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# Metrics and geodesics on fuzzy spaces

David Viennot

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David Viennot. Metrics and geodesics on fuzzy spaces. Journal of Physics A: Mathematical and Theoretical, 2024, 57 (42), pp.425201. 10.1088/1751-8121/ad7c9d. hal-04717062

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# Metrics and geodesics on fuzzy spaces

David Viennot

Institut UTINAM (CNRS UMR 6213, Université de Franche-Comté,
Observatoire de Besançon), 41bis Avenue de l'Observatoire, BP1615, 25010
Besançon cedex, France.

**Abstract.** We study the fuzzy spaces (as special examples of noncommutative manifolds) with their quasicoherent states in order to find their pertinent metrics. We show that they are naturally endowed with two natural “quantum metrics” which are associated with quantum fluctuations of “paths”. The first one provides the length the mean path whereas the second one provides the average length of the fluctuated paths. Onto the classical manifold associated with the quasicoherent state (manifold of the mean values of the coordinate observables in the state minimising their quantum uncertainties) these two metrics provides two minimising geodesic equations. Moreover, fuzzy spaces being not torsion free, we have also two different autoparallel geodesic equations associated with two different adiabatic regimes in the move of a probe onto the fuzzy space. We apply these mathematical results to quantum gravity in BFSS matrix models, and to the quantum information theory of a controlled qubit submitted to noises of a large quantum environment.

**Keywords:** fuzzy spaces, noncommutative geometry, quantum geometry, quantum gravity, matrix models, quantum information, coherent states

## 1. Introduction

Fuzzy spaces [1] are special cases of Connes' noncommutative geometry [2]; fuzzy spaces are inspired from the fundamental example of the fuzzy sphere [3]. They are mathematical framework for matrix models of quantum gravity (nonperturbative regime of type IIB string theory): BFSS (Banks-Fischler-Shenker-Susskind) [4] and IKKT (Ishibashi-Kawai-Kitazawa-Tsuchiya) [5] matrix models. Simple fuzzy spaces arise by considering a fermionic string linking a D2-brane and a D0-brane [6, 7] (the spacetime dimension could be reduced to 3+1 by considering a truncation by taking an orbifold [7]). The D2-brane is then assimilated to a noncommutative manifold. Quantum gravity is not the only domain of application of fuzzy spaces. Indeed adiabatic control of qubits entangled with an environment exhibits higher gauge structures similar to the ones of string theory [8], and the Hamiltonian in the interaction picture is similar to the geometric operator of a fuzzy space.

In the spirit of the Connes' noncommutative geometry, we can simply seen a fuzzy space as the quantisation of an extended “solid body”. Usual quantum mechanics (obtained by application of the canonical quantisation rules – first quantisation –) is the quantisation of classical mechanics of point systems. Quantum field theory is the quantisation of classical field theory (second quantisation, by application of the canonical quantisation rules onto conjugated fields – the four-potential vector field and the electric field for electrodynamics for example –); so the quantisation of the “scalar,
<table><caption>**Table 1.** Comparison between first and second quantisation and fuzzyfication. $\mathcal{H}$ is a Hilbert space, $\mathcal{F}_{\pm}$ a fermionic or bosonic Fock space. $\Psi^{+}(x) = \sum_{i} \phi_{i}(x) a_{i}^{\dagger}$ and $\Psi(x) = \sum_{i} \phi_{i}(x) a_{i}$ are the field operators of creation and annihilation of a particle at point $x$ ($\langle \phi_{i} \rangle_{i}$ being an orthonormal basis of the Hilbert space of a single particle and $a_{i}^{\dagger}/a_{i}$ being the creation/annihilation operators of a particle on the mode $i$). $(X^{i})_{i}$ is a set of coordinate observables of the noncommutative manifold, the superoperator $-i[X_{i}, \bullet]$ playing the role of the noncommutative tangent vector field in the $i$-direction and $dX^{i}$ (with $d$ the Koszul differential) playing the role of the noncommutative cotangent vector field (differential 1-form) in the $i$-direction.</caption><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>Classical point mechanics<br> x&#773; &isin; &#8477;<sup>3</sup><br> (x<sup>i</sup>, p<sup>i</sup>) &isin; &#8477;<sup>6</sup></td><td>first quantisation &rarr;</td><td>Quantum mechanics<br> &psi; &isin; L<sup>2</sup>(&#8477;<sup>3</sup>, d&#773;x) = &#8475;<br> x&#770;<sup>i</sup>, p&#770;<sup>i</sup> &isin; &#8475;(&#8475;)<br> [x&#770;<sup>i</sup>, p&#770;<sup>j</sup>] = &iota;&delta;<sup>ij</sup>id</td></tr><tr><td>Classical field theory<br> {&#773;x &mapsto; &psi;(&#773;x)} &isin; &#8455;&#8477;<sup>3</sup></td><td>second quantisation &rarr;</td><td>Quantum field theory<br> {&#773;x &mapsto; &Psi;(&#773;x), &Psi;<sup>+</sup>(&#773;x)} &isin; &#8475;(&#8475;<sub>&pm;</sub>)<sub>&#8477;<sup>3</sup></sub><br> [&Psi;(&#773;x), &Psi;<sup>+</sup>(&#773;x')]<sub>&pm;</sub> = &delta;(&#773;x - &#773;x')id</td></tr><tr><td>Classical solid mechanics<br> M = {&#773;x &isin; &#8477;<sup>3</sup> with f(&#773;x) = 0}</td><td>fuzzyification &rarr;</td><td>Fuzzy space theory<br> &#8475; = Lin&#8477;(<i>X</i><sup>1</sup>, <i>X</i><sup>2</sup>, <i>X</i><sup>3</sup>)<br> with &#773;X &isin; &#8475;(&#8475;)<sup>3</sup>, f(&#773;X) = 0<br> and [<i>X</i><sup>i</sup>, <i>X</i><sup>j</sup>] &ne; 0</td></tr><tr><td>&part;<br> d<i>x</i><sup>i</sup> &isin; TM</td><td></td><td>-&iota;[<i>X</i><sup>i</sup>, &bull;] &isin; Der(<&#8475;)<br> Der(<&#8475;) &ni; L &mapsto; d<i>X</i><sup>i</sup>(L) = L(<i>X</i><sup>i</sup>)</td></tr></tbody></table>

vector or spinor degrees of freedom" by sustaining classical the base points of the fields. We can imagine the fuzzy spaces as resulting from a noncanonical third quantisation ("fuzzyfication") of rigid or deformable bodies. Table 1 summarises briefly this idea for the fuzzyfication of a surface.

An important notion concerning the fuzzy space concerns their quasic coherent states [9, 10]. These ones are named by analogy with the Perelomov coherent states of a Lie algebra [11], and are the states such that the Heisenberg uncertainties concerning the coordinate observables are minimised. Moreover these ones are the ground eigenstates of the geometry operator $\mathcal{D}_{x}$ of the fuzzy space. In the BFSS matrix model [7], $\mathcal{D}_{x}$ is the Dirac operator of the fermionic string, and the quasic coherent states are the ones for which the displacement energy (the "tension energy" of the string) is zero. In quantum information theory, the interaction Hamiltonian of the qubit has a structure similar to $\mathcal{D}_{x}$, and the quasic coherent state is (in the language of ref. [8]) a $*$-eigenvector associated with a noncommutative eigenvalue, corresponding to a qubit control with zero energy uncertainty. In a recent work [12], it is proposed that the quasic coherent picture permits to define the emergent gravity at the Planck scale in the BFSS model. Indeed, we can see the BFSS model as the quantisation of a flat spacetime in which the noncommutativity induces a quantum non-zero (Weitzenböck) torsion (see ref. [13] for an introduction to Weitzenböck torsions). At the semi-classical thermodynamical limit (number of strings tending to infinity with constant density), gravity (spacetime curvature) emerges at the macroscopic scale from the noncommutativity at the microscopic one (more precisely, the noncommutativity relations define at the thermodynamical limit a Poisson structure which defines a spacetime effective metric). Quasic coherent states are associated with a classical "eigenmanifold" (as a quantum
object, a fuzzy space has for “eigenvalues” – measurement outputs – a classical quantity, here a classical manifold). In ref. [12] it is argued that this eigenmanifold is the emergent curved spacetime at the Planck scale, in the meaning where it is the classical manifold closest to the quantum geometry of the fuzzy space (since it is associated with the states minimising the Heisenberg uncertainties and with adiabatic transport which is the quantum dynamical regime closest to classical dynamics). This emerging spacetime is endowed with a natural metric and with a Lorentz connection (inducing curvature and torsion).

In the present paper, we want to analyse more precisely the proposition of ref. [12] with more general and mathematical point of view of the possibility to endow a fuzzy space and its eigenmanifold with metrics and geodesic equations. Different works explore the application of the Connes’ metric onto fuzzy spaces [14, 15, 16, 17, 18], but in this paper we present another way by considering metrics associated with the quasicoherent picture. We will show that a fuzzy space can be endowed with two different metrics, one being a quantised metric as an infinitesimal square length (as in the Connes theory) and the other one being a quantised metric as a field of inner product of tangent vectors. The presence of two metrics is in accordance with the “quantum fluctuations” onto a fuzzy manifold. “Paths” onto a fuzzy manifold are submitted to these quantum fluctuations, and so the square root of the averaging of the first quantum metric in quasicoherent states (which is the metrics onto the eigenmanifold of ref. [12]) is the length of the mean infinitesimal path, whereas the averaging of the square root of the second quantum metric is the mean length of the fluctuating infinitesimal quantum paths (the mean length of the paths being larger than the length of the mean path, as for a classical Brownian motion). This effect of the quantum uncertainty (of the quantum indeterminism) explains why the two notions of metric (infinitesimal square length and field of inner product), which are totally equivalent in classical geometry, are two distinct objects in quantum geometry. Moreover, in addition to the interpretation to quantum gravity, we want also here explore the interpretation of the quantum metrics for quantum information theory (as we will seen, they are associated with residual energy uncertainties associated with measurement misalignments, i.e. small errors in the application of adiabatic quantum control of the qubit onto the manifold of zero energy uncertainty).

