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arXiv:0804.3159 · Physical Review A

Maximally entangling tripartite protocols for Josephson phase qubits

Andrei Galiautdinov, John M. Martinis

Published Jul 9, 2008• 32 citations
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Abstract

We introduce a suit of simple entangling protocols for generating tripartite Greenberger-Horne-Zeilinger and W states in systems with anisotropic exchange interaction $g(XX+YY)+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}ZZ$. An interesting example is provided by macroscopic entanglement in Josephson phase qubits with capacitive $(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}=0)$ and inductive $(0<\ensuremath{\mid}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}∕g\ensuremath{\mid}<0.1)$ couplings.

DOI
10.1103/physreva.78.010305

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We introduce a suit of simple entangling protocols for generating tripartite Greenberger-Horne-Zeilinger and W states in systems with anisotropic exchange interaction g(XX+YY)+g\ifmmode~\else˜\fiZZg(XX+YY)+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}ZZg(XX+YY)+g\ifmmode~\else˜\fi​ZZ. An interesting example is provided by macroscopic entanglement in Josephson phase qubits with capacitive (g\ifmmode~\else˜\fi=0)(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}=0)(g\ifmmode~\else˜\fi​=0) and inductive (0<\ensuremath∣g\ifmmode~\else˜\fi∕g\ensuremath∣<0.1)(0<\ensuremath{\mid}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}∕g\ensuremath{\mid}<0.1)(0<\ensuremath∣g\ifmmode~\else˜\fi​∕g\ensuremath∣<0.1) couplings.
We generalize the recently proposed Greenberger-Horne-Zeilinger tripartite protocol [A. Galiautdinov and J. M. Martinis, Phys. Rev. A 78, 010305(R) (2008)] to fully connected networks of weakly coupled qubits interacting by way of anisotropic Heisenberg exchange $g(XX+YY)+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}ZZ$. Our model differs from the more familiar Ising-Heisenberg chain in that here every qubit interacts with every other qubit in the circuit. The assumption of identical couplings on all qubit pairs allows an elegant proof of the protocol for arbitrary NNN. In order to further make contact with experiment, we study fidelity degradation due to coupling imperfections by numerically simulating the N=3N=3N=3 and 4 cases. Our simulations indicate that the best fidelity at unequal couplings is achieved when (a) the system is initially prepared in the uniform superposition state (similarly to how it is done in the ideal case) and (b) the entangling time and the final rotations on each of the qubits are appropriately adjusted.
We consider the possibility of generating macroscopic entangled states in capacitively coupled phase qubits. First we discuss the operation of phase qubits and the implementation of the basic gate operations in them. We then analyze two possible procedures that can be used to generate n-qubit entangled states, such as the Greenberger-Horne-Zeilinger state for the case n = 3. The procedures we propose are constructed under the experimentally motivated constraint of trying to minimize the number of control lines used to manipulate the qubits.
Going beyond the entanglement of microscopic objects (such as photons, spins, and ions), here we propose an efficient approach to produce and control the quantum entanglement of three macroscopic coupled superconducting qubits. By conditionally rotating, one by one, selected Josephson-charge qubits, we show that their Greenberger-Horne-Zeilinger (GHZ) entangled states can be deterministically generated. The existence of GHZ correlations between these qubits could be experimentally demonstrated by effective single-qubit operations followed by high-fidelity single-shot readouts. The possibility of using the prepared GHZ correlations to test the macroscopic conflict between the noncommutativity of quantum mechanics and the commutativity of classical physics is also discussed.
We theoretically study the macroscopic quantum entanglement in superconducting flux qubits. A phase-coupling scheme is proposed to offer enough strength of interactions between qubits. It is shown that due to the two-qubit tunneling processes both the ground state and excited states of coupled two flux qubits can be Bell type states, maximally entangled, in experimentally accessible regimes. The parameter regimes for the Bell states are discussed in terms of external magnetic flux and Josephson coupling energies. We also investigate two types of genuine three-qubit entanglement, known as the Greenberger-Horne- Zeilinger(GHZ) and W states. While an excited state can be the W state, the GHZ state is formed at the ground state of the coupled three flux qubits. The GHZ and W states are shown to be robust against external flux fluctuations for feasible experimental realizations.
We investigate two types of genuine three-qubit entanglement, known as the Greenberger-Horne-Zeilinger (GHZ) and WWW 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 WWW state, the GHZ state is formed at the ground state of the three flux qubits. The GHZ and WWW states are shown to be robust against external flux fluctuations for feasible experimental realizations.
We propose a circuit quantum electrodynamics (QED) realization of a protocol to generate a Greenberger-Horne-Zeilinger (GHZ) state for NNN superconducting transmon qubits homogeneously coupled to a superconducting transmission line resonator in the dispersive limit. We derive an effective Hamiltonian with pairwise qubit exchange interactions of the XYXYXY type, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(XX+YY)$, that can be globally controlled. Starting from a separable initial state, these interactions allow us to generate a multi-qubit GHZ state within a time ${t}_{\mathrm{GHZ}}\ensuremath{\sim}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}^{\ensuremath{-}1}$. We discuss how to probe the nonlocal nature and the genuine NNN-partite entanglement of the generated state. Finally, we investigate the stability of the proposed scheme to inhomogeneities in the physical parameters.