Decoherence of many-body systems due to many-body interactions
aa r X i v : . [ qu a n t - ph ] J u l Decoherence of many–body systems due to many–body interactions
T. Carle, H. J. Briegel,
1, 2 and B. Kraus Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria Institute for Quantum Optics and Quantum Information,Austrian Academy of Science, Innsbruck, Austria
We study a spin-gas model, where N S system qubits are interacting with N B bath qubits viamany–body interactions. We consider multipartite Ising interactions and show how the effect of de-coherence depends on the specific coupling between the system and its environment. For instance,we analyze the influence of decohenerce induced by k –body interactions for different values of k .Moreover, we study how the effect of decoherence depends on the correlation between baths that arecoupled to different individual system qubit and compare Markovian with non–Markovian interac-tions. As examples we consider specific quantum many–body states and investigate their evolutionunder several different decoherence models. As a complementary investigation we study how thecoupling to the environment can be employed to generate a desired multipartite state. I. INTRODUCTION
In any realistic scenario the system of interest is nevercompletely decoupled from its environment. This cou-pling causes the system to decohere which is believed tobe the main reason why macroscopic quantum systemsdo not occur in nature. The effect of decoherence haswidely been investigated in the literature. There, dif-ferent decoherence models have been assumed based oninteractions with a harmonic oscillator bath [1], a spinbath [2] or a spin gas [3].We will consider here a microscopic decoherencemodel, where both, the system and the environment aredescribed by qubits. For a general spin model, where N S system ( S ) qubits are coupled to N B bath ( B ) qubits, theinteraction would be described by a general unitary, U SB ,acting on the whole system. Assuming that the initialstate of the bath would be described by the state ρ B , theevolution of the system qubits would be governed by thecompletely positive map E ( ρ S ) = tr B ( U SB ρ S ⊗ ρ B U † SB ).Due to the exponential scale of the dimension of theHilbert space the investigation of the evolution of thesystem qubits under such a general map is unfeasible.Thus, one has to restrict the coupling U SB to a certainclass of unitaries and therefore to a certain class of cou-pling Hamiltonians H SB . A reasonable choice is to con-sider only generalized Ising interactions, which describethe effect of multi-qubit collisions. The coupling Hamil-tonian for this choice H SB is of the form H SB = P i α i σ i z ,with i denoting a ( N S + N B )–bit string. Each term inthe Hamiltonian corresponds to a certain collision. Theunitary U SB = e − iH SB t , which governs the evolution ofthe state, is a phase gate, i.e. a unitary which is diag-onal in the computational basis. Due to the fact thatall terms in the Hamiltonian commute, U SB can be writ-ten as a product of phase gates, where each factor cor-responds to a certain collision. The order of the phasegates, i.e. the number of qubits on which the phase gatesacts non–trivially, is equivalent to the number of qubitswhich collided. The value of the phase in the phase gatedepends on the interaction strength. In this article we will describe the evolution of the system in terms of phasegates for arbitrary phases. That is, we will not considerthe evolution as a function of time, but will rather as-sume that due to a certain physical interaction a specificphase gate, U SB is given. The investigation of such ageneral model is only feasible because all the phase gatescorresponding to different collisions commute.In this article we investigate first the effect of decoher-ence induced by multi-qubit Ising interactions. Second,we will show how the coupling between the environmentand the system can be employed to prepare the systemqubits in a desired state. The first part of this articleconstitutes a generalization of the work presented in [4],where the decoherence of a spin gas has been investi-gated. In contrast to [4] we do not restrict ourself tobipartite interactions, but consider multipartite interac-tions and the resulting decoherence. Obviously, consid-ering many–qubit phase gates leads to a much bigger va-riety of different interactions. For two–qubit phase gates,the only relevant interaction is the one where one bathqubit is interacting with one system qubit. In the case ofthree qubit gates, we could for instance consider 3–qubitinteractions between either 2 system qubits and 1 bathqubit or between 1 system qubit and 2 bath qubits. Apartfrom that, considering several system qubits, which areinteracting for instance with the bath qubits via a three–qubit phase gate, the bath qubits could either be inde-pendent, or not. In this case, non, one, or both of thebath qubits could interact with both system qubits. Theeffect of decoherence will strongly depend on the corre-lations within the bath. Considering more general phasegates (acting non–trivially on k systems) allows for cor-respondingly more different kind of interactions.The aim of this paper is to get a better understandingof the effect of non–local decoherence on a multipartitesystem. This will be achieved by comparing the influ-ence of decoherence for different interaction models. Forinstance, we will compare the effect of two–qubit phasegates to the one of three and more qubit gates. Moreover,we will compare Markovian to non–Markovian processes.We will furthermore investigate how the correlation inthe bath qubits affect the decoherence. Even though wewill consider the most general dephasing maps, our mainfocus will be on those maps, where each system qubit isonly interacting with bath qubits, i.e. there is no inter-action between two, or more system qubits and the bath.We will call these maps purely dephasing maps. The rea-son for focusing on purely dephasing maps is that theycorrespond to the realistic situation where the processesin which several system qubits collide with bath qubitscan be ignored. Another reason for focusing on purelydephasing maps is that they cannot generate entangle-ment. Thus, they are perfectly suited to investigate theeffect of decoherence on the entanglement and coherenceof the system qubits.The outline of the paper is the following. First of allwe will review the notion of a certain class of multipar-tite states, called locally maximally entangleable (LME)states. As we will see, the reduced states of LME states(LMESs) play an important role in our considerationsof dephasing maps. The projected–entangled–pair state(PEPS) picture will be used to compute their reducedstates. In Sec. III we will consider the most general de-phasing maps and will derive a simple relation betweenthe maps and the corresponding LMES. After that, wewill focus (apart from Sec. V) on purely dephasing mapswhich result from interactions where each system qubitis only coupled to baths qubits and not to other systemqubits. We will show that all these maps are separable,i.e. they can be written as a convex combination of localmaps and will express them in terms of simple local maps.In order to study how the bath–correlation influences thedecoherence, we will derive a simple expression of themaps which describe the following scenario. We consider m bath qubits which are all interacting with each systemqubit. Each of the system qubits is also interacting withsome other bath qubits, which are coupled to the m bathqubits, but not to any other system qubit. The influenceof correlated noise will be investigated by considering themaps described above for different values of m . In Sec.IV we will analyze how the effect of decoherence dependson the various coupling scenarios. In particular, we willinvestigate the Markovian case and compare the effectof decoherence to the non–Markovian case for a generalinput state (Sec. IV A). Furthermore, we will comparethe effect of phase gates of order k for different values of k . As examples we will determine the effect of decoher-ence on arbitrary single system qubit state, two systemqubits states and certain multipartite states (Sec. IV B).We will show that the intuition, that the entanglementshared by the system qubits is less disturbed, if the bathsof the system qubits are stronger correlated, that is m islarge, is not true. It will be demonstrated that for cer-tain multipartite states the entanglement is more robustunder the decoherence of independent baths than undercorrelated ones. In Sec. V we will consider the com-plementary process to the one considered above. Therewe will employ the interaction between the system andbath qubits in order to prepare the system qubits in adesired multipartite state. In particular, we will show how LMESs can be generated via unitary interactionsbetween the system qubits and the bath qubits.Through out the paper we will use the following no-tation. The subscript of an operator will always denotethe system it is acting on, or the system it is describing.We denote by i the classical bit–string ( i , . . . , i n ) with i k ∈ { , } ∀ k ∈ { , . . . , n } and | i i ≡ | i , . . . , i n i denotesthe computational basis. Normalization factors as wellas the tensor product symbol will be omitted wheneverit does not cause any confusion and 1 l will denote thenormalized identity operator. II. LOCALLY MAXIMALLY ENTANGLEABLESTATES
