arXiv:1610.04503v2 [quant-ph] 9 Nov 2016 ‡ † ∗ tts[] tts[01] lse tts[,4 and 4] entanglement [3, inequivalent states 15]. from Cluster [14, come situa- classes [10–12], [13] the states states exam- Dicke W LOCC, For [9], under (GHZ) Greenberger-Horne-Zeilinger states qubits. states parties, more four for of for there ple, states dramatically class pure changes one bipartite tion each for communication only into While classical converted is and 16]. be [10, operations (LOCC) cannot entangled local that qubits by finding other three the of by states stimulated was glement the and entanglement. to states of resource important rela- classes equivalence inequivalent entangled is the different understand it among a to for network, as tions resource well a shared as optimal task, of the given prepare nodes make to the to and identify order of In use quantum distributed [6–8]. full of for computing proposed and nodes communication been the have network among a structures entanglement with resources sophisticated different on Recently, relying architectures [1–5]. network quantum tasks communica- quantum computing of and variety tion a for resource ementary lcrncades [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic rgesi h eea td fmliatt entan- multipartite of study general the in Progress el- an is particles more or two between Entanglement .Tashima, T. htncmliatt nageetcneso sn nonl using conversion entanglement multipartite Photonic 2 ASnmes 36.n 36.a 42.50.-p 03.67.-a, 03.67.Mn, numbers: PACS en variable discord. generate ca quantum to gate nonvanishing used the but how be entanglement show can without also and We networks, working state. state Dicke basic graph symmetric lo the other and show to GHZ state to the cluster order entangled c flexible In four-qubit quantum a a converting networks. distributed on extended for based is in suited It tion ideally states. are the which to photons, access full classical and require operations opera not local nonlocal under uses inequivalent scheme are The that access. restricted with work a-lnkIsiu ¨rQatnpi,Hans-Kopfermann f¨ur Quantenoptik, Max-Planck-Institut rdaeSho fEgneigSine sk University, Osaka Science, Engineering of School Graduate 4 epooeasml eu o h ovrino multipartite of conversion the for setup simple a propose We colo hmsr n hsc,Uiest fKwaZulu-Nata of University Physics, and Chemistry of School .INTRODUCTION I. 1 8 eatetfu hsk Ludwig-Maximilians-Universit¨a f¨ur Physik, Department hsc eatet nvriyo ihgn n ro,Mic Arbor, Ann Michigan, of University Department, Physics 9 ,2 ,4 3, 2, 1, 3 htnSineCne,TeUiest fTko Bunkyo-ku, Tokyo, of University The Center, Science Photon iiino aeil hsc,Dprmn fMtrasEng Materials of Department Physics, Materials of Division 5 7 ainlIsiuefrTertclPyis KwaZulu-Natal Physics, Theoretical for Institute National etrfrEegn atrSine IE,Siaa351-019 Saitama RIKEN, Science, Matter Emergent for Center ahntnUiest nS.Lus t oi,M 33 USA 63130 MO Louis, St. Louis, St. in University Washington .S Tame, S. M. 6 eateto lcrcladSsesEngineering, Systems and Electrical of Department ,5 4, ¸ .K. S¸ . Dtd etme 1 2018) 11, September (Dated: Ozdemir, ¨ 6 u oteoeaigrneo h ovringate. conversion the of networks range complex operating more overa the larger the to to study, due extended our we be in While could states principle and entangled applications. four-qubit (b), networking on panel quantum focus in for en shown multipartite states of as types gled network, different of the fro conversion con- the qubits of We enables on nodes operates access. that local gate restricted two conversion with nonlocal network a sider Quantum 1: FIG. omlre ttso h aetp uhta entangled to that type such type certain same a the of of states states larger entangled form of fuse toolboxes manipulate, and to complete gates expand forming and operations ion on realizable focused to physically Stud- also [18], 22]. have [21, resonance circuits ies superconducting magnetic and 20] nuclear [19, traps pho- and from ranging [5] these [17], tonics setups characterize physical and different prepare in states to schemes and methods .Nori, F. hscasfiainldt eiso ffrst find to efforts of series a to led classification This omncto,btms motnl does importantly most but communication, al nqiaetfu-ui tts uhas such states, four-qubit inequivalent cally Srse1 -54 acig Germany Garching, D-85748 1, -Strasse in oeal h rprto fstates of preparation the enable to tions ierotclcneso aeta uses that gate conversion optical linear muainadqatmcommunica- quantum and omputation ageetadqatmcorrelations quantum and tanglement ,D879M¨unchen, Germany D-80799 t, ,8 7, ooaa sk 6-51 Japan 560-8531, Osaka Toyonaka, nage ttsi unu net- quantum a in states entangled rnilso h ae efcson focus we gate, the of principles .Koashi, M. ,Dra 01 ot Africa South 4001, Durban l, eicroae noextended into incorporated be n ia 80-00 USA 48109-1040, higan ot Africa South , neigScience, ineering 1-66 Japan 113-8656, ,Japan 8, 9 n .Weinfurter H. and cloperations ocal ‡ † ∗ ,2 1, tan- m ll 2 networks can be formed [23–35, 39–46]. In parallel to this work, there has also been much interest in the use of entangled states in quantum information processing tasks, especially in finding tasks for which the states from one entanglement class may be more efficient than one from another class. For example, cluster states and graph states are universal