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We study the dynamics of entanglement of two electron spins in two quantumdots, in which each electron is interacting with its nuclear spin environment.Focusing on the case of uncoupled dots, and starting from either Bell or Wernerstates of two qubits, we calculate the decay of entanglement due to thehyperfine interaction with the nuclei. We mostly focus on the regime ofmagnetic fields in which the bath-induced electron spin flips play a role, forexample their presence leads to the appearance of entanglement sudden death atfinite time for two qubits initialized in a Bell state. For these fields theintrabath dipolar interactions and spatial inhomogeneity of hyperfine couplingsare irrelevant on the time scale of coherence (and entanglement) decay, andmost of the presented calculations are performed using the uniform-couplingapproximation to the exact hyperfine Hamiltonian. We provide a comprehensiveoverview of entanglement decay in this regime, considering both free evolutionof the qubits, and an echo protocol with simultaneous application of $\pi$pulses to the two spins. All the currently relevant for experiments bath statesare considered: the thermal state, narrowed states (characterized by diminisheduncertainty of one of the components of the Overhauser field) of twouncorrelated baths, and a correlated narrowed state with a well-defined valueof the $z$ component of the Overhauser field interdot gradient. While we mostlyuse concurrence to quantify the amount of entanglement in a mixed state of thetwo electron spins, we also show that their entanglement dynamics can bereconstructed from measurements of the currently relevant for experimentsentanglement witnesses, and the fidelity of quantum teleportation performedusing a partially disentangled state as a resource.

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