Measurement-erasure duality: Axiomatic uniqueness and exact thermodynamic identity

We establish a duality between quantum measurement and Landauer erasure at the level of quantum channels, using the Hilbert--Schmidt adjoint. For projective measurements on a finite-dimensional system, the quantum-to-classical measurement channel and the classical-to-quantum preparation channel are mutual adjoints, and the resulting involution exchanges the Schrodinger and Heisenberg pictures. The duality yields an exact thermodynamic identity: the total minimum work cost of a complete measurement-erasure cycle depends only on the Hilbert-space dimension and the temperature, independent of the input state or measurement basis. We prove that the Hilbert--Schmidt adjoint is the unique duality satisfying five physically motivated axioms when the dimension is three or greater; consequently, the total cycle cost is the only value consistent with these axioms. We show that measurement irreversibility equals the relative entropy of coherence in the input state, and that near-equilibrium work fluctuations scale with the same quantity. Combined with the Sagawa--Ueda generalized second law, these results yield a basis-independent bound on quantum Maxwell-demon work extraction that is strictly tighter than the standard constraint whenever the thermal state is not maximally mixed. The duality generalizes to arbitrary faithful reference states via the KMS dual, which for state-preserving channels coincides with the Petz recovery map of quantum error correction.