Dynamics of entanglement of two electron spins interacting with nuclear spin baths in quantum dots

We study the dynamics of entanglement of two electron spins in two quantum dots, in which each electron is interacting with its nuclear spin environment. Focusing on the case of uncoupled dots, and starting from either Bell or Werner states of two qubits, we calculate the decay of entanglement due to the hyperfine interaction with the nuclei. We mostly focus on the regime of magnetic fields in which the bath-induced electron spin flips play a role, for example their presence leads to the appearance of entanglement sudden death at finite time for two qubits initialized in a Bell state. For these fields the intrabath dipolar interactions and spatial inhomogeneity of hyperfine couplings are irrelevant on the time scale of coherence (and entanglement) decay, and most of the presented calculations are performed using the uniform-coupling approximation to the exact hyperfine Hamiltonian. We provide a comprehensive overview of entanglement decay in this regime, considering both free evolution of the qubits, and an echo protocol with simultaneous application of $π$ pulses to the two spins. All the currently relevant for experiments bath states are considered: the thermal state, narrowed states (characterized by diminished uncertainty of one of the components of the Overhauser field) of two uncorrelated baths, and a correlated narrowed state with a well-defined value of the $z$ component of the Overhauser field interdot gradient. While we mostly use concurrence to quantify the amount of entanglement in a mixed state of the two electron spins, we also show that their entanglement dynamics can be reconstructed from measurements of the currently relevant for experiments entanglement witnesses, and the fidelity of quantum teleportation performed using a partially disentangled state as a resource.