The study of feedback control inspired by Maxwell's demon has yielded fundamental insights into the relationship between thermodynamics and information. However, its fully quantum formulation, where the controller holds quantum side information that cannot be treated as a classical record, remains incomplete. In this paper, we establish fundamental no-go limits for thermodynamic state conversion while allowing quantum side information to be fed back to the system coherently. From two basic operational principles respected by all physically admissible feedback-control schemes, we derive the tightest possible bounds on the single-shot work of formation and extractable work for arbitrary quantum systems conditioned on arbitrary quantum side information. In the asymptotic limit, these bounds yield a generalized second law of thermodynamics with quantum feedback, governed by a conditional free energy. We further show that this generalized second law is consistent with the traditional second law once the acquisition, processing, and reset of side information are accounted for. These results extend information thermodynamics to the fully quantum regime and provide precise thermodynamic meanings for the negativity of single-shot conditional entropies.

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