Quantum limits to information about states for finite dimensional Hilbert space K. R. W. Jones H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, U.K. ∗ School of Physics, University of Melbourne, Parkville 3052, Melbourne, Australia. July 15, 2012 Abstract A refined bound for the correlation information of an N–trial ap- paratus is derived for Hilbert spaces of arbitrary finite dimensionality. This is then used to identify the optimal apparatus for large ensem- ble Quantum Inference, thereby solving the asymptotic Optimal State Determination Problem. In this way we are able to identify an alter- native Inferential Uncertainty Principle, which is then contrasted with the usual Heisenberg Uncertainty Principle. Short title: Information limits for quantum states PACS: 0365 Quantum theory; quantum mechanics 0250 Probability Theory * Present address 1