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The position and momentum distributions play an important role in thedetermination of classical and quantum states, which reveals a fundamental gapbetween classical and quantum world. For finite dimensional quantum system, aparticle moving in one spatial dimension is expressed as a pure quantum stateand projective measurements onto two special orthonormal bases are regarded asthe observations of position and momentum, which are not enough for theuniquely determination of pure qudit states. In this paper, we focus on thedetermination of all finite dimensional pure quantum states with projectivemeasurements onto two orthonormal bases. Firstly, a new orthonormal basis isconstructed for the replacement of discrete analogy of momentum, which can helpto filter out finite candidates among all pure qudit states with probability 1.Then, the analogy bases for position and momentum are enough to determinealmost all pure qudits when introducing an adaptive POVM.

Determination of finite dimensional pure quantum state by the discrete analogues of position and momentum