Entanglement spectrum What does it mean by the entanglement spectrum of a quantum system? A brief introduction and a few key references would be appreciated.
 A: 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/
A: 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. 
A: 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.
