### 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)

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:

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...

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 ...

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