Adapted operator representations: Selective versus collective properties of quantum networks Alexander Otte and Gu¨nter Mahler Institut fu¨r Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Quantum networks (composite systems) may be de- Basedonlocalunitaryoperatorsactingonan-dimensional scribedintermsofproduct-operators,Qˆ;ifeachlocalop- Hilbert-space,weinvestigateselectiveandcollectiveoperator eratoristakenfromthe respectiveunitarybasisset,also basis sets for N-particle quantumnetworks. Selective cluster theproduct-operatorsareunitary. Furthermore,theyare 1 operators are used to derive the properties of general cat- completelyspecifiedbythetypeof”action”tobeapplied 0 statesforanynandN. Collectiveoperatorsareconveniently on each subsystem µ, µ=1,2,...,N; these Qˆ-operators 0 used to account for permutation symmetry: The respective 2 Hilbert-space dimension is then only polynomial in N and will be termed ”selective”. Alternatively, we may introduce operators Eˆ which specify the action but not the n governed by strong selection rules. These selection rules can a be exploited for the design of decoherence-free subspaces as ”address”. Inthiscasewearenaturallyledto”collective J well asfor theimplementation of efficient routestoentangle- operators”, defining a specific action on a given number 0 ment if suspended switching between states of different sym- α of subsystems. Complete sets require the inclusion of 3 metry classes could be realized. phases. Collective as well as selective operator sets are equivalent: in particular, we can express one type by 1 the other. A subset of collective operators has permuta- v tionsymmetry: These are the only allowedoperatorsfor 8 fundamentally indistinguishable subystems (fermions or 3 I. INTRODUCTION bosons). 1 1 Typical scenarios realized, e.g. in nanostructures, 0 Problem-adapted representations are convenient tools will exhibit neither complete selectivity nor complete 1 fordealingwithconcretephysicalmodelsinvirtuallyany non-selectivity (permutation symmetry). However, sys- 0 branchofphysics: Examplesarethechoiceofcoordinates tems with weakly restricted selectivity should still more h/ in classical mechanics, of mode representations in linear efficiently be described by Qˆ-Operators, systems with p wave theory, of state vector- or matrix-representations weakly broken permutation symmetry by collective op- - and ofcomplete operatorbasis sets in quantummechan- erators. It is this latter theme of operational (partial) t n ics. Themodeofadaptionmayrefertotheinternalsym- indistinguishability which will be of central interest in a metry of the system under consideration and/or its cou- our present investigation. u pling to the outside world (means of measurement and Our paper is organized as follows: In section II we q control). Adapted representations - though in principle discuss local basis operators with special emphasis on : v equivalenttoanyother-areexpectedtosimplifynumer- unitary operators and introduce the complex coherence Xi ical calculations and to enhance insight. (cf. [1]) vectorasageneralizationtotheBlochvector. SectionIII Operators in quantum mechanics may represent ob- extends this concept to quantum networks by the use of r a servables, states or transformations. In any case it is index-selective cluster operators. Alternatively, section convenienttothink interms of”elements”,i.e. basicop- IV proposes the use of collective operators for which all