This paper is organised as follows. Section 2 sets definitions concerning fuzzy spaces and noncommutative geometry. The role of this section is essentially to fix some notations and some definitions which can be slightly change from an author to another one. Section 3 is a review about the eigen geometry of a fuzzy space (theory of quasicoherent states and eigenmanifolds). Section 4 introduces the two quantum metrics, their relations with the geometry of the eigenmanifold and their interpretations. Section 5 studies the adiabatic transport of a probe onto the eigenmanifold of a fuzzy space. The torsion is intimately related to this question, and we introduce then the associated auto-parallel geodesics and their interpretations. Section 6 explains how generalise the present discussion to a time-dependent fuzzy space. Finally in a concluding section, we discuss the generalisation to higher dimensional fuzzy spaces (the dimension is set to be 3 in this paper for the sake of simplicity), the analogy between quantum gravity and quantum information theory provided by the common model of fuzzy space; and we draw some futur research directions. Four appendices finish this paper. The first one introduces the perturbation theory of fuzzy spaces, which is needed to treat concrete examples. The second one presents a generalisation of section 4 which focuses only on fuzzy spaces homogeneous concerning their
entanglement properties. In the second appendix we relax this property. The third
appendix presents some computations associated with the Lorentz connection of a
fuzzy space, which are not necessary to the understanding of the main discussion of
this paper. And the last one presents the relation between the fuzzy geometry and
the category theory (the presence of two quantum metrics being related to the need of
defining a metric for the objects and another one for the arrows - morphisms - onto a
categorical manifold). The fuzzy space geometry is often illustrated in the literature
by the highly symmetric examples of the fuzzy sphere, the fuzzy plane, and the fuzzy
complex projective spaces. Throughout this paper, we illustrate the present results by
several examples, and especially the case of fuzzy surface plots (fuzzyfication of classi-
cal surfaces defined by a cartesian equation $z = f(x, y)$). The examples are splitted in
the whole of the paper in order to illustrate a notion immediately after its introduction.

Some notations are used throughout this paper. We adopt the Einstein notations concerning the repetition of an index at lower and upper positions which is equivalent to a summation. For a tensor $T$, we denote the symmetrisation and the anti-symmetrisation of two indices by $T_{...(a...b)...} \equiv T_{...a...b...} + T_{...b...a...}$ and $T_{...[a...b]...} \equiv T_{...a...b...} - T_{...b...a...}$. In a tensor, latin indices run from 1, whereas greek indices run from 0. For a linear set of operators (or matrices) $\mathcal{X}$, $\text{Env}(\mathcal{X})$ denotes its enveloping $C^*$-algebra and $\text{Der}(\mathcal{X})$ its linear space of inner derivatives (Lie derivatives $-i[X, \bullet]$). For a set of vectors $V$ in a vector space $E$ over $\mathbb{K}$, $\text{Lin}_{\mathbb{K}}(V)$ denotes the vector subspace of $E$ generated by $V$. For a manifold $M$, $T_x M$ denotes the set of tangent vectors at $x$, $\Omega^n M$ the set of differential $n$-forms and $\text{Diff} M$ its set of diffeomorphisms. $\simeq$ between two manifolds stands for “diffeomorphic to”. $E_M$ denotes the $E$-valued differentiable functions on $M$. For a category $\mathcal{C}$, $\text{Obj}\mathcal{C}$ denotes its set of objects, $\text{Morph}\mathcal{C}$ denotes its set of arrows, $s$, $t$ and $\text{id}$ denote their source, target and identity maps; the composition of arrows being denoted by $\circ$.

For the applications, we use the unit system such that $\hbar = c = G = 1$ ($\ell_P = m_P = t_P = 1$ Planck units) or $\hbar = \frac{c^2}{4\pi\epsilon_0} = 1$ (atomic units).

## 2. Fuzzy spaces

In this section we present the general concept of fuzzy spaces [1] and some related
notions about noncommutative geometry [2]. We present no new result, but this
section permits to fix some notations and some definitions for the sequel of this paper.

**Definition 1 (Fuzzy space)** $A$ (3D) fuzzy space is a noncommutative manifold defined by a spectral triple $\mathfrak{M} = (\text{Env}(\mathcal{X}), \mathbb{C}^2 \otimes \mathcal{H}, \mathcal{D}_x)$ where

• $\mathcal{H}$ is a separable Hilbert space.

• $\mathcal{X} = \text{Lin}_{\mathbb{R}}(X^1, X^2, X^3, \text{id})$ is a space generated by three self-adjoint linear operators $X^i$ of the Hilbert space $\mathcal{H}$ (and the identity operator on $\mathcal{H}$) and $\text{Env}(\mathcal{X})$ is the $C^*$-enveloping algebra of $\mathcal{X}$.

• $\mathcal{D}_x = \sigma_i \otimes (X^i - x^i)$ is the Dirac operator of the noncommutative manifold where $(\sigma_i)$ are the Pauli matrices and $x \in \mathbb{R}^3$ is a classical parameter.

Note that the case where one $X^i$ be the zero operator is not excluded. It can be useful
to introduce the non-self-adjoint coordinate observable $Z = X^1 + iX^2$ associated with
the complex parameter $z = x^1 + i x^2$ to write $\mathcal{D}_x = \begin{pmatrix} X^3 - x^3 & Z^\dagger - \bar{z} \\ Z - z & -X^3 + x^3 \end{pmatrix}$.

As usual in noncommutative geometry, the non-abelian $C^*$-algebra $\mathrm{Env}(\mathcal{X})$ plays the role of the space of functions of $\mathfrak{M}: \mathcal{C}_{\mathrm{n.c.}}^{\infty}(\mathfrak{M}) \sim \mathrm{Env}(\mathcal{X})$, and $\mathbb{C}^2 \otimes \mathcal{H}$ plays the role of a spinor field space on $\mathfrak{M}$. Let $\mathcal{Z}(\mathrm{Env}(\mathcal{X}))$ be the center of $\mathrm{Env}(\mathcal{X})$. The space of derivatives $\mathrm{Der}(\mathcal{X}) = \mathrm{Lin}_{\mathcal{Z}(\mathrm{Env}(\mathcal{X}))}(L_{X^1}, L_{X^2}, L_{X^3})$ (with $L_X(Y) = -i[X, Y]$, $\forall X \in \mathcal{X}, Y \in \mathrm{Env}(\mathcal{X})$) plays the role of the space of tangent vector fields on $\mathfrak{M}: T_{\mathrm{n.c.}}\mathfrak{M} \sim \mathrm{Der}(\mathcal{X})$. The set of noncommutative differential $n$-forms $\Omega_{\mathrm{Der}}^n(\mathcal{X})$ is the set of $\mathcal{Z}(\mathrm{Env}(\mathcal{X}))$-multilinear antisymmetric maps from $\mathrm{Der}(\mathcal{X})^n$ to $\mathrm{Env}(\mathcal{X})$.

The differential $\mathbf{d}: \Omega_{\mathrm{Der}}^n(\mathcal{X}) \to \Omega_{\mathrm{Der}}^{n+1}(\mathcal{X})$ is defined by the Koszul formula:

$$ \mathbf{d}\xi(L_1, \dots, L_{n+1}) = \sum_{i=1}^{n+1} (-1)^{i+1} L_i\xi(L_1, \dots, \tilde{L}_i, \dots, L_{n+1}) \\ + \sum_{1 \le i < j \le n+1} (-1)^{i+j} \xi([L_i, L_j], L_1, \dots, \tilde{L}_i, \dots, \tilde{L}_j, \dots, L_{n+1}) \quad (1) $$

where $\tilde{L}_i$ means “deprive of $L_i$”.

In contrast with a commutative manifold, the duality relation between the derivative with respect to $X^i$ and the differential of $X^j$ fails:

$$ \langle \mathbf{d}X^i, L_{X^j} \rangle_X = -i[X^j, X^i] \neq \delta^{ij} \quad (2) $$

where $\langle \bullet, \bullet \rangle_X$ stands for the duality bracket between $\Omega_{\mathrm{Der}}^1(\mathcal{X})$ and $\mathrm{Der}(\mathcal{X})$. $\Theta^{ij} = -i[X^i, X^j]$ defines a Poisson structure on $\mathfrak{M}$ with the following Poisson bracket $\Theta: \mathfrak{X}^2 \to \mathrm{Env}(\mathcal{X})$: $\Theta(f_i X^i, g_j X^j) = -i[f_i X^i, g_j X^j] = \Theta^{ij} f_i g_j$ (with $f_i, g_j \in \mathcal{Z}(\mathrm{Env}(\mathcal{X}))$ or often in $\mathbb{R}$). Let $\theta_i \in \Omega_{\mathrm{Der}}^1(\mathcal{X})$ be the 1-forms defined by duality with $L_{X^j}$:

$$ \langle \theta_i, L_{X^j} \rangle_X = \delta_i^j \quad (3) $$

By construction we have $\mathbf{d}X^i = \Theta^{ji}\theta_j$.

**Definition 2 (States of a fuzzy space)** We call normal states of a fuzzy space $\mathfrak{M}$ the normal states [19] of the $C^*$-algebra $\mathrm{Env}(\mathcal{X})$:

$$ \mathcal{E}(\mathfrak{M}) = \{ \omega : \mathrm{Env}(\mathcal{X}) \to \mathbb{R}, \exists \rho \in \mathcal{L}(\mathcal{H}), \rho^\dagger = \rho, \rho \ge 0, \mathrm{tr} \, \rho = 1, \text{ with } \omega(X) = \mathrm{tr}(\rho X) \} $$

We call pure states of $\mathfrak{M}$ the pure states [19] of the $C^*$-algebra $\mathcal{L}(\mathbb{C}^2) \otimes \mathcal{B}(\mathcal{H})$:

$$ \mathcal{P}(\mathfrak{M}) = \{ \omega : \mathrm{Env}(\mathcal{X}) \to \mathbb{R}, \exists P \in \mathcal{L}(\mathbb{C}^2 \otimes \mathcal{H}), P^\dagger = P, \mathrm{tr} \, P = 1, P^2 = P \text{ with } \omega(X) = \mathrm{tr}(PX) \} $$

The density matrices $\rho$ play the role of the density functions (or distributions) on $\mathfrak{M}$ and the states play the role of the integration on $\mathfrak{M}$:

$$ \int_{\mathfrak{M}}^{\mathrm{n.c.}} Y \rho \mathbf{d}X^1 \mathbf{d}X^2 \mathbf{d}X^3 \sim \mathrm{tr}(\rho Y) \quad \forall Y \in \mathrm{Env}(\mathcal{X}) \quad (4) $$

The choice to define pure and normal states with respect to two different $C^*$-algebras can be surprising. But we want to consider pure states of the bipartite system (spin+coordinate degrees of freedom). The normal states, the density matrices, onto the coordinate degrees of freedom must result from partial trace of pure states onto the spin degree of freedom. And so the character pure or mixed of the density matrices is related to the character separable or entangled of the pure states of the bipartite system.
The pure states replace the notion of points on $\mathfrak{M}$, in the meaning that $\omega_P(Y) = \text{tr}(PY)$ is equivalent to $P(f) = f(P)$ for $P$ a point in a commutative manifold and $f$ a function onto this manifold; a point $P$ on a commutative manifold defining a Dirac distribution density centred on itself. We do not have a notion of points on a noncommutative manifold, since two different pure states are not necessary separated $P_1P_2 \neq 0$ even if $P_1 \neq P_2$. Note that we define the pure states of $\mathfrak{M}$ by projectors of $\mathbb{C}^2 \otimes \mathcal{H}$ and not of $\mathcal{H}$. We have then $\omega_P(X) = \text{tr}(PX) = \text{tr}(\rho_P X)$ where $\rho_P = \text{tr}_{\mathbb{C}^2} P$ (with $\text{tr}_{\mathbb{C}^2}$ the partial trace onto $\mathbb{C}^2$, note that in $\text{tr}(PX)$ tr denotes the trace onto $\mathbb{C}^2 \otimes \mathcal{H}$ whereas in $\text{tr}(\rho_P X)$ it denotes the trace onto $\mathcal{H}$). So a pure state of $\mathfrak{M}$ defines a normal state $\text{tr}(\rho_P \bullet)$ (which is not pure for the $C^*$-algebra $\text{Env}(\mathcal{X}))$. $P$ defines also the (mean value) vector $\vec{n}(P) = (\text{tr}(P\sigma_1), \text{tr}(P\sigma_2), \text{tr}(P\sigma_3)) \in \mathbb{R}^3$. At this stage we can think $\vec{n}(P)$ as the mean local orientation of $\mathfrak{M}$ at the pure state $\omega_P = \text{tr}(P\bullet)$. This is the possible quantum entanglement between the local orientation degree of freedom $(\mathbb{C}^2)$ and the coordinate degrees of freedom $(\mathcal{H})$ which needs to define pure and normal states of $\mathfrak{M}$ with respect to two different $C^*$-algebras.

