What does it mean by the entanglement spectrum of a quantum system? A brief introduction and a few key references would be appreciated.
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2$\begingroup$ Beats me. I've certainly never heard of it before, and the easy references that pop up are already pretty deep into their topics. But it's an intriguing phrase, so I'm also curious now. Insights, anyone? $\endgroup$– Terry BollingerCommented Apr 23, 2012 at 3:43
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3 Answers
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If a system S is composed of two subsystems A and B, then a state of S is a vector $$|\Psi\rangle \in H_A\otimes H_B$$ Tracing over the "B degrees of freedom" allows you to define the reduced density matrix $\rho_A$ The entanglement entropy is defined as$$-Tr(\rho_Aln\rho_A)$$
I believe that the entanglement spectrum just refers to the spectrum of eigenvalues of $\rho_A$. Sorry I don't know any references though.
Edit to add: The entanglement entropy is a fairly crude measure of the entanglement present (just a single number). Knowledge of the entanglement spectrum provides further information on the entanglement properties - it includes much more information about the entire reduced density matrix $\rho_A$. This has been used, for example, in investigations of the scaling behaviour of extended quantum systems.
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$\begingroup$ I second the lack of references... it seems to just be used in a lot of places. Perhaps a few words about its use and relevance would suffice? $\endgroup$– gennethCommented Apr 23, 2012 at 9:57
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$\begingroup$ @genneth: I added a little more into and a reference. $\endgroup$ Commented Apr 23, 2012 at 12:02
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$\begingroup$ Indeed this answer is correct according to section 17.2 pg 758 of "Field theories of Condensed Matter physics" 2nd edition CUP by Eduardo Fradkin. $\endgroup$ Commented Sep 2, 2023 at 5:45
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For any equilibrium density matrix, we can recover the Hamiltonian if we know the temperature: $$\beta H = - \log \rho.$$ The motivation is what happens if we get $\rho$ by taking a sub-system (tracing over the rest)? In that case, we can still make up a Hamiltonian --- and the entanglement spectrum simply refers to the eigenvalues of that Hamiltonian (up to a scaling ambiguity in $\beta$). This is interesting because in some ways it contains the degrees of freedom on the boundary, which, for a topologically ordered bulk, will contain non-trivial (and usually gapless) degrees of freedom.
Haldane has some slides online about its appearance and use in quantum hall states: http://online.itp.ucsb.edu/online/lowdim_c09/haldane/
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$\begingroup$ I see you found a reference with some nice pictures! +1 $\endgroup$ Commented Apr 23, 2012 at 12:05
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The entanglement spectrum was first proposed by Li and Haldane in their paper "Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States" They calculated the entanglement spectrum for fractional quantum Hall states and show it can be a "fingerprint" to identify different topological order.
