Greenberger-Horne-Zeilinger generation protocol for N superconducting transmon qubits capacitively coupled to a quantum bus
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Greenberger-Horne-Zeilinger generation protocol for N superconducting transmonqubits capacitively coupled to a quantum bus Samuel Aldana, Ying-Dan Wang,
1, 2 and C. Bruder Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland Department of Physics, McGill University, Montreal QC, H3A 2T8, Canada
We propose a circuit quantum electrodynamics (QED) realization of a protocol to generate aGreenberger-Horne-Zeilinger (GHZ) state for N superconducting transmon qubits homogeneouslycoupled to a superconducting transmission line resonator in the dispersive limit. We derive aneffective Hamiltonian with pairwise qubit exchange interactions of the XY type, ˜ g ( XX + Y Y ),that can be globally controlled. Starting from a separable initial state, these interactions allow togenerate a multi-qubit GHZ state within a time t GHZ ∼ ˜ g − . We discuss how to probe the non-localnature and the genuine N -partite entanglement of the generated state. Finally, we investigate thestability of the proposed scheme to inhomogeneities in the physical parameters. PACS numbers: 03.67.Bg, 85.25.Cp, 03.67.Lx
I. INTRODUCTION
Entangled quantum states are one of the essential re-sources for quantum information processing. They arenecessary for the realization of quantum communicationand the most important computational tasks. Many ef-forts have been devoted to the elaboration of physicalsystems enabling the generation and the control of suchstates. In particular, different types of superconductingqubits are promising candidates to solve this problem.Until recently limited to two qubits , efforts to entanglesuperconducting qubits have lately reached a new mile-stone with the experimental demonstration of three-qubitentanglement .In the present paper we consider transmon qubits ina circuit quantum electrodynamics architecture andpresent a way to generate GHZ states , i.e., maximallyentangled states. Although the mathematical descrip-tion of multipartite entanglement for more than threequbits is still debated , GHZ states remain paradig-matic entangled states which are, in particular, usefulfor fault-tolerant quantum computing or quantum se-cret sharing . So far, many different protocols havebeen proposed to generate such states in circuit QEDsetups . Some of them are of probabilistic nature,i.e., if a measurement on the N -qubit system has a spe-cific result, the system is known to be in a GHZ state af-ter the measurement . In Ref. 23, a Mølmer-Sørensentype one-step scheme to generate GHZ states both forsuperconducting flux qubits and charge qubits was pro-posed. The procedure is independent of the initial stateof the quantum bus and works in the presence of mul-tiple low-excitation modes. However, higher excitationmodes of the quantum bus will introduce inhomogeneitybecause of the shorter wavelengths of the higher modesand decrease the GHZ fidelity. Moreover, uncontrolleddissipation might be coupled through the higher excita-tion modes and induce extra noise. It would be idealto devise a GHZ generation scheme that, while keepingthe one-step, deterministic nature, would involve only a single mode of the quantum bus mediating the qubit in-teraction.For this purpose, in the present paper, we consider N superconducting transmon qubits homogeneously cou-pled to a superconducting transmission line resonator inthe dispersive limit, i.e., the architecture realized in anumber of experiments . We show that the sys-tem is characterized by effective qubit exchange interac-tions of XY type that can be globally controlled. Start-ing from a separable initial state, these interactions allowto generate a GHZ state in a deterministic one-step pro-cedure. We discuss how to probe the non-local natureand the genuine N -partite entanglement of the gener-ated state and investigate the stability of the proposedscheme to inhomogeneities in the physical parameters. Incontrast to Ref. 23, the qubit-resonator interaction doesnot commute with the free Hamiltonian, and the qubitfrequencies are tuned close to one resonator mode. Thetime evolution of the system is described by an effectiveHamiltonian which allows a direct implementation of theMølmer-Sørensen idea. Our scheme is the first one-stepdeterministic generation protocol of GHZ states whichcould be possibly implemented in the currently availablecircuit QED design.The paper is organized as follows: in Section II wederive an effective Hamiltonian for N transmon qubitscapacitively coupled to a superconducting transmissionline resonator in the dispersive regime. In Section III wedescribe the protocol for generating GHZ states in oursystem. In Section IV, we discuss ways to confirm the N -partite nature of the entanglement in the generatedstates, and in Section V we study the effects of non-idealphysical parameters like inhomogeneities in the qubit-resonator coupling constants. II. FULLY CONNECTED NETWORK OFTRANSMON QUBITS IN THE DISPERSIVELIMIT
