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In classical theory (e.g., classical mechanics and electromagnetic theory), we introduce the complex values for the mathematical convenience, e.g., A=(a+a*)/2. There, the use of complex values (or vectors) is not necessary as we can in principle describe everything with real values (or vectors).
However, in quantum mechanics and quantum optics, why do we need to use the complex vector to express a quantum state? Would it possible to describe a quantum state with a real vector although it will not have a mathematically elegant forms?
Some book says it is just a postulate. Does this mean that we should not try to prove or demonstrate why the complex vector is a necessary since it is just correct as the complex vector based description fits experiment results very well?
Would there be any kind of reasonable explanation for why we describe a quantum state as a complex vector in Hilbert space?