# Implication of the area law of the entanglement entropy

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 $\rho_{AB}^{'}=\rho_A^{'}\times \rho_B^{'}$ ?

• 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.
– XXDD
May 25, 2017 at 2:03
• Rigorously? Intuitively? Under additional "typical" assumptions? Exact area law scaling, or with some corrections (e.g. topological order)? May 27, 2017 at 13:46
• 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. May 27, 2017 at 18:29
• 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. May 27, 2017 at 18:49
• @TuhinSubhraMukherjee Please use @[user] in your comments if you want others to be notified. May 30, 2017 at 10:05