We propose a solid-state implementation, based on anarchitecture of superconducting transmon qubits capaci-tively coupled to a quantum bus and derive an effectiveHamiltonian for the system, which exhibits the appropri-ate XY exchange interaction.Transmon qubits consist of a superconducting islandconnected to a superconducting electrode through aJosephson tunnel junction with capacitance C J and anextra shunting capacitance C B . A gate voltage V g is ap-plied to the island via a gate capacitance C g , allowingto tune the dimensionless gate charge n g = C g V g / (2 e ).The system is characterized by the charging energy E C = e / (2 C Σ ), where C Σ = C g + C J + C B is the total capac-itance of the island, and the Josephson energy E J of thetunnel junction.Such Josephson junction based qubits behave effec-tively as quantum two-level systems in different regimes,categorized by the ratio E J /E C . We will focus on theso-called transmon regime, when E J /E C ∼ − H qb can be expressed as H qb = 4 E C (ˆ n − n g ) − E J cos ˆ ϕ . (1)In the following we assume that the Josephson junc-tions form a dc-SQUID i.e., E J is tunable by an externalapplied magnetic flux Φ ext allowing to control indepen-dently each qubit. In this case C Σ = C g + 2 C (1) J + C B and E J = 2 ˜ E J cos( π Φ ext / Φ ) with C (1) J and ˜ E J the ca-pacitance and the Josephson energy of a single junction.If a qubit is capacitively coupled to a superconductingtransmission line resonator, C g is now the capacitance be-tween the superconducting island and the resonator. Inthat particular situation the gate voltage involves a dc-part and an extra term depending on the state of the res-onator, V g = V dc g + V ( x ). Therefore the interaction withthe resonator appears via the gate charge n g , which im-plicitly includes the voltage V ( x ). Transmon qubits aremore robust to 1 /f -noise than charge qubits due to theirexponentially suppressed charge dispersion . However,we assume that the gate of each qubit can be controlledseparately by microwave pulses in order to perform singlequbit quantum gates.For simplicity we consider the qubits to be coupled toa single mode of the resonator. This is a reasonable as-sumption if the qubits are nearly resonant with only onemode. Since higher modes have frequencies which aremultiples of the fundamental frequency, we can tune thequbit transition frequencies such that the detuning withrespect to one mode of the resonator is one order of mag-nitude smaller than the detuning to all the other modes.Under these conditions we can realize the dispersive limitfor a single mode of the resonator and neglect the influ-ence of higher modes, as is the case in experiments usingone transmon qubit . FIG. 1: (Color online) Sketch of a possible coplanar geometryfor the proposed device with N = 4 qubits. Qubits (small bluesquares) are placed around the maxima of the electrical fieldamplitude (red line), i.e. near the center and the ends of the(quasi-)one-dimensional resonator (gray strip). The secondmode of the electrical field (red arrows) mediates the qubit-qubit interaction. Input and output ports of the resonatorare drawn in black. For instance the qubits could be mainly coupled to thesecond mode if they are placed near the ends or the centerof the resonator, that is the positions where the electricalfield amplitude is maximal. Such a possible geometry issketched in Fig. 1. Following the procedure of canonicalquantization of a (quasi-) one-dimensional superconduct-ing resonator , the voltage across the resonator is givenby V ( x ) = r ω r L c cos (cid:18) πxL (cid:19) ( a + a † ) . (2)The length of the resonator is L and its resonance fre-quency ω r = 2 π/ p L lc depends on its capacity c andinductance l per unit length. The position along theresonator is denoted by x ∈ [ − L / , L /
2] and a ( a † ) rep-resent bosonic annihilation (creation) field operators.Following Ref. 7, the system, composed of the res-onator and N transmon qubits, can be described with ageneralized Jaynes-Cummings Hamiltonian. This Hamil-tonian is expressed in the basis of transmon eigenstates | j i q , where the indices q label the transmon qubits, andfor readability we define the operators Π ( q ) j = | j i q h j | q , σ ( q ) j, − = | j i q h j + 1 | q , σ ( q ) j, + = | j + 1 i q h j | q and set ~ = 1, H = ω r a † a + N X q =1 X j h ω ( q ) j Π ( q ) j + (cid:16) g ( q ) j a σ ( q ) j, + + H.c (cid:17)i . (3)The qubits frequencies ω ( q ) j are presumed to be tun-able through external magnetic fields Φ ( q )ext , changingthe effective Josephson energies of the qubits E ( q ) J =2 ˜ E ( q ) J cos( π Φ ( q )ext / Φ ), and the coupling frequencies g ( q ) j depend on the position of the qubits. Invoking therotating-wave approximation, we have neglected rapidlyoscillating terms. In the transmon regime, we can onlykeep transmon-resonator coupling coefficients for neigh-boring levels, since terms like | i i q h j | q for | i − j | > E J /E C limitasymptotic expression can be obtained for ω ( q ) j and g ( q ) j in first order perturbation theory , ω ( q ) j ≃ q E ( q ) C E ( q ) J (cid:18) j + 12 (cid:19) − E ( q ) C