eratorsets,outofwhichanyotheroperatorcouldbecon- subsystems aretreated onequalfooting. Applications of structed [2]: For a n-dimensionalHilbert-space there are theseconceptsareworkedoutinsectionV,startingwith n2 such (orthogonal) basis operators (defining Liouville- highly entangled states and generalized cat states. Per- space). Ifweprefertothinkintermsofbasicobservables mutation symmetry plays the key role in the remaining or states, these basis operators should be chosen hermi- applicationsshowingthefundamentaldifferencebetween tian; a pertinent example are the SU(n)-generators. If selective and collective treatment of quantum networks. we rather think in terms of basic ”actions”, the basis- operators should be unitary (thus defining basic unitary transformation). The latter approach has become the II. BASIS OPERATORS method of choice for investigations relating to quantum computation and quantum information processing [3]: A. Transition- and SU(n)-operators there we are typically concerned with sequences of unitary transformations. (Only for n = 2 are unitarity and The Hilbert-space of dimension n is taken to be hermiticitycompatiblerequirementsforacompletebasis spannedbyacomplete andorthonormalsetofstates j , set.) | i j =0,1,2,...,n 1. Theymaybeconsideredeigenstates − 1 of some operator Aˆ, ii. The operators based on projection operators Pˆ jj are kept separated from the others. n 1 Aˆ= − AjPˆjj (2.1) As a result of i., the product-operators of a composite j=0 system (cf. Sect. III,IV can be decomposed into a hier- X achyofm-cluster-operatorsQˆ,wheremisthe numberof werhaetroersP,îwjh=ich|iiahrje|;chaPrâi†jct=eriPˆzjeiddbeynottweons2tattreaninsidtiicoenso(p=- indAicsesasre6=su0lt. of ii., and if Aˆ = Hˆ, the Hamiltonian of quantum numbers = eigenvalues Aj); they are orthonor- thesystemunderconsideration,theexpectationvaluesof malized,tr PîjPˆk†l =δikδjl,(tr=traceoperation)and the Pˆjj orcombinationsof those areconstantsofmotion (under the unitary evolution generated by Hˆ). complete. Anny otheor operatorBˆ can then be written as s Bˆ = B Pˆ (2.2) s s,ij ij B. Unitary operators ij X where Bs,ij = tr BˆsPî†j . We require the Bˆs, (s = The operators [5,6] 0,1,... ,n2 1) tonform aocomplete orthogonal set, nor- n 1 malized to n−, Uâb := − ωnbk|k+aihk| ; ωn =e2nπi (2.8) k=0 X tr BˆsBˆs†′ = Bs,ijBs∗′,ij =nδss′ . (2.3) are defined in a double index notation a,b [0,n 1], n o Xij where addition modulo n is denoted by und∈erlining−and (Remember that the identity operator ˆ1 is also normal- the constant ωn is the n-th root of unity. We note for later reference, that ized to n.) Note, that the transformation eq. (2.2) does not change the representation of any given operator; rather we change the set of operators within n 1 n 1 tthrieceAsˆ)-.repAresweneltla-ktinoonwn(i.ee.xamthpeleseatreoftheelemn2enhtearrmyitmiaan- − ωnbk =nδ0b ; − ωnb(k−k0) =nδkk0 , (2.9) k=0 b=0 SU(n)-generators λˆ = λˆ ,s = 0,1,...,n2 1 = X X s ˆ1,uˆ ,uˆ ,...,uˆ ,...,vˆ{ ,...,wˆ ,...,wˆ − } where δij is the Kronecker-delta. The use of two in- 01 02 12 01 0 n 2 { − } dices leads to an easy interpretation of the action such n 1 an operator has on a state k , λˆ ˆ1= − Pˆ , (2.4) | i 0 jj ≡ Xj=0 Uâb|ki=ωnbk|k+ai , (2.10) n uˆ = (Pˆ +Pˆ ), (2.5) namely b induces a phase shift and a causes a state-shift ik ik ki r2 (for a concrete example see Appendix A). This set of n2 n operatorsdefines an orthonormaland complete operator vˆ = i(Pˆ Pˆ ), (2.6) ik 2 ik− ki