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

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"

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.

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

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"

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.

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 further specify that $\omega$ on $\mathcal{L}$ is the symplectic form that is specified by 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 $\mathcal{H}$.

A1  The answer is no (not in general). For example (as explained by Will Sawin) the manifolds $\mathcal{L}(a-a,\{a,a\})$ are determinatal, and thus are known to be generously endowed with singular points.

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 $\mathcal{H}$.

A1  The 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.

Update: Motivated by Will Sawin's explicitly constructive partial answer, the following LangreTangles have been computed:

1. a full (closed-loop) LangreTangle for the simplest-of-all determinantal varieties $\mathcal{L}(1,\{2,2\})$ ( image here );
2. a partial LangreTangle for the determinantal variety $\mathcal{L}(4,\{5,5\})$ ( image here )
3. 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.

Will Sawin's answer emphasizes that 2-qudit unit-zabacity varieties of the form $\mathcal{L}(a-1,\{a,a\})$ are determinantal; hence questions Q1-3 may be particularly tractable for 2-qudit state-spaces; hence answers/comments/citations relating to the symplectic geometry of the singularities of determinantal varieties are welcomed especially.

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.

The following questions generalize and naturalize the question that was originally asked, and a bounty is now offered.

Let $\mathcal{L}(r,\{a_i\}) \subset \ \mathbb{P}^{(\Pi_{i=1}^n a_i) -1}$ be the usual Segre immersion of the rank-$r$ secant join $\sigma_r\big(\text{Seg}(\mathbb{P}^{a_1-1}\times\cdots\times\mathbb{P}^{a_n-1})\big)$, as was discussed in the question originally asked.

  Regarding $\mathbb{P}^{(\Pi_{i=1}^n a_i) -1}$ as an $n$-qudit Hilbert space, viewed as a Kähler manifold and denoted $\mathcal{H}$ for concision, we further specify that $\omega$ on $\mathcal{L}$ is the symplectic form that is specified by 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 $\mathcal{H}$.

A bounty is offered for answers to these questions:

Q1  Is the Zariski closure of $\mathcal{L}(r,\{a_i\})$ endowed with a differential structure that is globally smooth?

Q2  In 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?

Q3  In 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 $\mathcal{H}$?

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.

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 further specify that $\omega$ on $\mathcal{L}$ is the symplectic form that is specified by 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:

Q1  Is the Zariski closure of $\mathcal{L}(r,\{a_i\})$ endowed with a differential structure that is globally smooth?

A1  The answer is no (not in general). For example (as explained by Will Sawin) the manifolds $\mathcal{L}(a-a,\{a,a\})$ are determinatal, and thus are known to be generously endowed with singular points.

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Q2  In 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?

A2  Formally the answer is no, in that $\mathcal{L}$ generically has no smooth differential structure, per answer A1.

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Q3  In 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 $\mathcal{H}$?

A3  Despite the algebraic singularities that are mentioned in A1–2, 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.

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:

1. a full (closed-loop) LangreTangle for the simplest-of-all determinantal varieties $\mathcal{L}(1,\{2,2\})$ ( image here );
2. a partial LangreTangle for the determinantal variety $\mathcal{L}(4,\{5,5\})$ ( image here )
3. 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.