Concluding Progressive Remarks
A new finding is Bates and Oeding's preprint "Toward a salmon conjecture" (arXiv:1009.6181), with its reference to the Salmon Prize.
The Salmon Prize (photo of the prize here) is offered by mathematician/biologist Elizabeth Allman, and can be appreciated in the broad mathematical context that is provided by Topics in Tensors: A Summer School by Shmuel Friedland.
Until such time as further comments are offered, a working answer is:
An introduction to multilinear varieties considered as algebraic/geometric objects may be found on page 99ff of Joe Harris' Algebraic Geometry: a First Course.
A recent comprehensive survey is chapter 7 of Joseph Landsberg's brand-new book Tensors: Geometry and Applications (December 2011), which begins (encouragingly)
"This chapter includes nearly all that is known about defining equations for secant varieties of Segre and Veranese varieties …"
At the conclusion of the meta thread that was started by Andy Putman (who wondered why this question was being downvoted) concrete examples now are given of higher-order 'almost-Hilbert' varieties. The following intent is posted there:
Last night I discovered a brand-new monograph by Joseph Landsberg that encompasses more-or-less the answer sought, and so I have amended the beginning of the question to be a pointer to Landsberg's monograph.
Sometime in the next week or two I will post a concrete mathematical question — framed within the context that Landsberg's monograph supplies — asking for a classification of all multilinear varieties having unit-dimension defect with respect to their natural Segre embedding.
At that time I will request closure of the original question, to be supplanted by this classification question.
A shorter, accessibly written, and free-as-in-freedom introduction to these multilinear varieties is Joseph Landsberg's relatively recent — and much-cited — Bulletin of the AMS survey article "Geometry and the complexity of matrix multiplication" (2008).
Please let me apologize for the deficiencies of original question in conveying to MOF readers the mathematical depth and breadth of these multilinear varieties, their implications for fundamental quantum physics, and their numerous practical applications in engineering fields ranging from the computational complexity matrix multiplication to the simulation of quantum transport.
Fortunately, Joseph Landsberg's recent writings have done an immensely better job of this than the original draft of my MOF question did!
My appreciation and thanks are extended to all who have provided comments, and in particular, sincere congratulations are extended to Theo Johnson-Freyd for providing an answer that has received MOF's first-ever Gold Reversal Medal. It has been great fun to help this happen!
The question asked (as clarified per Joseph Landsberg)
For the theorem of algebraic geometry that is specified below, please provide a reference (or references) that:
- states the theorem rigorously,
- proves the theorem explicitly,
- within a framework that extends naturally to multi-linear algebraic varieties
The theorem for which mathematical references are sought
Let $k\ge1$ be an integer and let $\boldsymbol{\psi}=\{\psi_{(mn)}\}$ and $\boldsymbol{\xi}=\{\xi_{(srm)}\}$ be vectors in $\mathbb{C}^{k^2}$ and $\mathbb{C}^{2(k-1)k}$ respectively. Here ${(}\dots{)}$ is a multi-index, repeated indices are summed, and the indices $\{s,r,m,n\}$ range over $s \in \{1,2\}$, $r \in \{1,\dots,k-1\}$, and $m,n \in \{1,\dots,k\}$. Then we have:
The Second-Hand Lion Theorem (SHLT) $$ %\forall\ \boldsymbol{\psi}\ \colon\ \ \det_{mn}\ \psi_{(mn)} = 0 \quad\Longleftrightarrow\quad \exists\ \boldsymbol{\xi}\ \colon\ \ \psi_{(mn)} = \xi_{(1rm)}\,\xi_{(2rn)}. $$
Context of the question in multilinear algebraic geometry
Primary consideration should be given to references that prove the theorem and/or discuss related theorems within a mathematical framework that extends naturally to multilinear algebraic varieties.
Secondary consideration should be given to references that are reasonably accessible to the (many) engineers and physicists for whom these multilinear varieties increasingly are finding practical applications.
In the context of algebraic geometry, $r$ may be regarded as an index over $(k{-}1)$ order-2 Segre varieties that enter in a rank-$(k{-}2)$ secant join having the natural Segre embedding in the tensor product space $\mathcal{H}_1 \otimes \mathcal{H}_2$.
