The measure of quantum noise is the power spectral density that is defined by the following expression:
Here is the quantum state of the light wave.In our review, we will encounter two types of quantum states that we have described above, i.e., vacuum and squeezed vacuum states. Let us show how to calculate the power (doublesided) spectral density of a generic quantity in a vacuum state. To do so, one should substitute Eq. (86) into Eq. (87) and obtain that:
where we used the definition of the power spectral density matrix of light in a vacuum state (65)^{7}. Similarly, one can calculate the spectral density of quantum noise if the light is in a squeezed state , utilizing the definition of the squeezed state density matrix given in Eq. (83):It might also be necessary to calculate also crosscorrelation spectral density of with some other quantity with quantum noise defined as:

Using the definition of crossspectral density similar to (87):
one can get the following expressions for spectral densities in both cases of the vacuum state: and the squeezed state:Note that since the observables and that one calculates spectral densities for are Hermitian, it is compulsory, as is well known, for any operator to represent a physical quantity, then the following relation holds for their spectral coefficients and :
This leads to an interesting observation that the coefficients and should be realvalued functions of variable .Now we can make further generalizations and consider multiple light and vacuum fields comprising the quantity of interest:
where stand for quadrature amplitude vectors of independent electromagnetic fields, and are the corresponding complexvalued coefficient functions indicating how these fields are transmitted to the output. In reality, the readout observable of a GW detector is always a combination of the input light field and vacuum fields that mix into the output optical train as a result of optical loss of various origin. This statement can be exemplified by a single lossy mirror I/Orelations given by Eq. (34) of Section 2.2.4.Thus, to calculate the spectral density for such an observable, one needs to know the initial state of all light fields under consideration. Since we assume independent from each other, the initial state will simply be a direct product of the initial states for each of the fields:

and the formula for the corresponding power (doublesided) spectral density reads:
with standing for the ith input field spectral density matrix. Hence, the total spectral density is just a sum of spectral densities of each of the fields. The crossspectral density for two observables and can be built by analogy and we leave this task to the reader.
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Living Rev. Relativity 15, (2012), 5
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