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Currently doing a PhD in Controlled Quantum Dynamics at Imperial College and the University of Oxford.


6h
comment Connections between Density Matrix Renormalization Group and Conformal Field Theory
Well they are both useful techniques in 1+1 dimensions. You might be interested in continuous matrix product states, which can represent quantum field states in 1D, presumably including those corresponding to a CFT.
6h
comment Unitary evolution operator
@ConstantineBlack I edited to try and make some sense of the question. Hopefully I interpreted what you were trying to say correctly. As it stands, the question still does not make much sense to me: the actual question at the end seems to have nothing to do with the rest of the text. In general the unitary time evolution operator $e^{-iHt}$ contains all information about the way different parts of the system interact, since this information is contained in the Hamiltonian $H$.
6h
revised Unitary evolution operator
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2d
comment System of two harmonic oscillators and its quantum partition function
@sandstone Yes, the oscillators are independent, as you can see from the Hamiltonian and from your expression for the energies. Your attempted solution is just the product of two very similar sums, which you can carry out in exactly the same way as your other question.
2d
comment Calculating quantum partition functions
@sandstone This answer explains how to carry out your attempted solution method to get the correct outcome. I am not sure what else I could explain. Your $\sum_\Gamma$ equates to my use of $\mathrm{Tr}$.
2d
comment System of two harmonic oscillators and its quantum partition function
This is essentially identical to this question, for the specific case of two modes. Note that the energies are labelled by two integers, $n_1$ and $n_2$, with energies $\epsilon(n_1,n_2) = \epsilon_{n_1} + \epsilon_{n_2}$. Summing over all values of $n_1$ and $n_2$ gives you the answer.
2d
comment Calculating quantum partition functions
@sandstone You do not need to evaluate the product. What you need is to calculate $-\partial \ln Z/\partial\beta = -\frac{1}{Z}\partial Z/\partial \beta$, right?
2d
comment Calculating quantum partition functions
@sandstone You're answer cannot depend on $m_\mathbf{k}$ (i.e. the number of bosons corresponding to a micro-state of mode $\mathbf{k}$) since you have already summed over this variable. You need the result for a geometric series. You are on the right track, keep at it.
2d
revised Calculating quantum partition functions
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2d
comment Integrating entropy on an arbitrary boundary
Substitutions followed by integration by parts should give you a solution in terms of the error function and some other terms.
2d
comment Integrating entropy on an arbitrary boundary
Are you familiar with the error function? I expect your solution will probably be in terms of it. You can't simply integrate $\mathrm{e}^{-x^2}$ over a finite interval and get out an answer in terms of elementary functions.
2d
revised Calculating quantum partition functions
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Apr
16
revised Calculating quantum partition functions
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Apr
16
answered Calculating quantum partition functions
Apr
16
comment White noise and Fourier transform
@David I have updated my answer. By the way, you can tag users by putting the arroba symbol (@) in front of their name. Otherwise they will not be notified of your response.
Apr
16
revised White noise and Fourier transform
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Apr
15
comment Difference between apply quantum gate and measure a qubit?
If you apply a two-qubit gate (i.e. if the qubit interacts with something else), then the superposition state of one qubit alone does "collapse", because the qubit is entangled with something else. Similarly if the qubit in a superposition interacts with a measuring device it becomes entangled with it.
Apr
15
comment White noise and Fourier transform
@pyramids I don't agree. The Fourier transform of each realisation does not vanish. However, the average value (average over realisations) of the Fourier transform at each frequency $\omega$ must be zero. What else could it be?
Apr
15
comment White noise and Fourier transform
@pyramids What do you mean the Fourier transform conserves power? To consider the power, aren't we then talking about the second moment anyway?
Apr
15
comment White noise and Fourier transform
@pyramids Nevertheless, I think the conclusion, that $\langle \int\mathrm{d}t\, \mathrm{e}^{\mathrm{i}\omega t} \xi(t) \rangle = 0$, should be correct, no? Since we are considring a bunch of delta-correlated random functions, I would expect the Fourier transform of each one of these functions, evaluated at some frequency $\omega$, to be distributed equally between positive and negative values.