Optimal finite-dimensional probe states for quantum phase estimation

Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the N00N state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.