Virtual Numbers to Represent Entangled Quantum States

In the existing formalism of quantum states, probability amplitudes of quantum states are complex numbers. A composition of entangled quantum states, such as a Bell state, cannot be decomposed into its constituent quantum states, implying that quantum states lose their identities when they get entangled. This is contrary to the observation that a composition of entangled quantum states decays back to its constituent quantum states. To eliminate this discrepancy, this paper introduces a new type of numbers, called virtual numbers, which produce zero upon multiplication with complex numbers. In the proposed formalism of quantum states, probability amplitudes of quantum states are general numbers made of complex and virtual numbers. A composition of entangled quantum states, such as a Bell state, can then be decomposed into its constituent quantum states, implying that quantum states retain their identities when they get entangled.