basis for the Liouville-space with r n wˆl =− (l+1)(l+2)(Pˆ00+... (2.7) tr UâbUˆc†d =nδacδbd . (2.11) r +Pˆ (l+1)Pˆ ), n o ll− l+1,l+1 A operator Aˆ can be expanded like gweintherattronrλˆssaλˆrse′ooft=ennnoδsr,ms′alfiozredaltlos2, rsa′.the(rTthheanSUn([n2)]-; Aˆ= n1 n−1 uabUâb ; uab :=tr Uâ†bAˆ . (2.12) the latter choice is more convenient for our purposes.) aX,b=0 n o For n = 2 the representation of the λˆ = λˆ ,uˆ ,vˆ ,wˆ j 0 01 01 0 Coming next we will state some of the basic properties leads to the well-known Pauli matrices. For n > 2 the these unitary operators obey. Because Uˆ = ˆ1, the or- SU(n)-algebratendstolooseitsconvenience,becausethe 00 thonormality relation (2.11) implies tr Uˆ = 0 for corresponding operators are no longer unitary and their ab definition becomes rather unwieldy (cf. [4]). a,b = 0. This tracelessness will be essenntial foor the def- 6 With the unitary operators to be discussed next, the inition of cluster-operators (c.f. Sect. III). The adjoint SU(n)-operators share two properties: operatorsareagain,barringaphase,membersoftheset, i. sTh=e0seatndcotnhseisutsniotf-onp2er−at1ortrBâce=lesˆ1s. operators Bˆs, Uâ†b =Uâ−b1 =ωnabUˆ−a,−b . (2.13) 0 6 2 A very useful property of these operators is their cyclic C. Complex coherence vector symmetry,implyingthatanyproductreducestojustone operator of the set: The general state of a quantum-mechanical system is specified by its density-operator ρˆ = ρ Pˆ with Uˆ Uˆ =ωbcUˆ (2.14) i,j ij ij ab cd n a+c,b+d UâbUˆc†d =ωncdωn−bcUâ−c,b−d (2.15) ρInijm=antyrnarρêPaîs†joofpdheyfisniicnsg,etshpeecrieaslplyecfotirvfiePnditeen-sdiitmyemnsaitornixa.l Uˆc†dUâb =ωncdωn−adUâ c,b d (2.16) state spaces, the description of ρûsing a coherence vec- − − Uˆ Uˆ Uˆ =ωbcω(b+d)eUˆ (2.17) tor (as the set of expectation values of the underlying ab cd ef n n a+c+e,b+d+f SU(n)-operator basis) has shown great power because It is remarkable to note that the cyclic property allows of its intuitive, almost ”classical” behaviour. For two- to use - instead of the complete set Uˆ - the two op- level systems the coherence vector lives in an ordinary ab erators Uˆ ,Uˆ only. All t{he o}thers can then 3-dimensional space. In general, the SU(n) basis, which be gener{at0e,dn−a1s spne−c1i,fi0c}product forms, e.g. for n = 3: is hermitian, leads to a (n2 1)-dimensional vector con- − Uˆ = (Uˆ )3,Uˆ = (Uˆ )2,Uˆ = Uˆ Uˆ ,Uˆ = sistingofrealvalues,while theunitaryoperatorsstudied 00 20 10 20 22 20 02 01 (Uˆ )2,Uˆ = (Uˆ )2 Uˆ , etc. This prop·erty is remi- here give complex elements. nes0c2ent o1f2the cre2a0tion·- a0n2d destruction operators,aˆ ,aˆ, For the operators Uâb we will henceforth use a single † index notation Uˆ with i [0,n2 1], interpreting ab as convenientlyappliedtoharmonicoscillatormodels;itin- i ∈ − the n-adic number representing i. troducesakindof”non-linearity”ifexpectationvaluesof Following eq. (2.12), we can expand the density oper- products (cf. eq. (2.28)) are approximated by products ator ρâs of expectation values. From eq. (2.17) and eq. (2.13) we conclude that n2 1 ρˆ= 1 − u Uˆ (2.26) Uˆ0†d·Uâb·Uˆ0d =ωn−daUâb (2.18) n i=0 i i Uˆd†0·Uâb·Uˆd0 =ωnbdUâb (2.19) and collect the coefficienXts to define the complex co- i.e. Uˆ (and their combinations) are invariant under herence vector a0 cyclicpermutationsofstates,theUˆ areinvariantunder 0a cyclic permutations of phases. u:= u , 01,lookingateach node separatelyis notenoughsince correlationsbetween nodesemerge–sotheconceptofclustershastobeintro- 2 Y(µν...