Recently a new classification of multipartite stateshas been proposed in order to study the properties ofpure quantum states describing n qubits [5]. There,the following question has been addressed. For whichstates, | Ψ i , do there exist local controlled operations, C l = P i U il ⊗ | i i l a h i | , where U il are unitary operationsacting on system l and | i i l a h i | is acting on the auxil-iary system attached to system l , such that the state C ⊗ C ⊗ . . . ⊗ C n | Ψ i S ( | i + | i ) ⊗ nA is maximally entan-gled between the system ( S ) and the auxiliary system( A ). States for which this is possible were called LocallyMaximally entangleable States (LMESs). Important ex-amples of these states are all stabilizer states. In [5] ithas been shown that any LMES is (up to local unitaryoperations) of the form | Ψ i = r n X i e iα i | i i ≡ U Ψ ph | + i ⊗ n , (1)where α i ∈ IR and U Ψ ph denotes the diagonal unitary oper-ator with entries e iα i (phase gate). Note that the 2 n realphases α i characterize the LMES (up to local unitaries).From Eq (1) it can be easily seen that any LMES can beprepared by applying generalized phase gates, acting onup to all qubits, to a product state, i.e. non–entangledpure state. That is, any LMES | Ψ i can be written as | Ψ i = U ,...,n Y U i k ,...,i kn − · · · Y U i | + i n , (2)where U i k ,...,i kl is a phase gate acting non–trivially on l qubits. For instance, U = 1 l − (1 − e iφ ) | i h | ,with φ ∈ IR is a three–qubit phase gate. It is straight-forward to see that in this hierarchical way the 2 n phases α i can be generated. Thus, any LMES can be preparedusing generalized phase gates, which could result froma generalized Ising interaction. If α ( i ) is a polynomialof degree k (as a function of i = ( i , . . . , i n )) then thecorresponding state can be prepared using only k –bodyinteractions. Thus, the correlations in the coefficients aredirectly related to a preparation scheme and therefore tothe entanglement contained in the state.Apart from those properties, LMESs have the followingproperties [5]: (i) According to their definition, LMESsare characterized by the fact that their global informationcan be washed out by maximally entangling the systemqubits to local auxiliary qubits. (ii) A state is LME iffit can be used to encode locally the maximum amountof n independent bits. (iii) For any LMES, | Ψ i , one canconstruct a complete set of commuting unitary observ-ables such that | Ψ i is the unique eigenstate with eigen-value one for all these observables (the so–called gener-alized stabilizer, see also Sec. V). (iv) LMESs can havean exponentially large quantum Kolmogorov complexity[6]. That is, for certain LMESs it requires exponentiallymany classical bits to describe the state.Note that this classification of multipartite states is in-teresting for several quantum information theoretic tasks.First of all, (i) is equivalent to the fact that all the infor-mation contained in a LMES can be coherently ”copied”into local auxiliary systems by local unitary operations.This seems to be a crucial property shared by those stateswhich are useful for quantum information tasks, like one–way quantum computing, quantum error correction andquantum secret sharing. Second, due to the preparationscheme [Eq. (2)], LMES can be entangled in many dif-ferent, but hierarchical ways. Product states but alsostabilizer states are all LME. Stabilizer states or moregenerally, weighted graph states [3, 4, 7] are those LMESwhich require only two–qubit phase gates for their prepa-ration. LMES, however, can be much more entangledthan the stabilizer states. Moreover, there exist, in con-trast to stabilizer states, complex LMES [(iv)], which arenecessarily highly entangled [6].In the following subsection we will consider the reducedstate of LMESs, which will be relevant for our consider-ations of dephasing maps. In Sec. V we will show howLMESs can be generated using either dissipation or uni-tary interactions. A. Reduced state of LMESs
In Sec. III we will show that dephasing maps arestrongly related to the reduced states of LMESs. Dueto this fact we will present here some methods for com-puting them.Let us consider an arbitrary n –qubit LMES, | Ψ i = U | + i ⊗ n = P i e iα i | i i , where i = i , . . . , i n and U denotesan arbitrary phase gate. We call a phase gate, U , purephase gate of order k if it can be written as U = 1 l +( e iφ − | i h | ⊗ k . A phase gate will be said to be of order k if it can be decomposed into pure phase gates which aremaximally of order k . Note that any phase gate can bedecomposed into pure phase gates. We will say that aqubit is overlapping if there exist at least two pure phasegates in the decomposition of U , which act non–triviallyon this qubit. If there are two or more qubits, say qubit1 and 2, for which there exists more than one pure phasegate, which act on them simultaneously, we say that the interaction possesses an overlapping edge between qubit1 and 2.Let us now consider an arbitrary bipartite splitting, A, B , where A ( B ) denotes a subsystem composed outof N A ( N B ) qubits respectively. We write the n –qubitLMES, | Ψ i as | Ψ i = U | + i ⊗ n , where n = N A + N B and U denotes a phase gate. Since U can be de-composed into pure commuting phase gates, we have U = U A U AB U B , where U A ( U B ) is only acting onthe qubits within A ( B ) respectively and U AB is act-ing non–trivially on both, A and B . Then, the re-duced state of system A can be written as ρ A = U A tr B ( U AB ( | + i h + | ) ⊗ ( n ) U † AB ) U † A . Thus, we will focusin the following on the computation of the reduced stateof | Ψ i = U AB | + i ⊗ n ≡ P i e iα i | i i . Writing | i i = | i A i | i B i we have ρ A = P i A , j A , k B e i ( α i A, k B − α j A, k B ) | i A i h j A | . De-pending on the correlations of α i the reduced state canbe computed efficiently, or not. We are going to con-sider now some simple examples of LMESs for whichthe reduced state can be computed efficiently. In gen-eral, however, this will not be possible, as will be ar-gued below. If | Ψ i is a product state, i.e. α i = P k α i k ,then ρ A = P i A , j A e i ( α i A − α j A ) | i A i h j A | is a pure productstate. The second class in the hierarchy of LMESs (seeEq (2)) constitute the so–called weighted graph states(see e.g. [3, 4, 7]), which include (up to local uni-taries) the well–known stabilizer states [8]. There, α i is a polynomial of degree two in i , i.e. α i = i T Γ i (up to local unitaries), where the n × n matrix Γ ≡| i h | ⊗ Γ A + | i h | ⊗ Γ B + ( | i h | + | i h | ) ⊗ Γ AB iscalled adjacency matrix. In this case, the reduced statehas the form (up to normalization) ρ A = X ~s A ,~s ′ A e i ( ~s TA Γ A ~s A − ( ~s ′ A ) T Γ A ~s ′ A ) c ~s A ,~s ′ A | ~s A i h ~s ′ A | , (3)where c ~s A ,~s ′ A = Q N B i =1 (1 + e i ( e Ti Γ AB ( ~s A − ~s ′ A )) ), with e i de-noting the standard basis. Note that c ~s A ,~s ′ A and therefore ρ A can be computed efficiently [9].In general, however, it will not be possible to de-termined the reduced state of an arbitrary LMES effi-ciently. The reason for this is the following. Let us con-sider the n –qubit LMES, | Ψ i = U | + i ⊗ n , where U is aproduct of pure phase gates of order three, i.e. α i isa monomial of degree three in i . Let us now computethe reduced state of qubit 1. Without loss of general-ity we write U = V V , with V acting as the identityon the first qubit, and V = | i h | ⊗ l + | i h | ⊗ U ,where U is a product of two–qubit phase gate. Thus,we have | Ψ i = V ( | i | + i ⊗ ( n − + | i U | + i ⊗ ( n − ).Writing the weighted graph state, | φ i = U | + i ⊗ ( n − as | φ i = P ~s e i~s T Γ ~s | ~s i and noting that ( n − ⊗ h + | ~s i = √ n − ∀ ~s , we see that we would need to efficiently com-pute P ~s e i~s T Γ ~s in order to efficiently compute ρ . It ishowever very unlikely that this expression can be com-puted efficiently for an arbitrary Γ, since it resemblesthe partition function, which is in general not efficientlycomputable.We will use now the discussion above to understand forwhich states it will be possible to compute the reducedstate efficiently. Suppose that Γ is block–diagonal, i.e.Γ = L Γ i , where the dimension of each of the block–diagonal matrices, Γ i is independent of n . Then, it iseasy to see that P ~s e i~s T Γ ~s can be computed efficientlyand therefore the reduced state can be computed effi-ciently. Note that for the state | Ψ i this means that qubit1 is independently coupled to several other qubits, butamong the set of coupled qubits there is no coupling.That is, the number of overlapping qubits which are cou-pled to qubit 1 is small. Similarly, one can see which m –partite reduced states of LMES with higher order canbe computed efficiently.Let us now compute the reduced state of certainLMESs, | Ψ i = U | + i ⊗ n , which will become rele-vant in the subsequent sections. For a single qubit,which we choose here without loss of generality tobe qubit 1, and an arbitrary interaction, U , whichwe write as before as U = V V , with V acting asthe identity on the first qubit, and V = | i h | ⊗ l + | i h | ⊗ U , where U = P i e iα i | i i h i | we find ρ = (1 l + | i h | ( n − ⊗ h + | U † | + i ⊗ ( n − + h.c. ) , where ( n − ⊗ h + | U † | + i ⊗ ( n − = P i e − iα i . Suppose now that U is a product of pure phase gates, U l , each acting on k l − φ l . Then, the reducedstate is given by ρ = 12 (1 l + | i h | Y l k l − (2 k l − − e − iφ l ) + h.c. ) . (4)As another example we compute the single qubit re-duced state of the state U ( φ ) | + i ⊗ n , where U ( φ ) is a purephase gate, i.e. U ( φ ) = 1 l + ( e iφ − | i h | ⊗ n . It is easyto show that ρ ( φ ) = trall but ( U ( φ ) | + i h + | n U ( φ ) † ) hasthe following form ρ ( φ ) = 12 n (cid:18) n − n − − e − iφ n − − e iφ n − (cid:19) . (5)Similarly the reduced state of several qubits can bedetermined.Another way to compute the reduced state is tomake use of the notion of projected–pair–entangled state(PEPS) [10, 11]. The idea is to rewrite the quan-tum state by substituting physical qubits on which sev-eral phase gates are acting on by a number of virtualqubits. The transfer from the virtual system to thephysical one is done by projectors acting on those vir-tual qubits. For example, if qubit i is part of k purephase gates of arbitrary order, we would introduce k vir-tual qubits, one per phase gate. Then i k denotes the k th virtual qubit of the physical qubit i . The followingsimple example of two three-qubit phase gates illustratesthe idea: U U | + i ⊗ = P U U | + i ⊗ with P = √ | i h | + | i h | ). The projector, P , is independent of the order of the phase gates appliedto the physical qubit and independent of the phases. Itonly depends on the number of pure phase gates whichare acting non–trivially on system 3. In general, theprojector associated to a physical qubit i on which k pure phase gates are acting on has the following form P i ,...i k i = √ k − ( | i i h . . . | i ,...i k + | i i h . . . | i ,...i k )[19].Let us now use the procedure described above topresent the reduced state ρ A of an arbitrary LMES | Ψ i AB = U AB | + i ⊗ ( N A + N B ) . Since the phase gatescommute, the reduced state can be determined froman interaction pattern without the need of consideringany time ordering. For each qubit in A and B , whichinteracts via several pure phase gates with other qubits,we introduce projectors and virtual qubits as explainedabove. Note that any pure phase gate is acting ondifferent virtual qubits. That is, there is no virtualqubit on which two or more phase gates are actingon. All information about the coupling is then cap-tured by the projectors. An easy example for 2–qubitphase gates would be: U U | + i h + | U † U † = P , ( U U | + i h + | U † U † ) P † , = P , ( ρ ⊗ ρ ) P † , , where P , = √ | i h | + | i h | ) and ρ ij = U ij | + i h + | U † ij with ( i, j ) ∈ { (1 , ) , (2 , } . In general, wedefine for any qubit b j in B , where m ( b j )pure phase gates are acting on, the projector P b j ,...b jm ( bj ) b j = √ m ( b j ) − ( | i b j h . . . | b j ,...b jm ( bj ) + | i b j h . . . | b j ,...b jm ( bj ) ) and use a similar definitionfor any system s j in A . Then the reduced state ofsystem A can be written as ρ A = tr B ( | Ψ i h Ψ | ) = P A tr B ( P B ⊗ i ρ i ⊗ { kl } ρ kl . . . ⊗ { o,...n } ρ o,...n P † B ) P † A , where P B = N j P b j ,...b jm ( bj ) b j and P A is defined analogously.The operators ρ denote, like in the example above, pureLMESs. For instance, ρ i denotes a pure state which isobtained by applying a 1-qubit phase gates on the state | + i state, the operators ρ kl result from 2–qubit phasegates applied to | ++ i and so forth.We will use this formlater to compute the reduced state of special cases. III. DECOHERENCE MAPS
