We consider a general and practical scenario of quantum key distribution (QKD) over an unknown, stationary, unital qubit channel. Furthermore, due to practical limitations, e.g., relative movement and rotation of communicating parties, a global shared reference frame cannot be established. This scenario can routinely appear in satellite QKD. We propose two methods to overcome the physical qubit noise and the lack of shared reference frame. The first proposed approach involves constructing the Pauli transfer matrix (PTM) description of the channel, which we achieve without requiring a shared reference frame, by absorbing the lack of shared reference frame in the channel definition. This is followed by the identification of singular vectors of PTM as the Bloch vectors for optimal signal states. In the optimized local bases, the resulting correlations are equivalent, up to outcome relabeling, to those of a Pauli channel, allowing us to show the optimality of the BB84 and six-state QKD protocols under these conditions. The second approach, called the sequential basis matching (SBM) involves sequentially identifying the channel-optimized local bases that enable QKD. We show that both of these approaches result in the same effective key exchange rate for QKD.

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