12 (6 j + 6 j + 3) ,g ( q ) j ≃ g ( q )0 p j + 1 cos (cid:18) πx q L (cid:19) ,g ( q )0 ≃ − i eC ( q ) g C ( q )Σ E ( q ) J E ( q ) C ! / r ω r L c . (4)This form of the coupling frequencies g ( q ) j describes thesituation shown in Fig. 1. The amplitudes of these cou-pling coefficients g ( q ) j can be assumed to be approxi-mately homogeneous if the positions x q of the qubitssatisfy | x q /L | ≃ /
2, since the electrical field am-plitude decreases quadratically with the distance from itsmaxima and since the size of the qubits is typically muchsmaller than the resonator wavelength in realistic sys-tems. However even if close to the center or the ends ofthe resonator, the qubits should be placed sufficiently farapart to reduce direct inductive or capacitive qubit-qubitcoupling. There are also other positions that the qubitscan be placed in (e.g. nodes of higher modes). However,the homogeneity of the coupling constants is importantin our approach and should be taken care of.In the so-called dispersive regime | g ( q ) j / ∆ ( q ) j | ≪ ω ( q ) j,j +1 are detuned from the resonator frequency ω r , ex-citations of the resonator are virtual and the latter willrather act as a quantum bus mediating effective qubit-qubit interactions. The transition frequencies of thetransmon qubits are defined as ω ( q ) j,j +1 = ω ( q ) j +1 − ω ( q ) j andtheir respective detuning as ∆ ( q ) j = ω ( q ) j,j +1 − ω r . In thisregime, eliminating the direct interaction between res-onator and transmon qubits to lowest order in g ( q ) j / ∆ ( q ) j ,we exhibit an effective qubit-qubit interaction. Thiscan be seen by performing the canonical transformation e S H e − S , where S = N X q =1 X j g ( q ) j ∆ ( q ) j a σ ( q ) j, + − H.c ! . (5) Keeping terms up to second order in g j / ∆ j , we obtain. e S H e − S ≃ ω r + N X q =1 − χ ( q )0 Π ( q )0 + X j ≥ (cid:16) χ ( q ) j − − χ ( q ) j (cid:17) Π ( q ) j a † a + N X q =1 ω ( q )0 Π ( q )0 + X j ≥ (cid:16) ω ( q ) j + χ ( q ) j − (cid:17) Π ( q ) j + N X q =1 aa X j η ( q ) j σ ( q ) j +1 , + σ ( q ) j, + + H.c + X q = q ′ X j,j ′ ˜ g ( qq ′ ) jj ′ (cid:16) σ ( q ) j, + σ ( q ′ ) j ′ , − + σ ( q ) j, − σ ( q ′ ) j ′ , + (cid:17) , (6)Here the ac-Stark shifts χ ( q ) j , the two-photon transitionrates η ( q ) j and the effective qubit-qubit coupling coeffi-cient ˜ g ( qq ′ ) jj ′ are given by χ ( q ) j = | g ( q ) j | ∆ ( q ) j ,η ( q ) j = 12 g ( q ) j g ( q ) j +1 ∆ ( q ) j ∆ ( q ) j +1 (cid:16) ω ( q ) j,j +1 − ω ( q ) j +1 ,j +2 (cid:17) , ˜ g ( qq ′ ) jj ′ = (cid:12)(cid:12)(cid:12) g ( q ) j g ∗ ( q ′ ) j ′ (cid:12)(cid:12)(cid:12) ∆ ( q ) j + ∆ ( q ′ ) j ′ ( q ) j ∆ ( q ′ ) j ′ . (7)Two-photon transitions can be safely neglected sincethe parameters η ( q ) j are small in the dispersive regime.An effective Hamiltonian H eff is now obtained by re-stricting our Hilbert space to the computational sub-space, that is the first two levels of each transmon qubit {| i , | i} ⊗ N . In principle, the qubit-qubit interactioncouples any states of the qubits with more than one exci-tations to states that do not belong to the computationalsubspace (e.g. for N = 3, the state | i or | i will becoupled to | i or | i ). However, the amplitudes forthese mixing processes of computational states with suchnon-computational states are of order g / ( E C ∆) and willbe neglected . Under these conditions, H eff = ω + X q χ ( q ) σ ( q ) z ! a † a + X q ˜ ω ( q )01 σ ( q ) z + X q,q ′ ˜ g ( qq ′ )00 (cid:16) σ ( q ) x σ ( q ′ ) x + σ ( q ) y σ ( q ′ ) y (cid:17) , (8)where χ ( q ) = χ ( q )0 − χ ( q )1 / σ z = Π − Π , σ x = σ + + σ − , σ y = − i ( σ + − σ − ). The resonator and qubit frequen-cies get slightly renormalized, that is ω = ω r − P q χ ( q )1 / ω ( q )01 = ω ( q )01 − χ ( q )0 . The Hamiltonian has the de-sired YX-form, provided that all qubits have identicalparameters: that is all qubit and coupling frequenciesare homogeneous, ˜ ω ( q )01 = Ω, | g ( q )0 | = g , ∆ ( q )0 = ∆ and˜ g ( qq ′ )00 = χ ( q )0 = ˜ g = g / ∆. Using Eq. (4) we infer that χ ( q ) = χ = − ˜ gE C / (∆ − E C ) < ˜ g , where E C = ω − ω isthe weak anharmonicity of the transmon qubits. As men-tioned