# Research Polymath

## July 28, 2009

### Proposal: Quantum Entanglement

Filed under: polymath proposals — ramanujantao @ 4:55 pm
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Polymath Proposal

The goal of this project is to study “entanglement” in finite vector spaces. Entanglement is the hallmark of quantum mechanical systems. Quantum mechanics is modelled by vector spaces over $\mathbb{C}$ the field of complexes. We propose to consider entanglement in finite dimensional vector spaces over $\mathbb{Z}_p$ ($p$ is prime). We will think of this as providing a test bed (or toy model) for general conjectures about entanglement. One way to characterize entanglement over any field is described as follows.

Suppose we have two non-interacting subsystems $A$ and $B$. Each of these subsystems is described by Hilbert Spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$. Then the Hilbert space for the composite system is $\mathcal{H}_{A} \otimes \mathcal{H}_{B}$. Now if the state of the composite system $AB$ is $|\psi \rangle_{A} \otimes |\phi\rangle_{B}$, then this is a product state. More generally, suppose we have different bases for $\mathcal{H}_A$ and $\mathcal{H}_B$. Call these $\{|i \rangle_A \}$ and $\{|j \rangle_B \}$. Then we can represent the state of the composite system as $\sum_{i,j} c_{ij}|i \rangle_{A} \otimes |j \rangle_{B}$.

Now if $c_{ij} \neq c_{i}^{A}c_{j}^{B}$, then we have an entangled state. In other words, you cannot assign pure states to either subsystem $A$ or $B$. The key problem for us is whether entanglement exists in finite vector spaces. We will analyze systems such as $\mathbb{Z}_3 \times \mathbb{Z}_3$. Simple systems like this will lead to further insight to vector spaces such as $\mathbb{Z}_p \times \mathbb{Z}_p$ (where $p$ is prime). More specifically, does entanglement exist in all finite vector spaces? .