Determining the minimum number of measurement bases required for the complete characterization of unknown quantum states is a central problem in quantum state tomography, especially in high-dimensional systems. For a $d$-dimensional quantum state, parameter counting implies that at least $d+1$ projective measurement bases are required for full reconstruction. However, explicit constructions achieving this information-theoretic minimum are not generally available beyond special cases based on mutually unbiased bases (MUBs). Here, we present a quantum state tomography scheme that reconstructs an arbitrary $d$-dimensional quantum state using exactly $d+1$ projective measurement bases. Our construction is explicit, applies to all finite dimensions, and does not rely on the existence of a complete set of MUBs. We further demonstrate the scheme on a silicon photonic chip by reconstructing six-dimensional quantum states, obtaining fidelities above 0.96 in a dimension where a complete set of MUBs is not known to exist. These results establish an explicit minimal-basis construction for high-dimensional quantum state tomography and provide a different perspective on the design of informationally complete quantum measurements at the information-theoretic limit.

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• Quantum foundations
• Quantum measurements
• Quantum parameter estimation
• Quantum tomography