We consider a system composed out of N S systemqubits, which will be denoted by s , . . . s N S , which areinteracting with N B bath qubits, which will be denotedby b , . . . b N B . The qubits interact via some arbitraryphase gate, which will be denoted by ˜ U SB . The evolu-tion of the system qubits is governed by the completelypositive map, ˜ E ( ρ S ) = tr B ( ˜ U SB ρ S ⊗| + i h + | ⊗ N B ˜ U † SB ). Inthis section we will first of all consider the most generaldephasing map and show how it can be represented interms of the reduced state of some LMES. After that wewill represent the dephasing map in terms of Pauli opera-tors. Since we are interested in the decoherence caused bysuch an interaction we will restrict ourselves in Sec. III Ato those interactions where non of the system qubits isdirectly interacting with another system qubit. As men-tioned before, we will call those maps purely dephasingmaps. We will show that all of them are separable andderive a simple expression for them. Then we will inves-tigate how the correlations among the bath qubits willaffect the evolution of the system qubits.As before, we decompose the general phase gate, ˜ U SB ,into ˜ U SB = U B U SB U S , where U B ( U S ) is acting ex-clusively on the bath (system) qubits respectively and U SB describes the interaction between the system andbath qubits. Thus, we have ˜ E ( ρ S ) = U S tr B ( U SB ρ S ⊗| + i h + | ⊗ N B U † SB ) U † S . In order to understand how de-coherence can affect the properties of the system stateit is therefore sufficient to consider the map E ( ρ ) =tr B ( U SB ρ S ⊗ | + i h + | ⊗ N B U † SB ). In the following we willfocus on this expression.Note that U SB can always be written as U SB = P k | k i B h k | ⊗ U k , where U k denotes a phase gate actingonly on the system qubits [20]. Using this decompositionit is easy to see that E ( ρ S ) = X k U k ρ S U † k = σ S K ρ S , (6)where σ S = P k U k | + i h + | ⊗ N S U † k and ⊙ denotes theHadamard product [12]. Note that σ S = E ( | + i h + | ⊗ N S )is the (unnormalized) reduced state of the LMES U SB | + i ⊗ n , with n = N S + N B . In order to obtain thesecond equality in Eq. (6) we made use of the fact that forany n –qubit state, ρ and any phase gate, U , the followingholds U ρU † = [ U | + i h + | ⊗ n U † ] J ρ . Maps which can bewritten as in Eq (6) are called diagonal or Hadamardmaps. For these maps several interesting properties inthe context of optimal output entropy and purity havebeen recently shown [13].It is important to note that the evolution of the sys-tem is completely determined by the reduced state σ S .Thus, all the correlations which are established betweenthe system and the bath during the evolution are com-pletely captured by the reduced state of the correspond-ing LMES, U SB | + i ⊗ n . As mentioned above, the com-putation of this reduced state gets more and more com-plicated as the number of overlapping edges, or qubitsincreases. Physically this corresponds to the situationwhere more and more bath qubits take part in joint col-lisions with system qubits.Another way of representing the dephasing map isto write it in terms of Pauli operators. To do so, werecall the so–called Choi–Jamiolkowski isomorphismbetween completely positive maps and positive semidef-inite operators (states). For a completely positivemap, E : B ( H ) → B ( H ), the corresponding state, E , is acting on the Hilbert space H ⊗ H . It isdefined as E = ( E ⊗ l )( P ), where P = | Φ + i h Φ + | with | Φ + i = P dim ( H ) i | ii i . On the other hand,we have, E ( ρ ) ∝ tr ( E ρ P ). In the case of de-phasing maps, the corresponding Choi–Jamiolkowskistate, E , has the simple form E s ,...s NS ,a ,...a NS = E s ,...s NS ⊗ l a ,...a NS ( N N S i =1 | φ + i s i a i h φ + | ) = N i ˜ P s i a i ( σ S ⊗ | + i h + | ⊗ N S ) N i ˜ P † s i a i , where˜ P s i a i = √ | i h | + | i h | ) and a i denote theauxiliary systems. That is, the Choi-Jamiolkowski state E can be rewritten in terms of the reduced state σ S ofthe LMES, U SB | + i ⊗ n .We will now use the operator E to rewritethe map in terms of the Pauli operators.We decompose E in the Bell basis, E = P i ,...,i n ,j ,...,j n λ j ,...,j n i ,...,i n | Ψ i , . . . , Ψ i n i h Ψ j , . . . , Ψ j n | ,where | Ψ i k i = σ i k ⊗ l | Φ + i , with i k ∈ { , . . . , } and σ = 1 l , σ = σ x , σ = σ y , σ = σ z denote the Paulioperators. In order to present the map in a compactform we use the notation | Ψ i i = | Ψ i , . . . , Ψ i n i . Usingnow that E ( ρ ) ∝ tr ( E ρ P ) we find E ( ρ ) ∝ X i , j λ ji ( n O k =1 σ i k ) ρ ( n O l =1 σ j l ) , (7)with λ ji = h Ψ i | E | Ψ j i . Due to the projector ˜ P occur-ring in the expression of E , which projects onto the sub-space spanned by {| Ψ i , | Ψ i} we find that λ ji = 0 unless i k , j l ∈ { , } for all 1 ≤ k, j ≤ n .Let us now consider some simple examples. First ofall we consider the simplest case where a single sys-tem qubit is coupled to several bath qubits via two–qubit phase gates. This scenario was already treatedin [4]. There, the decoherence map for the systemqubit, denoted here by 1, which interacts with some bathqubits b i ∈ B is given by E ( ρ ) = tr B ( Q b i ∈ B U b i ( ρ ⊗| + i b i h + | ) Q b i U † b i ) = λ ρ + λ σ z ρσ z + λ (1 l ρσ z − σ z ρ l ),with λ = (1 + r cos γ ) / λ = (1 − r cos γ ) / λ = ( ir sin γ ) / r = Q b i ∈ B cos( φ b i /
2) and γ = P b i ∈ B ( φ b i / b i b j . We find E ( ρ ) = tr B ( U b i b j ( ρ ⊗ | ++ i b i b j h ++ | ) U † b i b j )= λ ρ − λ σ z ρσ z + λ (1 l ρσ z − σ z ρ l ) (8)with λ = (7 + cos( φ )), λ = (1 − cos( φ )) and λ = i sin( φ ).Investigating now a sequence of interactions with three–qubit phase gates forces us to consider different cases,namely the cases of independent collisions or collisionswith overlapping qubits/edges. We will discuss the mapsdescribing the different coupling scenarios in Sec. IV B.For two system qubits where each of them couples tothe bath qubits via one three–qubit phase gate we endup with three different cases. When the bath qubits arenon-overlapping we have E ( ρ ) = E ⊗ E ( ρ ) with E i de-noting the single qubit map for a three qubit phase gate(see Eq. (8)). If the system qubit 1 is interacting withtwo bath qubits, say b b and system qubit 2 with b and b , i.e. b denotes an overlapping qubit we find E ( ρ ) =tr b b b ( U b b U b b ( ρ ⊗ ( | + i h + | ) ⊗ ) U † b b U † b b ) = λ ρ + λ ( ρσ z l − σ z l ρ ) + λ ( ρ l σ z − l σ z ρ ) + λ ( σ z l ρσ z l ) + λ (1 l σ z ρσ z l + σ z l ρ l σ z − σ z σ z ρ − ρσ z σ z ) + λ (1 l σ z ρ l σ z ) + λ ( σ z l ρσ z σ z − σ z σ z ρσ z l ) + λ (1 l σ z ρσ z σ z − σ z σ z ρ l σ z ) + λ ( σ z σ z ρσ z σ z ) with λ = (25 + 3 cos( φ ) + cos( φ )(3 + cos( φ ))), λ = i (3 + cos( φ )) sin( φ ), λ = i (3 +cos( φ )) sin( φ ), λ = (3 + cos( φ )) sin ( φ / λ = (sin( φ ) sin( φ )), λ = (3 +cos( φ )) sin ( φ / λ = i (sin ( φ /
2) sin( φ )), λ = i (sin( φ ) sin ( φ / λ = (sin ( φ /
2) sin ( φ / b coincides with b , i.e. when there isan overlapping edge, the structure of the map is the sameas in the overlapping qubit case. The coefficients howeverchange to λ = (13 + cos( φ ) + cos( φ )(1 + cos( φ )))and λ = i (cos ( φ / φ ) and λ = i (cos ( φ )) sin( φ ) λ = cos ( φ /
2) sin ( φ / λ = (sin( φ ) sin( φ )) λ = (cos ( φ / ( φ / λ = i (sin ( φ /
2) sin( φ )) λ = i (sin( φ ) sin ( φ / λ = (sin ( φ /
2) sin ( φ / σ S (see Eq. (6)) and in termsof the Pauli operators (see Eq. (7)). In the followingwe will mainly use Eq. (6) to investigate the effects ofdecoherence. A. Purely dephasing maps
Let us now focus on those interactions where eachsystem qubit is only interacting with bath qubits, butnot with another system qubit. We will call these mapspurely dephasing maps, because, as we will show, theyare all separable, i.e. they are of the from E ( ρ ) = P i p i A i ⊗ B i ⊗ · · · ⊗ N i ρA † i ⊗ B † i ⊗ · · · ⊗ N † i . Note thatthose maps cannot generate entanglement and can be im-plemented (at least probabilistically) by local operationsand classical communication. The reason, why we areespecially interested in those maps is because we are in-terested in the decoherence effect of multipartite phasegates and not in generating entanglement among the sys-tem qubits. Apart from that, considering those mapsallows us to make a fair comparison to the decoherencecaused by two-qubit phase gates, which always leads toseparable maps.In this section we will first show that any purely de-phasing map is separable. Then we will restrict ourselvesto those situations where m bath qubits are interactingwith all qubits in the system. The system qubits are also interacting with some additional, but not overlap-ping qubits. We will present a simple formula for thosemaps, which will allow us to study how bath correlationsaffect the evolution of the system. This kind of investi-gations will further allow us to determine the behavior ofthe system in the case of Markovian and non–Markovianinteractions.As before we consider the map E ( ρ ) = tr B ( U SB ρ S ⊗ ( | + i h + | ) ⊗ N B U † SB ). In order to deal with the various in-teractions we are going to introduce the following nota-tion. Let I = { i , . . . , i k } , denote a set of k indices forsome k ≤ N B and B I denote those k bath qubits, i.e. B I ≡ { b i , . . . , b i k } . Then, U B I ,s l denotes a pure phasegate acting on the bath qubits, B I and on the systemqubit s l . Furthermore, I denotes the set of all sets ofindices of bath qubits which are interacting with somesystem qubit, i.e. I = { I ∈ { , . . . , N B } × k , for some k ≤ N B s.t. ∃ l s.t. U B I ,s l = 1 l } . That is, for each I ∈ I there exist | I | bath qubits B I and the single systemqubit, s l which interact with each other via the non–trivial pure phase gate of order | I | + 1, U B I ,s l . Foreach system qubit, l , I l ⊂ I denotes the set of indices, I = { i , . . . , i k } , for some k ≤ N B such that the bathqubits B I = { b i , . . . , b i k } interact with the system qubit l . That is, I l = { I ∈ { , . . . , N B } × k , for some k ≤ N B s.t. U B I ,s l = 1 l } . Moreover, for each