The following questions generalize and naturalize the question that was originally asked. Provisional answers largely due to Will Sawin are now included.

As was discussed in the question originally asked, let $$\mathcal{L}(r,\{a_i\})\ {\colon}{=}\ \sigma_r\big(\text{Seg}(\mathbb{P}^{a_1-1}\times\cdots\times\mathbb{P}^{a_n-1})\big) \subset \ \mathbb{P}^{(\Pi_{i=1}^n a_i) -1}$$ be the usual Segre immersion of the rank-$r$ secant join. Regarding $\mathbb{P}^{(\Pi_{i=1}^n a_i) -1}$ as an $n$-qudit Hilbert space denoted $\mathcal{H}$ for concision, and furthermore regarding $\mathcal{H}$ as a Kähler manifold, we specify $\omega$ on $\mathcal{L}$ as the symplectic form associated to the Kähler potential $\kappa = \langle\psi|\psi\rangle$, given first as a real-valued (bilinear/biholomorphic) function $\kappa\colon \mathcal{H}\to\mathbb{R}$, and thus specified as $\kappa\colon \mathcal{L}\to\mathbb{R}$ by pullback onto the Segre immersion $\mathcal{L}\subset\mathcal{H}$; similarly let $g=\langle\psi|G|\psi\rangle$ be the real-valued (bilinear/biholomorphic) symbol function that is specified by a general hermitian operator $G$, with $g$ similarly pulled-back to $\mathcal{L}$ from $\mathcal{H}$.

Answers to these questions (or better-constructed questions) were sought:

Q1Is the Zariski closure of $\mathcal{L}(r,\{a_i\})$ endowed with a differential structure that is globally smooth?

A1The answer is no (not in general). For example (as explained by Will Sawin) the manifolds $\mathcal{L}(a-1,\{a,a\})$ are determinantal, and thus are known to be generously endowed with singular points.===

Q2In the event that $\mathcal{L}$ is equipped with (one or more) smooth differential structure(s), is it the case that $dg \in \text{span}\ \hat\omega$? Physically, are hermitian operators on $\mathcal{H}$ generically associated to singularity-free symplectomorphic flows on differentially smooth immersed $\mathcal{L}$-manifolds?

A2Formally the answer is no, in that $\mathcal{L}$ generically has no smooth differential structure, per answerA1.===

Q3In the event that $\mathcal{L}$ has no defect-free differential structure(s), what smoothness-defects appear in the symplectomorphic flows on $\mathcal{L}$ that are induced by pullback of (bilinear/biholomorphic) Hamiltonian potential functions from $\mathcal{H}$?

A3Despite the algebraic singularities that are mentioned inA1–2, the answer is conjectured to be "symplectic defects are dynamically occult", in a concrete sense that will be explained in an auxiliary answer (to be posted in the next day or two). In brief, the Hamiltonian structure that is pulled back from $\mathcal{H}$ is conjectured to induce dynamical maps $M(t)\colon \mathcal{L}\to\mathcal{L}$ that are exact symplectic isomormorphisms, despite the singular points of $\mathcal{L}$, and despite the rank-deficit of the pulled-back symplectic form $\omega$ at those singular points. Physically speaking this, means that the varietal singularities of $\mathcal{L}$ do not obstruct computational simulations of thermodynamical physics associated to symplectomorphic dynamical flow.

**Engineering motivation** Engineers "know" — from concrete computations — that $\mathcal{L}$ is richly endowed with singularities in Riemannian/Ricci/scalar curvature. And yet experience teaches that these curvature singularities are seldom (never?) associated to computational pathologies in simulating quantum trajectories as integral curves of Hamiltonian flows. Thus the questions asked are motivated by a postulate that the structure of $\mathcal{L}$ is differentially smooth with respect to symplectic forms and Hamiltonian potentials that are pulled-back from $\mathcal{H}$, even though $\mathcal{H}$ induces metric curvature on $\mathcal{L}$ that is singular.

**The broad question** In what mathematical sense(s) — if any — does algebraic geometry teach that $\mathcal{L}$'s symplectic geometry is differentially smooth, even though $\mathcal{L}$'s metric geometry is singular? In purely practical terms, why don't our quantum trajectory simulation codes break more often? Advice in framing these questions more clearly and naturally is very welcome.

