- #1

JorisL

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I'm investigating typical values of entropy for a subsystem of a 1D (non-interacting) spin chain.

Most of the problem is essentially solved

I've shown that a typical pure state of the entire chain is close (trace norm) to the state ##\Omega_S## when reduced.

[tex]\Omega_S = \text{Tr}_E \frac{1\!\!1}{d_U}[/tex]

Here ##d_U=\text{dim}\mathcal{H}_U## is the Hilbert space of the entire chain which we further on split as ##\mathcal{H}_U = \mathcal{H}_S\otimes \mathcal{H}_E##.

I've essentially followed Section V in the paper by S. Popescu et al. with some sidesteps to understand every single step.

By the Fannes-Audenaert inequality we have that the von Neumann entropy of ##\text{Tr}_E |\phi\rangle\langle\phi|## is close to ##S[ \Omega_S]##.

Now I'm not entirely sure how to find ##S[ \Omega_S]##. I think I've found 3 approaches to this.

Which would be the nicest to use? (it's from the perspective of mathematical physics)

1. Direct calculation of the partial trace ##\Omega_S## and diagonalize

2. Use the thermal canonical principle from the same paper

[tex]H_S = \sum\limits_{i=1}^N \sigma_i^{(z)}[/tex]Given that the total energy of the universe is approximately E, interactions between the system and the rest of the universe are weak, and that the energy spectrum of the universe is suffi- ciently dense and uniform, almost every pure state of the universe is such that the state of the system alone is approximately equal to the thermal canonical state ##e^ {− \frac{H_S}{k_BT}}## , with temperature T (corresponding to the energy E)

Now if ##S\ll U## I figured ##H_S\ll k_BT## and thus ##\Omega_S \approx \frac{1\!\!1_S}{d_S}## which leads to ##S[\Omega_S] = log{N}## with N the number of spins in the subsystem S.

I'm not entirely certain I can do this, but haven't found any immediate problems here

3. Describe the spins as fermions with ##|\downarrow\rangle## the absence of a particle while ##|\uparrow\rangle## the presence of a particle. Then we can use creation and annihilation operators and the machinery of the CAR-algebra description. I haven't entertained this method a lot but I'm wondering if I can use this?

So the main question is the following, is approach 2 valid? If it is I'm done.

If it is not, what can I best do. Direct calculation or mapping the system to a fermion chain?

Thanks,

Joris