It is important not to confuse with the notion of fuzzy manifolds [20, 21] which is related to the notion of fuzzy sets which are “sets” “containing” elements without certainty (each element has a classical probability to belong to the set). The two notions are close but fuzzy manifolds deal with classical probabilities whereas fuzzy spaces deal with quantum probabilities (noncommutative probabilities or free probabilities in the language of the mathematicians). To be more precise and avoid any confusion, it could be more convenient to call topological fuzzy manifolds the case associated with classical probabilities and quantum fuzzy manifolds or noncommutative fuzzy manifolds the case associated with quantum probabilities. But in the whole of this paper, the term fuzzy spaces refers only to this last case.

The operators $(X^i)$ play the role of noncommutative coordinate observables on $\mathfrak{M}$. In other words, for $\omega \in \mathcal{P}(\mathfrak{M})$, $\omega(X^i)$ is the mean value of the “i-th coordinate” of the pure state $\omega$ on $\mathfrak{M}$, and $\Delta_\omega X^i = \sqrt{\omega(X^i)^2 - \omega(X^i)^2}$ is the quantum uncertainty onto the $i$-th coordinate of $\omega$ on $\mathfrak{M}$. Due to the noncommutativity of the coordinate observable we have Heisenberg uncertainty relations: $\forall \omega \in \mathcal{P}(\mathfrak{M})$

$$ \Delta_{\omega} X^i \Delta_{\omega} X^j \geq \frac{1}{2} |\omega([X^i, X^j])| \qquad (5) $$

$\mathfrak{M}$ is said “fuzzy” since the coordinates of its pure states are “delocalised” as a quantum particle in the usual space. This is the existence of the quantum coordinate observables which is the main particularity of a fuzzy space with respect to any non-commutative manifold.

The choice of $\mathcal{D}_x$ is made to correspond to the string theory BFSS matrix model (the spacetime being reduced to 3+1 dimensions by truncation with a supersymmetric orbifold [7]). In this one, $\mathfrak{M}$ is a noncommutative D2-brane and the classical parameter $x$ is the classical coordinates of a probe D0-brane; a fermionic string (of spin state space $\mathbb{C}^2$) links the D2-brane and the probe D0-brane. $\mathcal{D}_x$ is the displacement energy observable (the “tension energy” of the fermionic string). If the D0-brane is “far away” from the D2-brane the tension energy of the fermionic string increases. From the point of view of the fuzzy geometry, $\mathcal{D}_x$ represents the minimal coupling between the spin (local orientation) observables ($\sigma_i$) and the coordinate observables
(X^i), responsible of the quantum entanglement. x ∈ ℝ³ is the position of the probe of the classical observer (this one has ℝ³ as the physical space in her mind). This probe is needed to define the observation of the fuzzy space by the observer (the observer realises experiments “somewhere” in the classical space of her mind). As in usual quantum mechanics, we cannot completely separate the “properties” of the quantum system from their observations. Here this does not imply that the Hilbert space is L²(ℝ³, dx) (not a quantum particle) but this implies the presence of x in D̃x (the geometry observable of X). In contrast with usual quantum mechanics where the quantum system and the observer share a common spacetime background, here the quantum system (the quantum spacetime itself) and the classical observer have not a common background. The geometry observable D̃x (which is related both to the quantum system and to the observer) depends on the space observables (X^i) but also on the classical space generated by (x^i) in which the observer places the measurement outcomes. The space generated by (x^i) is not physical for the point of view of M but is a parameter space (in the mind of the observer) permitting for example to define the change of the place where the measures are performed (by varying the values of x^i). The spacetime ℝ³ × ℝ permits to the observer to place the events corresponding to the measurement outcomes, permitting to easily endow this set of events with a causal structure [22]. ℝ³ × ℝ is not physical (it is the spacetime in the observer’s mind), but the causal structure is. We can also understand this by comparison with commutative geometry in the context of general relativity. To observe the geometry of the spacetime, an observer needs to measure the geodesic moves of a test particle. In a same way, to observe the fuzzy geometry of M, an observer needs to make measurement on a test classical particle (of position x in the classical space ℝ³ of the measurement outcomes of position).

The choice of $\mathcal{D}_x$ corresponds also to a problem of quantum information theory consisting to study a qubit (assimilated to a 1/2-spin system) manipulated by a magnetic field $\vec{B}$ and in contact with an environment inducing a decoherence phenomenon [23]. In the interaction picture, the Hamiltonian of the qubit is $\mathcal{D}_x$ where $x^i = \frac{\mu}{2}B^i$ are the parameters of the magnetic field controlling the qubit to realise single-qubit logical gates ($\mu$ is the magnetic moment magnitude), and $X^i = U_\varepsilon^\dagger(t)V^iU_\varepsilon(t)$ where $U_\varepsilon(t)$ is the evolution operator of the environment and $V = \sigma_i \otimes V^i$ is the interaction operator between the qubit and its environment. $X^i$ are then the environmental noise observables. In a first step, we treat the problem with time-independent observables $X^i$. The time-dependent case will be treated section 6, where we will see (property 13) that for the quantum information models, the time-dependent behaviour is deduced from the stationary case.

Some authors prefer to consider fuzzy spaces with $\Box_x = \delta_{ij}(X^i - x^i)(X^j - x^j)$ a noncommutative equivalent of a Laplace-Beltrami operator or of a D'Alembertian operator as fundamental geometry observable in place of $\mathcal{D}_x$ (noncommutative equivalent of a Dirac operator), see for example [9]. We have $\mathcal{D}_x^2 = \text{id} \otimes \Box_x + \frac{4}{3}\varepsilon_{ij}{}^k\sigma_k \otimes [X^i, X^j]$, and so $\Box_x = \text{tr}_{\mathbb{C}^2}\mathcal{D}_x^2$. It is clear that the geometric informations encoded in $\Box_x$ are also in $\mathcal{D}_x$, in contrast $\mathcal{D}_x$ includes informations concerning the local orientation which are lost with $\Box_x$ (these ones are clearly erased by the partial trace onto $\mathbb{C}^2$). In particular, possible entanglements between states of local orientation and states of location of $\mathfrak{M}$ is a priori ignored if we consider firstly $\Box_x$ in place of $\mathcal{D}_x$. The choice
of $\mathcal{D}_x$ as fundamental geometry observable is then more general.

### Example 1: Fuzzy sphere

$\mathcal{X}$ is generated by $X^i = rJ^i$ with $r \in \mathbb{R}^{+*}$ and $[J^i, J^j] = i\epsilon^{ij}J^k$ (Lie$(X^1, X^2, X^2)$ is the $\mathfrak{su}(2)$ algebra), $\mathcal{H} = \mathcal{H}_j$ is the space of the unitary irreducible representation of $\mathfrak{su}(2)$ of dimension $2j + 1$ ($j \in \frac{1}{2}\mathbb{N}$). This Fuzzy space is called Fuzzy sphere since the coordinate observables satisfy the following equation

$$ (X^1)^2 + (X^2)^2 + (X^3)^2 = r^2 j(j+1)\mathrm{id} \qquad (6) $$

mimicking the equation of a classical sphere of radius $r\sqrt{j(j+1)}$. In quantum gravity matrix model, the thermalisation of a Fuzzy sphere has been proposed as a model of event horizon of a quantum black hole [24, 25]. In quantum information theory, consider a qubit (assimilated to a $\frac{1}{2}$-spin system $\vec{S}$) controlled by a magnetic field $\vec{B}$ and interacting with $N$ other qubits $(\vec{S}_a)_{a=1,...,N}$ forming its environment (the quantum computer) [23]. The interaction Hamiltonian of the qubit is then

$$ \mathcal{D}_x = -\mu \vec{S} \cdot \vec{B} + \mathcal{J} \vec{S} \cdot \sum_{a=1}^{N} \vec{S}_a \qquad (7) $$

$$ = \sigma_i \otimes (X^i - x^i) \qquad (8) $$

($\mu$ is the magnetic moment magnitude of the spin system and $\mathcal{J}$ is the exchange integral) with $\vec{x} = \frac{\mu}{2}\vec{B}$ and

$$ X^i = \frac{\mathcal{J}}{4} \bigoplus_{j=0 \text{ or } 1/2}^{N/2} J_{(j)}^i \qquad (9) $$

where $(J_{(j)}^i)_i$ are the $(2j + 1)$ dimensional unitary irreducible representations of the $\mathfrak{su}(2)$ generators onto $\mathcal{H}_j$ ($\mathcal{H} = \bigoplus_{j=0 \text{ or } 1/2}^{N/2} \mathcal{H}_j$). The system can be viewed as $[N/2]$ concentric Fuzzy spheres of radii $\frac{\mathcal{J}}{4}\sqrt{j(j+1)}$.

### Example 2: Fuzzy surface plots

**Fuzzy plane:** Let $(a, a^+, \mathrm{id})$ be the generator of the CCR algebra $[a, a^+] = \mathrm{id}$ (annihilation and creation operators) onto the Fock space $\mathcal{F}$ (of canonical basis ($|n\rangle\rangle_{n \in \mathbb{N}}$). The Fuzzy space defined by $\mathcal{H} = \mathfrak{F}, Z = a$ and $X^3 = 0$ is a Fuzzy plane. In quantum gravity matrix model, it is the fundamental example of flat spacetime slice [26]. The geometry observable can be written in the canonical basis of $\mathbb{C}^2$ as

$$ \mathcal{D}_{\alpha} = \begin{pmatrix} -x^3 & a^+ - \bar{\alpha} \\ a - \alpha & x^3 \end{pmatrix} \qquad (10) $$

with $\alpha = x^1 + ix^2$. This operator can be viewed as the Hamiltonian of a qubit considered as an atomic two-level system controlled by a strong classical electric field $\vec{E}$ in the rotating wave approximation [27] ($\alpha = \frac{1}{2\Omega}\vec{\mu} \cdot \vec{\mathcal{E}}$ where $\vec{\mu}$ is the atomic dipolar moment and $\vec{\mathcal{E}}$ is the complex representation of the electric field, $\Omega x^3$ being the detuning between the atomic transition energy gap and the field frequency). The atom is also in interaction with a reservoir of single-mode bosons (described by $a$ and $a^+$) constituting its environment ($\Omega$ is the coupling strength between the atom and the bosons). These bosons can be for example photons of another electromagnetic field or phonons of a solid system in which the atom is included.

