Mutually Unbiased Bases: complementary observables in finite-dimensional Hilbert spaces

Rinie N. M. Nasir, Shamsul Shaari, Jesni, Stefano Mancini

Rinie N. M. Nasir, Jesni Shamsul Shaari, Stefano Mancini

Berge Englert

M. D. Sheppeard

Complementary observables in quantum mechanics may be viewed as Frobenius structures in a dagger monoidal category, such as the category of finite dimensional Hilbert spaces over the complex numbers. On the other hand, their properties crucially depend on the discrete Fourier transform and its associated quantum torus, requiring only the finite fields that underlie mutually unbiased bases. In axiomatic topos theory, the complex numbers are difficult to describe and should not be invoked unnecessarily. This paper surveys some fundamentals of quantum arithmetic using finite field complementary observables, with a view considering more general axiom systems.

M. Revzen

Analysis of state reconstruction both classical and quantum mechanical on equal footing is outlined. The meaning of "mutual unbiased bases" (MUB) of Hilbert spaces is explained in detail. An alternative quantum state reconstruction, that utilizes mutual unbiased bases (MUB), is given. The MUB approach is then used for state reconstruction in a finite, d, dimensional Hilbert spaces.