%% % %%%%%%%% Measurement/observation -of%%%%%%%%around%time%t% %%%%%%%%%%%is%an% empirical/objective-property -of%s-around%time%t-:- % 3)

13 Projective%measurements,%ctd.- """"State%%%%%%%is%an%incoherent-superposition-of%eigenstates%of%%%%%,%%%%%% ,-4)-- - %%%%%where%%%%%%%%%is%the%state%of%%s--right%before%measnt.%of%%%%%,%i.e.,%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%""5) % %%%%%Information-Loss is%usually%a%mixed%state%% %%%%%on%%%%%%%%%%even%if%the%initial%state%of%s-has%been%pure,%as%a%% %%%%%state%on%%%%%%%!%

14 Projective%measurements,%ctd.% Suppose,%for%simplicity,%that%%%%%%%%%%is%isomorphic%to%some% B t ),%i.e.,%%%%%%%%%is%of%%%%%%%%%%%%%%%%%%%%%%%%syst.%non'autonomous)%%% %%%%%%%%%%%%%%%%%%%%%%Eq.%4)6) De>inition- % %%%%%%%%%%%%%%%%%%is%measured/observed%around%time%t is%an% empirical/objective-prop. -of%s-around%time%t----iff" %%7) and%some%z-in%the%center%of%%%%%%%%%..%more%generally,%%%%%%%%%%% belongs%%to%the%center-of-the-centralizer-of-the-state , Note%that%R.S.%of%7)%is%cond.expectation%of%%%%%%%%%%%w.r.%to% %%%%%%%%%%%%Eq.%7)%%%%%%%%Eq.%4)!%%$%%TomitaRTakesaki%theory!)% " -

15 Projective%measurements,%ctd.- Axiom-A- If%%%%%%%is%measured%i.e.,%an%empirical/objective%prop.%of%S)% around%time%%t--then%%%%%%%has%a%value%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%around% time%%t %8) %%%%%%%%%%%%%%9) should%be%used%for%improved%predictions%of%future%after% time%t.-eq.8)isborn srule,eq.9) collapse-postulate.)% %%%%%%%%%%%%%%%%% -

16 Projective%measurements% %summary% 1) Given%the%initial-state-of%the%system%S,"time-evolution,% {Ut,s)},%determines%which%pot.%prop.%%%%%%%%%%%%%%%%%%%will%Tirst%be% measured%i.e.,%become%empirical/objective),%and%around% which%time!% 2) Measnt.%of%%%%%%%%%is%independent-of%an%earlier-measnt.%of%%%%%%%%% iff%%%%%%%%%%becomes%empirical/objective%after-time%of%measnt.% of%%%%%%%%%,%no%matter%what%the%outcome%of%measnt.%of%%%%%%%was,%% i.e.,%for-all-states ,-j=-1,- -,-k,-with%%%%%%%%%%%%%%%%as%in%9).% Decoherence,- consistent-histories.- 3) Time%of%measurement:--Time,%%t *-,%of%observation%of%%%%%%%%% %%%%%%%det.%by%minimizing%in%%t%%the%fu.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%where%%%%%%%%% is%the% cond.-exp. -of%%%%%%%%%%%%%onto%%%%%%%%%%%%%- 4) %%%%%%%%%%=%center%of%centralizer%of%%%%%%%%%%%)%contains%all%phys.%% %%%%%%%%quantities%observable%at%time%t.% %%%%%%% -