Since $a$ and $a^+$ are unbounded operators, $\mathcal{X}$ is not included in its $C^*$-enveloping algebra $\mathrm{Env}(\mathcal{X})$ which is generated by the elements of the form $\mathcal{D}(\alpha) = e^{\alpha a^+ - \bar{\alpha}a}$ (for $\alpha \in \mathbb{C}$). But if (by definition) $\mathrm{Env}(\mathcal{X})$ is closed for the norm topology, it is not for the strong topology. By the Stone theorem we have: $\lim_{s \to 0} \frac{1}{s}(e^{s(\alpha a^+ - \bar{\alpha}a)} - \mathrm{id})\psi = (\alpha a^+ - \bar{\alpha}a)\psi$ ($\forall \psi \in \mathfrak{F}$), and so $\mathcal{X}$ is included into the topological closure of $\mathrm{Env}(\mathcal{X})$ for the strong topology.
The fuzzy plane can be used to define the “fuzzyfication” of surface plots $x^3 = f(\alpha, \bar{\alpha})$ for $f$ a polynomial function or depending only on $|\alpha|$.

**Fuzzy elliptic paraboloid:** $Z = a$ and $X^3 = \epsilon(a^+a + \frac{1}{2}id)$ (with $\epsilon \in \mathbb{R}^*$). The coordinate observables satisfy $X^3 = \epsilon((X^1)^2 + (X^2)^2)$ mimicking the equation of a classical paraboloid. For the physical point of view, $\mathcal{D}_x$ is an effective Hamiltonian [27] where $X^3$ adds a perturbative nonlinear interaction of the two-level atom with its environment (boson scattering by the two-level atom).

**Fuzzy hyperbolic paraboloid:** $Z = a$ and $X^3 = \frac{\epsilon}{2}(a^2 + (a^+)^2)$, the coordinate observables satisfying $X^3 = \epsilon((X^1)^2 - (X^2)^2)$, another possible perturbative nonlinear interaction (absorption and emission of two bosons by the two-level atom).

**Fuzzy hyperboloid:** $Z = a$ and $X^3 = \epsilon\sqrt{(r^2 + 1/2)id + a^+a} = \epsilon\sum_{n=0}^{+\infty}\sqrt{r^2 + 1/2 + n|n\rangle\langle n|}$ with $r \in \mathbb{R}^*$. The coordinate observables satisfy then $(X^3)^2 - \epsilon^2(X^1)^2 - \epsilon^2(X^2)^2 = \epsilon^2 r^2$. Note that another way to define a Fuzzy hyperboloid consists to set $X^i = rK^i$ where $K^i$ are unitary irreducible representations of the generators of the su(1, 1) algebra, obtains a Fuzzy space of the same family than the Fuzzy sphere.

**Fuzzy Flamm's paraboloid:** $Z = a$ and $X^3 = 2\sqrt{r_S}\sqrt{a^+a - r_S} = 2\sqrt{r_S}\sum_{n=0}^{+\infty}\sqrt{\sqrt{n} - r_S|n\rangle\langle n|}$ with $r_S \in \mathbb{R}^*$. This system can be viewed in quantum gravity matrix model as a Fuzzy slice of a Schwarzschild spacetime. It has been numerically studied in a previous work as a toy model of quantum black hole [28].

$\mathbb{C}^2 \otimes \mathcal{H}$ can be viewed as a right $\mathcal{L}(\mathbb{C}^2)-C^*$-module [8] with the following inner product:

$$ \forall \Psi, \Phi \in \mathbb{C}^2 \otimes \mathcal{H}, \quad \langle \Psi | \Phi \rangle_* = \mathrm{tr}_{\mathcal{H}} |\Phi\rangle \langle \Psi| \in \mathcal{L}(\mathbb{C}^2) \quad (11) $$

where $\langle\langle\bullet|\bullet\rangle\rangle$ stands for the inner product induced by the tensor product of $\mathbb{C}^2 \otimes \mathcal{H}$ viewed as a Hilbert space. By construction the square $C^*$-norm of $\Psi \in \mathbb{C}^2 \otimes \mathcal{H}$, $\langle\Psi|\Psi\rangle_* = \mathrm{tr}_{\mathcal{H}}|\Psi\rangle\langle\Psi|$ is a density matrix onto $\mathbb{C}^2$. The reason of the consideration of this structure is the following. The choice of the local orientation of $\mathfrak{M}$ is arbitrary. It is a gauge choice. Let $u \in SU(2) \subset \mathcal{L}(\mathbb{C}^2)$ be a change of local orientation of $\mathfrak{M}$. Let $\omega_P \in \mathcal{E}(\mathfrak{M})$. The state $\omega_{uP_{u-1}}$ has then the same physical meaning than the state $\omega_P$ (in the same way than in commutative geometry, the change of the local orientation at the neighbourhood of a point on a classical manifold does not change the point). So $u$ for the fuzzy space is equivalent for a quantum particle to a phase change. If the phases for the states of the fuzzy space are elements of $U(2)$, $\mathbb{C}$ must be replaced by $\mathcal{L}(\mathbb{C}^2)$ as algebra of “scalers”. Note that $\langle\bullet|\bullet\rangle_*$ is not invariant under $SU(2)$-phase changes but is equivariant:

$$ \langle u\Psi|u\Phi\rangle_* = u\langle\Psi|\Phi\rangle_*u^{-1} \quad (12) $$

(in a commutative algebra as $\mathbb{C}$, equivariance and invariance are the same thing). For the BFFS matrix model, $u \in SU(2)$ is a rotation of the fermionic string spin. For the point of view of the quantum information model, the $C^*$-norm of a state is the density matrix of the qubit under the effects of the environment and the action of $u$, $\|u\Psi\|_*^2 = u\|\Psi\|_*^2 u^{-1}$ is a (local) unitary operation onto the qubit. These density matrices define then probability laws onto $\mathbb{C}^2$:

**Definition 3 (Almost surely properties)** Let $\rho = \|\Psi\|_*^2$ be a density matrix of $\mathbb{C}^2$. Let $p_F$ be a property satisfied by an observable $A \in \mathcal{L}(\mathbb{C}^2)$ if $F(A) = 0$ for $F: \mathcal{L}(\mathbb{C}^2) \to \mathcal{L}(\mathbb{C}^2)$ (not necessarily linear). We say that A satisfies $p_F$ almost surely
with respect to $\rho$ ($\rho$-a.s.), if $\mathrm{tr}(\rho F(A)) = 0$. If $\mathcal{F} = \{A \in \mathcal{L}(\mathbb{C}^2), \text{ such that } F(A) = 0\}$ we denote by $\mathcal{F}_{a.s.} = \{A \in \mathcal{L}(\mathbb{C}^2), \text{ such that } \mathrm{tr}(\rho F(A)) = 0\}$ the set of observables satisfying $p_F$ almost surely ($\mathcal{F} \subset \mathcal{F}_{a.s.}$).

For example, with $F(A) = A^\dagger - A$, $A$ is said almost surely self-adjoint if $\mathrm{tr}(\rho(A^\dagger - A)) = 0$.

The dynamics of $\Psi \in \mathbb{C}^2 \otimes \mathcal{H}$ is governed by the Schrödinger-like equation:

$$i\dot{\Psi} = \mathcal{D}_{x(t)}\Psi \qquad (13)$$

Note that $x$ can be time dependent if the observer moves its probe. This one is the Dirac equation of a massless fermionic string in the BFSS model in Weyl representation [7]. The inner dynamics of $\mathfrak{M}$ is then submitted to the following evolution of their pure states:

$$\omega_P \circ \mathrm{Ad}_{U_{\mathcal{D}_x}(t)} = \omega_{U_{\mathcal{D}_x}(t)PU_{\mathcal{D}_x}^{-1}(t)} \qquad (14)$$

(with $\mathrm{Ad}_u(X) = u^{-1}Xu$) where $U_{\mathcal{D}_x}(t) = \mathbb{T}e^{-i \int_0^t \mathcal{D}_{x(t)} dt} \in \mathcal{U}(\mathbb{C}^2 \otimes \mathcal{H})$ is the evolution operator ($\mathbb{T}e$ stands for the time ordered exponential - the Dyson series -, i.e. $i\partial_t U_{\mathcal{D}_x} = \mathcal{D}_{x(t)}U_{\mathcal{D}_x}$ with $U_{\mathcal{D}_x}(0) = \mathrm{id}$). Even if we consider a state without entanglement between the local orientation and the coordinate states, $U_{\mathcal{D}_x}(t)$ being not separable (in general) it induces the growing of the entanglement with the time. We are interested by the adiabatic regimes for this equation, because they are similar to semi-classical approximations for the dynamics (for the point of view of the evolution operator $\mathbb{T}e^{-i\frac{\hbar}{T}\int_0^t \mathcal{D}_{x(s)} ds}$ - by restoring the writing of the fundamental constants - the semi-classical limit $\hbar \to 0$ and the adiabatic limit $T \to +\infty$ are equivalent, with $s = t/T$, $T$ being the total duration of the dynamics). The adiabatic regimes are then associated with the situations closest to classical dynamics. Suppose that $|\Lambda(x)\rangle\rangle$ is an instantaneous eigenvector of $\mathcal{D}_x$, the dynamics is said:

* strongly adiabatic if [29]

$$|\Psi(t)\rangle\rangle \approx e^{i\varphi(t)}|\Lambda(x(t))\rangle\rangle, \qquad e^{i\varphi(t)} \in U(1) \qquad (15)$$

* weakly adiabatic if [23]

$$|\Psi(t)\rangle\rangle \approx u(t)|\Lambda(x(t))\rangle\rangle, \qquad u(t) \in U(2)_{a.s.} \qquad (16)$$

$(u$ is almost surely unitary with respect to $\rho_\Lambda = \mathrm{tr}_\mathcal{H} |\Lambda\rangle\rangle\langle\Lambda|)$. So, the regime is strongly adiabatic if it is adiabatic for whole bipartite system (the state remains on the instantaneous eigenvector up to a phase change), and weakly adiabatic if it is adiabatic for the subsystem described by $(X^i)$ but not for the one described by $(\sigma^i)$ (the state remains on the instantaneous eigenvector up to a local orientation change). For the point of view of quantum information theory, in the strong adiabatic cases, the control $t \mapsto x(t)$ onto the qubit $(\sigma_i)$ is sufficiently slow not to induce transition of its instantaneous state nor of the environment state. But in weak adiabatic cases, the control induces transition of the instantaneous qubit state, but the environment $(X^i)$ is sufficiently large to the evolution of the qubit does not induce transition of its instantaneous state via the couplings $\sigma_i \otimes X^i$. For the point of view of the quantum gravity, the strong adiabatic regime corresponds to move the probe sufficiently slowly not to induce transition of the instantaneous quantum spacetime state. In the weak adiabatic regime, the local orientation defined by the probe spin $(\sigma_i)$ is rotated by gravitational effects, but the move is sufficiently slow not to deform spacetime by transition of the instantaneous location state (associated with $(X^i)$).
### 3. The eigen geometry

#### 3.1. The eigenmanifold and the quasicoherent states

M is a quantum object, but the outcomes of the measurements are classical quantities. These outcomes are eigenvalues of pertinent observables. We can think that the spectral properties of the geometry observable $\mathcal{D}_x$ define a classical manifold close to M.