σ) := u(µν...σ) , (3.6) duced. A cluster operator acting on m particles is build m ij...k out of m one particle operators Uˆ(µ), i=0, where greek i,jX,...,k (cid:12) (cid:12) i 6 (cid:12) (cid:12) indices label different nodes of the network. All cluster mindices6=0(cid:12) (cid:12) operators with special cases Y =| {1za}nd Y(µ) = u(µ) 2. For a 0 1 Uˆ(µ) :=ˆ1 ... ˆ1 Uˆ ˆ1 ... ˆ1, (3.1) networkwith N nodes,these clustersums give2N scalar i ⊗ ⊗ ⊗ i ⊗ ⊗ ⊗ (cid:12) (cid:12) invariants under (products of) local unitar(cid:12)y tra(cid:12)nsforma- node µ tions. All the cluster sums together are subject to the Uˆ(µν) :=ˆ1 ... Uˆ .|.{.z} ˆ1 ... Uˆ ... ˆ1 (3.2) sum rule ij ⊗ i ⊗ ⊗ j ⊗ node µ node ν N N tr ρˆ2 n =Y + Y(µ)+ Y(µν)+ (3.7) etc. |{z} |{z} µ 0 1 2 µ=1 µ=1 µ<ν (cid:8) (cid:9)Y X X are unitary and orthonormal, and together span the Y(µνσ)+...+Y(123...N) . complete Liouville-space of the quantum network. Act- 3 N µ<ν<σ ing on only m nodes out of N means leaving the others X unaffected by choosing Uˆ0 = ˆ1 for them (Thus m is the For a pure state, tr ρˆ2 = 1 and for nµ = n the left numberofindices i,j,... unequalzero). Thegeneralde- hand side is nN. According to eq. (3.5), any m-cluster composition of ρˆ(like of any other operator Aˆ) for a N sum of a product stat(cid:8)e fa(cid:9)ctorizes into its 1-cluster-sum particle quantum network with node ν being a nν-level components. Agivenclusterofsizemcanthenbetested system reads asbeinginanon-product-state,ifthereissomepartition intosmallerclusters,the cluster-sumproductofwhichis smaller than Y , e.g. Y(1)Y(2) < Y(12). The cluster m 1 1 2 4 sums can be related to the ”purity” of a m particle clus- Toensurethatallsubsystemsaretreatedonequalfoot- ter by interpreting the cluster as a nm-level system with ing, the sum extends over all permutations p, weighted coherencevectorum. (Forsimplicityweassumenµ =n.) only with pure phase fators, where b [0,Ω 1] labels ∈ − Anormalized”purityfactor”cannowbedefinedfromthe the phase shift between ”neighbouring” p. (Here the respective coherence vector length as (m N) numbering of permutations is a matter ofchoice and the ≤ phase has no physical meaning.) The set of collective operators is orthonormal p := |um|2 = nmtr ρˆ2Cluster −1 (3.8) m u 2 nm 1 | m1|max m (cid:8) − (cid:9) Ω12N tr Eˆαβγ,bEˆα†′β′γ′,b′ =δαα′δββ′δγγ′δbb′ (4.3) = Y(µ)+ Y(µν)+...+Y(12...m) , n o nm 1 1 2 m and complete, so that the density operator ρˆ of the ! − µ=1 µ<ν X X network can be decomposed as characterizing the purity of a m-node cluster on the 1 scale (cf. eq. (2.32)) ρˆ= 2N Eαβγ,bEˆαβγ,b , (4.4) {αXβγ}Xb 1 0≤pm ≤1 (3.9) Eαβγ,b = Ωtr ρˆ·Eˆα†βγ,b . (4.5) pm =0 ⇐⇒ maximal mixed m-cluster The expectation vnalues Eαβγo,b are collective in the p =1 pure m-cluster. m sensethattheydo notrefertospecific subsystem-indices ⇐⇒ (as opposed to the Uˆ-operators, cf. eq. (3.1,3.2)). The Alternatively, the purity could be characterized by the inverse transformation is given by respective cluster-entropy [8]; p has the advantage of m being a simple algebraic function of the expectation val- Ω 1 Ω 1 Ω 1 ues (matrix-elements of the reduced density matrix). − ωΩ−bp0Eˆαβγ,b = − Cˆαβγ,b − ωΩ(p−p0)b =ΩCˆαβγ,p0 , b=0 p=0 b=0 X X X (4.6) IV. COLLECTIVE OPERATOR-BASIS FOR where we have made use of eq. (2.9). For example, the TWO-LEVEL SUBSYSTEMS selectiveoperatorCˆ λˆ(1) ˆ1 ... couldbewritten 100,0 ≡ 1 ⊗ ⊗ A. Definition and properties as Ω1 bΩ−1Eˆ100,b. Note