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When working on the lattice it is easy to define a trivial product state. A state $|\psi\rangle$ is a trivial product state if it admits the following tensor decomposition, \begin{equation} |\psi\rangle=\bigotimes_{i}^N |\psi_i\rangle, \end{equation} where $|\psi_i\rangle$ is the state of a single a lattice site and $N$ is the number of lattice sites. A trivial product state is one with no entanglement since every lattice site is essentially doing its own thing.

This definition works perfectly when our space is a lattice with separate sites, but is there an analogous definition of trivial product state that works well in the continuum without ever using the lattice?

Truth and Beauty and Hatred
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2 Answers2

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Your lattice apparently has finite size (and we could assume cyclic boundaries for convenience). If you let $N\rightarrow\infty$ but keep finite size the limit is a continuum with cyclic boundaries.

To keep the energy finite in the limit process, each site on the lattice should decrease it's particle number expectation, e.g. we can choose: $$ |\psi_i\rangle = \frac1{\sqrt{1+\frac1N}}\ \big(1+{ \frac{a_i^\dagger}{\small\sqrt N}}\big)\ |0\rangle $$ so a superposition of (mainly) the vacuum and (a little bit of) a single particle state. The total continuum state is then: \begin{align} |\psi\rangle&=\lim_{N\rightarrow\infty}\ \frac1{(1+\frac1N)^{N/2}}\ \otimes_{i}^N \big(1+{ \frac{a_i^\dagger}{\small\sqrt N}}\big)\ |0\rangle \\ &= \frac1{\sqrt e}\ \lim_{N\rightarrow\infty}\ \otimes_{i}^N \big(1+{ \frac{a_i^\dagger}{\small\sqrt N}}\big)\ |0\rangle \end{align}

A mathematical technicality is that we do not have a notation for the continuum limit of the product (like the integral is the continuum limit of a summation) so it is then merely a problem of how to write down the limit in a closed form expression.

Jos Bergervoet
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You can formally define such a state, but if the Hamiltonian has a term like $|\nabla \phi|^2$ in it, then that state will have an infinite energy. The reason is that a totally unentangled state will have no correlation between spatial regions, including infinitesimally separated ones. See this answer for more details.

11zaq
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