set of in-dices, I , we introduce the function d I ( k ) = Q i ∈ I k i .The pure phase gates U B I ,s l can be written as U B I ,s l =1 l + | , . . . , i h , . . . , | B I ⊗ ( U I,s l − l ) where U I,s l is a sin-gle qubit phase gate acting on the system qubit s l [21].Thus, we have U B I ,s l = P k | k i h k | B I ⊗ U d I ( k ) I,s l ⊗ l , where U d I ( k ) I,s l = 1 l if d I ( k ) = 0 and U d I ( k ) I,s l = U I,s l if d I ( k ) = 1.Since the global unitary, U SB can be written as a prod-uct of the unitaries U B I ,s l , i.e. U SB = Q l Q I ∈I l U B I ,s l we find U SB = X k | k i h k | B ⊗ U k S , (9)where U k S = O l Y I ∈I l U d I ( k ) I,s l ≡ U k ⊗ U k ⊗ · · · ⊗ U k N S , (10)with U k l = Q I ∈I l U d I ( k ) I,s l . The corresponding completelypositive (CP) map is given by E ( ρ ) = X k U k S ρ ( U k S ) † = σ S K ρ, (11)where all U k S given in Eq. (10) are local unitary phasegates and σ S = P k U k S ( | + i h + | ) ⊗ N S ( U k S ) † , is a convexcombination of product LMESs and is, in particular, sep-arable (see also Eq (14) below). It should be noted herethat since the bath qubits can be in an entangled state,there is no reason, in general, why the map should beseparable. However, in the case investigated here, all theinteractions commute, which implies that instead of con-sidering an entangled initial state of the bath qubits, wecould equally well consider the product state, | + i ⊗ N B asinitial state. This fact alone is however, not enough toconclude that the maps are always separable, since thebaths qubits couple to several system qubits.So far we have seen that, whenever each system qubitinteracts only with bath qubits any decoherence map, E can be expressed as a convex combination of local deco-herence maps, i.e. E ( ρ ) = P i p i E i ( ρ ). Here, E i denotes alocal unitary map on the system qubits. The structureof E i is determined by the number of qubits the phasegates are acting on and by the number of overlappingqubits/edges. As a next step we will investigate the in-fluence of the coupling between the different baths (foreach system qubit) on the evolution of the system qubits.The more bath qubits interact with all system qubits thestronger the correlation between the baths (for each sys-tem qubit) is. We will derive for any of these correlationsa simple expression for the corresponding map, which willallow us then to analyze this effect quantitatively.We denote by E m ( ρ ) the state of the system qubitsafter each of them interacted via a phase gate (not nec-essarily pure) of arbitrary order, n l + 1, with the bathqubits { b i , . . . , b i m } and some other n l − m bath qubits,which do not interact with any other system qubit. Thatis, E m denotes the map which results from an interaction U SB = Q l Q I ∈I l U B I ,s l , where I is such that the follow-ing condition is fulfilled. There exists some fixed set of m indices, { i , . . . , i m } , such thata) ∀ I ∈ I , { i , . . . , i m } ⊆ I andb) ∀ I ∈ I l , I ′ ∈ I l ′ with l = l ′ it holds that I ∩ I ′ = { i , . . . , i m } .For instance, E ( ρ ) will describe the system qubits af-ter all of them interacted with independent bath qubits. E ( ρ ) describes a situation where all the gates acting onthe system qubits are coupled to one bath qubit, etc..Considering these kind of maps allows us to study howthe effect of decoherence changes with the degree of de-pendency of the environments of the system qubits. Forinstance, how does the decoherence affect the behav-ior of the system if all system qubits interact with thesame bath, compared to the one, where each system in-teracts with its own bath. In order to study all thosecases we are going to derive now a simple expression forthe corresponding maps. To this end we introduce thefollowing notation. J l = { J ∈ { , . . . , N B } k s.t. ∃ I ∈I l s.t.J ∪ { , . . . , m } = I } , i.e. each J ∈ J l denotesthe bath qubits, B J which interact with the bath qubits { b , . . . b m } and the l –th system qubit via a pure phasegate. J denotes the union of all sets J l . Let J = { j , . . . j | J | } be some set of indices. For some vector k =( k , . . . , k N B ) we denote by k J the | J | –dimensional vectorwith entries k j for j ∈ J , i.e. k J = ( k j , . . . k j | J | ). More-over k J l will denote the set of entries of k , { k j , . . . , k j r } ,where for each j t with t ∈ { , r } there exists some J ∈ J l such that j t ∈ J . Using this notation we can now statethe following theorem. Theorem 1.
Let U SB = Q l Q I ∈I l U B I ,s l , where B I denotes | I | bath qubits and s l denotes a single systemqubit. Moreover, let I be such that the conditions (a)and (b) are fulfilled for some set of indices, { i , . . . , i m } .Then the corresponding CP map, E = tr B [ U SB ( ρ ⊗| + i h + | ⊗ N B ) U † SB ] is given by E ( ρ ) = 12 N B ((2 N B − N B − m ) ρ + ˜ E ⊗ . . . ⊗ ˜ E N S ( ρ )) , (12) where ˜ E l ( σ ) = P k Jl ˜ U k Jl s l σ ( ˜ U k Jl s l ) † . Here, ˜ U k Jl s l = Q J ∈J l U d J ( k ) s l ,J is a local unitary operation. Note that this implies that the number of bath qubitswhich are interacting with all system qubits, m , onlychanges the factor in front of ρ and the weight with which˜ U k s l is applied but leaves the rest unchanged. Proof.
We assume without loss of generality that theset of bath qubits with which any system qubit is in-teracting, { b i , . . . , b i m } , is { b , . . . , b m } (condition (a)).Then, due to the fact that { , . . . , m } ⊆ I ∀ I ∈I we have that d I ( k ) = Q mi =1 k i Q i ∈ I (cid:31) { ,...,m } k i ≡ c ( k ) Q i ∈ I (cid:31) { ,...,m } k i , where c ( k ) = Q mi =1 k i is indepen-dent of I . Thus, we find for the local unitary trans-formations, U k l in Eq (10), U k l = ( ˜ U k l ) c ( k ) , where˜ U k l = Q I ∈I l U d I (cid:31) { ,...,m } ( k ) I,s l . Using this expression inEq (11) and the fact that P k : c ( k )=0 N B − N B − m we find E ( ρ ) = NB [(2 N B − N B − m ) ρ + ˜ E ( ρ )], where˜ E ( ρ ) = P k =( k m +1 ,...,k NB ) N l ˜ U k l ρ N l ( ˜ U k l ) † . As men-tioned before, J denotes the set of indices of baths qubitswhich are interacting with the system qubits and thebath qubits b , . . . , b m . Hence, for each pair J ∈ J l and J ′ ∈ J l ′ with l = l ′ we have J T J ′ = ∅ . Thus, the sum inthe expression of ˜ E ( ρ ) can be decomposed as a sum overall bath qubits with are interacting with the first systemqubit, J , the second, J , and so on. Since non of the in-dices will occur twice, we have ˜ E ( ρ ) = ˜ E ⊗ . . . ⊗ ˜ E N S ( ρ )) , with ˜ E l as defined above.It should be noted here that this result is independentof the coupling between the bath qubits and a single sys-tem qubit. The influence of their couplings and orders isreflected in the details of the local maps ˜ E l , but does notchange the general form of the map.In summary, we have seen that whenever some ( m ) ofthe bath qubits are interacting with all system qubitsvia some arbitrary pure phase gates and if no otheroverlapping bath qubit exists, then the map correspondsto a convex combination of the identity (with a weightthat depends on m ) and a local map. The local mapcorresponds to the map one would get by condition-ing on the fact that all qubits b , . . . b m are in thestate | i . That is, ˜ E l ( σ ) = tr B ( U B I (cid:31) { ,...,m } ,s l [ σ ⊗ ( | + i h + | ) B I (cid:31) { ,...,m } ] U † B I (cid:31) { ,...,m } ,s l ),where U B I (cid:31) { ,...,m } ,s l = b ,...b m h , . . . , | Q I ∈I l U B I ,s l | , . . . i b ,...b m .Let us now consider some examples. We will assumethat any system qubit, l , is interacting with the bathqubits only via a single pure phase gate of order n l +1 andphase φ l . Suppose further that the conditions (a) and (b)are fulfilled for some set of m indices, { i , . . . , i m } . Let E n ,...,n NS m denote the corresponding map. It is straight-forward to see that the maps ˜ E l in Eq (12) are then givenby ˜ E l ( σ ) = 2 n l − m [(1 − m − n l ) σ + 2 m − n l U l ( φ l ) σU l ( φ l ) † ].Here and in the following U l ( φ l ) denotes the single qubitphase gate with phase φ l . Using then that Q l n l − m =2 N B − m we find E n ,...,n NS m ( ρ ) = 12 m [(2 m − ρ + N S O l =1 E l ( ρ )] , (13)where E l ( σ ) = (1 − m − n l ) σ + 2 m − n l U l ( φ l ) σU l ( φ l ) † .For instance, if all system qubits interact with thesame m bath qubits (extreme non–Markovian case,see Sec. IV A) we obtain E m,...,mm ( ρ ) = NB [(2 N B − N B − m ) ρ + 2 N B − m ( E ⊗ . . . ⊗ E N S )( ρ )] , where E l ( σ ) = U l ( φ l ) σU l ( φ l ) † ). We are going to show in Sec. IV B howthe effect of decoherence depends on the number of bathqubits which are interacting with all system qubits. Thatis, we will analyze there how the degree of correlationsbetween the environment of the various system qubits isaffecting the evolution of the system.Note that any map which results from a pure phasegate of arbitrary order can be reinterpreted as a mixtureof the identity map and a map resulting from a two–qubit phase gate acting on the system and the bath qubit,which is prepared in the state | Ψ i = √ α | i + √ − α | i ,for some appropriately chosen α ≥
0. That is, E l ( σ ) =(1 − m − n l ) σ +2 m − n l U l ( φ l ) σU l ( φ l ) † = tr B [ U SB ( | Ψ i h Ψ |⊗ σ ) U † SB ], where U SB denotes the two–qubit pure phasegate with phase φ l and α = 1 − m − n l .Let us now also express the maps studied above interms of the Hadamard product, E ( ρ ) = σ S J ρ . First ofall, we present the state σ S , for the situation when thereis no interaction between the system qubits (see Eq (11)).We start with the LMES | ψ i = U SB | + i N S + N B . As thenext step we use the PEPS picture and introduce vir-tual qubits b i , . . . , b i k for every bath qubit b i on which k pure phase gates are acting on. For the qubits which arenon-overlapping, which means just one pure phase gateis acting on them, the projector P b i ,...,b ik b i will simply bethe identity and for the overlapping bath qubits it willhave the form P b i ,...,b ik b i = √ k − ( | i b i h . . . | b i ,...b ik + | i b i h . . . | b i ,...b ik ). For I ∈ I l , B v I denotes the set ofvirtual bath qubits which interact with system qubit s l via a phase gate U s l B v I . The LMES in the PEPS picturecan be written as | ψ i = N P b i ,...,b ik b i N l N I ∈I l (cid:12)(cid:12) ψ s l ,B v I (cid:11) with (cid:12)(cid:12) ψ s l ,B v I (cid:11) = U s l B v I | + i | I | +1 . As a first step we trace over all qubits which are non-overlapping. Wedenote by F = { i : P b i ,...,b ik b i = 1 l b i } the set of non-overlapping qubits. Then we have σ S = tr B ( | ψ i h ψ | ) =tr B \ F [ N i F P b i ,...,b ik b i ⊗ l ρ s l ,B v l N i F P † b i ,...,b ik b i ] with ρ s l ,B v l = tr i ∈ F ( ⊗ I ∈ I l (cid:12)(cid:12) ψ s l ,B v I (cid:11) (cid:10) ψ s l ,B v I (cid:12)(cid:12) ). Performing thetrace over all coupled qubits leads to a summation overall operators N l ρ k s l where ρ k s l = h k | ρ s l ,B v l | k i , with | k i denoting the computational basis of the overlapping bathqubits. We end up with σ S = X k O l ρ k s l (14)Examples of the reduced state in this form are given inSec. IV. For the special case where all system qubitscouple to the same m bath qubits via one pure phase gate(see Eq. (13)) we find σ S = m [(2 m − | + i h + | ) ⊗ N S + N N S l =1 ((1 − m − n l ) | + i h + | + 2 m − n l U l | + i h + | U † l )][22]. Asmentioned before, for any n –qubit phase gate, U it holdsthat U | + i h + | ⊗ n U † J ρ = U ρU † . Thus, writing σ S as σ S = P k U k | + i h + | ⊗ n U † k as we did here, leads directlyto the map E ( ρ ) = P k U k ρU † k . Hence, E can be directlyread off given σ S and visa versa. Depending on whichproperty of the output state one is interested in, one, orthe other way of looking at the map is better suited. IV. COMPARISON OF DIFFERENTINTERACTIONS