earlier in Eq. (4) the qubit transition frequenciescan be made homogeneous by tuning the flux biases Φ ( q )ext .From now on we assume the g ( q ) j are homogeneous. Thisis motivated by a promising new transmon architecturewith tunable coupling that has been proposed recently .Inhomogeneous coupling constants will be discussed inSec. V.Previous GHZ state generation protocols based on ho-modyne measurement of the transmission line ne-glected the effective exchange interaction because of thelarge differences in qubit frequencies. In our case, thequbit frequencies ω ( q )01 are tuned to be identical using theflux biases, and this effective interaction plays a signifi-cant role in the generation of the GHZ state in a one-stepprocedure as shown below.If the qubit and coupling frequencies are homogeneous,the total spin operators ˆ J x,y,z = P q σ ( q ) x,y,z and their cor-responding Casimir operator ˆ J = ˆ J x + ˆ J y + ˆ J z can beused to write the effective Hamiltonian in a very conve-nient form, H eff = ωa † a + ˜ g ˆ J + (Ω + 2 χa † a ) ˆ J z − ˜ g ˆ J z . (9)Evidently, H eff is diagonal in the basis | J, J z i⊗| n i , wherethe states | J, J z i , describing the states of the N qubits,are the eigenstates of the operators ˆ J and ˆ J z with re-spective eigenvalues J ( J + 1) and J z . The states | n i , de-scribing the state of the resonator, are eigenstates of a † a with eigenvalue n ≥
0. Since [ H , ˆ J ] = 0, any eigenstatesof ˆ J will remain so under the action of this Hamiltonian.In the following, we will restrict ourselves to such stateswith J = N/
2. For example states with all spins alignedin a particular direction belong to this type and are there-fore an appropriate choice for the initial state. Setting J = N/ | J = N/ , J z i by | J z i .The eigenstates of H eff are | J z i ⊗ | n i with eigenvalues ε ( n, J z ) = ωn + ˜ g ( N/ N/ χn ) J z − ˜ gJ z . III. PROTOCOL FOR GENERATING GHZSTATES
The effective Hamiltonian H eff allows to produce GHZstates by turning on the interaction for a definite dura-tion t GHZ . It was shown in Refs. 22,24 that a Hamil-tonian of the type ˜ g ˆ J x will produce a GHZ state af-ter the time π/ (2˜ g ), starting for instance from the state N q | i q . Implementation of such scheme in other qubitsystem has also been proposed . The multi-qubit gate exp( iπ ˆ J x /
2) is sometimes referred to as Mølmer-Sørensengate.We conveniently choose an initial state with all thequbits aligned in the same direction, that is the maximalsuperposition state | ψ i = N q ( | i q + | i q ) / √
2. We as-sume that the qubits and the resonator are initially in aproduct state and the state of the resonator at t = 0 isdenoted ρ res , ρ ( t = 0) = | ψ ih ψ | ⊗ ρ res . (10)Moreover | ψ i = | J x = N/ i and can be expressed as alinear superposition of the states | J z i (see Appendix ), | ψ i = 1 √ N N X k =0 q(cid:0) Nk (cid:1) | J z = k − N/ i . (11)We define ρ ( t ) as the density matrix evolving un-der the action of the time-evolution operator U ( t ) =exp( − i H eff t ), where H eff is the effective HamiltonianEq. (9), ρ ( t ) = U ( t ) ρ ( t = 0) U † ( t ) . (12)We assumed that g/ ∆ ≪ e S onthe state vector. This turns out to be particularly useful,since U ( t ) is diagonal in the basis | n i , we can describe di-rectly the time evolution of the reduced density matrix ofthe qubits ρ qb ( t ), obtained by tracing over the resonatorstate, ρ qb ( t ) := Tr res [ ρ ( t )]= 12 N X n,k,k ′ h n | ρ res | n i q(cid:0) Nk (cid:1)(cid:0) Nk ′ (cid:1) e − i ( ϕ k,n ( t ) − ϕ k ′ ,n ( t )) | J z = k − N/ ih J z = k ′ − N/ | , (13)where ϕ k,n ( t ) = k (Ω t + 2 χtn + ˜ gt ( N − k )).The Greenberger-Horne-Zeilinger (GHZ) states we aimto produce are of the following form, | GHZ ± i = 1 √ N O q =1 | i q + | i q √ ± i N O q =1 | i q − | i q √ ! , (14)which are standard GHZ states up to single qubit rota-tions. These states can be expressed as a linear superpo-sition of the states | J z i as well (see Appendix ): | GHZ ± i = N X k =0 ± i e iπk √ N √ q(cid:0) Nk (cid:1) | J z = k − N/ i . (15)To see why a GHZ state is produced after some time t GHZ we consider the effects of either exp( i ˜ gt ˆ J z ) orexp[ i ˜ gt ( ˆ J z − ˆ J z )] (for N either even or odd) on the state | J z = k − N/ i . We establish that one of the two possi-ble GHZ states Eq. (14) is produced when ˜ gt = π/ ie iπ ( k + N − √ e − i π + i π ( k − N ) , N even , ie iπ ( k + N − ) √ e − i π + i π [ ( k − N ) − ( k − N ) ] , N odd . Thus, a GHZ state is produced for every odd multipleof time t GHZ . The shortest preparation time is t GHZ = π/ (2˜ g )However the remaining term of the effective Hamil-tonian in Eq. (9), the one which is proportional to ˆ J z ,induces a collective rotation of the final state. The ro-tation angle depends again on N and the state of theresonator. The state ρ qb ( t GHZ ) is, ρ qb ( t GHZ ) = X n h n | ρ res | n i | GHZ( α n ) ih GHZ( α n ) | . (16)Here, | GHZ( α ) i = e − iα ˆ J z √ N O q =1 | i q + | i q √ e iπ N − N O q =1 | i q − | i q √ ! , (17)and 2 α n /π = (Ω + 2 nχ ) / ˜ g for N even. For N odd,2 α n /π = (Ω + 2 nχ ) / ˜ g −