resources for quantum comput- ing [3, 4], GHZ states have been proposed as resources for achieving consensus in distributed networks without classical post-processing [47] and W states have been proposed as resources for leader election in anonymous quantum networks [47] and asymmetric telecloning [48]. From a network perspective, the qubits are distributed to the nodes where the users have access only to a limited FIG. 2: Nonlocal gate using linear optics. The gate is used in our restricted-access network scenario to convert between number of qubits. Typically they make use of classical different types of entanglement structures. communication channels to perform individually or col- lectively an assigned task [47–49]. In some cases, two nodes of a network may be close enough to each other full access to all of the resource state’s constituent ele- such that joint operations can be carried out with a small ments. We call the operations bilocal, as they involve a overhead in communication, as shown in Fig. 1. Such a nonlocal two-qubit gate and local one-qubit operations. scenario could also describe the case where a node of a Here, the two qubits operated on by the nonlocal gate network is also the node of another network and holds need to be in close proximity, whereas the other qubits two or more qubits belonging to different networks. This do not. This is in contrast to a fully nonlocal operation ‘nonlocal’ node could manipulate qubits from different that would require all qubits to be in close proximity for networks for the purpose of fusing those networks into a it to be applied. Our scheme employs a flexible linear larger merged network. This and similar tasks are of con- optical conversion gate. The gate can be used to gen- siderable importance as they help us to understand how erate variable entanglement, as well as quantum corre- to efficiently exploit multipartite entanglement for quan- lations without entanglement captured by the quantum tum communication protocols and how to design practi- discord. While we focus on four-qubit entangled states in cal applications. our study, e.g. cluster, GHZ and Dicke states, the over- Recent work has focused on the transformation of dif- all principle can be extended to larger and more complex ferent types of entangled states into each other by lo- networks due to the flexible operation of the gate. We cal one and two-qubit operations while allowing classi- highlight this extension briefly in our work. cal communication among all the nodes. For example, The paper is organized as follows: In Sec. II, we de- Kiesel et al. [50] demonstrated that a four-partite sym- scribe the principles of the nonlocal gate used in our con- metric Dicke state can be used to prepare a W state version scheme and highlight its overall range of opera- among three parties if one of the parties in the network tion. We also show how it can prepare states that are projects their qubit onto the computational basis. The either entangled or have nonzero discord. In Sec. III, we initial symmetric Dicke state can be used in certain quan- show that the gate can be used to convert a four-qubit tum versions of classical games whereas the final W state cluster state into other four-qubit entangled states by cannot be used [51, 52]. On the other hand, the final W operating nonlocally on only two qubits. In Sec. IV we state can be used for asymmetric telecloning of a qubit show some examples of incorporating the nonlocal gate [48]. Along the same lines, another interesting work is in an extended graph state network. Finally, in Sec. V that of Walther et al. [53], who showed that by the ap- we provide a brief summary and outlook. plication of appropriate generalized measurements on a tripartite GHZ state, one can obtain a state whose fidelity to a tripartite W state approaches one, when the proba- bility of success approaches zero. Tashima et al. [31], on II. NONLOCAL GATE FOR PHOTONIC STATE the other hand, have proposed a series of optical gates CONVERSION that can prepare W states of arbitrary size by accessing only one qubit of the W state. They also introduced Let us start by introducing the nonlocal two-qubit gate a scheme in which two parties sharing a pair of Bell for photonic state conversion. The gate shown in Fig. 2 states can prepare tripartite W states and expand them is composed of two polarizing beamsplitters (PBSs) and to larger sizes with the help of ancillary qubits, again by four half-wave plates (HWPs). At the most basic level accessing only one qubit in the network [31, 32, 35]. it functions as a tunable polarization-dependent beam- In this work we go beyond those concepts by introduc- splitter (PDBS). However, its advantage over the stan- ing a scheme that uses ‘bilocal’ operations to enable the dard PDBS already used in experiments [32, 36, 37] is preparation of resource states in a network that are in- that as the operation of the individual components can equivalent under LOCC, but crucially does not require be adjusted easily the configuration is far more flexible 3 and readily implemented experimentally. This is in di- π π 2 2 rect contrast to a bulk PDBS, whose operation is set once 1.0 3π 3π fabrication has been completed and cannot be changed. 8 8 0.8 The flexibility of this tunable PDBS complements well π π 0.6 the work on a tunable polarization-independent beam- 2 2

