It is believed that the ground state entanglement entropy for a local gapped Hamiltonian satisfies an area law. That is, if we divide such a system in two parts A and B, then $S(\rho_A)=-Tr\left[\rho_A\log(\rho_A)\right]$ is proportional to the area of the boundary separating A and B, where $\rho_A=Tr_B(\rho_{AB})$ and $\rho_{AB}$ is the ground state density matrix. Does it mean that we can perform unitary operations localized near the boundary and make the ground state a product state of the form $\rho_{AB}^{'}=\rho_A^{'}\times \rho_B^{'}$ ?
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asked May 21, 2017 at 13:32
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$\begingroup$ I think the area law says, the entanglement between A,B is concentrated near the boundary and fades away with distance. So local operations near the boundary can reduce the entanglement efficiently but may not necessarily reduce it to 0. It also depends on how you define 'local operation' relative to the correlation range of the system. $\endgroup$– XXDDMay 25, 2017 at 2:03
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$\begingroup$ Rigorously? Intuitively? Under additional "typical" assumptions? Exact area law scaling, or with some corrections (e.g. topological order)? $\endgroup$– Norbert SchuchMay 27, 2017 at 13:46
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$\begingroup$ I mean, just from the fact that the $S(\rho_A)$ is proportional to the boundary, can we conclude that the ground state can be reduced to product state by applying unitary transformations localized at the boundary ? If additional qualifications (assumptions) are needed then I already know that the answer is No. $\endgroup$– Tuhin Subhra MukherjeeMay 27, 2017 at 18:29
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$\begingroup$ I found one example where that is not true. It is the "GHZ" state $\psi= 1/\sqrt{2}(0^{\bigotimes N} + 1^{\bigotimes N})$, on an one dimensional lattice with $N$ cites, each having a 2- dimensional Hilbert space associated with them. This state satisfies the area law, but cannot be reduced to a product state by applying unitary transformations near the boundary. However, I think the GHZ state cannot be a ground state of any local Hamiltonian. $\endgroup$– Tuhin Subhra MukherjeeMay 27, 2017 at 18:49
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$\begingroup$ @TuhinSubhraMukherjee Please use @[user] in your comments if you want others to be notified. $\endgroup$– Norbert SchuchMay 30, 2017 at 10:05
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1 Answer
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we have recently shown that the area law implies that one can apply a unitary in region A such that after the application of the unitary the state almost factorizes between the bulk of the region of A and the rest. Bulk of A here means that it contains all of A but a "thickened boundary" of A whose thickness depends on the error that one wants to allow (i.e. the distance to the nearest product state). In other words one can approximately localize the information in a region "on its boundary". Note however that the unitary in general has to act on the whole region A, since the entanglement between A and its complement is not somehow "localized at the boundary" even if an area law holds. For example, you could have a single EPR pair across the cut, where one half of the EPR pair is deep inside of A and the other half far away from A.
Best, Henrik
Single-shot holographic compression from the area law
Henrik Wilming, Jens Eisert
arxiv:1809.10156