1, and the relative phaseexp( iπ ( N − /
2) in Eq. (17) is changed to exp( iπN/ ρ ( t GHZ ) is not ex-actly the state depicted in Eq. (14) and therefore certainconstraints on the angles α n in Eq. (16) are required togenerate the proper state | GHZ + i . At low temperature,only the ground state of the resonator is significantly pop-ulated and h | ρ res | i ≫ h n | ρ res | n i for n ≥
1. Thus we canrestrict our considerations to α n =0 and this translates tosome condition on the ratio Ω / ˜ g .To illustrate this we consider the resonator to be ini-tially in its ground state ρ res = | n = 0 ih n = 0 | . The state | GHZ + i is indeed produced at t GHZ , provided we cantune the frequencies Ω and ˜ g such thatΩ˜ g = 4 m + 2 − N , m ∈ Z . (18)If the above condition cannot be satisfied, some correct-ing pulse exp( iδ N ˆ J z ) can be applied to the final state ρ qb ( t GHZ ) to obtain a proper | GHZ + i state. The appro-priate pulse length δ N depends on N and the ratio Ω / ˜ g , δ N = π (cid:20)(cid:18) Ω˜ g + N − (cid:19) mod 4 (cid:21) . (19)Furthermore δ N = 0 implies Eq. (18).If not only the ground state of the resonator ispopulated, higher photon numbers n produce ro-tated GHZ states, according to Eq. (16). We no-tice that h GHZ( α n ) | GHZ( α n + k ) i = cos N ( kπχ/ (2˜ g )), which means that if a | GHZ + i state is producedfor excitation number n , a slightly rotated stateexp( − iπχ ˆ J z / ˜ g ) | GHZ + i is produced for n + 1 (since χ < ˜ g ). Assuming some correcting pulse exp( iδ N ˆ J z ) has beenapplied, the reduced density matrix of the qubits ρ qb is amixture of rotated GHZ states with classical probabilitiesdepending only on the initial state of the resonator, e iδ N ˆ J z ρ qb ( t GHZ ) e − iδ N ˆ J z = h | ρ res | i| GHZ + ih GHZ + | + X n> h n | ρ res | n i e − iπn χ ˜ g ˆ J z | GHZ + ih GHZ + | e iπn χ ˜ g ˆ J z . (20)We will now show that it is possible to choose realisticphysical parameters in agreement with our assumptions.Transmon qubits have typical frequencies Ω / π around10 GHz and coplanar waveguide resonators (the quan-tum bus) can be realized with frequencies ω/ π of theorder of 10 GHz with high quality factors . Transmon-resonator coupling frequencies g/ π around 200 MHz isa reasonable assumption. Detuning the qubits from theresonator such that g/ ∆ ≃ /
10 would lead to an effec-tive qubit-qubit coupling of the order of ˜ g = g/
10 and topreparation time t GHZ of approximately 12.5 ns.
IV. MEASURING THE GENERATED GHZSTATES
The question of detecting and probing the states gen-erated in our scheme naturally arises. For N ≥
4, thereis no unique way to quantify entanglement. We will focuson a measurement of the Bell-Mermin operator definedas B = e iπN i " N O q =1 (cid:16) σ ( q ) z − iσ ( q ) y (cid:17) − N O q =1 (cid:16) σ ( q ) z + iσ ( q ) y (cid:17) = 2 N − (cid:0) | GHZ + ih GHZ + | − | GHZ − ih GHZ − | (cid:1) , (21)whose expectation value for N -qubit quantum states isbounded by |h B i| ≤ N − , and the extremal values ± N − are reached for the states | GHZ ± i . The maximalvalue predicted by local hidden-variable theory is √ N ( √ N − ) for N even (odd), leading to an exponentiallyincreasing violation for the states | GHZ ± i with N , thenumber of qubits. Therefore, a measurement of the Bell-Mermin operator leading to a result greater than √ N ( √ N − ) indicates the non-local nature of the generatedquantum states.Other bounds can be derived for this operator: e.g.,any separable state ρ S satisfies | Tr( ρ S B ) | ≤
1. A signifi-cant bound can also be derived if the state is m -separable,i.e. describes a system which is partitioned in m sub-systems that only share classical correlations. In otherwords, a pure state is called m -separable, for 1 < m ≤ N ,if it can be written as a product of m states, | ψ m i = m O i =1 | ψ i i P i , (22)where the { P i } describe a partition of the N qubits.Thus, a separable state in the traditional sense is N -separable. A mixed m -separable state ρ m is defined as aconvex sum of pure m -separable states, which might be-long to different partitions . Such an m -separable statesatisfies Tr( ρ m B ) ≤ N − m . Thus, any measurement of B with outcome above 2 N − indicates that the state isnot even biseparable ( ) and demonstrates theexistence of genuine N -partite entanglement.The Bell-Mermin operator expectation value can inprinciple be obtained experimentally. This operator canbe expressed as a sum of parity operators, and inferringits expectation value would require 2 N − parity measure-ments, h B i = N X l =1 odd X p ( − N − l +12 * N − l O q =1 σ p ( q ) z N O q ′ = N − l +1 σ p ( q ′ ) y + . (23)For each term, l is the number of factors σ y and P p stands for the sum over the (cid:0) Nl (cid:1) permutations p that givedistinct products. The states | GHZ ± i defined in Eq. (14)are those that give exactly ± N − terms.There are therefore 2 N − parity measurements to real-ize which is possible only if one is able to generate GHZstates with high accuracy in a repeated way. FollowingRef. 