How many quantum noises need to be added to make an LTI system physically realizable?

In M. R. James, H. I. Nurdin, and I. R. Petersen, “H∞ control of linear quantum stochastic systems,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1787-1803, 2008, the authors showed that an arbitrary LTI system can be made physically realizable if additional quantum noise components are permitted in the implementation. However, incorporating additional noise is undesirable, and this paper refines their method in order to provide a concrete expression for the number of additional quantum noise components needed to guarantee physical realizability.