  Entanglement is one of the most fundamental properties of quantum mechanics,
and is the key resource for quantum information processing. Bipartite entangled
states of identical particles have been generated and studied in several
experiments, and post-selected or heralded entangled states involving pairs of
photons, single photons and single atoms, or different nuclei in the solid
state, have also been produced. Here, we use a deterministic quantum logic gate
to generate a "hybrid" entangled state of two trapped-ion qubits held in
different isotopes of calcium, perform full tomography of the state produced,
and make a test of Bell's inequality with non-identical atoms. We use a
laser-driven two-qubit gate, whose mechanism is insensitive to the qubits'
energy splittings, to produce a maximally-entangled state of one Ca-40 qubit
and one Ca-43 qubit, held 3.5 microns apart in the same ion trap, with 99.8(6)%
fidelity. We test the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's
inequality for this novel entangled state and find that it is violated by 15
standard deviations; in this test, we close the detection loophole but not the
locality loophole. Mixed-species quantum logic is a powerful technique for the
construction of a quantum computer based on trapped ions, as it allows
protection of memory qubits while other qubits undergo logic operations, or are
used as photonic interfaces to other processing units. The entangling gate
mechanism used here can also be applied to qubits stored in different atomic
elements; this would allow both memory and logic gate errors due to photon
scattering to be reduced below the levels required for fault-tolerant quantum
error correction, which is an essential pre-requisite for general-purpose
quantum computing.
