# Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, June 26, 2009

### M Theory Lesson 280

According to Oeding, the Lagrangian Grassmanian of an even dimensional symplectic space $V$ is the image of a map $f$ that takes a symmetric matrix and gives a vector of minors. There is a projection from the Grassmanian onto the variety of principal minors of all $n \times n$ matrices.

This is interesting because minors are a natural way to describe pure states in quantum mechanics. Consider a three qubit state with $8$ amplitudes. Forgetting about $a_{000}$, which we can set to $1$ projectively speaking, and letting $a_{111}$ be related somehow to the determinant of a matrix, it turns out that the other six amplitudes should be expressed as the principal minors of the matrix which has full determinant given by the entanglement measure (Cayley's hyperdeterminant)

#### 2 Comments:

PhilG said...

By the spookiest of coincidences I am in the middle of writing something very similar on my hyperdeterminant blog, but I think I can go a little further...

June 26, 2009 4:48 AM
Kea said...

That's great, Phil! The more the merrier, I say. Actually, I spoke about this at PI recently, but I haven't been keeping up with the blogging. More to come ...

June 26, 2009 8:04 PM