θ 4 θ 4 splitter [38]. In this section we consider the individual 0.4 π π components of the gate in order to derive the neces- 8 8 0.2 sary expressions to demonstrate its working principles 0.0 0 0 and range of operation. We then discuss how it can be 0 π π 3π π 0 π π 3π π 8 4 8 2 8 4 8 2 used to generate entanglement and more general quan- θ1 θ1 tum correlations. FIG. 3: (a): Success probability for the input state ψin = + + for the conversion gate operating with HWPs| asi po- | i| i A. Working principle of the conversion gate larization rotators at angles θ1 and θ2. (b): Entanglement generated from this input as quantified by the concurrence. The conversion gate is shown in Fig. 2. Its successful operation is based on postselection such that one photon is detected in each of the output modes, labeled 5 and 6. operator E0 corresponds to a successful gate operation with probability ps, where each of the output ports The Kraus operator E0 of the gate transforms the input (modes 5 and 6) has one photon. It is clear that by tuning state ρin = ψin ψin according to | ih | the rotation angles of HWP1 and HWP2, the gate acts

E0 ψin ψin E0† as a tunable PDBS, enabling the construction of different ρin | ih | (1) Kraus operators. This makes it possible to perform dif- → ps ferent tasks using this simple linear optical construction. where ps = Tr(E0 ψin ψin E0†) is the success probabil- ity. Here, the action| ofih the HWPs| in modes 3 and 4 for the polarization degree of freedom of one photon is given by B. Entanglement generation

H j cos(2θl) H j + sin(2θl) V j | i → | i | i We now show some basic examples that demonstrate V j sin(2θl) H j cos(2θl) V j . (2) the entangling power of the conversion gate and its PDBS | i → | i − | i ability. In Fig. 3, we show the success probability and a where j =3, 4 labels the modes of the HWPs and l =1, 2 range of states with different amounts of entanglement labels the HWPs. When the input state is HV j , the generated by tuning the angles θ1 and θ2, as quantified action of the HWPs in modes 3 and 4 is given| byi by the concurrence [54]. Here, the input state is ψin = | i + + , where + = ( H + V )/√2. Applying the HV j √2 cos(2θl) sin(2θl) 2H j | i → | i | i| i | i | i | i 2 2 Kraus operator E0 of Eq. (4) on this input state yields (cos (2θl) sin (2θl)) HV j − − | i √ 1 + 2 sin(2θl) cos(2θl) 2V j , (3) E ψin = (α β ) HH + (α β ) V V | i 0| i 2 1 − 1 | i 2 − 2 | i where we have used the definition HV j for one hori- √ + + 2(µ1 µ2) Ψ  (5) zontal polarized photon and one vertical| i polarized pho- − | i ton in the same mode j, and 2H j ( 2V j ) for two where Ψ+ = ( HV + VH )/√2. | i | i horizontal (vertical) polarized photons in mode j. In First,| ini order| toi prepare| i the Bell state Ψ+ from addition, the first PBS applies the unitary transforma- the state space of the output in Eq. (5) we need| i to set tions: H 1 H 4, H 2 H 3, V 1 V 3 and α1 β1 = 0 and α2 β2 = 0 by rotating the angles | i → | i | i → | i | i → | i π V 2 V 4, and the second PBS applies similar trans- − − of the HWPs in modes 3 and 4 as θ1 = (2k + 1) 8 and | i → | i π formations: H 3 H 6, H 4 H 5, V 3 V 5 θ = (2n + 1) , respectively, for k and n taking non- | i → | i | i → | i | i → | i 2 8 and V 4 V 6. By combining all the unitary opera- negative integer values. The success probability is max- | i → | i tions of the HWPs and PBSs, the Kraus operator of the imized when µ µ = cos2(θ + θ ) = 1, leading to 1 − 2 1 2 ∓ nonlocal gate can be written as θ1 + θ2 = mπ/2 for some integer m. These three equa- π tions are satisfied simultaneously when (2k +1) 8 +(2n+ E = (α β ) HH HH π π 0 1 − 1 | i56 12h | 1) = m . This can be reformulated to k + n +1=2m, +(α β ) V V V V 8 2 2 − 2 | i56 12h | implying that k + n should be an odd number. In other +µ HV HV µ VH HV words, when k is even n is odd and vice versa. Thus the 1| i56 12h |− 2| i56 12h | +µ VH VH µ HV VH (4) solution we are looking for is µ1 = µ2 = 1/2 resulting 1 56 12 2 56 12 + + − ± | i h |− | i h | in E0 = Ψ Ψ . Consequently, the Kraus opera- 2 2 ±| ih | + + where αl = cos (2θl), βl = sin (2θl), µ = tor E prepares the output state ρout = Ψ Ψ with 1 0 | ih | cos(2θ1) cos(2θ2) and µ2 = sin(2θ1) sin(2θ2). The Kraus the success probability ps = 1/2. In order to give an 4