20, these parity operators could be measured by dis-persive readout. Since the frequency of the resonator isac-Stark shifted ω → ω + χ P q σ ( q ) z , it is possible to accessthe value of the operator ˆ J z . The value of the parity op-erator N q σ ( q ) z can then be unambiguously deduced from J z = h ˆ J z i , * N O q =1 σ ( q ) z + = ( − N − J z . (24)Hence, we can measure all the needed parities by rotatingthe operators σ ( q ) y appearing in Eq. (23) to σ ( q ) z usingsingle-qubit rotations.By means of Eq. (13), we can give an expression forthe time evolution of the expectation value of the Bell-Mermin operator, h B ( t ) i = Tr [ Bρ qb ( t )]. For this pur-pose we can express the matrix elements of B in the basisof the states | J z i , which diagonalizes the effective Hamil-tonian, B = N X k,k ′ =0 b k,k ′ | J z = k ′ − N/ ih J z = k − N/ | , (25)where b k,k ′ = 12 i q(cid:0) Nk (cid:1)(cid:0) Nk ′ (cid:1) h ( − k − ( − k ′ i . (26) - Π (cid:144) -Π (cid:144) -Π (cid:144) Π (cid:144) Π (cid:144) Π (cid:144) - WΤ G N H t G H Z + Τ L N = = = = FIG. 2: Behavior of the function G N ( t GHZ + τ ) for different N , assuming for simplicity that δ N = 0. Hence, h B ( t ) i can be expressed as a sum of oscillatingfunctions G nN , indexed by the photon number n , h B ( t ) i = 2 N − ∞ X n =0 h n | ρ res | n i G nN ( t ) . (27)The functions G nN are Fourier series over a finiterange of frequencies ˜ ω nk,k ′ defined as ˜ ω nk,k ′ = ( k − k ′ ) [( k + k ′ − N )˜ g − Ω − nχ ], G nN ( t ) = N X k,k ′ =0 a k,k ′ sin(˜ ω nk,k ′ t ) , (28)where a k,k ′ = 2 − N (cid:0) Nk (cid:1)(cid:0) Nk ′ (cid:1) h ( − k − ( − k ′ i . (29)Equation (27) shows that h B ( t ) i is characterized bymany oscillations on timescales ∼ t GHZ , since the ˜ ω nk,k ′ are of the same order as Ω ≫ ˜ g, χ . However, the en-velope indeed reaches its maximum at t GHZ , providedthat only the ground state of the resonator is significantlypopulated. These fast oscillations are the manifestationof local rotations of the qubits, Eqs. (16-17). We haveseen that this issue can be solved equivalently in two dif-ferent ways and that the state | GHZ + i is indeed gener-ated after t GHZ , either by applying some correcting pulseexp( iδ N ˆ J z ), defined in Eq. (19), or by tuning the frequen-cies Ω and ˜ g to satisfy the condition Eq. (18). Assumingfor simplicity that δ N = 0, we have then G nN ( t GHZ ) = cos N (cid:18) n π χ ˜ g (cid:19) − sin N (cid:18) n π χ ˜ g (cid:19) . (30)The fast oscillations of h B ( t ) i around t GHZ becomesharper as the number of qubits N increases, as shown inFig. 2. In the simpler case δ N = 0, the behavior of G N around t GHZ is given by G N ( t GHZ + τ ) ≃ − τ N Ω , | τ | ≪ , (31) w − N m a x Æ B æ −3 T/ w e [ % ] N = 4 W / w = 1.11g/ w = 0.011 FIG. 3: Temperature dependence of the maximum max h B i of h B ( t ) i , for t ∼ t GHZ (squares), normalized by 2 N − . The solidline shows the theoretical bound tanh( ω/ (2 T )) for a resonatorinitially in the thermal state Eq. (33). Inset: relative devi-ation ε = t max /t GHZ − t max at which max h B i is realized compared to the predicted time t GHZ = π/ (2˜ g ).Here we considered N = 4 qubits and the parameters areΩ /ω = 1 . g/ω = 0 . g/ ∆ ≃ .
1. Localhidden-variable theory only allows values of h B i below thedashed line. For N = 4 this value also corresponds to theupper bound for biseparable states. and that also means that we need a higher precision, forlarger N , in controlling either the protocol time t GHZ orthe correcting pulse.Finally, the maximal value h B ( t GHZ ) i can reach de-pends only on the initial state of the resonator ρ res , pro-vided the above considerations have been taken into ac-count. Equations (20) and (21) show that2 − N Tr h B ( e iδ N ˆ J z ρ qb ( t GHZ ) e − iδ N ˆ J z ) i = ∞ X n =0 h n | ρ res | n i (cid:20) cos N (cid:18) n π χ ˜ g (cid:19) − sin N (cid:18) n π χ ˜ g (cid:19)(cid:21) . (32)For instance, we assume ρ res to be a thermal state char-acterized by a temperature T , ρ res = (cid:16) − e − ω/T (cid:17) X n e − nω/T | n ih n | . (33)In this simple case, the outcome of the Bell-Merminoperator measurement h B ( t GHZ ) i should be at least2 N − tanh( ω/ (2 T )).A numerical evaluation of h B ( t ) i , using the Jaynes-Cummings Hamiltonian Eq. (3), shows good agreementwith our theoretical analysis. We consider the ideal caseof homogeneous qubit and coupling frequencies and wechoose frequencies satisfying Eq. (18) such that δ N = 0.We look for the maximal value of h B ( t ) i around t GHZ ,that is for | t − t GHZ | < π , and for the time t max at which this maximal value is realized. The results for N = 4 qubits are shown in Fig. 3. V. INHOMOGENEOUS COUPLINGFREQUENCIES
