Generating genuine multipartite entanglement via XY-interaction and via projective measurements
aa r X i v : . [ qu a n t - ph ] O c t Generating genuine multipartite entanglement viaXY-interaction and via pro jective measurements
Mazhar Ali , Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße3, 57068 Siegen, Germany Department of Electrical Engineering, COMSATS Institute of InformationTechnology, 22060 Abbottabad, PakistanE-mail: [email protected]
Abstract.
We have studied the generation of multipartite entangled states forthe superconducting phase qubits. The experiments performed in this directionhave the capacity to generate several specific multipartite entangled states for threeand four qubits. Our studies are also important as we have used a computablemeasure of genuine multipartite entanglement whereas all previous studies analyzedcertain probability amplitudes. As a comparison, we have reviewed the generation ofmultipartite entangled states via von Neumann projective measurements.PACS numbers: 03.67.-a, 03.67.Bg enerating multipartite entanglement
1. Introduction
Quantum entanglement has been recognized as a resource with applications in theemerging field of quantum information and quantum computation [1, 2]. The creationand measurement of entangled states is crucial for the various physical implementationsof quantum computers [3, 4]. Therefore it is of much interest and importance to generateentanglement in experiments and also characterize it in theory. The description ofquantum entanglement for bipartite quantum systems is relatively simple as any givenquantum state is either entangled or separable. However for quantum systems withmore than two subsystems, this problem becomes richer and also more difficult. For thesimplest multipartite quantum system of three qubit system, it was shown that threequbits can be entangled in two fundamentally different ways under stochastic localoperations and classical communication (SLOCC) [5]. In contrast, such inequivalentSLOCC-classes of entangled states for four qubits are already infinite [6, 7].One of the basic task in a quantum computer is the implementation of a set ofuniversal gates usually a two-qubit gate such as controlled-NOT (CNOT) gate andsingle qubit rotations [8]. However, it is also possible that a three-qubit gate like Toffoligate achieves universality [9, 10, 11]. Therefore it is desired to design experiments witha direct implementation of multi-qubit gates. Recently, such implementations havebeen successfully performed on the superconducting phase qubits [12, 13, 14, 15]. Inthese experiments, the phase qubits were coupled by connecting them with a capacitorto generate multi-qubit interactions leading to multi-qubit gates rather than designingthem from more elementary two-qubit gates. Moreover, it was shown that one can alsouse two qubit gates to create | GHZ i states and a more efficient entangling protocol basedon a single three qubit gate to create | W i state [13]. We have extended these studies forthree and four qubits and have shown that this experimental setup can offer some uniquepossibilities which are not explored till now. We have also proposed an experimentalarchitecture for generating four qubit | χ i state which might be implemented with smallmodification in the original experiment.Another approach to create SLOCC-inequivalent multipartite entangled states is toapply von Neumann projective measurements on some of the subsystems [16, 17, 18, 19].This way to create entangled states is due to the property of certain symmetricmultipartite entangled states that allow a more flexible preparation of families ofSLOCC-inequivalent entangled states by projective measurements on small subsystems.We have reviewed this method to compare the results. We have shown that it is possibleto create both GHZ -type and W -type states of three qubits by applying projectivemeasurements on a single qubit of four qubit states. We have shown that in creating | χ i state, we have used GHZ -type entanglement and W -type entanglement. Althoughthese three states are inequivalent, nevertheless, in the reverse process of projectivemeasurements, we can extract GHZ -type and W -type entanglement from | χ i state.This process of creating inequivalent entangled states has increased our understandingabout multipartite entangled states. enerating multipartite entanglement
2. Genuine multipartite entanglement