In the notation of Joe Harris' Algebraic Geometry: a First Course, the theorem asserts the identity of the preceding Segre embedding with what is called the generic determinantal variety $\mathcal{M}^{(kk)\!}_{k{-}1}$ that comprises (by definition) the set of $k\times k$ complex matrices having matrix rank $k-1$.
Attention is directed particularly to a passage in Harris (page 100) that states:
"We should draw a fundamental and important distinction between bi- and tri- or multilinear objects […] whose invariants are far from being completely understood."
Thus, although the theorem stated can be solved via specialized techniques that apply uniquely to bilinear varieties, a broader and deeper grounding is sought for this theorem within the context that modern algebraic geometry provides, with regard especially to techniques that extend naturally to generic multilinear algebraic varieties. To borrow a phrase from Richard Hamming, "The purpose of the question is insight, not theorems."
Quantum physics and engineering applications
In the context of quantum physics, $s$ may be regarded as an index over two $k$-dimensional Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$, each equipped with an $k$-element orthonormal basis, such that $(mn)$ is a multi-index over the quantum amplitudes $\psi_{(mn)}$ that are naturally associated to the $k^2$ orthonormal basis vectors of the bipartite Hilbert space $\mathcal{H}_1 \otimes \mathcal{H}_2$.
In quantum systems engineering (QSE), determinantal varieties are the bread-and-butter state-spaces of large-scale quantum simulations, because they support both the natural geometric pullback of thermodynamical relations and conservation laws and the numerically efficient integration of dynamical trajectories that respect these relations.
The origin of the SHLT name
The name "SHLT" is a homage to the following dialog line in the film Second Hand Lions:
Uncle Garth (Michael McCaine): This lion's no good … it's … it's … defective.
The word defective refers specifically to a $\mathcal{M}^{(kk)\!}_{k{-}1}$ determinantal variety's one-dimensional (nonlinear) rank-defect as a quantum state-space, relative to the $k^2$-dimensional Hilbert space in which it is immersed (see below).
As a starting-point, the SHLT is mentioned — but regrettably only in passing and with no derivation given — in the paragraphs following Example 9.2 on page 99 of Harris.
Two quantum physics conjectures
By definition, let a quantum state $\psi \in \mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$ (so that $\dim \mathcal{H} = k^2$) be called $k$-Lion iff $\psi \in \mathcal{M}^{(kk)\!}_{k{-}1}$ (in Harris' notation for $\mathcal{M}\,$).
Then we have:
The Weak $k$-Lion Hypothesis
There exists a finite integer $k\lesssim 2^5$ such that no practicable quantum experiment can observationally disprove the hypothesis that the state-space of a symmetrically bipartite dynamical system is $k$-Lion rather than Hilbert.
Physically, the limit $k\lesssim 2^5$ corresponds to the case of quantum entanglement in a bipartite dynamical system having $5+5=10$ qubits in total.
The Weak $k$-Lion Hypothesis is sufficiently difficult to test — although by construction the required tests are far easier than demonstrating fault-tolerant quantum computing (FTLC) — that it is reasonable to suppose that Weak $k$-Lion Hypothesis cannot be feasibly be disconfirmed even for for quite small values of $k$. Hence it is both mathematically and physically natural to conjecture:
The Strong $k$-Lion Hypothesis
There exists a constant of Nature in the form of a finite integer $k$, such that no experiment can observationally disprove the hypothesis that the state-space of a symmetrically bipartite dynamical system is $k$-Lion rather than Hilbert, for the fundamental reason that the dynamical state-space of Nature is a determinantal variety rather than a Hilbert space.
This question's three four five-level reward structure
Associated to this question is a three four five-level reward structure:
Gain MathOverflow reputation by providing good math literature references.
Contribute to the (wonderful) ongoing GLL debate between Aram Harrow and Gil Kalai, and thereby help also to accelerate the medical goals of the UW/ISH Naturality and Guidance Seminar.
Demonstrate either $k$-Lion Hypothesis to $100K wager on quantum computing">win $100,000 from Scott Aaronson.
Receive MOF's first-ever award of the Gold Reversal Medal. Congratulations, Theo!
Best of all, the Salmon Prize is offered by mathematician/biologist Elizabeth Allman.
