# Research Polymath

## August 2, 2009

### Application

Filed under: Uncategorized — ramanujantao @ 2:17 am
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Hardy used to say that number theory is the purest subject. He was proud of the fact that it had no applications in that time. Now however, it has many applications to cryptography, code breaking etc.. In the same way, quantum entanglement does not seem to have any real practical applications. However this following problem and thought process helped someone solve a problem he was having trouble on involving statistical entanglement. So it goes to show you, many subjects, no matter how pure they are, can have real world applications.

Question. Consider three cities: Philadelphia, New York City, and Trenton. These can be represented by subsystems $A$, $B$ and $C$. Now if we measure $A$, $B$ and $C$ separately, it is impossible to obtain information about the entire system. The information is encoded in the nonlocal correlations between the subsystems.

This problem helped someone in a real world setting.

## July 31, 2009

### Principles of Mathematical Analysis by Walter Rudin

Filed under: Analysis — ramanujantao @ 4:56 pm
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Chapter 1: The Real and Complex Number Systems

One thing I notice about Rudin in general is that he tends to “pull rabbits out of hats.” For instance, on pg.2, he does the following: Let $A = \{p: p^2<2, \ p \in \mathbb{Q}, \ p>0 \}$ and $B = \{p: p^2>2, \ p \in \mathbb{Q}, \ p>0 \}$. He wants to show that $A$ contains no largest element and $B$ contains no smallest. Then all of a sudden he does the following: For every $p>0$ associate $q = p-\frac{p^2-2}{p+2} = \frac{2p+2}{p+2}$. Then consider $q^2-2$ to show that $q \in A$ or $q \in B$.

### Collaborative Learning

Filed under: Uncategorized — ramanujantao @ 4:43 pm
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As I mentioned in my previous post, I want to experiment with collaborative learning as well. My plan is to start with a Rudin’s Principles of Mathematical Analysis and apply a “polymath approach.” Although I think each post will be a new chapter. Unlike lectures, I think this approach will allow people to learn actively and with questions. Also, one can get many different perspectives of the material.

## July 28, 2009

### Quantum Entanglement: Discussion Thread

Filed under: Uncategorized — ramanujantao @ 7:48 pm
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First of all, we need to gather some resources to tackle this problem. I found John Preskill’s notes on quantum computation to be very valuable. But what about $\mathbb{Z}_3$ is special? Moreover, why are we considering spaces of the form $\mathbb{Z}_p$ where $p$ is prime? As mentioned before here, quantum entanglement can be modelled more generally than tensor products in hilbert spaces.  We can consider cartesian products of various sets. But will this general view help us tackle our more specific problem?

### 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? .

### Welcome

Filed under: Uncategorized — ramanujantao @ 4:52 pm

The purpose of this blog is to encourage collaborative research on tractable research problems. This is much in the spirt of Terence Tao’s polymath blog. I might experiment with “collaborative learning” as well. For example, suppose we want to go through Rudin’s Principles of Mathematical Analysis. Perhaps we can apply a polymath approach to this task as well.

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