To estimate whether our scheme is robust against smallrandom deviations in the physical parameters, we con-sider small inhomogeneities in the coupling strengths g ( q ) j . This effect will be investigated numerically and,for this purpose we compute the real-time evolution ofthe Bell-Mermin operator, using the Jaynes-CummingsHamiltonian Eq. (3), truncated to the two lowest lev-els of the transmon qubits. This should capture themain features of this effect, since in our effective de-scription of the system Eq. (8), the third levels of thetransmon qubits only affect the ac-Stark shifts χ ( q ) andrenormalize the resonator frequency. Assuming the qubittransition frequencies are still homogeneous ω ( q )01 = Ω,the inhomogeneity of the coupling frequencies g ( q )0 pro-duces inhomogeneous qubit-qubit couplings coefficients˜ g ( qq ′ )00 = | g ( q )0 g ( q ′ )0 | / ∆.The coupling constants g ( q )0 are assumed to be Gaus-sian distributed with mean g and standard deviation δg .The notation { g q } denotes a particular set of couplingfrequencies g ( q )0 . The real-time evolution of the Bell-Mermin operator for one set of coupling frequencies { g q } is denoted h B { g q } ( t ) i .For a given number n r of random realizations { g q } ( n r around 200) with fixed δg , we first calculate the meanvalue, h ¯ B ( t ) i = 1 n r X { g q } h B { g q } ( t ) i . (34)Then, the maximal value h ¯ B ( t max ) i defined by h ¯ B ( t max ) i = max t ≥ h ¯ B ( t ) i (35)is found. Finally the variances, above and belowthe maximal mean value h ¯ B ( t max ) i , of the particu-lar set (cid:8)(cid:10) B { g q } ( t max ) (cid:11)(cid:9) are calculated. The variancesare calculated separately above and below, because the h B { g q } ( t max ) i are not Gaussian distributed. We also cal-culate the median among the h B { g q } ( t max ) i and noticethat the distribution is strongly asymmetric.Results for N = 4 and δg/g between 0 to 20 % areshown in Fig. 4. The time at which the maximum is at-tained is generally in good agreement with the predictedvalue t GHZ = π/ (2˜ g ), as long as g/ ∆ is small. The valueof h ¯ B ( t max ) i remains close to the ideal one for δg/g ofthe order of a few percents and thus we notice that ourscheme can tolerate some inhomogeneity in the couplingconstants. d g/g [%] − N Æ B ( t m a x ) æ N = 4 W / w = 1.11g/ w = 0.011T/ w = 0.11 FIG. 4: Effect of inhomogeneous coupling frequencies g ( q )0 with mean g and standard deviation δg . We show the depen-dence of the maximal mean value h ¯ B ( t max ) i of h B { g q } ( t ) i on δg/g for t ∼ t GHZ (squares). The error bars show the stan-dard deviation of h B { g q } ( t max ) i above and below the meanvalue. The median of h B { g q } ( t max ) i (circles) is clearly abovethe mean value. Local hidden-variable theory only allows val-ues of h B i below the dashed line. For N = 4 this value alsocorresponds to the upper bound for biseparable states. VI. CONCLUSION
To conclude, we have shown that it is possible to gen-erate multipartite Greenberger-Horne-Zeilinger states ona set of transmon qubits in a circuit QED setup in aone-step deterministic protocol. In the dispersive limit g ≪ ∆, such a system behaves as a fully connectedqubit network with exchange interactions proportionalto ˜ g = g / ∆. The preparation time of the protocolis inversely proportional to ˜ g . The non-local nature ofthe generated state can be investigated using a Bell-Mermin inequality. Moreover, we have derived and ap-plied bounds on the expectation value of the Bell-Merminoperator as a detection criterion for genuine N -partiteentanglement. Finally we have shown that our scheme isrobust against small inhomogeneities in the coupling fre-quencies. The implementation proposed here looks likea promising way to generate GHZ states, and hopefullycan be experimentally realized in a circuit QED setup. VII. ACKNOWLEDGMENT
We would like to thank S. Filipp and J. Koch for dis-cussions and correspondence about the qubit-qubit inter-action of transmon qubits in a circuit-QED setup, and forsending unpublished notes and M. Pechal’s Master the-sis. We would also like to thank L. DiCarlo, S. Chesi,F. Pedrocchi, and G. Str¨ubi for discussions. This workwas financially supported by the EC IST-FET project SOLID, the Swiss SNF, the NCCR Nanoscience, and theNCCR Quantum Science and Technology.