idea of the robustness of the success probability to vari- π ππ ations in the optical components, as in an experimental 2 22 1.0 implementation, we performed a Monte Carlo simulation, 3π 3π 0.8 8 8 varying the angles θ1 and θ2 of the wave plates around 0.6 π ππ 2 2 2

their ideal values. We set a range of variation for the θ 4

θ 4 θ 4 0.4 angles of 10% from the ideal value. Performing 5000 π ππ runs (to reach± asymptotic behaviour) we find an average 8 88 0.2 success probability of p = 0.509 0.010, which clearly 0.0 s 0 0 ± 0 π π 3π π 0 π π 3π π shows that the success probability is quite robust to re- 8 4 8 2 8 4 8 2 θ1 θ alistic perturbations. π 1 π 1 Similarly, we see from Eq. (5) that if we were able to set 2 2 3π 3π3π µ1 µ2 = 0 and α1 β1 = (α2 β2) this would prepare 3π 8 − − ± − 88 8 the Bell state Φ = [ HH V V ]/√2. For the former ± π π 2 π 2

π 2 | i | i±| i π 2

θ 4

θ 4

θ 4 equality, we find θ1 + θ2 = (2m + 1) 4 for non-negative θ 4 integer numbers m. The latter equality is satisfied when π π 8π 8 8 8 4θ2 =4θ1+2nπ and 4θ2 =4θ1+(2k+1)π, respectively for + and signs. It is easy to show under these conditions 0 00 − 0 π π 3π π 0 π π 3π π that the case with the + sign gives α β = α β =0 8 4 8 2 8 4 8 2 1 − 1 2 − 2 which is not desirable. On the other hand, the case with θ1 θ1 π π the sign leads to θ1 = (m k) 4 and θ2 = (m + k + 1) 4 − − FIG. 4: Generation of nonclassical correlations by the con- for which α1 β1 = (α2 β2) is always satisfied. The − − − version gate that are not due to entanglement. (a): Success above implies that we can prepare only the state Φ− with the success probability p = 1/2. In this case,| wei probability of the conversion gate. Here points (i) and (ii) cor- s respond to the output state 5 H H φ φ + 2 V V obtain α = β = µ = µ = 0 and α = β = 1, 7 | ih | ⊗ | ih | 7 | ih |⊗ 1 2 1 2 2 1 + + , where φ = (2 H V )/√5. Points (iii) and (iv) leading to E0 = HH HH V V V V and the output | ih | | i | i − | 1 i | ih |−| ih | correspond to the output state 2 11 + + . (b): Entangle- state ρout = Φ− Φ− with success probability ps = ment generated as quantified by the⊗ | concurrence.ih | (c): Dis- 1/2. A Monte| Carloih simulation| with a 10% variation ± cord generated with respect to measurements performed on on the wave plate angles gives a success probability of the second qubit. (d): Discord generated with respect to ps =0.481 0.021. measurements performed on the first qubit. ± It is clear that starting with the input state ψin = | i + + (or ψin = , where = ( H | i| i | i | −i| −i | −i | i − V )/√2)), the proposed gate can prepare a range of I(A, B) and J(A B) given by | |statesi with various amounts of entanglement as seen I(A, B) = S(ρA)+ S(ρB) S(ρAB) (6) in Fig. 3, including separable and maximally entangled − states, by simply setting the rotation angles of HWP 1 J(A B) = S(ρA) min X pbS(ρA b) | − Πb | and HWP2 correctly. It is also noted that the output { } state resulting from the input state ψin = + or | i | i| −i where S(ρ) is the von Neumann entropy S(ρ) = ψin = + can be obtained from the input state | i | −i| i Trρlogρ, ρA = TrBρAB, ρB = TrAρAB, Πb is a pos- ψin = + + by locally compensating the phase shift − { } |in eitheri | modei| i 5 or 6. itive operator valued measure on system B, and pb = Tr[ρΠb] is the probability of obtaining measurement out- come b that leaves A in the conditional state ρA b. In general the discord is not symmetric, δ(A B) = δ(|B A), C. Discord generation but if it is non-zero with respect to measurements| 6 | on at least one system, then nonclassical correlations exist. We now give a brief example of how the conversion gate In addition, when the entanglement is zero and the dis- can be used to generate quantum correlations different cord is non-zero with respect to measurements on one or from entanglement. Here, the quantum discord [55–57] both of systems A and B (δ(B A) = 0 or δ(A B) = 0), allows the quantification of nonclassical correlations that then there are quantum correlations| 6 present that| are6 not exist between quantum systems even when the entangle- due to entanglement. In order to show that the con- ment is zero. Recently it has been shown that discord version gate generates such quantum correlations from can be used as a resource for various types of quantum initial product states without entanglement we input the 1 protocols that are useful in a quantum network scenario, state ρin = 11 + + . In Fig. 4 (a) the success prob- 2 ⊗ | ih | including remote state preparation [58], encoding infor- ability is shown as the angles θ1 and θ2 of the gate are mation that only coherent interactions can extract [59] modified. In Fig. 4 (b) the corresponding entanglement and verification that untrusted parties can