**Definition 4 (Eigenmanifold)** We call eigenmanifold $M_\Lambda$ of a fuzzy space M the subset of $\mathbb{R}^3$ defined by

$$M_{\Lambda} = \{x \in \mathbb{R}^3, \ker(\mathcal{D}_x) \neq \{0\}\}$$

Note that $\dim M_\Lambda \le 2$ and $M_\Lambda$ is not necessary connected.

**Definition 5 (Quasicoherent state)** Let $x \in M_\Lambda$ a point on the eigenmanifold of a fuzzy space M. We call quasicoherent state of M at $x$ a normalised vector $|\Lambda(x)\rangle\rangle \in \mathbb{C}^2 \otimes \mathcal{H}$ such that

$$\mathcal{D}_x |\Lambda(x)\rangle\rangle = 0 \qquad (17)$$

The reason of these definitions is the following. Let $\omega_x = \text{tr}(|\Lambda(x)\rangle\rangle\langle\Lambda(x)|\cdot) \in \mathcal{P}(M)$ be the pure state associated with $|\Lambda(x)\rangle\rangle$. We have $\omega_x(X^i) = x^i$ (because $\frac{1}{2}\{\sigma^i, \mathcal{D}_x\} = X^i - x^i$) and we can prove that $\omega_x$ minimises the Heisenberg uncertainty relation $\Delta_{\omega_x} \vec{X}^2 = \omega_x(\vec{X}^2) - \omega_x(\vec{X})^2$ [9] as the Perelomov coherent states of an usual quantum system [11]. Coherent states being the quantum states closest to classical ones (since the quantum delocalisation is minimised), the quasicoherent states $\omega_x$ are the pure states of M closest to classical pure states of a classical (commutative) manifold. So, $\omega_x$ are the pure states of M closest to the classical notion of points. We can then think $\omega_x$ as a “quantum point” of M, which is naturally labelled by a point $x$ of the classical space of the observer. Nevertheless, in general $\langle\Lambda(y)|\Lambda(x)\rangle\rangle \neq \delta(x - y)$, the pseudo-points $\omega_x$ and $\omega_y$ are not separated (if the system is in the state $|\Lambda(x)\rangle\rangle$) the probability that the outcome of the location measured by the observer be $y$ is not zero: $|\omega_x(|\Lambda(y)\rangle\rangle\langle\Lambda(y)|)|^2 = |\langle\Lambda(y)|\Lambda(x)\rangle\rangle|^2$. From the viewpoint of string theory, $|\Lambda(x)\rangle\rangle$ being the eigenvector of $\mathcal{D}_x$ associated with the zero eigenvalue, it is the state which minimises the displacement energy (the “tension energy” of the fermionic string). The fermionic string having no tension, its two ends are “at the same place”. So the probe D0-brane is closest as possible to the D2-brane. These three arguments (minimal quantum delocalisation, pure states closest to points, minimal “distance” between the probe and the noncommutative manifold) show that the eigenmanifold $M_\Lambda$ is the classical (commutative) manifold closest to M (the geometry defined by $M_\Lambda$ is the classical geometry which looks the most the quantum geometry of the fuzzy space). $M_\Lambda$ is the slice of space which is the more representative of the geometry of M.

By construction, in quantum information theory, restricting $x$ to be on $M_\Lambda$ consists to adiabatically control the qubit by keeping constant at 0 its energy dressed by the environment with zero energy uncertainty. If we denote by $H_{int}$ any interaction Hamiltonian, $\mathcal{D}_x = H_{int} - \text{tr}_{\mathbb{C}^2} H_{int}$ is the part which induces the state transitions and which is leave invariant by changes of the potential energy origin. The eigenstate of zero eigenvalue of $\mathcal{D}_x$ is then the state minimising the energy exchanges between the qubit and the environment. Moreover, in contrast with the other eigenstates of $\mathcal{D}_x$,
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Eigenmanifold $M_\Lambda : \zeta \mapsto \langle\Lambda(\zeta)|\vec{X}|\Lambda(\zeta)\rangle$ (yellow surface) for the Fuzzy sphere (with $j = 2$) with the quantum uncertainty clouds $\frac{1}{2}\Delta X^i = \frac{1}{2}\sqrt{\langle\Lambda|(X^i)^2|\Lambda\rangle - (\langle\Lambda|X^i|\Lambda\rangle)^2}$ (blue ellipsoids) of some points, in the target space $\mathbb{R}^3$ of the probe.</figcaption></figure>

the quasicoherent state minimises the quantum uncertainties $\Delta X^i$ and so minimises the noises coming from the environment and perturbing the qubit (and responsible of the decoherence of qubit mixed state $\rho$). Adiabatically controlling the qubit on $M_\Lambda$ permits then to minimise the effects of the environment noises onto the qubit.

**Example 1: Fuzzy sphere**

The eigenmanifold of a Fuzzy sphere is the classical sphere $M_\Lambda = \{\vec{x} \in \mathbb{R}^3, \|\vec{x}\| = rj\}$ and its quasicoherent state is [12]

$$|\Lambda(x)\rangle\rangle = |\zeta\rangle_{1/2} \otimes |\zeta\rangle_j \qquad (18)$$

with $\zeta = e^{i\varphi} \tan \frac{\theta}{2}$ for $x = rj(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)$. $|\zeta\rangle_j$ are the Perelomov SU(2) coherent states [11] for the unitary irreducible representation of dimension $2j + 1$. The eigenmanifold is plotted fig.1.

We see with this example that the quasicoherent states of the Fuzzy spaces are strongly related to Perelomov coherent states of Lie algebras [11]. This is confirmed by the following other examples.

**Example 2: Fuzzy surface plots**

**Fuzzy plane:** The eigenmanifold of the Fuzzy plane is just the complex plane $M_\Lambda = \{(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha), 0)\}_{\alpha \in \mathbb{C}}$ with the quasicoherent state in the canonical basis of $\mathbb{C}^2$ [12]:

$$|\Lambda(\alpha)\rangle\rangle = \begin{pmatrix} |\alpha\rangle \\ 0 \end{pmatrix} \qquad (19)$$

where $|\alpha\rangle$ is the Perelomov coherent state of the quantum harmonic oscillator algebra. The quantum uncertainties are $\Delta X^1 = \Delta X^2 = 1/2$ ($\Delta X^3 = 0$). To treat the other examples of fuzzy surface plots, it is interesting to consider all eigenvectors of $\mathcal{D}_\alpha$ [12, 28]:

$$\mathcal{D}_\alpha |\lambda_{n\pm}(\alpha)\rangle\rangle = \pm \sqrt{n} |\lambda_{n\pm}(\alpha)\rangle\rangle \qquad (20)$$
with

$$|\lambda_{n\pm}(\alpha)\rangle\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} |n\rangle_\alpha \\ \pm|n-1\rangle_\alpha \end{pmatrix} \qquad (21)$$

where $|0\rangle_\alpha \equiv |\alpha\rangle$ and

$$|n\rangle_\alpha = \frac{(a^+ - \bar{\alpha})^n}{\sqrt{n!}} |0\rangle_\alpha \qquad (22)$$

($|n\rangle_\alpha$)$_{n \in \mathbb{N}}$ form a “translated” basis of $\mathcal{F}$: $a_\alpha|n\rangle_\alpha = \sqrt{n}|n-1\rangle_\alpha$ ($a_\alpha|0\rangle_\alpha = 0$) and $a_\alpha^+|n\rangle_\alpha = \sqrt{n+1}|n+1\rangle_\alpha$ with $a_\alpha \equiv a - \alpha$ and $a_\alpha^+ \equiv a^+ - \bar{\alpha}$. In the sequel, the quasi-coherent state of the Fuzzy plane will be denoted by $|\lambda_0\rangle\rangle$.

Let $U = e^{i\vartheta a^+ a}$ be the rotation operator in the complex plane: $Ua^+U^\dagger = e^{i\vartheta}a^+$, $U|\alpha\rangle = |e^{i\vartheta}\alpha\rangle$. Since $U(a^+ - \bar{\alpha})U^\dagger = e^{i\vartheta}(a^+ - e^{-i\vartheta}\bar{\alpha})$, we have $U|n\rangle_\alpha = e^{in\vartheta}|n\rangle_{e^{i\vartheta}\alpha}$, and then $|\langle\lambda_{n\pm}(\alpha)|U|\lambda_{n\pm}(\alpha)\rangle|^2 = \frac{1+\cos(\vartheta)}{2}$. For $n > 0$, the eigenvectors of the fuzzy plane are not invariant by rotation around 0. So the perturbed quasicoherent states of the fuzzy plane are not invariant by rotation around 0 even if the perturbation operator is.

**Fuzzy elliptic paraboloid:** For $\epsilon \ll 1$ and $|\alpha|^2$ not too large (in order to $\epsilon|\alpha|^2 \ll 1$) we can treat the system by using the perturbation theory Appendix A. The vertical deformation of the plane is then

$$\delta x^3(\alpha) = \langle \alpha | X^3 | \alpha \rangle \qquad (23)$$

$$= \epsilon(|\alpha|^2 + 1/2) \qquad (24)$$

and so $M_\Lambda = \{(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha), \epsilon(|\alpha|^2 + 1/2))\}_{\alpha \in \mathbb{C}}$. Since $\alpha \langle n|(a^+a - |\alpha|^2)|0\rangle_\alpha = \alpha\delta_{n,1}$ we have

$$|\Lambda(\alpha)\rangle\rangle = |\lambda_0\rangle\rangle - \sum_{n>0, \zeta \in \{\pm\}} \frac{\langle\langle \lambda_{n\zeta} | \sigma_3 \otimes (X^3 - \delta x^3) | \lambda_0 \rangle\rangle}{\zeta\sqrt{n}} |\lambda_{n\zeta}\rangle\rangle + \mathcal{O}(\epsilon^2) \qquad (25)$$

$$= \begin{pmatrix} 1 \\ -\epsilon\alpha \end{pmatrix} \otimes |0\rangle_\alpha + \mathcal{O}(\epsilon^2) \qquad (26)$$

The quantum uncertainties are $\Delta X^1 = \Delta X^2 = 1/2$ and $\Delta X^3 = \epsilon|\alpha|$.