17 Projective%measurements% %summary% 5)%A%state%is%called% passive -iff%the%center,%%%%%%%%%,%of%the%% %%%%%%%centralizer%of%%%%%%%%%%%is%time'independent.%there%are%plenty%of%% %%%%%%%examples%of%passive%states:% %%%%%%%Equilibrium%KMS)%states%at%positive%temperature%in%% %%%%%%%%%%%QFT;%KMS%states%of%a%QFT%in%the%spaceRtime%of%a%static% %%%%%%%%%%%black%hole.% %%%%%%%Perturbations%of%the%vacuum%state%by%coherent%clouds%of%% %%%%%%%%%%%massless%particles%e.g.,%of%photons% %courtesy%of%d.%%%% %%%%%%%%%%%Buchholz).% Passive%states%have%the%property%that%they%do%not%admit-anyprojective-measurements/observations%of%any%physical%quantities% % besides%measurements%of%time'independent-parameters% characteristic%of%the%state%in%question,%e.g.,%the%temperature%or%a% chemical%potential%of%an%equilibrium%state,%which,%indeed,%are% timerindependent%quantities).%

18 4.%Conclusions% Wehavetolearnmoreaboutwhichstatesandwhichtypesof timeevolutionsofisolatedsystemsallowfornon1trivial measurements/observationsoftime1dependentphysical quantitiessatisfyingaxioma. Needforilluminatingexamplesandmodels! Ex.:Dynamics ofatwoglevelatominaheatbathmonitoredbyphoton detectors,see,e.g.,deroeckgderezinski,bauergbernardg ): P t : occupation prob. of upper level, dw t : Brownian motion;, p: consts.) Statesofquantumsystemsdonothaveany ontological signiticance.the ontology ofqtresidesintimegordered sequencesof events =resultsofprojectiveorindirect observationsaccompaniedbyinfo.lossandentanglement withunobservabledegreesoffreedom).

19 EffectiveDynamics JürgFröhlich ETHZurich&IHES Paris,March2015! Why!should!I!blame!anyone!but!myself!if!I!cannot!understand! what!i!know!nothing!about?! PabloPicasso!

20 CreditsandContents Credits: Abou!Salem,!Albanese,!Bach,!Ballesteros,!Bauerschmidt,!Benettin,!!!! Blanchard,!De!Roeck,!Faupin,!Fraas,!Gang,!Giorgilli,!Griesemer,!Knowles,! Merkli,!Pizzo,!Schenker,!Schlein,!Schnelli,!Schubnel,!Schwarz,!Sigal,! Spencer,!Ueltschi,!Wayne.!! Today,tworecentexamplesfromquantumtheory,with:Schenker;! Ballesteros,!Fraas,!Schubnel!! Contents:! 1. Introduction surveyofexamples 2. ThermalnoisekillsAndersonlocalization 3. DynamicsofaQuantumSystemunderrepeatedmeasurements 4. Indirect+measurements+&+ pointer+observables 5. Analogy+to+classical+statistical+mechanics+ 6. Effective+dynamics+of+pointer+observables 7. Conclusions

21 1.!Introduction!!survey!of!examples!! Returntoequilibrium Masterequations,kineticlimit,etc. examples:equilibrationofspinsoroscillators,thermalionization, easyhalfof0 th!law, ;withb&sig;m seealsojac&pil) Relaxationtoagroundstate scatteringtechniques,cluster expansionsinreal[timevariable[derexample:decayofexcited atomormolecule;preparationofstatesinqm,etc.;withgr,s;schub) ApproachtoaNESSortoaNEtime[periodicstate scattering techniques,resonancetheoryforliouvillians,dynamicswithentropy! production,!onsagerrelationsapplication:e.g.,2 nd! Law!of! Thermodynamics;withM&Ue,A[S) AndersonLocalization multi&scale+analysis,smalldenominators examples:absenceofdiffusionforaquantumparticleinarandom[ or1dqppotential,absenceofheattransportindisorderedarraysof anharmonicoscillators, ;withsp;sp&w;a;b&g)