implement en- generated by the gate is shown, as quantified by the con- tanglement operations [60]. Discord represents the quan- currence. tum component of correlations between two systems, A In Figs. 4 (c) and 4 (d), we give the discords δ(A B) and B, and is defined as δ(A B)= I(A, B) J(A B) with and δ(B A) generated in the output state with respect| to | − | | 5 measurements performed on the second and first qubit π 2 in the conversion gate, where A corresponds to the first 1.0 qubit in output mode 5 and B corresponds to the second 3π 0.8 qubit in output mode 6. When θ1 = 0 and θ2 = π/3 8 (point (i)) or θ1 = π/3 and θ2 = 0 (point (ii)) the output state from the gate has no entanglement in Fig. 4 (b), 0.6 2 π but interestingly has non-zero discord (See in Fig. 4 (c)). θ 4 5 0.4 The output state for these points is 7 H H φ φ + 2 | ih | ⊗ | ih | V V + + , where φ = (2 H V )/√5 and π 0.2 7 8 has| aih discord| ⊗ | ofihδ(A| B) 0.|082.i The| successi − | probabilityi | ≃ is 0.438 and a Monte Carlo simulation with a 10% vari- 0.0 ± ation on the wave plate angles gives ps =0.446 0.023. 0 ± 0 π π 3π π This way the conversion gate can also be used to gener- 8 4 8 2 ate states with quantum correlations that are not due to θ1 entanglement and may be used for various quantum tasks in a network scenario [58–60], most notably as resources FIG. 5: The success probability for nonlocal state conversion in quantum cryptography [61]. from the linear cluster state C4 to various four-qubit entan- gled states. See Tab. I for the| labelledi states and their success probabilities. III. STATE CONVERSION can be achieved with an operation element with compo- One of the powerful features of the nonlocal gate is the nents HH HH and V V V V . Thus, for the output ability to convert quantum states belonging to one type | ih | | ih | state in Eq. (7) to be GHZ4 , we should set αj and βj of state into another that is not LOCC equivalent. | i such that (α2 β2) = (α1 β1) and µj = 0. The lat- Starting with a four-qubit linear cluster state given by − − − π π ter equality is satisfied for (θ1,θ2) = (m 2 , (2n + 1) 4 ) or (θ ,θ ) = ((2n + 1) π ,m π ) for non-negative integers m 1 1 2 4 2 C = ( HHHH + HHVV and n. Substituting these in the former equality leads to | 4i 2 | i | i α1 β1 = 1 and α2 β2 = 1. Choosing these coef- + VVHH VVVV ), − ± − ∓ | i − | i ficients yields the Kraus operator of the evolution given by we will show how the action of the conversion gate shown in Fig. 2 on two qubits of the state ρin = C4 C4 will E0 = ( HH HH V V V V ) (8) prepare various LOCC inequivalent states,| likeih a| four- ± | ih | − | ih | qubit GHZ and Dicke state, as well as into two bipartite which achieves the task of transforming C4 to GHZ4 maximally entangled states, i.e. a product of Bell states. | i | i (up to a global phase) with a success probability of ps = In general, applying E0 given in Eq. (4) on the second 1/2. A Monte Carlo simulation with a 10% variation and third qubits of C , we find the general output ± | 4i on the wave plate angles gives a success probability of ps =0.457 0.027. 1 ± E C = (α β ) HHHH (α β ) VVVV 0| 4i 2 1 − 1 | i− 2 − 2 | i +µ HHVV + µ VVHH 1| i 1| i B. Identity operator: Keeping the cluster state µ HVHV µ VHVH , (7) intact − 2| i− 2| i which by tuning the different coefficients enables the gen- If we do not wish to convert C4 to any other state eration of a variety of states as described in the next sub- but keep it intact, while still using| i the nonlocal gate sections. In Fig. 5, we show a selection of output states (for practical reasons re-routing the photons may not that can be obtained by operating the conversion gate be a straightforward procedure), we see from Eq. (7) on qubits 2 and 3 of the cluster state together with their that in order to obtain E0 = 11 the coefficients µ2 = 0 success probabilities. and α β = α β = µ = 1 are to be satis- 1 − 1 2 − 2 1 ± fied for ps = 1. The former equality has the solutions π π (θ1,θ2) = (m 2 ,n 2 ), with m and n as non-negative inte- A. Transforming the cluster state into a GHZ state n+m gers. We find αj βj = 1 and µ1 = ( 1) from which − −π we obtain θj = kπ and θj = (2k + 1) 2 , respectively, for In order to obtain a four-qubit GHZ state, GHZ4 = even and odd m. Then the output state of Eq. (7) will | i ( HHHH + VVVV )/√2, from C we should de- be equivalent to C if n + m is an even number with a | i | i | 4i | 4i sign the evolution operator such that it discards the success probability of ps = 1. With a 10% variation on ± HHVV and VVHH components of C4 and flips the wave plate angles, ps = 0.909 0.056. In this case, |the minusi sign in| front ofi the VVVV component.| i This the Kraus operator for the successful± transformation is | i 6