**Fuzzy hyperbolic paraboloid:** for the same conditions ($\epsilon|\alpha|^2 \ll 1$), we have

$$\delta x^3(\alpha) = \epsilon \mathrm{Re}(\alpha^2) \qquad (27)$$

The eigenmanifold is then $M_\Lambda = \{(\mathrm{Re}(\alpha), \mathrm{Im}(\alpha), \epsilon(\mathrm{Re}(\alpha)^2 - \mathrm{Im}(\alpha)^2))\}_{\alpha \in \mathbb{C}}$. Since $\alpha \langle n|\frac{a^2+(a^+)^2}{2}|0\rangle_\alpha = \bar{\alpha}\delta_{n,1} + \frac{\sqrt{2}}{2}\delta_{n,2}$ the quasi-coherent state is

$$|\Lambda(\alpha)\rangle\rangle = \begin{pmatrix} |0\rangle_\alpha \\ -\epsilon\bar{\alpha}|0\rangle_\alpha - \frac{\epsilon}{2}|1\rangle_\alpha \end{pmatrix} + \mathcal{O}(\epsilon^2) \qquad (28)$$

In contrast with the previous examples, the quasicoherent state is entangled. The model being as simple as the previous ones, this entanglement results probably from the hyperbolicity of the geometry. The quantum uncertainties are $\Delta X^1 = \Delta X^2 = 1/2$ and $\Delta X^3 = \epsilon\sqrt{|\alpha|^2 + 1/2}$.

**Fuzzy hyperboloid:** we have no a simple expression of the vertical deformation of the plane, but (for $\epsilon|\alpha| \ll 1$) we have

$$\delta x^3(\alpha) = \epsilon e^{-|\alpha|^2} \sum_{n=0}^{+\infty} \frac{|\alpha|^{2n}}{n!} \sqrt{r^2 + 1/2 + n} \qquad (29)$$

$\delta x^3$ is in fact very close to the hyperboloid $\alpha \mapsto \sqrt{r^2 + 1/2 + |\alpha|^2}$ as we can see this fig.2. The quasi-coherent state is

$$|\Lambda(\alpha)\rangle\rangle = \begin{pmatrix} |0\rangle_\alpha \\ 0 \end{pmatrix} - \sum_{n=1}^{\infty} \frac{\alpha \langle n | X^3 | 0 \rangle_\alpha}{\sqrt{n}} \begin{pmatrix} 0 \\ |n-1\rangle_\alpha \end{pmatrix} + \mathcal{O}(\epsilon^2) \qquad (30)$$
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Comparison between $\alpha \mapsto \delta x^3(\alpha) = \langle\Lambda(\alpha)|X^3|\Lambda(\alpha)\rangle = \epsilon e^{-|\alpha|^2} \sum_{n=0}^{+\infty} \frac{|\alpha|^{2n}}{n!} \sqrt{r^2 + 1/2 + n}$ for the fuzzy hyperboloid with the classical hyperboloid $\alpha \mapsto \sqrt{r^2 + 1/2 + |\alpha|^2}$ with $r = 1$.</figcaption></figure>

with

$$ \alpha \langle n | X^3 | 0 \rangle_{\alpha} = \epsilon e^{-|\alpha|^2} \sum_{q=0}^{+\infty} \sum_{p=0}^{+\infty} \max(q,n) \frac{(-1)^{n-p} \alpha^{n-p+q} \alpha_q q-p}{(q-p)!} \sqrt{\frac{n!}{(n-p)!p!}} \sqrt{r^2+1/2+q} \quad (31) $$

The quantum uncertainties are $\Delta X^1 = \Delta X^2 = 1/2$ and $\Delta X^3 = \sqrt{\epsilon^2(r^2 + 1/2 + |\alpha|^2) - (\delta x^3(\alpha))^2}$.

The different eigenmanifolds are plotted fig.3.

**Property 1** $\vec{n}_x = \text{tr}(|\Lambda(x)\rangle\langle\Lambda(x)|)$ is a normal vector at $x$ of $M_\Lambda$ in $\mathbb{R}^3$

*Proof:* We denote by $d$ the differential on $M_\Lambda$. $\tilde{D}_x|\Lambda(x)\rangle = 0 \Rightarrow \sigma_i dx^i|\Lambda(x)\rangle = \tilde{D}_x d|\Lambda(x)\rangle$. So $\langle\Lambda(x)|\sigma_i|\Lambda(x)\rangle dx^i = 0$. $dx^i = \frac{\partial x^i}{\partial s^a} ds^a$ (for $(s^1, s^2)$ a local curvilinear coordinate system on $M_\Lambda$). So $\vec{n}_x \cdot \frac{\partial \vec{n}}{\partial s^a} = 0 (\forall a)$. $\vec{n}_x$ is then normal to any tangent vector of $M_\Lambda$ at $x$. $\square$

As normal vector to the surface $M_\Lambda$, $\vec{n}_x$ defines a local orientation of this manifold by the right hand rule. This is the reason why we assimilate the spin degree of freedom ($\mathbb{C}^2$) as a quantum local orientation on $\mathfrak{M}$.

Now we want to interpret the norm of $\vec{n}_x$. Let $(|0\rangle, |1\rangle)$ be the canonical basis of $\mathbb{C}^2$: $|\Lambda(x)\rangle = |0\rangle \otimes |\Lambda_0(x)\rangle + |1\rangle \otimes |\Lambda_1(x)\rangle$. By construction the normal vector is

$$ \vec{n}_x = \langle\Lambda(x)|\vec{\sigma}|\Lambda(x)\rangle = \begin{pmatrix} 2\operatorname{Re}\langle\Lambda_0|\Lambda_1\rangle \\ 2\operatorname{Im}\langle\Lambda_0|\Lambda_1\rangle \\ \langle\Lambda_0|\Lambda_0\rangle - \langle\Lambda_1|\Lambda_1\rangle \end{pmatrix} \quad (32) $$

Moreover we have in the same basis $(|0\rangle, |1\rangle)$:

$$ \rho_{\Lambda}(x) = \text{tr}_{\mathcal{H}} |\Lambda(x)\rangle\langle\Lambda(x)| = \begin{pmatrix} \langle\Lambda_0|\Lambda_0\rangle & \langle\Lambda_1|\Lambda_0\rangle \\ \langle\Lambda_0|\Lambda_1\rangle & \langle\Lambda_1|\Lambda_1\rangle \end{pmatrix} \quad (33) $$

$\langle\Lambda_0|\Lambda_0\rangle = p_0$ and $\langle\Lambda_1|\Lambda_1\rangle = p_1$ are the populations of the states $(|0\rangle, |1\rangle)$ in the eigengeometry and $\langle\Lambda_0|\Lambda_1\rangle = c$ is its quantum coherence. The normal vector is then

$$ \vec{n}_x = \begin{pmatrix} 2\operatorname{Re}c \\ 2\operatorname{Im}c \\ p_0 - p_1 \end{pmatrix} \quad (34) $$
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Eigenmanifolds $M_\Lambda : \alpha \mapsto \langle\Lambda(\alpha)|\vec{X}|\Lambda(\alpha)\rangle$ (yellow surfaces) for the Fuzzy surface plots with the quantum uncertainty clouds $\Delta X^i = \sqrt{\langle\Lambda|(X^i)^2|\Lambda\rangle} - (\langle\Lambda|X^i|\Lambda\rangle)^2$ (blue ellipsoids) of some points, in the target space $\mathbb{R}^3$ of the probe. $r=1$ for the Fuzzy hyperboloid.</figcaption></figure>

Finally we have

$$ \|\vec{n}_x\| = \sqrt{(p_0 - p_1)^2 + 4|c|^2} \qquad (35) $$

$$ = \sqrt{(\operatorname{tr} \rho_{\Lambda})^2 - 4 \det \rho_{\Lambda}} \qquad (36) $$

$$ = s_+(x) - s_-(x) \qquad (37) $$

where $s_\pm = \frac{\operatorname{tr} \rho_\Lambda \pm \sqrt{(\operatorname{tr} \rho_\Lambda)^2 - 4 \det \rho_\Lambda}}{2}$ are the eigenvalues of $\rho_\Lambda$. $\|\vec{n}_x\|$ is then a measure of the purity of $\rho_\Lambda$, since $\rho_\Lambda$ is pure state if $s_+ = 1$ and $s_- = 0$ ($\|\vec{n}_x\| = 1$) and is maximally mixed if $s_+ = s_- = \frac{1}{2}$ ($\|\vec{n}_x\| = 0$, microcanonical distribution). In other words, $1 - \|\vec{n}_x\|$ is a measure of the entanglement of $|\Lambda\rangle$, this one is separable if $\|\vec{n}_x\| = 1$ and is maximally entangled if $\vec{n}_x = \vec{0}$. In that case, in the eigenbasis of $\rho_\Lambda$ we have $|\Lambda(x)\rangle = |s_+(x)\rangle \otimes |\Lambda_+(x)\rangle + |s_-(x)\rangle \otimes |\Lambda_-(x)\rangle$ with $\langle\Lambda_\alpha|\Lambda_\beta\rangle = \frac{1}{2}\delta_{\alpha\beta}$ ($|\Lambda\rangle$ is
a Bell state).

For these reason we can refer to $\vec{n}_x$ as being the “purity normal vector” at $M_\Lambda$.

## 3.2. Degeneracy of the eigenmanifold

We say that $M_\Lambda$ is strongly non-degenerate at $x$ if dim ker($\mathcal{D}_x$) = 1. This is the usual definition of the non-degeneracy of an eigenvalue in the Hilbert space $\mathbb{C}^2 \otimes \mathcal{H}$. But it can be more natural to consider this space with its $C^*$-module structure. So if there exists two quasicoherent states at $x$ related by a SU(2)-phase change, we can consider that 0 is not degenerate in the $C^*$-module. Note that if $|\Lambda(x)\rangle\rangle$ and $u|\Lambda(x)\rangle\rangle$ are two orthogonal quasicoherent states of ker($\mathcal{D}_x$), $(\alpha + \beta u)|\Lambda(x)\rangle\rangle$ (with $|\alpha|^2 + |\beta|^2 = 1$) is also a quasicoherent state, but in general $(\alpha + \beta u)$ is not unitary: $(\alpha + \beta u)^\dagger(\alpha + \beta u) = |\alpha|^2 + |\beta|^2 u^\dagger u + \bar{\alpha}\bar{\beta}u + \alpha\bar{\beta}u^\dagger = 1 + \bar{\alpha}\bar{\beta}u + \alpha\bar{\beta}u^\dagger$. But $\langle\Lambda(x)|u|\Lambda(x)\rangle\rangle = 0 \iff \text{tr}(\rho_\Lambda(x)u) = 0$ with $\rho_\Lambda = \text{tr}_{\mathcal{H}}|\Lambda\rangle\rangle\langle\Lambda|$ the eigen density matrix. We have then $\text{tr}(\rho_\Lambda(\alpha + \beta u)^\dagger(\alpha + \beta u)) = 0$ and so $(\alpha + \beta u) \in U(2)_{a.s.}$.

**Definition 6 (Weakly non-degenerate fuzzy space)** A fuzzy space $\mathfrak{M}$ is said weakly non-degenerate at $x \in M_\Lambda$ if $\exists|\Lambda(x)\rangle\rangle \in \mathbb{C}^2 \otimes \mathcal{H}$ and $\exists u_i \in U(2)_{a.s.}$ such that $\text{ker}(\mathcal{D}_x) = \text{Lin}(|\Lambda(x)\rangle\rangle, u_1|\Lambda(x)\rangle\rangle, \dots, u_n|\Lambda(x)\rangle\rangle)$. Moreover:

* $\mathfrak{M}$ is said strongly non-degenerate at $x$ if $\text{dim ker}(\mathcal{D}_x) = 1$.

