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.
Key amended elements now are italicized; these amendments
improve the notation of the question, andclarify the nature of the reference(s) requested.Please note too that the request for references remains unanswered.
proves the theorem explicitly, andapplies the theorem in concrete examples.CAVEAT The key elements of the theorem are pairs of complex vectors $\{\boldsymbol{\psi},\boldsymbol{\xi}\}$ that satisfy a specified pair of algebraic relations. Because these algebraic relations are nonlinear, the set of solution pairs $\{\boldsymbol{\psi},\boldsymbol{\xi}\}$ does not form
within a natural vector space.This absence of natural linear structure in the solution space is why the sole answer suggested (up to the present time) — which appeals to the fundamental theorem of linear algebra — does not constitute an answer to the question asked (AFAICT, and I would be delighted framework that extends naturally to be corrected in this regard).
For this reason, a fairly careful reading of the question as-asked is necessary: there are good reasons why Joe Harris' graduate-level text Algebraic Geometry: a First Course analyzes this class of problems in great depth. multi-linear algebraic varieties
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 define four the indices $\{s,r,m,n\}$ that range over $s \in \{1,2\}$, $r \in \{1,\dots,k-1\}$, and $m,n \in \{1,\dots,k\}$. Let ${(}\dots{)}$ be a multi-index and let repeated indices be summed. Then for $\boldsymbol{\psi}=\{\psi_{(mn)}\}$ and $\boldsymbol{\xi}=\{\xi_{(srm)}\}$ as vectors in $\mathbb{C}^{k^2}$ and $\mathbb{C}^{2(k-1)k}$ respectively, we have:
%\forall\ \boldsymbol{\psi}\ \colon\ \ \det_{mn}\ \psi_{(mn)} =
0 \
xi_{(1rm)}\,\xi_{(2rn)}\quad\Longleftrightarrow\quad\det_{mn}\ quad\Longleftrightarrow\quad \exists\ \boldsymbol{\xi}\ \colon\ \ \xi_{(1rm)}\,\xi_{(2rn)}.In the context
Context of quantum physics, $s$ may the question in multilinear algebraic geometry
Primary consideration should 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 given to references that $(mn)$ is a multi-index over prove the quantum amplitudes $\psi_{(mn)}$theorem and/or discuss related theorems within a mathematical framework that are extends naturally associated to the $k^2$ orthonormal basis vectors of multilinear algebraic varieties.
Secondary consideration should be given to references that are reasonably accessible to the bipartite Hilbert space $\mathcal{H}_1 \otimes \mathcal{H}_2$(many) engineers and physicists for whom these multilinear varieties increasingly are finding practical applications.
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 $Y^{(kk)\!}_{k{-}1}$\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.
Note: the original request for concrete examples and practical applications has been split-off to a separate MOF question that focuses on practical applications, accompanied by a request for a concrete theorem (as contrasted with references).
The word defective refers specifically to a $Y^{(kk)\!}_{k{-}1}$\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).
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 $\psi \in Y^{(kk)\!}_{k{-}1}$\mathcal{M}^{(kk)\!}_{k{-}1}$ (in Harris' notation for $Y\,$\mathcal{M}\,$).
This question's three-level three four-level reward structure
Demonstrate either $k$-Lion Hypothesis to win $100,000 from Scott Aaronson.
Win MOF's first-ever Gold Reversal Medal!
(Just to be clear, it is reasonably easy to prove the SHLT via brute-force methods; the question asked seeks references that provide an illuminating mathematical context)context that extends naturally to the higher-order multilinear varieties that have come to play a vital practical role in modern quantum systems engineering).