that the operators Eˆ are, in general, no longer unitary; for b = 0 they are hermitian if thePoperators Cˆ are. For now we will stick to quantum networks build out αβγ,p of N two level systems, knowing that a generalizationto n-level systems is straight forward. First one needs to B. Alternative sets specify which single particle operator basis is used, e.g. uˆ = σˆ , vˆ = σˆ , wˆ = σˆ (which is hermitian and un01itary)xorσˆ01 =σˆ y iσˆ0,σˆ ozranyothercompletebasis. The operatorsEˆ stilldistinguish betweenallthree ba- x y z Wethenrela±belthe±clusteroperatorsUˆ(12...N) asCˆ , sic operators. Further reductions are possible: one such ij...k αβγ,p variant is (based on σˆ , σˆ , σˆ ) meaning an operator of dimension 22N, where α,β and + − z γ specify the multiplicity of σˆ , σˆ , and σˆ , respectively x y z Ω(z,γ)α β=z (α + β + γ N). Index p specifies a permutation of − ≤ Fˆ = ωbp Cˆ (4.7) these among the N subsystems. The number of such z,γ,b Ω(z,γ) αβγ,p permutationsandhencetheindexrangeforp [0,Ω 1] Xp=0 Xα,β ∈ − is which has been considered for nuclear spin-networks. In NMR the Fˆ -terms in a Hamiltonian are said to induce z N! atotalofz quantum-”flip-flops”[9];anFˆz-termentering Ω(α,β,γ)= . (4.1) thedensity-operatorρˆdescribestherespectivecoherence α!β!γ!(N α β γ)! − − − order z . | | These operators Cˆ , again, span the whole Another possibility is αβγ,p Liouville-space by defining all subsystem specific properties. We now go on to define collective operators Eˆ Ω(m)α+β+γ=m by Gˆ = ωbp Cˆ (4.8) m,b Ω(m) αβγ,p p=0 α,β,γ X X Ω 1 where p are permutations of the m operators (of any Eˆαβγ,b = − ωΩpbCˆαβγ,p ; ωΩ =e2Ωπi . (4.2) type) on the N subsystems, m = 0,1,...,N. Also this p=0 set is still complete. X 5 V. APPLICATIONS properties of unitary cluster operators lead to the com- mutator relation A. Commuting operator sets and generalized cat basis Uˆ Uˆ ...,Uˆ Uˆ ... =0 a1b1 ⊗ a2b2 ⊗ c1d1 ⊗ c2d2 ⊗ ⇐⇒ 1. Commuting sets of cluster operators h a d b c +a d b c +i.−..=0, (5.1) 1 1 1 1 2 2 2 2 − − and from this we find the following statements: Cluster sums and purity factors can be used to clas- sify states by their non-local properties. Highly entan- A There exists a set of(n 1)N completely commut- gled states tend to share correlations among all nodes − ing cluster operators of size N. This set can be rather than between only a few. Bell states [10] (N =2) constructed from all combinations of n 1 com- are perfect in this sense, because they are in a totally − pletely commuting one particle cluster operators. mixed state (Y = p = 0) locally and pure otherwise 1 1 (Y =3,p =1). Generalizingthis point, onemaydefine B There is a set of 2 2 ”highly entangled states” as states that have a cluster (n2 1)N2, N even (5.2) sum distribution with a strong focus on multi-particle − correlations. This definition does not give us a quantita- (n2 1)N2−1 (n 1), N odd (5.3) − − tive measure for multi-particle entanglement, but it can assistus in the searchfor new states with correlationsof completelycommutingN particleclusteroperators high order. Using the unitary operators defined above, of the form wecangiveaconstructivewayoffindingsuchstates. We Uˆ Uˆ Uˆ Uˆ .... (5.4) restrict ourselves to nµ =n. a1b1 ⊗ b1a1 ⊗ a2b2 ⊗ b2a2 C There can be larger sets of completely commuting Lemma 1 Let ψ be an eigenstate of nN completely commuting operators than given by eq. (5.3). However, no | i generalconstructivemethodto find themis known unitary operators out of which there are q N-cluster- N to us so far. For n > 2 we restrict ourselves to opererators. Then its highest order cluster sum is given N 3. by Y =q . N N ≤ D The maximum order of any such set is constrained Notethatthemodulusofanyeigenvalueofanycluster- by nN 1. This maximum value is typically not operator Ui(jµ..