* if $\mathfrak{M}$ is not strongly non-degenerate at $x$ and if $|\Lambda(x)\rangle\rangle$ is separable, then $\text{dim ker}(\mathcal{D}_x) = 2$ and $\text{ker}(\mathcal{D}_x) = \text{SU}(2) \cdot \text{Lin}(|\Lambda(x)\rangle\rangle)$.

* if $\mathfrak{M}$ is not strongly non-degenerate at $x$ and if $|\Lambda(x)\rangle\rangle$ is entangled, then $\text{dim ker}(\mathcal{D}_x) \le 4$.

*Proof:* We suppose that the quasicoherent state is separable $|\Lambda(x)\rangle\rangle = |O_x\rangle\otimes|\Omega_x\rangle$ and that $u|\Lambda(x)\rangle\rangle = u|O_x\rangle\otimes|\Omega_x\rangle$ is another quasi-coherent state orthogonal to the first one ($u \in U(2)_{a.s.}$). We have then $\langle I_x|O_x\rangle = 0$ with $|I_x\rangle = u|O_x\rangle$. So $(|O_x\rangle, |I_x\rangle)$ is an orthonormal basis of $\mathbb{C}^2$, and finally $\text{ker}(\mathcal{D}_x) = \{(\alpha|O_x\rangle + \beta|I_x\rangle) \otimes |\Omega_x\rangle, \alpha, \beta \in \mathbb{C}\} = U(2) \cdot |\Lambda(x)\rangle\rangle$ (we cannot find a third quasi-coherent state $u'|\Lambda(x)\rangle\rangle$ orthogonal to the first two ones, since $u'|O_x\rangle$ cannot be orthogonal to $|O_x\rangle$ and $|I_x\rangle$ except if it is zero).

We suppose that the quasicoherent state is entangled $|\Lambda(x)\rangle\rangle = |O_x\rangle\otimes|\Lambda_x^0\rangle + |I_x\rangle\otimes|\Lambda_x^1\rangle$ for some orthonormal basis $(|O_x\rangle, |I_x\rangle)$ of $\mathbb{C}^2$ ($|\Lambda_x^0\rangle \neq |\Lambda_x^1\rangle$). Let $u|\Lambda(x)\rangle\rangle = |O_x'\rangle\otimes|\Lambda_x'\rangle + |I_x'\rangle\otimes|\Lambda_x'\rangle$ be another quasi-coherent state orthogonal to the first one ($u \in U(2)_{a.s.}$), with $|O_x'\rangle = u|O_x\rangle$ and $|I_x'\rangle = u|I_x\rangle$.

$$ \langle\Lambda(x)|u|\Lambda(x)\rangle\rangle = 0 \iff \langle O_x|O_x'\rangle\langle\Lambda_x^0|\Lambda_x^0\rangle + \langle O_x|I_x'\rangle\langle\Lambda_x^0|\Lambda_x^1\rangle + \langle I_x|O_x'\rangle\langle\Lambda_x^1|\Lambda_x^0\rangle + \langle I_x|I_x'\rangle\langle\Lambda_x^1|\Lambda_x^1\rangle = 0 \quad (38) $$

$$ \iff ( \langle O_x | \langle I_x | ) \rho_\Lambda^T \otimes \text{id} \begin{pmatrix} |O_x'\rangle \\ |I_x'\rangle \end{pmatrix} = 0 \quad (39) $$

with $\rho_\Lambda = \text{tr}_{\mathcal{H}}|\Lambda\rangle\rangle\langle\Lambda|$. $\rho_\Lambda^\dagger = \rho_\Lambda$, $\rho_\Lambda \ge 0$ and $\det \rho_\Lambda \ne 0$ (otherwise $|\Lambda\rangle\rangle$ would be separable), $\langle\bullet|\rho^T \otimes \text{id}\bullet\rangle_{\mathbb{C}^4}$ is then a scalar product of $\mathbb{C}^4$ (a definite positive sesquilinear form). Two couples $(|O_x\rangle, |I_x\rangle)$ and $(|O_x'\rangle, |I_x'\rangle)$ (viewed as vectors of $\mathbb{C}^4$) define two orthogonal quasicoherent states if they are orthogonal with respect to $\langle\bullet|\rho^T \otimes \text{id}\bullet\rangle_{\mathbb{C}^4}$. We can then find at most four independent quasicoherent states. $\square$

In the following we consider for the whole of this paper only the case of eigenmanifolds at least weakly non-degenerate at all points (the case of a strongly non-degenerate eigenmanifold being not excluded).
If $\mathfrak{M}$ is not strongly non-degenerate at $x$ with separable quasienergy states, then these ones have the form $|\zeta\rangle \otimes |\Omega_x\rangle$ for some $|\Omega_x\rangle \in \mathcal{H}$ and for any $|\zeta\rangle \in \mathbb{C}^2$. It follows that any vector $\langle\zeta|\vec{\sigma}|\zeta\rangle$ is normal to $M_\Lambda$ at $x$. $x$ is then an isolated point (dim $M_\Lambda = 0$ at $x$). Moreover $\mathcal{D}_x|\zeta\rangle \otimes |\Omega_x\rangle = 0$ for any $|\zeta\rangle$ implies that $|\Omega_x\rangle$ is a common eigenvector of $X^1, X^2$ and $X^3$. $M_\Lambda$ is then a discrete set of points if $\mathfrak{X}$ is abelian.

**Example 3: “Fuzzy” circle**

Let $\mathcal{H} = \mathbb{C}^N$ be endowed with an orthonormal basis $(|n\rangle)_{n=0,...,N-1}$, $Z = \sum_{n=0}^{N-1} |n\rangle\langle n+1|$ with by convention $|N\rangle \equiv |0\rangle$) and $X^3 = 0$. We can view this model as the Fuzzy unit circle since $Z^\dagger Z = (X^1)^2 + (X^2)^2 = \text{id.}$ [$Z^\dagger, Z] = 0$, $\mathfrak{X}$ is then abelian.

$$ \begin{pmatrix} 0 & Z^\dagger - \bar{z} \\ Z - z & 0 \end{pmatrix} \begin{pmatrix} |\Lambda^0\rangle \\ |\Lambda^1\rangle \end{pmatrix} \iff \begin{cases} Z|\Lambda^1\rangle = z|\Lambda^1\rangle \\ Z^\dagger|\Lambda^0\rangle = \bar{z}|\Lambda^0\rangle \end{cases} \quad (40) $$

Since $Z^N = \text{id.}$, $z$ must be a $N$-th root of the unity, and then $M_\Lambda = \{(\cos(\frac{2\pi q}{N}), \sin(\frac{2\pi q}{N}), 0)\}_{q \in \{0,...,N-1\}}$, the quasicoherent states being weakly degenerate of the form

$$ |\Lambda_{\psi}(e^{i\frac{2\pi q}{N}})\rangle = |\psi\rangle \otimes \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{i\frac{2\pi q}{N}n} |n\rangle \quad (41) $$

for all $|\psi\rangle \in \mathbb{C}^2$. For finite values of $N$, the model is a Fuzzy regular cyclic $N$-gone with the circumscribed unit circle. For $N \to +\infty$, $M_\Lambda$ becomes the unit circle $S^1$ and the quasicoherent states tend in the weak topology to the singular distributions $|\Lambda_\psi(e^{i\varphi})\rangle = |\psi\rangle \otimes |\delta_\varphi\rangle$ with $\langle\theta|\delta_\varphi\rangle = \delta(\theta-\varphi)$ (Dirac distribution), the Hilbert space being assimilated to $\mathcal{H} = L^2(S^1, \frac{d\theta}{2\pi})$. Due to the abelianity of $\mathfrak{X}$ and to the zero quantum uncertainty, the system is not really fuzzy, it is more a matrix embedding of the classical circle.

If $\mathfrak{M}$ is not strongly non-degenerate at $x$ with entangled quasicoherent states, several situations can occur depending on the number of independent non-trivial purity normal vectors in $\text{Lin}(\vec{n}_x^\alpha)_{\alpha=0,...,n \le 3}$ with $\vec{n}_x^\alpha = \langle\langle \Lambda(x)|u_\alpha^\dagger \vec{\sigma} u_\alpha|\Lambda(x)\rangle\rangle$ ($u_\alpha \in U(2)_{\text{a.s.}}$). This number can be smaller than $n = \text{dim}(\mathcal{D}_x)$. Indeed, some quasicoherent states can be maximally entangled (Bell states), and their associated purity normal vectors are $\vec{0}$. If $u_\alpha$ belongs to the isotropy group of $\rho_\Lambda$: $u_\alpha^\dagger \rho_\Lambda u_\alpha = \rho_\Lambda$, then $\vec{n}_x^\alpha = \vec{n}_x^0$. In a same way, if $u_\alpha$ belongs to the isotropy group of $u_\alpha^\dagger \rho_\Lambda u_\beta$, then $\vec{n}_x^\alpha = \vec{n}_x^\beta$. We then have four possible situations:

* All purity normal vectors at $x$ are zero (ker $\mathcal{D}_x$ is a subspace of maximally entangled states). In this case we cannot conclude anything about the geometry of $M_\Lambda$ at $x$.

* We have only one independent non-trivial purity normal vector, then this one provides the normal direction to $M_\Lambda$ at $x$ as for the strongly non-degenerate case.

* We have only two independent non-trivial purity normal vectors, $M_\Lambda$ has then a normal plane at $x$, then $x$ is a pinch point of $M_\Lambda$ (dim $M_\Lambda = 1$ at $x$).

* We have at most three independent non-trivial purity normal vectors, the situation is similar to the case of a separable not strongly non-degenerate quasi-coherent state, $x$ is an isolated point (dim $M_\Lambda = 0$ at $x$).