In this section we briefly review the concept of genuine entanglement. We consider threequbits as an example to explain the concept. A state is separable with respect to somebipartition, say, A | BC , if it can be written as ρ = X k q k | φ kA ih φ kA | ⊗ | ψ kBC ih φ kBC | , (1)where q k form a probability distribution. We denote these states as ρ sepA | BC . The twoother possibilities are ρ sepB | AC and ρ sepC | AB . Then a state is called biseparable if it can bewritten as a mixture of states which are separable with respect to different bipartitions,that is ρ bs = p ρ sepA | BC + p ρ sepB | AC + p ρ sepC | AB . (2)Any state which is not biseparable is called genuinely entangled. The description ofgenuine entanglement is quite challenging. Considerable efforts have been devoted forits characterization, quantification, detection and preparation [20, 21, 22, 23, 24, 25, 26].Biseparable states can be created by entangling any two of three particles and thenone can create a statistical mixture by forgetting to which pairs this operation wasapplied. To detect genuine entanglement, it is not enough to apply bipartite criterion toevery partition, instead one has to show that it can not be written in form of Eq.(2). Asthere is no efficient way to search through all possible decompositions, we can considera superset of the set of separable states which can be characterized more easily thanthe set of separable states. As a superset of states that are separable with respect to,say, partition A | BC , we select the set of states that have a positive partial transpose P P T with respect to partition A | BC . A state ρ = P ijkl ρ ij,kl | i ih j | ⊗ | k ih l | is PPT if itspartially transposed matrix ρ T A = P ijkl ρ ji,kl | i ih j | ⊗ | k ih l | has no negative eigenvalues.The convex set ρ sepA | BC is contained in a larger convex set of states which has a positivepartial transpose. The benefit of doing so is the easy characterization of PPT set. Awell known fact is that separable states are always PPT [27]. We denote the stateswhich are PPT with respect to fixed bipartition by ρ pptA | BC , ρ pptB | AC , and ρ pptC | AB , and askthe question that whether a state can be written as ρ pptmix = p ρ pptA | BC + p ρ pptB | AC + p ρ pptC | AB . (3)Such mixing of PPT states is called PPT-mixture and it can be characterized moreeasily than biseparable states. This method allows use of semidefinite programming enerating multipartite entanglement negativity [28]. For multipartite systems,this monotone may be called genuine negativity. For multiqubits the value of genuinenegativity can be at most 1 /
3. Generating entanglement via XY-interactions among phase qubits
There are several physical systems such as spins, atoms, or photons, that might be usedas quantum information processing devices. Another promising candidate has emergedin recent years as superconducting qubit. Instead of relying on fundamental quantumsystems, these devices are engineered circuits that consist of many constituent atomsexhibiting collective quantum behavior. The two key features are superconductivity,which is a collective quantum behavior of many electrons that allows the entire circuitto be treated quantum mechanically, and the Josephson effect, which gives the strongnon linearity required to make an effective two-level system or qubit [30]. As we studyphase qubits in this paper and the circuit diagram for phase qubit is shown in Figure 2 . Let us first consider the three qubits case. An arbitrary initial pure state for three qubitscan be written as | ψ (0) i = P ijk =0 C ijk (0) | ijk i , where we have expressed the state inthe computational basis { | i , | i , . . . | i} and C ijk (0) are the initial probabilityamplitudes. To create | GHZ i = 1 / √ | i + | i ) state, one can design the quantumcircuit diagram [13] shown in Figure 1. This Figure utilizes a more natural universal gatecalled iSWAP gate [31]. The iSWAP gate can be generated by applying the interactionHamiltonian for the superconducting phase qubits written as [13, 31] H ijint = ~ g (cid:0) σ ix σ jx + σ iy σ jy (cid:1) , (4)for interaction time t iSW AP = π/ (2 g ), where g is the coupling strength and σ x , σ y arethe Pauli operators on qubits i and j . Alternatively, this coupling Hamiltonian can bewritten as H ijint = ~ g (cid:0) σ i + σ j − + σ i − σ j + (cid:1) , (5)where σ + and σ − are the operators which create and destroy an excitation in each qubitrespectively. This form reflects the fact that the interaction leads to excitation swappingback and forth between the two coupled qubits. Under the specified interaction time,qubit states are transform as | i 7→ − i | i and | i 7→ − i | i whereas | i and | i invariant. It is simple to figure out that the circuit shown in Figure 1 produce | GHZ i state provided all three initial qubits are in their ground state | i [13]. This means enerating multipartite entanglement Y ( π/
2) achieved byshining a laser on all three qubits, after that we first turn on the interaction for time t iSW AP between qubits 1 and 2 and then we turn on the interaction between qubits 2 and3 again for time t iSW AP , and finally we apply the rotation X ( − π/
2) again by shininglaser, then we get the GHZ state. We note that | GHZ i is maximally entangled statemeasured by genuine negativity, that is, E ( | GHZ i ) = 1 / Figure 1.
