# Is an immersed Kronecker join always a multilinear variety on a Hilbert space?

Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?

This is related to another MathOverflow question

In quantum dynamical simulations, what is the symmetric (Riemannian) analog of a Poisson bracket?

Examples: Let ${\mathcal{H}}$ be a Hilbert space of vectors ${\psi}$ that is spanned by a preferred basis that is given as an orthonormal set of ${n}$-spin (equivalently, ${n}$-qudit) Kronecker products, with the individual spin bases having possibly varying dimension.

Define the Kronecker join ${\mathcal{K}_{r} \subseteq \mathcal{H}}$ to be the rank-$r$ join of the (nonlinear) subspace of vectors in ${\mathcal{H}}$ that are given as Kronecker products in the preferred basis. Formally speaking, $\mathcal{K}_{r}$ is the naturally immersion in ${\mathcal{H}}$ of the rank-$r$ secant variety of the $n$-factor Segre variety (see Definitions and synopsis, below).

For example, let ${\mathcal{H}}$ be spanned by the Kronecker basis associated to two spin-${1/2}$ particles, such that the rank-1 states ${\psi\in \mathcal{K}_1}$ have the parametric representation

$\displaystyle \left[\begin{array}{c} \psi_{1}\\ \psi_{2}\\ \psi_{3}\\ \psi_{4} \end{array}\right] = \left[\begin{array}{c} \xi_{a,1}\\ \xi_{a,2} \end{array}\right] \otimes \left[\begin{array}{c} \xi_{b,1}\\ \xi_{b,2} \end{array}\right] = \left[\begin{array}{c} \xi_{a,1}\,\xi_{b,1}\\ \xi_{a,1}\,\xi_{b,2}\\ \xi_{a,2}\,\xi_{b,1}\\ \xi_{a,2}\,\xi_{b,2} \end{array}\right]$

Then we easily verify that ${\mathcal{K}_1}$, viewed as an projective algebraic variety immersed in ${\mathcal{H}}$ is represented implicitly by the multilinear variety $0 = {\psi_{1} \psi_{4}-\psi_{2}\psi_{3}}$.

Less obviously, let ${\mathcal{H}}$ be spanned by the Kronecker basis associated to two spin-${1}$ particles, such that the rank-2 Kronecker join states ${\psi\in \mathcal{K}_2}$ are represented by

$\displaystyle \left[\begin{array}{c} \psi_{1}\\ \vdots\\ \psi_{9} \end{array}\right] = \left[\begin{array}{c} \xi^1_{a,1}\\ \xi^1_{a,2}\\ \xi^1_{a,3} \end{array}\right] \otimes \left[\begin{array}{c} \xi^1_{b,1}\\ \xi^1_{b,2}\\ \xi^1_{b,3} \end{array}\right] + \left[\begin{array}{c} \xi^2_{a,1}\\ \xi^2_{a,2}\\ \xi^2_{a,3} \end{array}\right] \otimes \left[\begin{array}{c} \xi^2_{b,1}\\ \xi^2_{b,2}\\ \xi^2_{b,3} \end{array}\right] = \left[\begin{array}{c} \xi^1_{a,1}\,\xi^1_{b,1} + \xi^2_{a,1}\,\xi^2_{b,1} \\ \vdots\\ \xi^1_{a,3}\,\xi^1_{b,3} + \xi^2_{a,3}\,\xi^2_{b,3}\end{array}\right]$

Then we again verify that ${\mathcal{K}_2}$ has a multilinear implicit representation, namely $0 = { \psi_{2} \psi_{6} \psi_{7} + \psi_{3} \psi_{4} \psi_{8} + \psi_{1} \psi_{5} \psi_{9} - \psi_{3} \psi_{5} \psi_{7} - \psi_{1} \psi_{6} \psi_{8} - \psi_{2} \psi_{4} \psi_{9} }$.

Least obviously (to me), it is empirically the case that every ${n}$-spin Kronecker join, of every join rank (that can be computed in a reasonable time via Groebner basis implicitization methods) has a multilinear implicit representation on ${\mathcal{H}}$.

That is why I would be very grateful for a reference and/or a proof that all immersed Kronecker joins have a multilinear implicit representation …or for a counterexample.

Practical Motivation: Kronecker joins are ubiquitous in modern algorithms for large-scale quantum simulation, as they are the main building block for almost all state spaces (tensor network state-spaces, for example). Physically speaking, Kronecker joins loosely "fill up" Hilbert space with a geometry "foamy" algebraic manifold that nonetheless retains sufficient linear and symplectic structure that thermodynamical constraints and standard quantum limits are adequately respected.

Issues relating to multilinearity arise naturally when we seek to describe quantum flows in the language of geometric dynamics, specifically in the context of the following theorem:

In particular, even though in practical calculations we always use multilinear parametric representations of $\mathcal{K}$, nonetheless it would be very nice to know (as a fundamental mathematical insight) whether parametric multilinearity implies implicit multilinearity too (as seems empirically to be the case).

More broadly, any and all references and/or remarks regarding similar theorems in algebraic geometry, dynamics, etc., and/or regarding better ways of stating this theorem (and similar ones) would be very welcome.

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Further numerical experiments have (so far) found no counterexamples, and so perhaps in the next few days I will post these numerical examples as a partial answer, together with a narrative regarding why quantum systems engineers care about multilinear varieties ... in the hope that the examples and/or the narrative will assist algebraic geometers in providing a rigorous answer. – John Sidles May 11 '11 at 11:57
As a further followup, Joseph Landsberg has written a very readable Bull. AMS review titled "Geometry and the complexity of matrix multiplication" (2008) that provides a broad context for these questions ... I am still decoding this review. – John Sidles May 16 '11 at 18:02
Thanks to Joseph Landsberg's review "Geometry and the complexity of matrix multiplication", it is clear that the question asked broadly concerns the dimensionality and representation of k'th secant varieties of n-factor Segre varieties (and the quantum pullback theorem given above now employs the terminology of Landsberg's review). The lack of responses (to date) perhaps reflects the imprecision of the question as originally asked, and perhaps reflects too, the mathematical difficulties associated to these varieties. So I still intend to post a better-posed version of this question! – John Sidles May 17 '11 at 16:32