Quantifying the amount of quantum entanglement in a many-body state is a difficult problem that has stimulated a lot of papers but a limited amount of progress. This is largely because there are few good measures of multi-party entanglement. Even before finding a good measure one needs to decide how to partition your Hilbert space. (i.e., who are the parties: Alice, Bob, Charles, David, Erwin, Felix, Gerard,...?)

Why should we care?

Two ambitious outcomes one might aim for are:

-use the entanglement in chemical bonds as a resource to perform quantum information processing tasks

-use insights from quantum information theory to develop new quantum chemistry algorithms

I recently stumbled across a nice paper by Ziesche, Gunnarsson, John, and Beck, that was written in 1997, before quantum information became fashionable. Hence, the word entanglement is actually not in the paper! But, they do calculate it. They consider the one-particle density matrix for the two-site Hubbard model and the BCS ground state. They then calculate the "correlation entropy" of the ground state. This is essentially the von-Neumann entropy and corresponds to the entanglement of one electron with all the other electrons in the system. They reference earlier papers that perform similar calculations for the hydrogen molecule and several other systems.

The main goal of the paper is to test a conjecture of Collins that this entropy should be proportional to the "correlation energy" in quantum chemistry. This is usually the difference in energy between the true ground state energy and that calculated from a Hartree-Fock wave function, i.e., a single Slater determinant.

I really like the paper because it uses a simple model to illustrate not just entanglement but also issues such as correlations, natural orbitals, Heitler-London vs. Hartree-Fock.

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