The circuit diagram for generating | GHZ i state using two qubit iSW AP gate, which can be generated directly by capacitive coupling in phase qubits. Singlequbit rotations are applied before and after the action of gate. Another inequivalent entangled state which has been created in this setup is the | W i = 1 / √ | i + | i + | i ) state [13], which do not have maximum amountof entanglement as measured by genuine negativity, that is E ( | W i ) ≈ . | W i state is shown in Figure 2. It canbe seen that first middle qubit is excited by applying a π -pulse to qubit B to excite itwith single photon. The effect is the creation of state | i . After that the interactionamong all qubits can be turned on by applying the interaction Hamiltonian H int = H ABint + H ACint + H BCint , (6)for the interaction time t W = (4 / t iSW AP [13], which corresponds to gt W = (2 π ) / ≈ .
7. This action leaves all qubits in an equal superposition of single excitation. Finallya Pauli Z matrix is applied to correct the phase of qubit B . The Pauli rotations X and Z on second qubit can be manipulated by focusing a laser before and after turningon interaction for time t w . The tomographic data [13] yields a very efficient state | W i as a result. In the following, we reconfirm this result using computable entanglementmonotone E . In addition, we further investigate this setup and suggest that it can offersome more possibilities to create multipartite entangled states. Figure 2.
The circuit diagram for generating | W i state of three qubits using singleentangling step with simultaneous coupling between all qubits. Single qubit rotationsare applied on the middle qubit before and after such interaction. enerating multipartite entanglement { C ( t ) , C ( t ) , C ( t ) } and { C ( t ) , C ( t ) , C ( t ) } ,with C (0) and C (0) as invariants. The key observation is that if one has anyone of nonzero amplitudes in any set then one can generate either | W i state or | f W i = 1 / √ | i + | i + | i ) state. In Figure 3 we plot E ( | ψ ( t ) i ) against pa-rameter gt with initial condition C (0) = 1 and all other C ijk (0) = 0. We observe thecreation of | W i as predicted [13] at gt W ≈ . gt W ≈ .
4. However, tocorrect the phase of | W i state in second peak, we need to apply Z (8 π/
3) rotation. g t E ( | ψ ( t ) > ) | W > state C (0) = 1 ; Figure 3.
Generalized negativity E ( | ψ ( t ) i ) is plotted against the parameter gt forinitial condition C (0) = 1. It can be seen that | W i state is created at gt ≈ . gt ≈ . We investigate the possibility to create the | G i states [32] which are defined as | G ± N i = ( | W N i ± | f W N i ) / √ . (7)One interesting feature of | G i states is the possibility of transforming them directlyinto | GHZ i state via local filters given by [18] f + = H n h(cid:16) √ (cid:17) I + (cid:16) √ − i (cid:17) σ z io H ,f − = H n h(cid:16) √ (cid:17) σ x + i (cid:16) √ − i (cid:17) σ y io H , (8)where σ i are the Pauli matrices and H is the Hadamard transformation. As a result,we get ( f + ⊗ f + ⊗ f + ) | G +3 i = 13 | GHZ i , ( f − ⊗ f − ⊗ f − ) | G − i = 13 | GHZ i , (9)with probability 1 /
9. Genuine negativity for | G i is E ( | G i ) ≈ . | G i seems straight forward from two independent sets of probability amplitudes.By having excitations in each one of this set, the experiment performed for | W i state enerating multipartite entanglement | G i state at gt ≈ .
