Consider a quantum chain in its ground state and then take a subdomain of this system with natural truncated hamiltonian. Since the total hamiltonian does not commute with the truncated hamiltonian the subsystem can be in one of its eigenenergies with different probabilities. Since the global energy eigenstates are locally close to diagonal in the local energy eigenbasis we argue that the Shannon(Renyi) entropy of these probabilities follows an area-law for the gapped systems. When the system is at the critical point the Shannon(Renyi) entropy follows a logarithmic behaviour with a universal coefficient.
Our results show that the Shannon(Renyi) entropy of the subsystem energies closely mimics the behaviour of the entanglement entropy in quantum chains. We support the arguments by detailed numerical calculations performed on the transverse field XY-chain.
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