Quantum-Enhanced Parameter Estimation Without Entanglement

Entanglement is generally considered necessary for achieving the Heisenberg limit in quantum metrology. We construct analogues of Dicke and GHZ states on a single $N+1$ dimensional qudit that achieve precision equivalent to symmetrically entangled states on $N$ qubits, showing that entanglement is not necessary for going beyond the standard quantum limit. We define a measure of non-classicality based on quantum Fisher information and estimate the achievable precision, suggesting a close relationship between non-classical states and metrological power of qudits. Our work offers an exponential reduction in the physical resources required for quantum-enhanced parameter estimation, making it accessible on any quantum system with a high-dimensional Hilbert space.