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Suppose we construct a quantum coding scheme in which Alice and Bob agree on a number of things in advance, including, say, a bijection $\gamma$ between two sets of vectors (which live in different Hilbert spaces) which Bob has to apply once he receives a message/state from Alice. Is it inherent to quantum information theory that $\gamma$ needs to be linear (or even unitary), or doesn't such an "agreement map" need to be constrained ?

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It depends on the different Scenarios for 𝛾.

Encoding and Decoding: If γ is part of an encoding/decoding scheme where Alice encodes information into quantum states and Bob decodes it, the operations involved typically need to be unitary to ensure that the information can be perfectly retrieved without loss. Non-unitary operations could introduce errors or loss of information.

Quantum Channels: When dealing with quantum communication through a noisy channel, γ might represent a more general quantum operation or channel (described by a completely positive trace-preserving CPTP map). While not necessarily unitary, these operations still need to be linear. The channel can include effects like decoherence and noise, but the linearity constraint still holds.

Classical Agreement Map: If γ represents a pre-agreed classical mapping (e.g., a classical bijection between sets of basis vectors or labels for states), it need not be linear in the sense of quantum operations. It could be any bijection that both parties agree upon for labeling purposes. However, when this map is applied in the quantum context (e.g., changing the basis or labeling of quantum states), the corresponding quantum operation derived from this map must still adhere to linearity constraints.

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