22 Survey!of!examples,!ctd.! QuantumBrownianMotion,position[spacedecoherence, equipartition derivationofdiffusivemotion from scratch example:atomwithdinitelymanyinternalstatescoupledto aqmheatbathandhoppingonalattice;withder,piz) Staticdisorderand!thermalnoise:Weakdiffusivetransport examples:dyn.ofabec,electrontransportinasparselypopulated conductionbandofadisorderedsemi[conductor:withj.schenker) Quantum[andHamiltonianFriction frictionthroughemissionof Cherenkov!radiation!examples:heavyatommovingthroughaBose gasexhibitingbec;dipolemovingthroughopticallydensemedium; withgangzhou&sof) Dynamicsofquantumsystemssubjectedtorepeatedmeasurements, emergenceoffactsinqm statisticsofmeasurementprotocols, stochasticevolutioneqs.,largedeviations; theoryofknowledge acquisition.

23 Survey!of!examples,!ctd. example:experimentsofharoche[raimondgroup,electron conductioninpresenceofcoulombblockade,etc.;withb,f,sch) Dynamics+in+limiting+regimes+ mean[dieldlimit,kineticlimit Gross[PitaevskiidescriptionofBosegases,Bogliubov[)Hatree[Fock: e.g.,neutronstars;point[particlelimitofnlhartreeeq.,etc.;with Schwarz,Kn.andothers) Postmodern!examples!of!effective!dynamics:! Quantumchaosvs.quantumintegr.behavior;e.v.statistics, Quantumquenches,dynamicalproblemsrelatedtohardhalfof0 th! Law!of!Thermodynamicse.g., ETH ) Many[bodylocalizationseealsoresultsw.A;Sp&W;B&G) Veryfastprocesses,suchasionization,involvinglaser)lightF[P[S) Dynamicsofinvertedpopulationsdynamicsofnegative[ temperatureinitialstates;entanglementdynamics; ) Unfortunately,Idon thavemuchtosayabouttheseexamples,yet who!has? Iwilllimitmyattentiontoquantum!systems!

24 2.!Thermal!noise!kills!Anderson!localization!! with!j.!schenker)! Inthissectionweconsideraquantumparticlehopping onthesimplecubiclattice,d!=2,3,underthe indluenceofarandom!potential!andcoupledtoaheat! bathatsomepositivetemperature.thestateofthe particleisdescribedbyaone[particledensitymatrix,.1) ThedynamicsofisdescribedbyaLiouvilleequation 2) wheret!denotestime,andisthe Liouvillian givenby a Lindblad!generator!oftheform

25 Thermal!noise,!ctd.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!!!!!!!!!!!!!!!!!!3) wherethehamiltonian 4) isastandardrandom!schrödinger!op.!anderson Hamiltonian),witharandom potentialtheareiidrv swith,e.g.,bd.distr.,,of cpt.support),g!!isa gain!term!andl!!isa loss!term. Introducingthevariables, wecanwritetheintegralkernelofasafunctionof,i.e.,as.theng!andl!aregiven

26 Thermal!noise,!ctd.! bytheformulae 5) wherethekernelsatisdiesadetailed[balance conditionatthetemperatureoftheheatbathw.r.t. thekineticenergy,,oftheparticle). Ourmainresultisthefollowingtheorem.

27 Thermal!noise,!ctd. Theorem.ConsiderEq.2)for,with. Ifthecouplingconstantg!>0thenthediffusion!constant!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! wheredenotesanexpectationw.r.totheproduct measureexistsandsatisdies

28 Thermal!noise,!ctd.!! existsandisdinite. Theproof!ofthistheoremmakesuseofthefollowing formalism.letωdenotethespaceofrandomvariables ω[).wededinethehilbertspace Fouriertransformationofvectors,Ψ,inisdedinedby LetdenotetheFouriertransformof.

29 Thermal!noise,!ctd. Thensatisdiestheequation where= Fouriertransformof seeblackboard). ThenthediffusionconstantD/d!isgivenbytheexpression Dediningthevector,wedindthat TheR.S.canbeestimatedusingtheFeshbach[Schurmap. Well,thisisalittlesketchy!Butdetailsarefairlyeasy.)