Appendix: Schwinger representation of total spinoperators
We present briefly the Schwinger representation ofthe total spin operators ˆ J x,y,z = P q σ ( q ) x,y,z . Thisturns out to be particularly useful for calculations inthe subspace of ˆ J -eigenstates with maximal eigenvalue N (cid:0) N + 1 (cid:1) where N is the number of spins. From nowon we set J = N/ | J = N/ , J x,y,z i by | J x,y,z i .States like | J z i are sometimes referred to as Dickestates , they form a complete basis of symmetric N -qubit states, i.e., states invariant under any permuta-tion of qubits. We use for each qubit the standard ba-sis {| i , | i} with the convention σ ( q ) z | i q = | i q and σ ( q ) z | i q = −| i q , | J z = k − N/ i = 1 q(cid:0) Nk (cid:1) X p | i p (1) · · · | i p ( k ) | i p ( k +1) · · · | i p ( N ) , (A.1)with 0 ≤ k ≤ N and where the sum is taken over the (cid:0) Nk (cid:1) = N ! k !( N − k )! nonequivalent possible permutations p that give different product states.The operators ˆ J i are defined by means of two inde-pendent bosonic operators a and b , with commutationrelations [ a, a † ] = [ b, b † ] = 1 and [ a, b ] = [ a, b † ] = 0,ˆ J x = 12 ( b † a + a † b ) , ˆ J y = 12 i ( b † a − a † b ) , ˆ J z = 12 ( b † b − a † a ) , (A.2)fulfilling the SU(2) algebra [ ˆ J l , ˆ J m ] = iǫ lmn ˆ J n . Eigen-states of ˆ J z can be expressed as | J, J z i = (cid:0) b † (cid:1) J + J z | (cid:0) a † (cid:1) J − J z p ( J + J z )!( J − J z )! | n a = 0 , n b = 0 i , (A.3)where | n a = 0 , n b = 0 i is the vacuum state of the operators a and b . Since the choice of the operators a and b isnot unique, we can equivalently introduce the operators c = ( a − b ) / √ d = ( a + b ) / √
2, leading to ˆ J x = ( d † d − c † c ) and | J, J x i = (cid:0) d † (cid:1) J + J x | (cid:0) c † (cid:1) J − J x p ( J + J x )!( J − J x )! | n a = 0 , n b = 0 i . (A.4)We straightforwardly obtain the decomposition of thestates | J, J x i in terms of | J, J z i and in particular | J x = ± N/ i = N O q =1 | i q ± | i q √ (cid:0) a † ± b † (cid:1) N √ N N ! | n a = 0 , n b = 0 i (A.5)= 12 N/ N X k =0 ( ± k q(cid:0) Nk (cid:1) | J z = k − N/ i . Defining the ladder operators ˆ J ± = ˆ J x ± i ˆ J y of the total spins, the Dicke states can also be expressed as | J z = k − N/ i = (cid:16) ˆ J + (cid:17) k k ! q(cid:0) Nk (cid:1) N O q =1 | i q = (cid:16) ˆ J − (cid:17) N − k ( N − k )! q(cid:0) Nk (cid:1) N O q =1 | i q . (A.6) M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz,E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N.Cleland, and J. M. Martinis, Science , 1423 (2006). J. H. Plantenberg, P. C. de Groot, C. J. P. M. Harmans,and J. E. Mooij, Nature , 836 (2007). L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop,B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frun-zio, S. M. Girvin, et al., Nature , 240 (2009). M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz,E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Wei-des, J. Wenner, et al., Nature , 504 (2009). L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M.Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H.Devoret, and R. J. Schoelkopf, Nature , 574 (2010). M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero,M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang,M. Weides, J. Wenner, et al., Nature , 570 (2010). J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schus-ter, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 042319 (2007). J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R.Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frun-zio, M. H. Devoret, et al., Phys. Rev. B , 180502 (2008). A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J.Schoelkopf, Phys. Rev. A , 062320 (2004). A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature , 162 (2004). J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R.Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A.Houck, A. Wallraff, et al., Nature , 443 (2007). R. J. Schoelkopf and S. M. Girvin, Nature , 664 (2008). D. M. Greenberger, M. A. Horne, A. Shimony, andA. Zeilinger, Am. J. Phys. , 1131 (1990). F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde,Phys. Rev. A , 052112 (2002). L. Lamata, J. Le´on, D. Salgado, and E. Solano, Phys. Rev.A , 022318 (2007). L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, andW. Rubens, Phys. Rev. Lett. , 100507 (2010). M. Hillery, V. Buˇzek, and A. Berthiaume, Phys. Rev. A , 1829 (1999). D. I. Tsomokos, S. Ashhab, and F. Nori, New J. Phys. , 113020 (2008). F. Helmer and F. Marquardt, Phys. Rev. A , 052328(2009). C. L. Hutchison, J. M. Gambetta, A. Blais, and F. K.Wilhelm, Can. J. Phys. , 225 (2009). L. S. Bishop, L. Tornberg, D. Price, E. Ginossar, A. Nun-nenkamp, A. A. Houck, J. M. Gambetta, J. Koch, G. Jo-hansson, S. M. Girvin, et al., New J. Phys. , 073040(2009). A. Galiautdinov, M. W. Coffey, and R. Deiotte, Phys. Rev.A , 062302 (2009). Y.-D. Wang, S. Chesi, D. Loss, and C. Bruder, Phys. Rev.B , 104524 (2010). K. Mølmer and A. Sørensen, Phys. Rev. Lett. , 1835(1999). S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti,J. M. Fink, M. G¨oppl, L. Steffen, J. M. Gambetta, A. Blais,et al., Phys. Rev. Lett. , 200402 (2009). P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen,S. Filipp, and A. Wallraff, Phys. Rev. Lett. , 100504(2010). J. M. Chow, L. DiCarlo, J. M. Gambetta, A. Nunnenkamp,L. S. Bishop, L. Frunzio, M. H. Devoret, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 062325 (2010). L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H.Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf,Nature Phys. , 105 (2009). This can be seen by applying perturbation theory in ˜ g ( qq ′ ) jj ′ for | j − j ′ | > M. Goppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp,J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wall-raff, J. Appl. Phys. , 113904 (2008). S. J. Srinivasan, A. J. Hoffman, J. M. Gambetta, and A. A.Houck, Phys. Rev. Lett. , 083601 (2011). K. Helmerson and L. You, Phys. Rev. Lett. , 170402(2001). S.-B. Zheng, Phys. Rev. Lett. , 230404 (2001). N. D. Mermin, Phys. Rev. Lett. , 1838 (1990). O. G¨uhne and G. T´oth, Phys. Rep. , 1 (2009). L. You, Phys. Rev. Lett. , 030402 (2003). R. H. Dicke, Phys. Rev.93