**The questions originally asked** The questions asked concern one-dimensional closed-loop submanifolds of Hilbert space called *LangreTangles*. The specific questions asked are:

- What is the large-$n$ probability $P(n)$ that an $n$-dimensional Hilbert space supports LangreTangle trajectories?
- What is the large-$n$ length $l(n)$ of a generic LangreTangle trajectory?

Numerical evidence suggests $P(n) \gtrsim 5/(\ln n)$ and $\ln l(n)\sim \mathcal{O}(n)$.

In physical terms LangreTangles are emergently-quantum trajectories on "foamy" dynamical manifolds; thus natural physics-oriented questions include "What experimental techniques are suited to the observation of LangreTangle trajectories?"

*Caption* A numerically-integrated portion of a typical LangreTangle trajectory $L(\psi_0,4,\{3,3,3\})$, drawn as a projective map $S^3\,{\to}\,R^3$. It is conjectured that the full LangreTangle would fill $R^3$ with a closed-loop "tangle". Physically this particular LangreTangle is associated to the rank-$4$ secant join of the product state of three spin-$1$ particles. Further LangreTangles have been computed as follows:

- a full (closed-loop) LangreTangle for the simplest-of-all determinantal varieties $\mathcal{L}(1,\{2,2\})$ ( image here );
- a partial LangreTangle for the determinantal variety $\mathcal{L}(4,\{5,5\})$ ( image here )
- a partial LangreTangle for the unit-zabacity (is it nondeterminantal?) variety $\mathcal{L}(9,\{2,2,2,2,2,2\})$ ( image here ); this is an example of a nontrivial unit-zabacity variety that is nondefective in the sense of Landsberg.

**Remark** The $\mathcal{L}(a-1,\{a,a\})$ representation of determinantal varieties is defective for all $a\gt2$, and the $\mathcal{L}(4,\{3,3,3\})$ variety of the original question is defective also; the latter by a result of Strassen per Landsberg p. 130.

**Background** With reference to a recent much-discussed preprint *Scattering Amplitudes and the Positive Grassmannian( by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka (arXiv:1212.5605 [hep-th]), coauthor Arkani-Hamed asserted the following research objective in an associated lecture *The Amplituhedron*:

We can't just keep making equivalences between ideas that were essentially handed to us from the early part of the 20th century. We have to find really new things!

As a conjectural candidate for "a really new thing," Arkani-Hamed *et al.* describe a class of non-Hilbert dynamical systems in which unitarity and locality are encoded emergently, in geometric objects called *amplituhedrons* — for details see Gil Kalai's *MathOverflow* question "What is the amplituhedron?" — that have been depicted in the non-specialist literature as "a jewel at the heart of quantum physics"

**Engineering considerations** Quantum systems engineers are naturally sympathetic to the notion of dynamical systems in which unitarity and spatial locality are emergent rather than fundamental, in that large-scale quantum systems commonly are simulated by algorithms in which unitarity is only approximate (in that trajectories are integrated on non-Hilbert tensor network manifolds) and spatial locality is emergent (via Lindblad/Carmichael trajectory unravelings that dynamically quench macroscopic Schröedinger cats).

**Nomenclature** To concretely visualize these non-Hilbert quantum trajectories, we adopt the notation of Robert Landsberg's text *Tensors: Geometry and Applications*, in which the geometric objects of primary interest are *secant varieties of Segre varieties*

$$ \mathcal{L}(r,\{a_i\}) \equiv \sigma_r\big(\text{Seg}(\mathbb{P}^{a_1-1}\times\cdots\times\mathbb{P}^{a_n-1})\big) \subset \mathbb{P}^{(\Pi_{i=1}^n a_i) -1} $$

Here the manifold $\mathcal{L}(r,\{a_i\})$ is regarded as Segre-embedded in a Hilbert space of (normalized) states $\mathbb{P}^{(\Pi_{i=1}^n a_i)-1}$. Then the *zabacity* function $\text{zab}(r,\{a_i\})$ is defined to be the dimensional deficit

$$ \text{zab}(r,\{a_i\}) = \big((\Pi_{i=1}^n a_i)-1\big) - \dim \mathcal{L}(r,\{a_i\}) $$

