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We derive a general expression of the quantum Fisher information for aMach-Zehnder interferometer, with the port inputs of an \emph{arbitrary} purestate and a squeezed thermal state. We find that the standard quantum limit canbe beaten, when even or odd states are applied to the pure-state port. Inparticular, when the squeezed thermal state becomes a thermal state, all theeven or odd states have the same quantum Fisher information for given photonnumbers. For a squeezed thermal state, optimal even or odd states are needed toapproach the Heisenberg limit. As examples, we consider several common even orodd states: Fock states, even or odd coherent states, squeezed vacuum states,and single-photon-subtracted squeezed vacuum states. We also demonstrate thatsuper-precision can be realized by implementing the parity measurement forthese states.

Enhanced interferometry using squeezed thermal states and even or odd states