We quantify the influence of non-adiabatic leakage and system dissipation on the transfer fidelity with generic counter-intuitive pulses. By including the dissipation of all the parties, we find that the competition between non-adiabaticity and decoherence leads to an upper bound of the transfer fidelity. Using a systematic expansion valid in the desired high-fidelity limit, we derive the upper bound as a simple function of the system cooperativity. This approach also indicates how to reach the upper bound efficiently. Our results are widely applicable to quantum state engineering and are particularly significant for solid-state devices, which typically have non-negligible dissipation of the source and target systems.
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