7. However, this is not what we actually observewith initial conditions C (0) = C (0) = 1 / √
2. Although at gt ≈ . | W i and | f W i are formed, nevertheless their respective phases also mix which we were able tocorrect only in case of | W i state creation. Hence the state generated at gt ≈ . | ψ ( gt ≈ . i = 1 √ i φ ( φ ) | W i + e i φ ( φ ) | f W i ) . (10)Due to this phase dependence, the perfect | G i can not be generated in this method.Another method is to create | W i and | f W i separately as shown above and then preparethe superposition. Four qubits setup is an extension of three qubits setup which offers some uniquepossibilities to generate important multipartite entangled states. The interactionHamiltonian for four body interactions can be written as H int = H ABint + H ACint + H ADint + H BCint + H BDint + H CDint . (11)As a result of this Hamiltonian the following three sets of coupled differential equationsfor probability amplitudes emerge, that is, { C ( t ), C ( t ), C ( t ), C ( t ) } , { C ( t ), C ( t ), C ( t ), C ( t ) } , and { C ( t ), C ( t ), C ( t ), C ( t ), C ( t ), C ( t ) } . Whereas C (0) and C (0) are invariant. We recognize thatthe first two sets provide the possibility to generate | W i and | f W i states, respectively.Whereas the third set can be utilized to create the singlet state of four qubits as wedemonstrate below.We start with | GHZ i = 1 / √ | i + | i ) state with genuine negativity E ( | GHZ i ) = 1 /
2. The circuit diagram in this case is simple extension of Figure 1leading to desired state. Therefore, we do not repeat the diagram and arguments forthe preparation of GHZ state here.Next we consider the | W i state of four qubits given as | W i = 1 / | i + | i + | i + | i ), which is not a maximally entangled as measured by genuine negativity,that is E ( | W i ) ≈ . | W i is a generalization of Figure 2.We observed that with initial condition C (0) = 1, | W i state is created at gt W = π/ Z ( π ) rotation, by turning on a laser.Four qubits | G i = 1 / √ | W i + | f W i ) state is interestingly a maximally entangledstate as measured by genuine negativity, that is, E ( | G i ) = 1 / | G i = 12 √ | i + | i + | i + | i + | i + | i + | i + | i ) . (12)To generate this state, we start with initial condition C (0) = C (0) = 1 / √ | G i state to be created at gt = π/
4. However, after correcting phase, the statewe obtain is | ψ G ( gt = π/ i = 12 √ | i + | i + | i + | i enerating multipartite entanglement | i − | i − | i + | i ) . (13)This state is also maximally entangled as measurement by genuine negativity, that is, E ( | ψ G i ) = 1 / | GHZ i state,the second one is the cluster state, and the third one is the state | χ i = √ | i + | i + | i + | i + | i√ . (14)Surprisingly, it turns out that all these states are also maximally entangled measuredby genuine negativity, that is E ( | χ i ) = 1 /
2. It has been observed that | χ i state is thesymmetric four-qubit state that maximizes certain bipartite entanglement properties[36]. In addition, it was shown [35] that one can generalize this state to five and sixqubits, where it is also a maximally entangled state for some comb measure. Thegeneration of this state is still an experimental challenge [33]. We propose a simplemodification of current experimental setup which seems quite feasible and would leadto creation of | χ i state. To this end we realize that the state | χ i has a large overlapwith | W i state with additional component | i . It is easy to see that | χ i can bewritten as | χ i = α | i + √ − α | W i , (15)with α = 1 / √
3. It is interesting to note that | W i is not maximally entangled, howeverits entanglement can be maximized by mixing the component | i . One can also tryto maximize entanglement of | W i by mixing the component | i . It turns out thatone can do that and the resulting state | χ i = ( | i + | i + | i − | i ) / E ( | χ i ) = 1 /
2, however | χ i is equivalent to | GHZ i state [37], whereas | χ i is inequivalent to | GHZ i . We propose the circuit diagram forthe experiment to generate | χ i in Figure 4. Figure 4.