In this section we will use the result derived above inorder to compare different decoherence models with eachother. Due to the large variety of different interactionsthere is of course no way to make a complete comparison.However, we will focus here on certain relevant aspectsof the dependency of the evolution of the system qubitson various couplings. In Sec IV A and Sec IV B we in-vestigate how the decoherence depends on the degree ofbath–correlations. In Sec IV A, we will compare the casewhere there is no dependency, that is, each qubit inter-acts with an independent bath, and all pure phase gatesare applied to fresh bath qubits and the system qubits(Markovian case) to the other extreme case, where thesystem qubits are always interacting with the same bathqubits (extreme non–Markovian case). We will deter-mine the coherence time for an individual system qubitand compare the two extreme cases for different ordersof the applied phase gates. In Sec IV B we will first de-termine the evolution of a single system qubit and willinvestigate how the decoherence depends on the degreeof correlation of the baths. In contrast to Sec IV A wekeep the order of the phase gate fixed and vary the num-ber of qubits which are interacting more than once withthe system qubit. Then we will consider 2–qubit statesand certain n –qubit states and will show how the evo-lution of the entanglement shared by the system qubitsdepends on the coupling within the bath. Due to Eq.(12) one might expect that the system gets less disturbedthe larger m is. However the examples investigated herewill show that this is not the case. In fact we will showthat for certain states the entanglement is more robustagainst decoherence induced by individual baths ( m = 0)than against the one where m is very large. In Sec IV Cwe will discuss the effect of the order of the applied phasegates on the evolution of the system. A. Markovian scenario compared to (extreme)non–Markovian scenario
In the following we will call an interaction, U SB ,Markovian, if non of the qubits is interacting more thanonce with any other qubit via a pure phase gate, i.e. ∀ I, I ′ ∈ I , I ∩ I ′ = Ø. We will compare the decoherenceinduced by a Markovian process to the one induced by anon–Markovian process, where some bath qubits interactmore than once with some system qubits. This compar-ison will be performed for several different orders of theapplied phase gates.In the Markovian case we have that each system qubitinteracts independently from the other system qubitswith its own bath. Thus, we have that E ( ρ ) = E ⊗ . . . ⊗ E N S ( ρ ). In order to gain some insight into howthis coupling affects the evolution of the system we studyhow the local maps, E i acts on a single qubit. The sin-gle qubit is, at each step, interacting with n differentbath qubits. We consider here the case where the sys-tem qubit is interacting k times via a ( n + 1)–qubitphase gate with its environment. The phases of thephase gates are denoted by φ i , with i = 1 , . . . , k . Theevolution is governed by E ( ρ ) = E k ◦ . . . ◦ E ( ρ ), where E i ( ρ ) = α i ρ + β i U i ( φ i ) ρU i ( φ i ) † , with U i a single qubitphase gate with phase φ i and α i = 1 − − n , β i = 2 − n .The relevant parameter to be considered here is the off–diagonal element of ρ , ρ = h | ρ | i , which allows us todetermine the coherence time of the system [23]. Dueto Eq. (6) it is easy to see that after the evolution,the off– diagonal element of the output state will be ρ σ S , where σ S = h | σ S | i . For the sake of simplic-ity we chose now φ i = φ ∀ i ∈ { , . . . , k } . We obtain σ S = E ( | + i h + | ) = E k ( | + i h + | ) = J kl =1 σ S , where σ S corresponds to E and is given in Eq. (5) with n = n − ρ will be multiplied by thefactor | σ S | = h n − n q n − cos( φ ) + n − i k .Let us now investigate how this factor and thereforethe coherence time depends on the order of the appliedphase gates. The case where n = 1, i.e. the inter-action is described by 2–qubit phase gates, has beenstudied in [4]. There, | σ S | = [cos( φ/ k , which canbe approximated by e − kφ / for small values of φ k hasbeen obtained. In case the system qubit interacts viathree–qubit phase gates with its bath we have | σ S | =[ p
10 + 6 cos( φ )] k ≈ e − kφ . For four–qubit phasegates we get | σ S | = [ p
50 + 14 cos( φ )] k ≈ e − kφ / . The coherence in the Markovian process for a single qubitfalls therefore off exponentially with respect to the num-ber of occurred collisions, k . The coherence time in-creases as the order of the phase gates get larger. Inparticular the ratio of the coherence times between the3–qubit phase gate case and the 2–qubit phase gate caseis / ≈ , / ≈ , n bath qubits. It iseasy to see that in this case the evolution is governed bythe completely positive map E ( ρ ) = ρ ( φ + . . . + φ n ) J ρ ,where ρ ( φ ) is given in Eq. (5). There, the coher-ence term, | ρ | , gets multiplied by the factor | σ S | = n − n q n − cos( P ki =1 φ i ) + n − . Like in theMarkovian case we investigate how the coherence timedepends on the order of the applied phase gate. Again,in order to make a simple comparison we choose φ i = φ ∀ i . The 2-qubit phase gate case was already discussed in[4] where | σ S | = cos( kφ/ ≈ e − k φ / has been found.The approximation is valid for small values of kφ . Sincein the non-Markovian process the sum of all phases ap-pears in a cosine we get an oscillating behavior for largevalues of k . For the three– and four–qubit interactionswe find | σ S | = [ p
10 + 6 cos( kφ )] ≈ e − k φ / and | σ S | = [ p
50 + 14 cos( kφ )] ≈ e − k φ / , respectively.The coherence time is longer if we go to collisions withphase gates of higher order. This is similar to the Marko-vian case. Within the range of the approximations thetype of decay of the coherence in the non-Markovian caseis Gaussian whereas the decay in the Markovian case isexponential with respect to the number of collisions k .Note that the coefficients in the exponent coincide.In summary, we see that for both cases, the Markovianand the extremely non–Markovian case, the coherencetime increases with the order of the applied phase gates.This can be understood by noting that the higher the or-der is the more qubits get entangled to the system qubitduring the interaction. Since this entanglement is gen-erated by pure phase gates, the entanglement between asingle qubit and the rest gets smaller. More precisely, ifthe single system qubit is interacting with its bath viaa pure phase gate of order n + 1, we have E n l ( ρ ) = ρ ,where ρ is given in Eq. (5). It is easy to see that if n = 1and φ = π , the system qubit is maximally entangled toits bath. However, for ( n + 1)–qubit phase gates with n >
1, this is no longer possible since the off–diagonalterm of E ( ρ ), 2 − ( n +1) ( − n + e − iφ ), is non–vanishingfor any value of φ . As mentioned already, the only dif-ference between the two extreme cases investigated hereis the dependency on the number of collisions (in the0region, where the approximations are valid). Whereasthe coherence term in the Markovian case decays expo-nentially with the number of collisions it does so with aGaussian function in the extreme non–Markovian case. B. Comparing different bath dependencies
In this section we discuss how the induced decoherencedepends on how strongly the baths for the individual sys-tem qubits are coupled with each other. To this end wekeep the order, n + 1, of the applied pure phase gatesfixed and the same for each system qubit. That is, eachsystem qubit is interacting with n bath qubits. Out ofthese n baths qubits m of them interact with all sys-tem qubits. Using the notation from before, we have ∀ I, I ′ ∈ I I ∩ I ′ = { b i , . . . , b i m } for some fixed set of m bath qubits, { b i , . . . , b i m } . The aim of this subsection isto investigate how the evolution of the system changes asa function of m , the degree of bath–correlation. We willfirst work out in detail how the decoherence affects theevolution of a single and two system qubits. Then we willconsider the general case of N S system qubits and inves-tigate the influence of the degree of baths–correlation onthe evolution. That is, we will use Eq. (12) and Eq. (13)to determine the evolution of the system for the variousscenarios. For a single system qubits we consider the sit-uation where the system is interacting with the bath viaseveral phase gates. The number of overlapping qubits, m , will denote in this case the number of qubits which arecoupled to the system qubit in each phase gate. Physi-cally this means that those m bath qubits take part inany collision with the system qubit. We will investigatehere how the decoherence of the system qubit depends onthis coupling. For more than a single system qubit we in-vestigate the scenario where each system qubit interactswith its environment via a single pure phase gate. There, m denotes, as explained above, the number of bath qubitswhich interact with all system qubits. We will also inves-tigate the evolution of entanglement shared between thesystem qubits.As mentioned before, depending on which property ofthe evolved state one is interested in, the presentation ofthe decoherence map in terms of the Kraus operators, orin the form E ( ρ ) = σ S ⊙ ρ is better suited. This is whywe will give both representations in the examples inves-tigated here. Whenever we talk about the correspond-ing reduced state in this context, we refer to σ S , whichdenotes the reduced state of the LMES corresponding tothe evolution (see Sec. III). The phase gates, acting on n qubits will be denoted by U n ( φ i ) = 1 l − (1 − e iφ i ) | i h | ⊗ n and U i = U ( φ i ) will denote the single qubit phase gateswith phase φ i .
