Considerations on Coupled Oscillator Entanglement Entropy

  In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency  w of sqrt(w+ w-) (where w+ and w- are defined later). We find  this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. 

   Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. 

  We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating  Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2.

Only one set w,T for a single oscillator should map into the integrated result.