Circuit diagram is shown for generating | χ i state. | GHZ i state is createdin the first three qubits before applying a local filter and four body interactions withfourth qubit in excited state. Phases are corrected by Z ( γ ) rotations at the end. enerating multipartite entanglement | GHZ i state in the first three qubits using the methoddescribed in Figure 1. Once | GHZ i state is created then one can apply local filter (LF)on the first qubit to transform | GHZ i state into non-maximally entangled state | ψ ABC i = p / | i + p / | i . (16)The precise form of the filter f which does this job, is given as f = diag { a, /a } with a = (2) / . After that an excitation is created on the fourth qubit and then finally thisinput state is allowed to interact as four body interaction for same amount of time whichis required to create | W i state, that is, gt W = π/
4. It is not difficult to see that thisprocedure is equivalent to produce the initial conditions on probability amplitudes suchthat C (0) = p / C (0) = 1 / √
3. In Figure 5, we have plotted E ( | ψ ( t ) i )against parameter gt with these initial conditions. It can be seen that | χ i state iscreated as genuine negativity achieves its maximum value at gt = π/ ≈ . g t E ( | ψ ( t ) 〉 ) C (0) = sqrt(2/3) ; C (0) = sqrt(1/3) ; Figure 5. E ( | ψ ( t ) i ) is plotted against parameter gt with initial conditions C (0) = p / C (0) = 1 / √ correct phases, we first need to apply Z ( π ) rotation on the fourth qubit as we did in | W i case. As | i component is in the superposition, so this operation introducessome additional phases and to correct them we need to apply Z ( − π/
4) rotations onall four qubits. The resultant rotation on the fourth qubit is Z ( π − π/
4) = Z ( − π/ | Ψ i which are invariant under a simultaneous unitaries on all qubits.Such states only exist for an even number of qubits. For two qubits, singlet state isgiven as | ψ − i = 1 / √ | i − | i ). For four qubits, singlet state [33, 38] is given as | Ψ S, i = 1 √ | i + | i − / | i + | i + | i + | i ) ] , (17)with E ( | Ψ S, i ) = 1 /
2, which means that this state is also maximally entangled stateaccording to genuine negativity. The possibility to create this state appears natural asthe third set contains all relevant probability amplitudes for singlet state. One could enerating multipartite entanglement E ( | ψ ( t ) i ) against parameter gt with initial condition C (0) = 1.It can be seen that | Ψ S, i is created at gt ≈ . gt ≈ .
54. We emphasize at thispoint that although we have plotted the numerical value of E ( | ψ ( t ) i ) for all the casesstudied in this paper, however the detailed analysis of probability amplitudes confirmthat the corresponding states are created at specific instances of time with specific initialconditions. g t E ( | ψ ( t ) > ) singlet state C (0) = 1 ; Figure 6. E ( | ψ ( t ) i ) is plotted against parameter gt with initial conditions C (0) =1. The singlet state is generated at gt ≈ . gt ≈ .
4. Generating multipartite entanglement via von Neumann projections
In this section, we review the idea of generating multipartite entanglement via applyingvon Neumann projective measurements on some of the n -qubit state to get the m -qubit state with n > m . We stress here at this point that this idea is not new andextensive work has been done in linear optical quantum computation [17, 18, 19]. Ourmain purpose here in this paper is to compare this method with previous section inorder to show the interesting correspondence between inequivalent genuine multipartiteentangled states. Let ρ n be an initial n -qubit density matrix and ρ m be the densitymatrix of m qubits after applying projective measurements on the n − m qubits.Mathematically, one can write this operation as ρ m = 1 N Tr ij [ ( I ⊗ . . . M i ⊗ . . . M j ⊗ . . . I ) ρ n ( I ⊗ . . . M † i ⊗ . . . M † j ⊗ . . . I ) ] , (18)where N is the normalization factor, Tr ij is the partial trace over qubits being measured,and M i = V Π i V † is the von Neumann projection operator on i th qubit, with Π i = | i ih i | enerating multipartite entanglement V = t I + i ~y · ~σ is the unitary matrix, that is, V ∈ SU (2), suchthat t, y i ∈ R and t + y + y + y = 1 and σ is a vector of Pauli matrices.In recent experimental work [17, 18, 19], the authors mainly considered thisproblem for Dicke states of four, five and six qubits and experimentally demonstratedthe possibility of obtaining inequivalent multipartite entangled states by projectivemeasurements. In this work, we examine the several inequivalent multipartite entangledstates for four qubits and investigate the possibility to obtain ‡ inequivalent