1. A single system qubit
We consider a single system qubit which is interact-ing with its environment via k different ( n + 1)–qubitphase gate. We denote by m the number of bath qubits, { b , . . . , b m } , which take part in all interactions withthe system qubit. That is, ∀ I, I ′ ∈ I with I = I ′ it holds that I ∩ I ′ = { b , . . . , b m } . Then we have E ( ρ ) = 2 − m [(2 m − ρ + ◦ ki =1 E i ( ρ )], where ◦ denotes thecomposition and E i ( ρ ) = n − m ((2 n − m − ρ + U i ρU † i ) . Let us now explicitly determine the evolution as a func-tion of m . In order to see the effect of the coupling withinthe bath it suffices to consider only two different phasegates. We start with the Markovian case, where the phasegates are acting on independent bath qubits. The corre-sponding decoherence map is E ( ρ ) = σ S ⊙ ρ = E ◦ E ( ρ ),where E j ( ρ ) = λ ρ + λ U j ρU † j with λ = 1 − − n , λ = 2 − n . Note that the reduced state, σ S , is given by σ S = ρ ⊙ ρ where ρ i is the single qubit reduced stateof U n +1 ( φ i ) | + i ⊗ n +1 (see Sec II A).Next, we investigate the scenario in which one bathqubit is overlapping. The map can be written as E ( ρ ) = σ S ⊙ ρ = [ ρ + E ◦ E ( ρ )], where E j ( ρ ) = λ ρ + λ U j ρU † j with λ = 1 − − n and λ =2 − n . The reduced state for this case is given by σ S = 2 P k =0 ρ k ⊙ ρ k , with ρ i = | + i h + | and ρ i = n ((2 n − − | + i h + | + U i | + i h + | U † i ), for i ∈ { , } .The decoherence map for the case of two overlappingqubits is given by E ( ρ ) = σ S ⊙ ρ = [3 ρ + E ◦ E ( ρ )],where E j ( ρ ) = λ ρ + λ U j ρU † j and λ = 1 − − n , λ = 2 − n . The corresponding reduced state is σ S =4 P k ,k =0 ρ k k ⊙ ρ k k with ρ i = | + i h + | = ρ = ρ and ρ i = n ((2 n − − | + i h + | + U i | + i h + | U † i ), for i ∈ { , } . In the case where all qubits are overlapping,i.e. the gates act on the same bath qubits, we obtain E ( ρ ) = σ S ⊙ ρ = 2 − n [(2 n − ρ + U U ρU † U † ]. Thereduced state σ S is then the single qubit reduced state ofthe state U n +1 ( φ ) U n +1 ( φ ) | + i ⊗ n +1 = U n +1 ( φ + φ ) | + i ⊗ n +1 .It can be seen from the equation above that for a fixedorder of the phase gates, n , the coefficient in front of ρ gets larger as m increases. Thus, the system gets lessdisturbed if many of the bath qubits are overlapping.In order to investigate how the coherence for the singlesystem qubit is affected by the number of coupled qubits, m , we plot the absolute value of the off-diagonal elementof σ S for different values of m . Fig IV B 1 shows thevalue of | σ S | for 5-qubit phase gates for different numberof coupled qubits m = 0 , , , φ = φ = φ .The extreme cases where no bath qubits are overlapping, m = 0, and the case where all bath qubits are coupled, m = 4, correspond to the Markovian and extreme non-Markovian case as already studied in Sec IV A. One maindifference between those two cases is that the the non-Markovian case shows an oscillating behavior in contrastto the Markovian one. When both phases are π , the value1 FIG. 1: For a 5–qubit phase gate the absolute value of h | σ S | i is plotted for different values of m. The continu-ous line shows the Markovian case where the two phase gateshave no overlapping qubit, i.e. m = 0. The dashed-dottedline corredsponds to one overlapping qubit ( m = 1) and thedashed line to three overlapping qubits ( m = 3). The dot-ted line refers to the extreme non-Markovian case with fouroverlapping qubits ( m = 4). of | σ S | is maximal for the non-Markovian case, sincethere the map acts as the identity map. However, it iseasy to see that in the Markovian case the value of | σ S | is minimal then. From the cases m = 1 and m = 3 onecan infer the transition between the extreme cases. The m = 1 case looks very like the Markovian case, whereasthe m = 3 case shows already an oscillating behavior. Forsmall phases the Markovian case is less damped than thenon-Markovian one. This can be explained as follows. Inthe Markovian case the value of | σ S | is multiplied k timeswith itself, where k denotes the number of phase gatesapplied to the qubit. Since k equals only 2 in our case,the value remains relatively large since | σ S | ≈
1. Thevalue of | σ S | for the non-Markovian case is decreasing,because we add the two phases of the two applied gatesand therefore the value of the cosine appearing in theformula gets smaller. This effect is for small phases largerthan in the Markovian scenario.
2. Two system qubits
In this subsection we consider two system qubits whichare both interacting via a ( n + 1)–qubit phase gate withthe environment. Similarly to the single qubit case we in-vestigate here how the degree of dependency in the bathsinfluences the evolution of the system. Here, we will alsostudy how the entanglement between the system qubitsis affected by the different couplings within the bath.The decoherence map for the case where two inde-pendent ( n + 1)-qubit phase gates are acting on a twoqubit initial state ( m = 0) has the following form: E ( ρ ) = σ S ⊙ ρ = E ⊗ E ( ρ ) with E j ( σ ) = (1 − − n ) σ + 2 − n U j σU † j ). The reduced state is given by σ S = ρ ⊗ ρ with ρ i being the single qubit reducedstate of U n +1 ( φ i ) | + i ⊗ n +1 (see Sec II A). For m = 1, i.e. one bath qubit is overlapping, the decoherence mapis given by E ( ρ ) = σ S ⊙ ρ = ρ + E ⊗ E ( ρ ) with E j ( ρ ) = (1 − − n ) σ + 2 − n U j σU † j . The correspondingreduced state is σ S = 2 P k =0 ρ k ⊗ ρ k , with ρ = | + i h + | and ρ = n ((2 n − − | + i h + | + U i | + i h + | U † i ). Thecase where two qubits are overlapping, i.e. m = 2, isdescribed by E ( ρ ) = σ S ⊙ ρ = [3 ρ + E ⊗ E ( ρ )] with E j ( σ ) = (1 − − n ) σ + 2 − n U j σU † j ). The correspond-ing reduced state is given by σ S = 4 P k ,k =0 ρ k k ⊗ ρ k k with ρ j = | + i h + | = ρ = ρ and ρ j = n ((2 n − − | + i h + | + U j | + i h + | U † j ). In the extremenon-Markovian case where both system qubits interactwith the same n bath qubits, the map can be written as E ( ρ ) = (1 − − n ) ρ + 2 − n U U ρU † U † with the reducedstate given by σ S = 2 n P k =0 ρ k k ...k n ⊗ ρ k k ...k n =(1 − − n ) | ++ i h ++ | + 2 − n U U | ++ i h ++ | U † U † ) with ρ k k ...k n j = n | + i h + | ∀ k · k · . . . k n = 0 and ρ k k ...k n j = n U j | + i h + | U † j ∀ k · k · . . . k n = 1. Sim-ilar to the single qubit case investigated before, we seethat the factor in front of ρ increases as m increases.Thus, the system gets less distorted the more bath qubitsare overlapping.Let us now study the evolution of the entanglementbetween the system qubits for several different couplingschemes. As examples we consider n = 2, that is thesystem qubits are interacting with the environment via3–qubit phase gates. We compare three different typesof coupling scenarios: 1) the case where each gate actsindependently on one of the system qubits ( m = 0); 2)the case where one bath qubits overlaps ( m = 1 = n − m = 2 = n ). We compute the entanglement of forma-tion of the output state E ( ρ ) for the different couplings.The entanglement of formation is numerically optimizedwith respect to a maximally entangled input state. Fig.IV B 2 shows the entanglement of formation of the op-timized input state as a function of the two phases, φ and φ , of the three–qubit phase gates. In case 1), where m = 0, the minimum at φ = φ = π , can be explainedas follows. As explained in Sec IV A considering a singlequbit interacting with its bath via a pure ( n + 1)–qubitphase gate with phase φ generates a maximally entan-gled state only if φ = π and n = 1. For larger valuesof n , the entanglement is still optimal, however it is nolonger maximal, for φ = π . Thus, if both system qubitsare interacting with their independent baths, the entan-glement between them is minimized if both phases are π due to entanglement monogamy.In the second case, it can be seen in Fig IV B 2 thatthe entanglement is minimized if one of the phases is π .This fact might be explained by the following observa-tion. The map which is applied to the input state is givenby E ( ρ ) = ρ + E ⊗ E ( ρ ) with E j ( σ ) = σ + U j σU † j .As mentioned in Sec III A we can interpret E j ( σ ) as themap corresponding to the interaction of the system qubit2 FIG. 2: entanglement of formation optimized with respect toa maximally entangled input state for three different couplingscenarios a) two three qubit phase gates U b b U b b actingindependently on the two system qubits 1 and 2 b) one bathqubit is overlapping within the phase gates U b b U b b c)the phase gates U b b U b b have one overlapping edge with a single bath qubit via a two–qubit phase gate, U ,i.e. E j ( σ ) = tr B ( U | + i h + | ⊗ σU † ). Using this fact it isnow easier to see why there is a minimum if one of thephases is π . Similar to the reasoning above the two qubitphase gates have its maximal entangling capability be-tween the system and the bath for φ = π and thereforethe entanglement between the system qubits after apply-ing the map E ⊗ E is minimized due to entanglementmonogamy.In the third case, where m = n = 2, we obtain a max-imally entangled state whenever φ = φ or φ + φ = 2 π .This can be easily explained by noting that the outputstate is of the form E ( ρ ) = αρ + βU ⊗ U ρU † ⊗ U † , forsome values of α and β . Now, if φ = φ , which impliesthat U = U the singlet state, | Ψ − i = 1 / √ | i + | i )is left invariant under this map. Therefore, the outputstate is maximally entangled. In the other case, where φ + φ = 2 π , which implies that U = U † , | Φ + i is leftinvariant. Thus in both cases the optimal entanglementof formation, optimized with respect to the maximallyentangled input state, is maximal.
