Due to the exponential growth of the state space of coupled quantum systems it is not possible, in general, to numerically store the state of a very large number of quantum systems within a classical computer. We demonstrate a method for modelling the dynamical behaviour of measurable quantities for very large numbers of interacting quantum systems. Our approach makes use of a symbolic non-commutative algebra engine that we have recently developed in conjunction with the well-known Ehrenfest theorem. Here we show the possibility of determining the dynamics of experimentally observable quantities, without approximation, for very large numbers of interacting harmonic oscillators. Our analysis removes a large number of significant constraints present in previous analysis of this example system (such as having no entanglement in the initial state). This method will be of value in simulating the operation of large quantum machines, emergent behaviour in quantum systems, open quantum systems and quantum chemistry to name but a few.
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