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We`ve known how to calculate entanglement entropy from a given ground state: make an entanglement cut (that divide system into subsystems $A$ and $B$), take the partial trace and $$S=-\operatorname{Tr}(\rho_A\cdot \log\rho_A).$$

But is it possible to start from entanglement entropy (ie: a function $S: \text{cuts} \rightarrow R$), as long as it is physical, to reconstruct a state that exhibits the entanglement pattern we demand?

Of course, there are examples that such a reconstruction will not be unique. But given knowledge of what local degrees of freedom are, is there any way to find at least one such state?

Any reference will be appreciated. And please feel free to edit my question if it is needed.

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  • $\begingroup$ To clarify: Are you assuming a fixed factorization of the Hilbert space, as in $$ \mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2\otimes\cdots\otimes \mathcal{H}_N, $$ and assuming that we are given the von Neumann entanglement entropy $S(\Omega)$ associated with every subset $\Omega$ of the factors? And then the question is how to construct an example of a single density matrix on $\mathcal{H}$ that is compatible with this set of $N$-choose-$2$ von Neumann entropies, when such a density matrix exists? $\endgroup$ – Chiral Anomaly Sep 20 '19 at 15:46
  • $\begingroup$ @ChiralAnomaly yes. And physically I think the density matrix has to be a pure state corresponding to some ground state of an almost local Hamiltonian. But anyway, let`s see if it is generally possible. $\endgroup$ – SSSSiwei Sep 20 '19 at 16:28
  • $\begingroup$ "reconstruct a state" -- as long as you impose any conditions the state must satisfy: Obviously yes. Otherwise, you will have to say which conditions (a) the entanglement spectrum satisfies (or whatever else -- do you only want to obtain a certain value for the entropy? for one/all/... cuts??) and (b) the bulk state should have. $\endgroup$ – Norbert Schuch Sep 21 '19 at 12:02
  • $\begingroup$ @NorbertSchuch What I`m looking for is actually an algorithm that starts from entanglement and ends up with a certain state. $\endgroup$ – SSSSiwei Sep 21 '19 at 17:38
  • $\begingroup$ Like "The entanglement is 5, then the state must be this and that?" How would that be possible??? Could you please clarify your question, either by being more specific or at least providing an illustrating example? $\endgroup$ – Norbert Schuch Sep 21 '19 at 18:37

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