In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. The Segre embedding is the categorical product in the category of projective varieties. As such, it becomes the natural product for the discussion of entangled states in the product of projective Hilbert spaces in quantum mechanics.

Definition

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The Segre map may be defined as the map

$\backslash sigma:\; P^n\; \backslash times\; P^m\; \backslash to\; P^\{(n+1)(m+1)-1\}$

taking a pair of points $(*,*)\; \backslash in\; P^n\; \backslash times\; P^m$ to their product

$\backslash sigma:([Y\_0:Y\_1:\backslash cdots:Y\_m)\; \backslash mapsto$

X_0Y_1: \cdots :X_iY_j: \cdots :X_nY_m

Here, $P^n$ and $P^m$ are projective vector spaces over some arbitrary field, and the notation

$*$

is that of homogeneous coordinates on the space. The image of the map is a variety, called a Segre variety. Notationally, it is sometimes written as $\backslash Sigma\_\{n,m\}$.

Discussion

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In linear algebra terms there is for given vector spaces U and V, over the same field, a natural way to map

U × V

to the tensor product space W. This not in general injective, because it takes the pair

(u,v)

to the pure tensor w formed from u and v. For any non-zero scalar c, the image of

(cu,c⁻¹v)

will also be w. In co-ordinate terms, w has co-ordinates formed of all products of a co-ordinate of u with a co-ordinate of v.

Considering now the underlying projective spaces P(U) and P(V), the mapping passes to a morphism of varieties

σ: P(U) × P(V) → P(W).

This is not only injective in the set-theoretic sense: it is a closed immersion in the sense of algebraic geometry. That is, one can give a set of equations for the image. Except for notational trouble, it is easy to say what such equations are: they express two ways of factoring products of co-ordinates from W, obtained in two different ways as something from U times something from V.

This mapping or morphism σ is the Segre embedding. Counting dimensions, it shows how the product of projective spaces of dimensions m and n embeds in dimension

(m + 1)(n + 1) − 1 = mn + m + n.

Classical terminology calls the co-ordinates on the product multihomogeneous, and the product generalised to k factors k-way projective space.

Categorical product

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The Segre embedding is the categorical product on the category of projective varieties. This may be expressed in fuller detail. Let $X\backslash subset\; P^n$ and $Y\backslash subset\; P^m$ be two projective varieties. Then the Segre embedding of $X\backslash times\; Y$ is also a variety. Furthermore, given maps $\backslash eta:Z\backslash to\; X$ an $\backslash zeta:Z\; \backslash to\; Y$, there is a unique map $Z\backslash to\; \backslash sigma(X\backslash times\; Y)$ such that the mapping commutes with the projection maps $\backslash pi\_X\; :X\backslash times\; Y\; \backslash to\; X$ and $\backslash pi\_Y\; :X\backslash times\; Y\; \backslash to\; Y$.

Properties

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The Segre variety is an example of a determinental variety; it is the zero locus of the 2×2 minors of the matrix $(Z\_\{i,j\})$. That is, the Segre variety is the common zero locus of the quadratics

$Z\_\{i,j\}\; Z\_\{k,l\}\; -\; Z\_\{i,l\}\; Z\_\{k,j\}$

Here, $Z\_\{i,j\}$ is understood to be the natural coordinate on the image of the Segre map. The fibers of the product are linear subspaces. That is, let

$\backslash pi\_X\; :P^\{(n+1)(m+1)-1\}\; \backslash to\; P^n$

be the projection to the first factor; and likewise $\backslash pi\_Y$ for the second factor. Then the image of the map

$\backslash sigma\; (\backslash pi\_X\; (\backslash cdot),\; \backslash pi\_Y\; (p)):P^\{(n+1)(m+1)-1\}\; \backslash to\; P^\{(n+1)(m+1)-1\}$

for a fixed point p is a linear subspace of the codomain.

Examples

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Quadric

For example with m = n = 1 we get an embedding of the product of the projective line with itself in P³. The image is a quadric, and is easily seen to contain two one-parameter families of lines. Over the complex numbers this is a quite general non-singular quadric. Letting

$*$

be the homogeneous coordinates on P³, this quadric is given as the zero locus of the quadratic polynomial given by the determinant

= Z_0Z_3 - Z_1Z_2

Segre threefold

The map

$\backslash sigma:\; P^2\; \backslash times\; P^1\; \backslash to\; P^5$

is known as the Segre threefold. It is an example of a rational normal scroll. The intersection of the Segre threefold and a three-plane $P^3$ is a twisted cubic curve.

Veronese variety

The image of the diagonal $\backslash Delta\; \backslash subset\; P^n\; \backslash times\; P^n$ under the Segre map is the Veronese variety of degree two $\backslash nu\_2:P^n\; \backslash to\; P^\{n^2+2n\}$.

Applications

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Because the Segre map is the categorical product on projective spaces, it is the natural mapping for describing entangled states in quantum mechanics and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert spaces. A short sketch equating ideas from quantum mechanics with algebraic geometry can be found in the article Fubini-Study metric.

References

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• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3

Algebraic varieties | Projective geometry