Macroscopic Greenberger-Horne-Zeilinger andWstates in flux qubits

Abstract

We investigate two types of genuine three-qubit entanglement, known as the Greenberger-Horne-Zeilinger ͑GHZ͒ and W states, in a macroscopic quantum system. Superconducting flux qubits are theoretically considered in order to generate such states. A phase coupling is proposed to offer enough strength of interactions between qubits. While an excited state can be the W state, the GHZ state is formed at the ground state of the three flux qubits. The GHZ and W states are shown to be robust against external flux fluctuations for feasible experimental realizations. As a macroscopic quantum system, superconducting qubit systems have been investigated intensively in experiments because their system parameters can be controlled to manipulate quantum states coherently. Indeed, the entanglements between two charges, 7 phase, 8,9 and flux qubits 10,11 have been reported. While the timely evolving states in the experiments of charge qubits 7 exhibit a partial entanglement, the excited level ͑eigenstate͒ of capacitively coupled two phase qubits 9 shows higher fidelity for the entanglement. The experiments in Ref. 10 show a possibility that two flux qubits can be entangled by a macroscopic quantum tunneling between two-qubit states, flipping both qubits. Actually, the higher fidelity in the capacitively coupled two phase qubits is caused by the two-qubit tunneling processes. 9 In a very recent study, the two-qubit tunneling process was theoretically shown to play an important role in generating the Bell states, maximally entangled, in the ground and excited states. 12 For multipartite entanglements in superconducting qubit systems, there have been few studies. To produce the GHZ state in three charge qubits, only a way of doing a local qubit operation via time evolutions was suggested. 13 As one of the possible directions to produce such multipartite entanglements, then it is natural to ask how to create the W state as well as the GHZ state in the eigenstates of superconducting three-qubit systems. Here, we consider three flux qubits. Normally, the interaction strength between inductively coupled flux qubits 14 is not so strong that the controllable range of interaction is not sufficiently wide. To control a wide range of interaction strengths in the qubits, we use the phase-coupling scheme [15][16][17][18][19] for three qubit ͓see Fig. 1͑a͔͒ which enables one to generate the GHZ and W states and to keep them robust against external flux fluctuations for feasible experimental realizations.We start with the model shown in Fig. 1͑a͒. The Hamiltonian is written by Ĥ = 1 2 P i T M ij −1 P j + U eff ͑ˆ͒, where The periodic boundary conditions involved in the qubit loops and the connecting loops can be written aswhere q = a, b, and c is qubit index and r, s, and n q integers. Here, f q ϵ ⌽ q / ⌽ 0 with external flux ⌽ q and the superconducting unit flux quantum ⌽ 0 = h / 2e. Two independent conditions in Eqs. ͑2͒ and ͑3͒ are the boundary conditions for connecting loops. For simplicity, we consider E J2 = E J3 = E J and C 2 = C 3 = C,...

Tripartite interactions between two phase qubits and a resonant cavity

“…Genuine tripartite entanglement is delineated by two inequivalent classes of states 22 : Greenberger-Horne-Zeilinger and W, where the W state involves only a single photon shared amongst three systems. Using multipartite entanglement in a solid-state-qubit system has only recently received theoretical attention [23][24][25] . Thus far in superconducting systems, bipartite entanglement has been verified by two-qubit quantum state tomography 13 and used to carry out a quantum algorithm 15 .…”

“…Genuine tripartite entanglement is delineated by two inequivalent classes of states 19 : GHZ (Greenberger-Horne-Zeilinger) and W, where the W-state involves only a single photon shared amongst three systems. Utilizing multipartite entanglement in a solid-state-qubit system has only recently received theoretical attention [20][21][22] .…”

Fast generation of multiparticle entangled state for flux qubits in a circle array of transmission line resonators with tunable coupling

“…Generation of the GHZ state in superconducting quantum circuits is consequently a highly important issue. While 14particle and 10-particle entanglement have been experimentally demonstrated in trapped-ion systems 4 and photonic systems 5 , respectively, theoretical studies have focused on generating the three-qubit GHZ state in superconducting charge 6 , flux 7 and phase 8 qubit circuits with direct qubit-qubit interaction. Experimental demonstrations of entanglement have also been limited to the cases of the two [9][10][11][12] or three [13][14][15][16][17] particles so far.…”

Greenberger-Horne-Zeilinger states in a quantum dot molecule

“…10 The GHZ states have been experimentally realized using photons, 11 in atomic systems using three Rydberg atoms, 12 and very recently maximally entangled GHZ states have been realized in a solid state system using superconducting qubits. 13,14 We refer the reader to Refs.…”

“…Theoretical proposals to realize GHZ states in solid state systems include, e.g., spin systems 1 , excitons in coupled dots 8 , two-level atoms in a non-resonant cavity 9 and superconducting flux qubits 10 . The GHZ states have been experimentally realized using photons 11 , in atomic systems using three Rydberg atoms 12 and very recently maximally entangled GHZ states have been realized in solid state system using superconducting qubits 13,14 .…”