3. n system qubits
We consider the evolution of N S system qubits, whereeach system qubit, s l is interacting with n bath qubitsvia a pure ( n + 1)–phase gate with phase φ l . The de-gree of bath–correlation is quantified by the number ofoverlapping bath qubits, m , which we again denote by b , . . . , b m . Those bath qubits are interacting with allsystem qubits and no other bath qubit is interacting withmore than one system qubit (see conditions (a) and (b)).The evolution of the system is governed by the completelypositive map given in Eq (12). Let us now explicitly writedown the completely positive map for the three cases: 1) m = n , 2) m = n −
1, and 3) m = 0. 1) For m = n all system qubits are coupled to the same m bath qubitsvia a ( n + 1)–phase gate, i.e. the baths of all systemqubits are maximally overlapping. In this case we have E ( ρ ) = αρ + β N i U i ρU † i with α = 1 − − n and β = 2 − n .2) For m = n − { b , . . . b n − } and one additional bath qubit,which is not connected to any other qubit. In this casewe have E ( ρ ) = αρ + β ( N E i )( ρ ) with α = 1 − − n +1 and β = 2 − n +1 . Here, E l ( σ ) = α l σ + β l U l σU † l with α l = β l = . 3) For m = 0 each system qubit is interact-ing with its own bath. In this case the evolution if gov-erned by the completely positive map E ( ρ ) = ( N E l )( ρ ),where E l ( σ ) = α l σ + β l U l σU † l with α l = 1 − − n l and β l = 2 − n l . As can be seen above (see also Eq (13)) thedecoherence map is always of the form E ( ρ ) = αρ + β ˜ E ( ρ ),where ˜ E ( ρ ) denotes a local map. Since the coefficient α is given by 1 − − m , one might expect that the systemgets less disturbed the larger m gets, that is the strongerthe coupling among the baths is. However, as we will seebelow, this is not the case in general. For instance, theentanglement of a state can be decreased faster when thesystem qubits are interacting with a maximally depen-dent bath than it is if each system qubit is interactingwith its own bath.As example we consider the input state | Ψ i = √ p | i ⊗ N S + √ − p | i ⊗ N S with p >
0. We assume thatall interactions with the environment are equivalent, i.e. φ l = φ , ∀ l . It is straightforward to compute the outputstate for the different couplings mentioned above. Let uscompare here the two extreme cases, m = 0 and m = n .In order to quantify the entanglement of the output statewe consider here the bipartite splitting of one systemqubit versus the remaining N S − E , between qubit 1 and the rest, whichwe measure with the log–negativity [14]. It can be easily3shown that for the case, where each system qubit is in-teracting with its own bath we find E indep = log (1 +2 |√ p √ − p p (1 − − n + e iφ − n ) N S | ) whereas in theextreme non–Markovian case we have E dep = log (1 +2 |√ p √ − p √ − − n + e iN S φ − n | ). The coupling ofeach system qubit to its individual bath destroys less en-tanglement than a correlated bath if E indep > E dep . Weare going to show next that such a scenario is possible foran arbitrary number of system qubits, N S . Consideringthree–qubit phase gates, n = 2, with φ = πN S we find E indep > E dep iff [ ( + cos( π/N S ))] NS <
1. Sincethe left hand side of this inequality can be easily upperbounded by [ ] NS this shows that in general it is nottrue that the entanglement is less distorted if the degreeof bath–correlation is larger. C. Comparison order of gates
In this section we investigate the effect of the order n l of the phase gates on the evolution of the system. We willfix the number of overlapping bath qubits m . The influ-ence of n l can be seen by looking at the decoherence mappresented in Eq. (13), i.e. E n ,...,n NS m ( ρ ) = NB [(2 N B − N B − m ) ρ + 2 N B − m N N S l =1 E l ( ρ )] , where E l ( σ ) = ((1 − m − n l ) σ + 2 m − n l U l ( φ l ) σU l ( φ l ) † ). The order of the phasegates, n l , does not affect the coefficient in front of ρ , butchanges only the weights in the maps E l . The larger n l the larger is the coefficient in front of σ . Therefore theinitial state is less altered by phase gates of higher order.This coincides with the fact that the probability of ap-plying a phase gate on the system qubit gets smaller thelarger the order of the phase gate is. For example, if thesystem interacts with the bath via a 5–qubit phase gate,all four bath qubits have to be in the state | i in order toapply the phase gate to the system qubit; for a 2–qubitgate, however, just one bath qubit has to be in the state | i . V. GENERATING LMESS
In this section we will consider the complementary pro-cess to the one investigated so far. We will use the cou-pling of the system qubits to the bath in order to generatea state of interest. This can then be used to either pre-pare the state using dissipation, or a unitary evolution.We call a complete set of commuting unitary observ-ables, { U i } ni =0 generalized stabilizer if the eigenvalues of U i are ± − n − –fold degenerate[24]. Furthermore, we call a n partite state, | Ψ i a gen-eralized stabilizer state, if there exists a generalized sta-bilizer, { U i } ni =0 , such that U i | Φ i = | Φ i ∀ i iff | Φ i = | Ψ i .Examples of generalized stabilizer states are all LMESs.For an arbitrary n –qubit LMES, | Ψ i = U Ψ ph | + i ⊗ n , thegeneralized stabilizer are of the form U k = U Ψ ph X k ( U Ψ ph ) † where X denotes here and in the following the Pauli op-erator σ x . It is easy to verify that U k | Φ i = | Φ i ∀ k iff | Φ i = | Ψ i . Depending only on the phases α i , whichdefine the LME, | Ψ i , the generators of the generalizedstabilizer can be quasi–local, i.e. act non trivially ona small set of (neighboring) qubits. In this case, themethods developed in [15] can be employed to derive aquasi–local dissipative process for which the unique sta-tionary state is | Ψ i . Apart form that, one can also easilyconstruct a frustration free Hamiltonian for which theunique ground–state is | Ψ i [5].In the following we show how generalized stabilizerscan be prepared by a unitary evolution. Therefore, weuse n auxiliary qubits which are all prepared in the state | + i . The unitary operation is composed out of controloperations, where the auxiliary systems act as controlqubits. The following theorem shows that a generalizedstabilizer state can always be prepared by applying thisunitary to an arbitrary input state of the system. Theorem 2. If | Ψ i is a n –qubit generalized stabilizerstate then there exists a complete set of commuting uni-tary observables, { U i } ni =0 (the generalized stabilizer) anda set { V i } ni =0 with { V i , U i } = 0 such that for all n –qubitstates | Φ i there exists a n –qubit state | φ i such that | Ψ i s | φ i a = U c U c . . . U cn | Φ i s | + i ⊗ na . (15) The control operations U ci are of the form U ci = ¯ U i ˜ U i ,where ˜ U i = 1 l ⊗ | i a i h | + U i ⊗ | i a i h | and ¯ U i = 1 l ⊗| + i a i h + | + V i ⊗ |−i a i h−| . Thus, in order to prepare a generalized stabilizer state,one only has to prepare the auxiliary systems in the prod-uct state | + i ⊗ n and apply certain control gates. Depend-ing on the properties of the state | Ψ i these gates might bequasi–local, i.e. act only on a few (neighboring) qubits.Note that this theorem can also be used to derive a pro-cess using either dissipation, or a completely positive mapin order to prepare the state, | Ψ i [15]. Proof.