Converted state θ1 θ2 ps D. Transforming the cluster state into a Dicke (i) Cluster state 0 0 1 state π/2 π/2 1 (ii) GHZ state 0 (π/2) π/4 1/2 Finally, we show that the nonlocal gate can be used to π/4 0 (π/2) 1/2 convert the linear cluster state C4 into the four-qubit (iii) Dicke state θ+ θ 3/10 | i − Dicke state, θ θ+ 3/10 − (iv) Two Bell states 3π/8 π/8 1/4 1 D(2) = ( HVVH + VHHV + HVHV π/8 3π/8 1/4 | 4 i √6 | i | i | i + VHVH + HHVV + VVHH ). (10) | i | i | i TABLE I: The success probability for nonlocal state conver- sion from the linear cluster state C4 to a four-qubit GHZ Consider rotating the polarization of the qubits in this | i and Dicke and two Bell states. The angle θ is found from state locally via the operation σz σx σzσx 11 to give ± ⊗ ⊗ ⊗ the relation, sin 2θ = (5 √5)/10. the following state, ± q ± (2) 1 D′ = ( HHHH + VVVV HHVV | 4 i √6 | i | i − | i the desired identity operator VVHH + HVHV + VHVH ). (11) −| i | i | i

It is now clear that if we can set β2 α2 = α1 β1 = E0 = HH HH + V V V V | ih | | ih | µ = µ in Eq. (7), the coefficients− will be equal− and + HV HV + VH VH . − 1 2 | ih | | ih | the final state will be locally equivalent to a Dicke state. π From µ1 = µ2, we find θ1 θ2 = (2k + 1) 4 which leads − − k to the relations sin 2θ1 = ( 1) cos2θ2 and cos2θ1 = k+1 − ( 1) sin 2θ2. Using these relations in α2 β2 = µ1, C. Transforming the cluster state into two Bell − 4 2 − we obtain 5 sin 2θ2 5 sin 2θ2 + 1 = 0, whose roots states 2 − satisfy sin 2θ2 = (5 √5)/10. Using the expression 2 2 ± 2 for sin 2θ2 in sin 2θ1 + sin 2θ2 = 1, which is derived 2 With a proper choice of the coefficients, we can disen- from the equality α2 β2 = β1 α1, we find sin 2θ1 = − 2 − tangle the four qubit entanglement in the cluster state to (5 √5)/10. Then, setting sin 2θ1 = (5 √5)/10 gives prepare two Bell states, one shared between modes 2 and ∓2 ± sin 2θ2 = (5 √5)/10 and leads to α1 β1 = 1/√5 3, and the other between the modes 1 and 4. Note that ∓ − ∓ and α2 β2 = µ1 = µ2 = 1/√5. Inserting these values only the photons in modes 2 and 3 enter the conversion for the− coefficients in Eq. (7),± we arrive at gate. It is easy to see that among all possible settings of the coefficients the setting α1 β1 = α2 β2 = 0 and 1 − − E0 C4 = [ HHHH + VVVV HHVV µ2 = µ1 leads to the desired transformation. Imposing | i ∓2√5 | i | i − | i these− conditions results in VVHH + HVHV + VHVH ] −| i | i | i µ1 √6 (2) E0 C4 = ( HHVV + VVHH = D′ (12) | i 2 | i | i ∓2√5| 4 i + HVHV + VHVH ) | + i + | i (2) = µ Ψ Ψ . (9) implying that we can convert C4 into D with a suc- 