multipartiteentangled states for three qubits. We have chosen only those examples which are notconsidered before.Our first example is the cluster state which is one of the two independent graphstates for four qubits. The other independent graph state for four qubits is | GHZ i state [33]. The cluster state can be written in the computational basis as | CL i = 12 ( | i + | i + | i − | i ) . (19)Interestingly, it turns out that the cluster state can only be mapped to either one of the | GHZ i state of three qubits, that is | CL i 7−→ X k =1 α k | GHZ k i , (20)where any other three qubit GHZ state is locally equivalent to | GHZ i state. It isknown that | GHZ i is the only independent graph state for three qubits [33].The next example is the four qubit singlet state | Ψ S, i which can also be mappedto superposition of | W i and | f W i state, that is | Ψ S, i 7−→ α | W i + β | f W i , (21)which can further be converted into | G i state and subsequently into | GHZ i state asdiscussed before.Our final example is the | χ i state which is the most interesting and illustrative case.We saw in the previous section that we combined GHZ -type entanglement and W -typeentanglement to create this state. In the reverse process of projective measurements, wefind that | χ i state can be mapped to superposition of these two types of states, that is | χ i 7−→ α | W i + β | ^ GHZ i , (22)where one can transform the non maximally entangled | ^ GHZ i = p / | i + p / | i into | GHZ i by local filtering. All these examples have increased ourunderstanding about the structure of genuine multipartite entangled states. Wehave seen that | χ i , | GHZ i , and | W i states are all inequivalent states, however,it is interesting to find that one can create another inequivalent entangled states bycombining two different inequivalent states. ‡ Here the word obtain means that we get the result after the arrow by applying projectivemeasurements and tracing out the qubit being measured. enerating multipartite entanglement
5. Conclusions
We have studied the generation of multipartite entangled states for superconductingphase qubits. We have shown that this model has the capacity to generate manyimportant multipartite entangled states. For three qubits, we have reconfirmed theprevious studies on creating | W i state and have examined the possibility to generate | G i states. We have examined the possibility to create several entangled states for fourqubits. In fact, the experimental setup has already been designed for four qubits withstudies only on three qubits [13]. All cases studied in this paper may be performedin exactly the same experimental setup designed for generating | W i state [13]. Wehave shown that four qubit setup is richer than three qubits and offer some uniquepossibilities to create several important entangled states. Particularly, we have proposedthe experimental architecture for generating | χ i state, which seems quite feasible inthis setup. Indeed the experimental creation of | χ i state is challenging and suchverification would be of much interest and importance for applications of this statein quantum information. We have also reexamined an alternative approach to generategenuine multipartite entangled states via von Neumann projective measurements. Inthis technique, one can apply projective measurements on one or more qubits of a higherdimensional density matrix before tracing them out to obtain multipartite entangledstates in a lower dimensional space. We have only restricted ourselves to four qubits byapplying projective measurements only on a single qubit. We have shown that we cangenerate both GHZ -type and W -type entangled states on a three qubit space. We havealso shown that | χ i state can be mapped to superposition of GHZ -type and W -typeentanglement which is intuitive as we actually utilized both these types of entanglementto generate it. Hence this study has increased our understanding of the structure ofgenuine multipartite entangled states. One of the future avenues is to investigate therobustness of this setup against decoherence. Acknowledgments
The author would like to thank Andreas Osterloh for discussions, Otfried G¨uhne for hisgenerous hospitality at the University of Siegen, and Geza T´oth for his kind hospitalityat the University of Basque Country, Bilbao, where this work was orally presented.This work has been supported by the EU (Marie Curie CIG 293993/ENFOQI) and theBMBF (Chist-Era Project QUASAR). The author is also grateful to referees for theirpositive comments which improved the earlier manuscript.
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