Complex numbers are necessary for the continuous time evolution of quantum gates, since real special orthogonal operators cannot represent gates with determinant -1 and real-to-complex mappings remain isomorphic to complex representations.
Quantum mechanics on a real Hilbert space
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abstract
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular frequency $\omega$ has any given finite or infinite set of discrete energy eigenvalues, limited below by $\hbar\omega/2$.
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This review compiles fourteen equivalent formulations of the open existence problem for maximal mutually unbiased bases in composite dimensions and summarizes known analytic, computer-aided and numerical results along with potential solution strategies.
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Time evolution of quantum gates and the necessity of complex numbers
Complex numbers are necessary for the continuous time evolution of quantum gates, since real special orthogonal operators cannot represent gates with determinant -1 and real-to-complex mappings remain isomorphic to complex representations.
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Mutually Unbiased Bases in Composite Dimensions -- A Review
This review compiles fourteen equivalent formulations of the open existence problem for maximal mutually unbiased bases in composite dimensions and summarizes known analytic, computer-aided and numerical results along with potential solution strategies.