1| i14| i23 | i | 4 i cess probability ps =3/10. The Monte Carlo simulation with a 10% variation on the wave plate angles gives a Thus, if we can find the angles satisfying the above ex- ± success probability of ps = 0.303 0.013. The Kraus pressions for the coefficients, the desired task will be ac- ± 2 operator for the successful transformation is given by complished with the success probability ps = µ1. Here we note that the operation required is the same as the 1 one of the first example given in Section II B. Thus, sub- E0 = ( HH HH V V V V HV HV ∓√5 | ih | − | ih | − | ih | stituting the angles for this example into the expressions VH VH + VH HV + HV VH ). given in Section II B gives µ1 = µ2 = 1/2. The suc- −| ih | | ih | | ih | 2 1 − ± cess probability is ps = µ = . With a 10% variation 1 4 ± on the wave plate angles, ps =0.270 0.017. The Kraus operator that performs this task is ±

IV. NONLOCAL GATE IN A MULTI-QUBIT 1 E = [ HV HV + VH HV + VH VH NETWORK 0 2 | ih | | ih | | ih | + HV VH ]. | ih | We now briefly discuss the integration of the nonlocal gate in a multi-qubit network in the form of a graph state. 7

FIG. 6: Nonlocal gate used in a multi-qubit network taking the form of a graph state. Here, vertices are qubits initialized to the state 1 ( H + V ) and edges correspond to the application of a controlled-Z (CZ) operation between the vertices: √2 | i | i CZ = H H 11 + V V σz. In the first step of each panel, a linear cluster state has local operations H 11 11 H | ih |⊗ | ih |⊗ ⊗ ⊗ ⊗ applied (H is the Hadamard operation) to take it from its canonical graph state form to the state C4 . It is then converted into various four-qubit entangled states using the nonlocal gate, as summarized in Tab. I. In the second| i step of each panel, these states are then connected to the rest of the graph representing the network via CZ gates (and local operations). See main text for details. The dashed edges represent CZ gates applied between the qubits shown and other qubits in the total graph (not shown). (a): Cluster state converted into a star graph (locally equivalent to the GHZ state), which is then connected to the rest of the graph of the network. (b): Cluster state remains intact as a cluster state and is connected to the graph (after local operations). (c): Cluster state converted into two 2-qubit graph states (each locally equivalent to a Bell state). (d): Cluster state converted into a Dicke state and connected to the graph to form a hybrid quantum network. Here, the red dashed edges signify the state is entangled and are not CZ operations.