**Unit-zabacity manifolds** Henceforth we focus our attention upon unit-zabacity manifolds (that is, $\text{zab}(r,\{a_i\})=1$). Physically these are tensor network manifolds that *almost* fill the embedding Hilbert space, being dimensionally deficient by precisely *one* complex dimension, which is to say, deficient by precisely *two* real dimensions. Among the most celebrated unit-zabacity manifolds is $\mathcal{L}(4,\{3,3,3\})$ (see for example Theorem 5.5.2.1 of Landsberg's text) which physically is the rank-$4$ secant join of the product state of three spin-$1$ particles.

**Visualizing unit-zabacity geometry** The geometry of $\mathcal{L}(4,\{3,3,3\})$, regarded as a submanifold of complex dimension 25 immersed in a normalized Hilbert space $\mathbb{P}^{3^3-1}$ of dimension $3^3-1=26$, can be concretely visualized by intersecting $\mathcal{L}(4,\{3,3,3\})$ with any convenient three-dimensional submanifold of the immersing $\mathbb{P}^{3^3-1}$ Hilbert space; an obvious and geometrically natural candidate for such a three-dimensional submanifold is the rotation group $\text{SO}(3)\sim S^3$. Then we have:

DefinitionTheLangreTangle trajectory$L(\psi_0,r,\{a_i\})$ that is associated to a unit-zabacity manifold $\mathcal{L}(r,\{a_i\})$, and to a fiducial point $p_0\in \mathcal{L}(r,\{a_i\})$, with $\psi(p_0)=\psi_0$, is by definition the one-dimensional submanifold of points $q\in S^3$ such that the rotated state $\psi(q)= R(q)\circ\psi_0$ satisfies $\psi(q) = \psi(p)$ for some $p\in \mathcal{L}(r,\{a_i\})$.

**Numerical and graphical considerations** Closed-form arbitrary-dimension representations of the quantum rotation matrix $R(q)$ (which are due to Wigner) are given in standard textbooks such as Gottfried's *Quantum Mechanics* (1966); quaternionic coordinates on $S^3$ are well-suited to efficient trajectory integration; graphical representations of the Hopf fibration commonly employ the simple-to-compute projective map $S^3\to R^3$; thus the exhibition of concrete LangreTangle trajectories is computationally straightforward.

**LangreTangled trajectories** For generic starting-states $\psi_0$, numerical integration of the LangreTangle trajectories for $L(\psi_0,4,\{3,3,3\})$ shows a striking "tangle", and higher-dimension unit-zabicity manifolds (not shown) are found to support even more tightly tangled trajectories. This tangled geometry motivates both the name of the *zabacity* dimension-measure (from the Persian *zabd* meaning "foam-like," and *zubdat* meaning "cream-like," as skimmed from a body of liquid) and the portmanteau word *LangreTangle*, which derives from "Robert LANdsberg" and "Corrado SeGRE*" and "quantum enTANGLEment".

**Open questions** The questions asked in the beginning then follow naturally from the preceding considerations (along with many more mathematical questions, needless to say). In particular, the strikingly high prevalence of unit-zabicity large-dimension Hilbert spaces apparently is associated to the numeric properties of Cunningham numbers, which cause unit-zabacity Hilbert spaces of large dimension to be exponentially more common than heuristic factor-counting arguments would suggest (in that, for various number-theoretic reasons, the Cunningham integers $a^b\pm 1$ commonly have small factors); the question-asker would be grateful for mathematical illumination in regard to this connexion.

**Physics perspectives** From a physics perspective, a natural question is simply: "By what observations, either in principle or in practice, can (flat) Hilbert-space dynamical manifolds be experimentally distinguished from ("foamy") finite-zabacity dynamical manifolds, even for systems as simple (seemingly) as three interacting spin-$1$ particles?" Again, references and/or experimental suggestions are welcome.

**Acknowledgements** The continuing stimulus of comments and questions by Gil Kalai]in regard to issues of quantum unitarity, locality, and algebraic geometry is gratefully acknowledged.

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