ν....) is exactly 1; the qN commuting N- reached−for N >2. cluster-operators thus imply Y = q provided all N N the non-commuting N-cluster-operatorscontribute zero. The resultsofA,B,C andDaresummarizedinTable This must be the case, however, if there is a total of nN I. commuting operators exploiting the sum rule eq. (3.7). A set of completely commuting operators is a set in n=2 n=3 which each operator commutes with all others. This N A B C D Cat N A B C D Cat lemma reduces the problem of finding a state with max- 1 1 1 1 1 1 1 2 2 2 2 2 imum cluster sum Y to the problem of finding a max- N 2 1 3 3 3 3 2 4 8 8 8 8 imum set of commuting N-cluster-operators. Before we 3 1 3 4 7 4 3 8 16 20 26 20 can make statements about cluster operators, the com- 4 1 9 9 15 9 4 16 64 ? 80 60 muting relations of single particle unitary operators Uî 5 1 9 16 31 16 5 32 128 ? 242 172 needtobeexamined. Since Uâ†b,Uˆcd =0iffad−bc= 6 1 27 33 63 33 6 64 512 ? 728 508 0, one can show: h i− n=4 N A B C D Cat Lemma 2. (based on theory of congruency classes) 1 3 3 3 3 3 EachoperatorUˆ commuteswithngcd(a,b,n)otherba- ab 2 9 15 15 15 15 sisoperatorsandtoeachoperatorUˆ thereexistsatriv- ab 3 27 45 54 63 54 ial set of n completely commuting basis operators. (gcd 4 81 175 ? 255 213 = greatest common divisor) 5 243 525 ? 1023 828 One of the n commuting operators in the completely commuting set is Uˆ = ˆ1, so the maximal number of TABLEI. Cluster sums YN for eigenstates corresponding 0 tosets ofcompletely commutingN particle clusteroperators commuting one particle operators is n 1. − calculatedbymethodsA,BandC.ColumnDgivestheupper For a network of N particles with n-levels each, the limitandcolumCatstatestheclustersumYN forgeneralized cat states as introduced in Sect. VA2. 6 2. Generalized cat states n 1 ofAtwsoasfiprinst1e/x2ampaprlteiclleets.usThloeokmaaxtimthuemweolrldkenrofwornccoamse- |Catic := √1n − ωnjc1 |jik⊗N=2 j+ck , (5.6) j=0 pletely commuting sets of 2-cluster-operators is 3 and X (cid:12) (cid:11) (cid:12) there are 6 such sets: isanexplicitdefinitionofthecatstatebasis. Theindex c = c , c [0,n 1] labels the nN states and ω is i i n { } ∈ − given by eq. (2.8). Orthonormality and completeness Uˆ Uˆ , Uˆ Uˆ , Uˆ Uˆ , (5.5) read 01 01 10 10 11 11 ⊗ ⊗ ⊗ n o Uˆ Uˆ , Uˆ Uˆ , Uˆ Uˆ , 01 01 10 11 11 10 nUˆ01⊗⊗Uˆ10, Uˆ10⊗⊗Uˆ01, Uˆ11⊗⊗Uˆ11o , dhCat|Catic =δc,d , Xc |CaticchCat|=ˆ1. (5.7) n o Uˆ Uˆ , Uˆ Uˆ , Uˆ Uˆ , InthelowdimensionalcaseofN =n=2thedefinition 01 10 10 11 11 01 ⊗ ⊗ ⊗ (5.6) reduces to the bell basis and for N = 3, n = 2 the nUˆ Uˆ , Uˆ Uˆ , Uˆ Uˆ o , GHZstate 1 (000 + 111 )isonememberofsuchaset. 