Mutually Unbiased Bases in Infinite Dimension Hilbert Spaces
While there has been investigation into mutually unbiased bases in infinite dimension Hilbert space, their existence remains an open question. It is conjectured that in a continuous Hilbert space, two orthonormal bases and are said to be mutually unbiased if
For the generalized position and momentum eigenstates and, the value of k is
The existence of mutually unbiased bases in a continuous Hilbert space remains open for debate, as further research in their existence is required before any conclusions can be reached.
Position states and momentum states are eigenvectors of Hermitian operators and, respectively. Weigert and Wilkinson were first to notice that also a linear combination of these operators have eigenbases, which have some features typical for the mutually unbiased bases. An operator has eigenfunctions proportional to with and the corresponding eigenvalues . If we parametrize and as and, the overlap between any eigenstate of the linear combination and any eigenstate of the position operator (both states normalized to the Dirac delta) is constant, but dependent on :
where and stand for eigenfunctions of and .
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