Graph states can be used for a wide variety of quantum star cluster state shown in step 2 of Fig. 6 (a) under local networking purposes, including in quantum communica- operations H 11 H H. Once these bilocal operations tion and distributed quantum computation [4, 64]. In have been performed,⊗ ⊗ ⊗CZ operations are then applied to this setting an important task is to ‘rewire’ the network connect the star cluster state into the total graph state, by changing the entanglement structure in order to carry as shown in step 3. This rewires the network. Note that out a specific protocol between selected parties. From one can choose any qubit to be the central node of the the previous section it is clear that the nonlocal gate can star cluster in step 2 (11 is applied to the central node convert a cluster state into a number of entangled states and H to the outer nodes), giving four possible rewiring with different entanglement structures, as summarized in configurations. It is also interesting to note that the star Tab. I. We now show that as a result the nonlocal gate cluster state and linear cluster state are not equivalent enables the rewiring of a graph state network. under local operations and classical communication [64], Consider the initial cluster state as part of a multi- making the use of the nonlocal conversion gate necessary qubit graph state, as depicted in the first step of in order to rewire the network in this case. Fig. 6 (a). Here, the cluster state is shown in its canonical (b) Keeping the initial wiring.– The nonlocal gate is applied between qubits 2 and 3 of the cluster state C , graph state form, where vertices correspond to qubits in | 4i the state 1 ( H + V ) and solid edges correspond to the which remains as C4 . This state is equivalent to the √2 | i | i linear cluster state| showni in step 2 of Fig. 6 (b) under application of a controlled-Z (CZ) operation between the local operations H 11 11 H. Once these local operations vertices: CZ = H H 11+ V V σz. The dashed ⊗ ⊗ ⊗ | ih |⊗ | ih |⊗ have been performed, CZ operations are then applied to edges represent CZ gates applied between the qubits connect the linear cluster state into the total graph state. shown and other qubits in the total graph state (not This keeps the initial wiring of the network. shown). The canonical cluster state can be converted H H (c) Rewiring into two graphs.– The nonlocal gate is into the state C4 with local operations 11 11 , applied between qubits 2 and 3 of the cluster state C , where H is the| Hadamardi operation. Once these⊗ ⊗ local⊗ op- 4 which becomes a product of two Bell states. Each of| thesei erations have been performed, the nonlocal gate is then states are equivalent to a two-qubit graph state, as shown applied. Here, there are four rewiring cases: in step 2 of Fig. 6 (c), under local operations 11 Hσx. (a) Rewiring into a star cluster.– The nonlocal gate is Once these local operations have been performed,⊗ CZ applied between qubits 2 and 3 for the cluster state C4 operations are then applied to connect the graph states to become a GHZ state. This state is equivalent to| thei into the total graph state. This rewires the network. 8

(d) Rewiring into a hybrid network.– The nonlocal gate sult, it can be used to implement various different types of is applied between qubits 2 and 3, and the correspond- fusion operation in a number of quantum state expansion ing local operations are performed, for the cluster state schemes [23, 24, 32]. Indeed, recent work has also shown (2) how to scale quantum networks by fusing small multipar- C4 to become a Dicke state D4 . This state is shown in| stepi 2 of Fig. 6 (d), where| the internali entanglement tite entangled states containing four-qubit entanglement connections of the Dicke state (not CZ gates) are rep- [62].Thus, by placing the gate in either a large-scale quan- resented by dashed red edges. CZ operations are then tum network [6] or small-scale on-chip network [63], we applied to connect each qubit of the Dicke state indi- envisage that it could play an important role in efficiently vidually into the total graph state. This is a standard preparing and converting other types of larger multipar- method for encoding qubits into graph states [4]. The tite entangled states where there may be restricted access operations produce a hybrid network of a graph state to a given resource. and a Dicke state that may be more efficient for certain distributed protocols, such as in telecloning and quantum secret sharing [44]. Acknowledgments

V. CONCLUSION This work was supported by a JSPS Postdoctoral Fel- lowship for Research Abroad. This work was also sup- In this work we proposed a simple and powerful linear ported by EU-FET project QWAD, the German excel- optical quantum gate that can be used to prepare entan- lence research cluster NIM, the South African National glement, or to observe states without entanglement but Research Foundation and the South African National nonzero discord. In addition, we showed how to convert Institute for Theoretical Physics. TT is supported by a four-qubit linear cluster state into a series of relevant Grant-in-Aid for Young Scientists (B), no.15K17729. FN multipartite entangled states, such as a four-qubit GHZ, was partially supported by the RIKEN iTHES Project, Dicke state and two bipartite maximally entangled states, the MURI Center for Dynamic Magneto-Optics, the IM- by acting on only two qubits. The gate can perform a PACT program of JST, a Grant-in-Aid for Scientific Re- range of nonlocal operations based on its functionality as search (A), and a grant from the John Templeton Foun- a tunable polarization dependent beamsplitter. As a re- dation.

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