01⊗ 11 10⊗ 01 11⊗ 10 √2 | i | i Furtherexamplesforcat-statesaregiveninAppendixC. n o Uˆ01 Uˆ11, Uˆ10 Uˆ10, Uˆ11 Uˆ01 . The characterizationof these generalized cat states in ⊗ ⊗ ⊗ terms of cluster sums shows (cf. Table I) n o The eigenstates to the first set form the Bell basis 00 11 , 01 10 , the second set has i 00 11 , |i 0i1±|10i a|seiig±en|staitesandthethirdgives |00i+±0|1 +i Y1 =0, (5.8) | i±| i | i | i 10 11 , 00 + 01 10 + 11 , 00 01 + 10 + 11 , (n 1)m+( 1)m(n 1) | i−| i | i | i−| i | i | i−| i | i | i Y = − − − , 1 m D+1 (in D dimensions) however, the interac- knownspin-echoes[17]areformallybasedontime rever- tion distances cannot all be the same. A partial remedy sal, thus undoing a unitary evolution over a time period is the introduction of a ”quantum bus”: In this case the ∆t/2 within the next period ∆t/2: nodes do not interact directly but only indirectly via a commondegreeoffreedom. Thisdegreeoffreedomcould be a central spin, but typically is implemented as a col- Uˆj(−∆t/2)=Uˆj†(∆t/2) (5.41) lective mode (like the phonon mode of a cold ion trap [14]). Thistime-reversalcannotbeimplementedeither;however, its effect can be simulated in any discrete Hilbert- spaceofdimensionnby meansofthe cyclicpermutation D. Symmetry breaking and irreversibility operations Uˆn 1,0, as introduced in section IIB. Let us consi−der the time-independent Hamiltonian Hˆ. Constrained operations will lead, quantum mechani- It is convenient to require tr Hˆ =0 so that cally, to selection rules accompanied by a tremendous n o reduction of the state space available to the system dy- namics starting from a giveninitial state. This situation n−1 E =0 (5.42) needs to be distinguished from lack of control (measure- k mentdata)implyinglackofinformation(entropyS >0). Xk In so far as this lack of control refers to expectation val- withE denotingtheeigenvaluesofHˆ. Thentheeffect ues b=0 (which would be absent for strict permutation k 6 of the unitary time evolution generated by Hˆ symmetry), ”uncontrolled” symmetry breaking may be said to lead to an ensemble description and irreversibility: Thetruestatespacehasbeenreducedtothesmaller ψ(∆t) =Uˆ (∆t) ψ(0) (5.43) one defined by the assumed permutation symmetry. In | i Hˆ | i thiscase,however,the”decoherence”doesnotreflectthe can be suppressed for any given ∆t by the iteration influence of an external bath but is rather of ”internal” origin. Phenomenologically ony may try to model these n effects, as usual, via some decoherence times; however, Uˆ (∆t):= Uˆ Uˆ (∆t/n) (5.44) it is not clear yet under what conditions such a proce- eff n−1,0 Hˆ (cid:16) (cid:17) dure would be appropriate. Clearly it should, at most, Here we have assumed for simplicity that the appli- work for sufficiently large networks and only to the ex- cation of Uˆ (cf. eq. (2.8)) does not consume any tentthatthereductionisreallysubstantial(cf. ref.[15]). additional tni−m1e,0. Uˆ generates cyclic permutations Thedecoherencetimewouldthen,inturn,allowtoassess n 1,0 between all the stat−es E of the spectrum. Any ini- symmetry breaking effects in a global way. k tialeigenstatethusvisitsallthe othereigenstatesforthe In the following, we will restrict ourselves to unitary sametime ∆t/n sothat, due to the constrainteq. (5.42) transformations. thetotalacquiredphaseaddsuptozero. Thisinvariance propertyholdsforanyinitialstate ψ(0) ,assuchastate | i canbe writtenasasuperpositionofeigenstates,andany E. Special unitary transformations eigenstate returns to its initial phase separately. This mapping, eq. (5.44), can be repeated to get the 1. Cyclic permutations: generalized echoes stroboscopic invariance (cf. ref. [16]) The pulse-like manipulation of Hamiltonians, nm Uˆ (m∆t)= Uˆ Uˆ (∆t/n) =ˆ1, (5.45) eff n 1,0 0 − Hˆ(t)=Hˆ , for t t

Adapted-operator representations: Selective versus collective properties of quantum networks

Similar Adapted-operator representations: Selective versus collective properties of quantum networks