First of all, note that the set { U i } does notonly define the state | Ψ i , but a whole set, the com-mon eigenbasis of the commuting observables. Each ele-ment of this basis is uniquely defined by its eigenvalues.We use the notation | Ψ i ,...,i n i for the basis elementswith eigenvalues (( − i , . . . ( − i n ), with i k ∈ { , } .Let us denote by S i ≡ s pan {| Ψ k ,...,k i =0 ,...,k n i k l ∈{ , } ( S i ≡ s pan {| Ψ k ,...,k i =1 ,...,k n i k l ∈{ , } ) the range of 1 l + U i (1 l − U i ) respectively. That is any state which is aneigenstate of U i with eigenvalue 1 ( −
1) is within S i ( S i ) respectively and vice versa. Note that both sub-spaces have dimension 2 n − . We define V i as the unitarywhich exchanges S i with S i . To be more precise, V i = P k ,...k n ∈{ , } | Ψ k ,...,k i =0 ,...,k n i h Ψ k ,...,k i =1 ,...,k n | + h.c. .Now, we decompose | Φ i into a state which is in S n andone which is in S n , i.e. | Φ i = (1 l + U n ) | Φ i +(1 l − U n ) | Φ i ≡ (cid:12)(cid:12) Φ n (cid:11) + (cid:12)(cid:12) Φ n (cid:11) . Applying ˜ U n to the state | Φ i | + i a n leadsto ˜ U n | Φ i | + i a n = (cid:12)(cid:12) Φ n (cid:11) ( | i + | i ) a n + (cid:12)(cid:12) Φ n (cid:11) ( | i − | i ) a n .4Applying now ¯ U n leads to U cn | Φ i | + i a n = (cid:12)(cid:12) Φ n (cid:11) | + i a n + V n (cid:12)(cid:12) Φ n (cid:11) |−i a n , where both, (cid:12)(cid:12) Φ n (cid:11) and V n (cid:12)(cid:12) Φ n (cid:11) are ele-ments of S n . Continuing in this way, we find that all theterms which occur are elements of S T S T . . . T S n = {| Ψ i} . This is due to the fact that the operators V i onlychange a state in S i to a state in S i and vice versa, but ifthe resulting state is an eigenstate of U k with eigenvalue1 ( − U k with the sameeigenvalue.Theorem 2 can also be used to show how an arbi-trary state can be constructed using a generalized sta-bilizer state as a recourse. This can be seen as fol-lows. Due to Theorem 2 we have that for any n –qubit state, | Φ i there exists a n –qubit state | φ i suchthat | Ψ i s | φ i a = U c U c . . . U cn | Φ i s | + i ⊗ na . Thus, we have | φ i a = h Ψ | s U c U c . . . U cn | Φ i s | + i ⊗ na . Note that U c U c . . . U cn | Φ i s | + i ⊗ na = (16) X i ,...i n X ( i )1 · · · X ( i n ) n | Φ i H ⊗ n | i , . . . i n i . Here, H denotes the Hadamard transformation, and X (0) j = 1 l + U j , and X (1) j = V j (1 l − U j ) = (1 l + U j ) V j .Since U j is hermitian, we have that h Ψ | X ( i )1 · · · X ( i n ) n = h Ψ i ,...,i n | . Denoting by U Ψ the unitary which trans-forms the computational basis into the basis | Ψ i ,...,i n i ,i.e. U Ψ = P i ,...i n | Ψ i ,...,i n i h i , . . . , i n | we have | φ i = H ⊗ n U † Ψ | Φ i . (17)Thus, an arbitrary state | Φ i can be prepared by apply-ing the controlled operations ( U ci ) † to the state | Ψ i | φ i ,where | φ i is given in Eq. (17). For any LMES wehave | Ψ i = U Ψ | , . . . , i . As before, one can definea phase–gate, U Ψ ph , which is diagonal in the computa-tion basis, such that, | Ψ i = U Ψ ph | + i ⊗ n , and, more gen-erally, U Ψ = U Ψ ph H ⊗ n . In this case we would have | φ i = ( U Ψ ph ) † | Φ i . Choosing | Φ i = | + i ⊗ n we obtain | φ i = | Ψ ∗ i , where | Ψ ∗ i denotes the complex conjugationof | Ψ i in the computational basis. Hence we have, | Ψ i | Ψ ∗ i = U c U c . . . U cn | + i ⊗ ns | + i ⊗ na . (18)It should be noted here that for an arbitrary numberof qubits, n , there exists a n –qubit LMES, | Ψ i , suchthat | Ψ i and | Ψ ∗ i are neither LU–equivalent, nor canbe transformed into each other by more general oper-ations, namely local operations and classical communi-cation [16]. For Graph states, into which any stabilizerstate can be transformed by local unitary operations, andalso for certain LMESs the phase gate is hermitian, whichimplies that | Ψ i = | Ψ ∗ i . Therefore, we find for thosestates | Ψ i | Ψ i = U c U c . . . U cn | + i ⊗ ns | + i ⊗ na . (19) Note that this is a new method to generate two copiesof an entangled state, where both of them are generatedin one step. That is, the states are not generatedindependently. One might compute the entanglement offormation using this procedure and compare it to twicethe entanglement of formation of a single copy of | Ψ i . VI. CONCLUSION AND OUTLOOK
We investigated the effect of decoherence induced bymany–body interactions. In our decoherence model weconsidered multi-qubit Ising interactions between the sys-tem and bath qubits. These interactions could resultfrom multipartite collisions. The corresponding unitarywas modeled as a multipartite phase gate. Due to thelarge variety of different interaction patterns we focusedhere on some relevant aspects of decoherence. We mainlyconsidered purely dephasing maps, where each systemqubit is coupled to another system qubit only via theenvironment, i.e. there is no collision between systemqubits. We showed that all purely dephasing maps areseparable. In order to analyze the effect of correlationswithin the bath on the evolution of the system we consid-ered those situations where m bath qubits are connectedto all system qubits, which can also interact with other,non–overlapping bath qubits. We derived a simple ex-pression for those maps and showed how the effect ofdecoherence depends on the number of commonly sharedbath qubits, m . This enabled us to compare all the cases,from the one where each system is coupled to its own en-vironment to the one where all system qubits are coupledto the same environment. Furthermore, we compared theMarkovian to the Non–Markovian processes. We alsoshowed how the evolution of the system is influenced bythe order of applied phase gates. Note that these resultscan also be generalized to higher dimensional systems.The coupling to the environment can not only destroythe coherence and entanglement in the system, but canalso be used to accomplish certain tasks [15]. We showedhere how the bath qubits can be used to prepare a de-sired state. In particular, we derived a unitary opera-tion, acting on the bath and system qubits which can beemployed to generate certain multipartite states. Thisinvestigation also lead to a protocol which prepares twocopies of certain multipartite states in a single step.An other interesting question in this context is howthe entanglement established between the system andbath qubits is related to the induced decoherence. Forinstance, the dephasing map E ( ρ ) = σ S J ρ is a com-pletely dephasing map, i.e. E ( ρ ) = ρ D , where ρ D is adiagonal matrix whose diagonal is the same as the oneof ρ , for any state ρ , iff σ S = 1 l . That is the map de-stroys all the coherence in the system iff the correspond-ing LMES is maximally entangled between the systemand the bath qubits. A simple example would be thestate | Ψ i SB = U SB | + i ⊗ n = | Φ + i ⊗ nSB . Here, | Φ + i de-5notes a maximally entangled state shared between a bathand a system qubit. Obviously, for states, ρ , whose diag-onal elements are all equal, such a map would completelydepolarize the state. Examples of such states are arbi-trary convex combination of product LMES (all in thecomputational basis).An other interesting topic is the characterization of de-coherence free subsets, i.e. the characterization of thosestates which are left unchanged under certain dephas-ing map. For instance, if we consider N S system qubits,which are all coupled to the same bath composed of m qubits, i.e. n l = m + 1 for all l we have (see Eq.(13)) E ( ρ ) = αρ + βU ⊗ . . . ⊗ U N S ρU † ⊗ . . . ⊗ U † N S .Thus, a state ρ is left invariant under this coupling iff U ⊗ . . . ⊗ U N S ρU † ⊗ . . . ⊗ U † N S = ρ . Choosing the phasessuch that all local unitaries coincide, we see that the state ρ will be invariant iff U ⊗ N S ρ ( U ⊗ N S ) † = ρ . States whichfulfill the above condition for any unitray U are calledsuper–singlets [17]. Since we consider here only dephas- ing maps, the state must only be invariant under localphase gates, which implies that the decoherence–free sub-set contains more states than the super–singlets. Forinstance, in the case of two qubits, the so–called Wernerstates, which are of the form ρ = (1 − p ) / l + p | Ψ − i h Ψ − | ,where 1 ≥ p ≥ | Ψ − i denotes the singlet state,characterize all states invariant under U ⊗ U , for anyunitary U [18]. However, the decoherence free subspaceof the map described above contains any state of the form ρ = α | i h | + β | i h | + γ | Ψ + i h Ψ + | + δ | Ψ − i h Ψ − | ,for α, β, γ, δ ≥ VII. ACKNOWLEDGMENT
B.K. would like to thank Peter Zoller for fruitful dis-cussions and T.C. and B.K. are grateful to Wolfgang D¨urfor helpful discussions. This work has been supported bythe FWF (Elise Richter Program, SFB FOQUS). [1] A. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.Fisher, A. Garg, W. Zwerger, Rev. Mod. Phys. ,1(1987).[2] Prokof’ev and P. Stamp, Rep. Prog. Phys. , 669 (2000).[3] J. Calsamiglia, L. Hartmann, W. D¨ur, H.J. Briegel, Phys.Rev. Lett. , 180502 (2005)[4] L. Hartmann, J. Calsamiglia, W. D¨ur, H.J. Briegel, Phys.Rev. A , 052107 (2005).[5] C. Kruszynska and B. Kraus, Phys. Rev. A, , 052304(2009).[6] C.-E. Mora, H. J. Briegel, B. Kraus, Int. J. Quant. Inf. , 5, 729-750 (2007).[7] M. Hein, W.D¨ur, J. Eisert, R. Raussendorf, M. Vanden Nest, H. J. Briegel, Proceedings of the InternationalSchool of Physics Enrico Fermi on Quantum Computers,Algorithms and Chaos, Varenna, Italy, July, 2005; seealso e-print quantph/ 0602096[8] D. Gottesman, Ph.D. Thesis, quant-ph/9705052.[9] W. D¨ur, L. Hartmann, M. Hein, M. Lewenstein, and H.-J. Briegel, Phys. Rev. Lett. , 097203 (2005)[10] for a recent review see F. Verstraete, V. Murg, J. I. Cirac,J. Mod. Opt. , 143 (2008) and references therein.[11] F. Verstraete, J. I. Cirac, arXiv: cond-mat/0407066v1;G. Vidal, Phys. Rev. Lett. , 147902 (2003).[12] R. A. Horn, Ch. R. Johnson, Topics in Matrix Analy-sis (Cambridge University Press, Cambridge, England,1991).[13] C. King, K. Matsumoto, M. Nathanson and M.B. Ruskai,Markov Process and Related Fields , 391-423 (2007).[14] G. Vidal, R.F. Werner, Phys. Rev. A, , 032314 (2002).[15] F. Verstraete, M. M. Wolf, J. I. Cirac, Nature Physics , 633 - 636 (2009); B. Kraus, H. P. B¨uchler, S. Diehl,A. Kantian, A. Micheli, and P. Zoller, Phys. Rev. A ,042307 (2008).[16] B. Kraus, Phys. Rev. Lett. , 020504 (2010).[17] A. Cabello, J. Mod. Opt. , 10049 (2003).[18] R. F. Werner, Phys. Rev. A , 4277 (1989).[19] Note that using the PEPS–formalism, a LMES can alsobe written in terms of maximally entangled states andprojectors which depend on the phases. For instance, the3–qubit LMES, | Ψ i = U ( φ ) | + i ⊗ can be writtenas | Ψ i = p ( φ )( (cid:12)(cid:12) φ + (cid:11) (cid:12)(cid:12) φ + (cid:11) ) with p i i i ( φ ) = √ | i h ++ | + | i h ++ | ) − (1 − e iφ ) √ | i h | .[20] Note that in order to make the corresponding completelypositive (CP) map trace preserving we would simply needto consider 1 / N B U SB . In the following we will howevernot explicitly write this normalization factor.[21] Note that any phase gate can be decomposed in this form.[22] Note that this corresponds to a map, which is a convexcombination of the identity map and the one which re-sults from 2–qubit phase gates, as it should be.[23] As explained in the introduction, the phase gates cor-respond to a evolution due to the Hamiltonian of theform H SB = P i α i σ i z . The time dependence of the uni-tary U SB = e − iH SB t is hidden in the phases of the phasegates. This is why we speak of coherence times, eventhough we deal with time independent gates.[24] Note that for stabilizer states, the stabilizers U ii