### 3.3. Internal and external gauge choices

The quasicoherent states are defined up to local $C^*$-phase changes: $\forall x \in M_\Lambda$

$$ |\tilde{\Lambda}(x)\rangle = h(x)u(x)|\Lambda(x)\rangle \quad h(x) \in U(1), \quad u(x) \in U_x \quad (42) $$
$U_x$ being the subgroup of $SU(2)_{a.s.}$ such that $U_x \cdot |\Lambda(x)\rangle\rangle = \ker(\tilde{\mathcal{D}}_x)$ (the quasicoherent state being supposed weakly nondegenerate). Note that even if the quasicoherent state is strongly nondegenerate (dim $\ker \tilde{\mathcal{D}}_x = 1$), $U_x$ can be nontrivial as being the isotropy subgroup of $|\Lambda(x)\rangle\rangle$ in $SU(2)$. $h$ corresponds to an usual phase change (it is the single phase change existing for a strongly nondegenerate quasienergy state), whereas $u$ is a local “orientation” change (a noncommutative phase change). We call this phase choice the internal gauge choice. We have also an external gauge choice associated with the coordinate changes (the change of the basis generating $\mathcal{X}$):

$$Y^i - y^i = J^i_j(X^j - x^j) \qquad J \in SO(3) \qquad (43)$$

$(Y^i)$ (resp. $(y^i)$) constitute new quantum (resp. classical) coordinate observables of $\mathcal{X}$ (resp. $\mathbb{R}^3$). The coordinate changes must leave invariant the geometry operator by inducing a spin rotation:

$$\tilde{\mathcal{D}}_y = \hat{\sigma}_i \otimes (Y^i - y^i) \qquad (44)$$

$$= J^i_j \hat{\sigma}_i \otimes (X^j - x^j) \qquad (45)$$

$$= u_J^{-1} \hat{\sigma}_j u_J \otimes (X^j - x^j) \qquad (46)$$

$$= \sigma_j \otimes (X^j - x^j) \qquad (47)$$

$$= \tilde{\mathcal{D}}_x \qquad (48)$$

$\hat{\sigma}_j = u_J \sigma_j u_J^{-1}$ being the Pauli matrices after the spin rotation $u_J \in SU(2)$ associated with the 3D-vector rotation $J$. But consider $\tilde{\mathcal{D}}_y = \sigma_i \otimes (Y^i - y^i) = u_J^{-1} \tilde{\mathcal{D}}_{J^{-1}y} u_J$ the geometry observable in the new coordinates without spin rotation and $|\hat{\Lambda}(y)\rangle\rangle$ an associated quasicoherent states $(\tilde{\mathcal{D}}_y |\hat{\Lambda}(y)\rangle\rangle = 0)$. $|\hat{\Lambda}(y)\rangle\rangle$ is related to $|\Lambda(x)\rangle\rangle$ by

$$|\hat{\Lambda}(y)\rangle\rangle = u_J^{-1} |\Lambda(J^{-1}y)\rangle\rangle \qquad (49)$$

From the point of view of quantum gravity matrix model, $J$ constitutes a frame change of the observer (it is the “observer” which is rotated by $J^{-1}$ and not the quantum spacetime!). This is the reason for which we consider this gauge change as being “external”.

## 4. Metrics on a Fuzzy space

To translate the notions of the usual geometry into the fuzzy space theory, we want to endow fuzzy spaces with metrics compatible with the quasicoherent geometry. For applications to quantum gravity, the goal of introducing these metrics is obvious (general relativity dealing with manifolds endowed with metrics). But we want also understand these metrics in the abstract geometric context and find their interpretations in the applications to quantum information theory. In this section we are interested with metrics associated with the quasicoherent picture which is different from the original Connes metric (for this one applied on fuzzy spaces we can see ref. [14, 15, 16, 17, 18]).

### 4.1. The natural metrics of $\mathfrak{M}$ and of $M_\Lambda$

$M_\Lambda$ is naturally endowed with the metric induced by its immersion in $\mathbb{R}^3$:

$$\gamma_{ab} = \delta_{ij} \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} \qquad (50)$$

where $(s^1, s^2)$ is a local curvilinear coordinate system on $M_\Lambda$.
**Property 2** The natural metric of $M_Λ$ can be written as $\forall x \in M_Λ$:

$$ \gamma_{ab}(x) = \frac{1}{2} \langle \partial_{(a}\Lambda(x) | \mathcal{D}_x^2 | \partial_{b)}\Lambda(x) \rangle \quad (51) $$

*Proof:* $\mathcal{D}_x|\Lambda\rangle = 0 \Rightarrow -\sigma_i \frac{\partial x^i}{\partial s^a}|\Lambda\rangle + \mathcal{D}_x\partial_a|\Lambda\rangle = 0$. It follows that

$$ \begin{aligned} \langle \partial_a \Lambda | \mathcal{D}_x^2 | \partial_b \Lambda \rangle ds^a ds^b &= \langle \Lambda | \sigma_i \sigma_j | \Lambda \rangle \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} ds^a ds^b \\ &= \delta_{ij} \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} ds^a ds^b \end{aligned} \quad (52) $$

$$ + \iota \varepsilon_{ij}^k \langle \Lambda | \sigma_k | \Lambda \rangle \underbrace{\frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} ds^a ds^b}_{=0} \quad (53) $$

$$ = \gamma_{ab} ds^a ds^b \quad (54) $$

□

The triads of $M_Λ$ ($e_a^i$) are $e_a^i = \frac{\partial x^i}{\partial s^a}$ in order to have $\gamma_{ab} = \delta_{ij}e_a^i e_b^i$.

To be consistent with the quasicoherent metric, the metric of $\mathfrak{M}$ as a square length observable is defined by $d\ell_x^2 = \mathcal{D}_x^2$ (and not $\mathcal{D}_x^{-2}$ as in usual noncommutative geometry [2], this is consistent with the fact that $\mathcal{D}_x$ and $-\iota[\mathcal{D}, \bullet] = \sigma_i \otimes L_{X^i}$ are homogeneous to a length and not to the inverse of a length). The definition of $\gamma$ can be then rewritten as

$$ d\ell_s^2 = \gamma_{ab}(x(s))ds^a ds^b = \omega_{x(s+ds)}(d\ell_x^2) \quad (55) $$

We recall that the geometric operator $\mathcal{D}_x$ can be interpreted as an energy observable (displacement energy in string theory, or energy of the qubit dressed by the environment in quantum information theory). By construction, the mean energy in a quasi-coherent state is zero with no uncertainty: $\omega_x(\mathcal{D}_x) = 0$ and $\Delta_x \mathcal{D}_x^2 = \omega_x(\mathcal{D}_x^2) - \omega_x(\mathcal{D}_x)^2 = 0$. But consider a “slipping” on $M_Λ$, for which the state of the quantum system is $\omega_{x(s+ds)}$ but the energy measurement is preformed with $\mathcal{D}_{x(s)}$ (the measurement is performed with an infinitesimal “misalignment” $ds$). The mean energy is then $\omega_{x(s+ds)}(\mathcal{D}_{x(s)}) = 0$ (at the first order of approximation), but the square deviation is $\Delta_{x(s+ds)}\mathcal{D}_{x(s)}^2 = \omega_{x(s+ds)}(\mathcal{D}_{x(s)}^2) - \omega_{x(s+ds)}(\mathcal{D}_{x(s)})^2 = d\ell_s^2$. $d\ell_s$ is then the energy uncertainty. $\ell(\mathcal{C}) = \int_{\mathcal{C}} d\ell$ is the cumulated energy uncertainty during a slipping along the path $\mathcal{C}$, and so the geodesics of $\gamma$ are the path minimising the energy uncertainty due to the infinitesimal misalignment of the probe.

$d\ell_x^2$ as square length observable induces a quantisation of the measure of the infinitesimal lengths. In contrast with a classical manifold for which we have a continuum of infinitesimal lengths, onto a fuzzy space these ones belong to $\text{Sp}(\sqrt{d\ell_x^2}) = \text{Sp}(|\mathcal{D}_x|)$. For example, for the fuzzy plane $\text{Sp}(|\mathcal{D}_x|) = \{\sqrt{n}, n \in \mathbb{N}\}$ which can be interpreted in quantum gravity by a quantisation of the infinitesimal lengths which are multiples of the Planck length (with factors being square roots of integers).

To continue the analysis of the relation between the geometry of $\mathfrak{M}$ and the one of $M_Λ$, we need maps permitting to transform the (co)tangent vectors of one into those of the other. Let $\omega_{x*}$ and $\omega_x^*$ be the push-forward (tangent map) and the pull-back (cotangent map) of $\omega_x$ (for $x \in M_Λ$)

$$ \begin{aligned} \omega_{x*} : \text{Der}(\mathcal{X}) &\rightarrow T_x \mathbb{R}^3 = T_x M_\Lambda \oplus N_x M_\Lambda \\ L &\mapsto \omega_x(L(X^i))\partial_i \end{aligned} \quad (56) $$
$T_x M_\Lambda$ and $N_x M_\Lambda$ being the tangent and the normal vector spaces at $x$ on $M_\Lambda$.

$$ \begin{aligned} \omega_x^* : \Omega_x^1 \mathbb{R}^3 & \to \Omega_{\text{Der}\mathfrak{X}}^1 \mathfrak{X} \\ \eta_i(x)dx^i & \mapsto \eta_i(x)\mathbf{d}X^i \end{aligned} \qquad (57) $$

These maps are defined to satisfy the following duality consistency condition: $\langle \eta(x), \omega_{x^*}(L) \rangle_{\mathbb{R}^3} = \omega_x(\langle \omega_x^* \eta(x), L \rangle_x) (\langle \bullet, \bullet \rangle_{\mathbb{R}^3}$ standing for the duality bracket between the tangent vectors and the differential 1-forms of $\mathbb{R}^3)$. Moreover let $\pi_x$ be the orthogonal projection onto $T_x M_\Lambda$:

$$ \begin{aligned} \pi_x : T_x \mathbb{R}^3 & \to T_x M_\Lambda \\ v^i \partial_i & \mapsto v^i \delta_{ij} \gamma^{ab} \frac{\partial x^j}{\partial s^b} \partial_a \end{aligned} \qquad (58) $$

and $\pi_x^*$ be its dual map $(\langle \pi_x^* \bullet, \bullet \rangle_{\mathbb{R}^3} = \langle \bullet, \pi_x \bullet \rangle_{\mathbb{R}^3})$:

$$ \begin{aligned} \pi_x^* : \Omega_x^1 M_\Lambda & \to \Omega_x^1 \mathbb{R}^3 \\ \eta_a ds^a & \mapsto \eta_a \gamma^{ab} \delta_{ij} \frac{\partial x^j}{\partial s^b} dx^i \end{aligned} \qquad (59) $$

$\omega_x^* \pi_x^*(\gamma_{ab}(x)ds^a ds^b) \in (\Omega_{\text{Der}\mathfrak{X}}^1)^{\otimes 2}$ defines then a metric on $\mathfrak{M}$ as a symmetric bivector.

**Property 3** The metric of a fuzzy space $\mathfrak{M}$ as a bivector is

$$ \gamma = \omega_x^* \pi_x^* (\gamma_{ab}(x) ds^a ds^b) = \delta_{ij} \mathbf{d}X^i \otimes \mathbf{d}X^j \qquad (60) $$

Proof:

$$ \omega_x^* \pi_x^* (\gamma_{ab}(x) ds^a ds^b) = \gamma_{ab} \gamma^{ac} \gamma^{bd} \delta_{ik} \delta_{jl} \frac{\partial x^k}{\partial s^c} \frac{\partial x^l}{\partial s^d} \mathbf{d}X^i \otimes \mathbf{d}X^j \qquad (61) $$

$$ = \gamma^{bd} \delta_{ik} \delta_{jl} \frac{\partial x^k}{\partial s^b} \frac{\partial x^l}{\partial s^d} \mathbf{d}X^i \otimes \mathbf{d}X^j \qquad (62) $$

$$ = \delta_{ij} \mathbf{d}X^i \otimes \mathbf{d}X^j \qquad (63) $$

$$ \text{because } \gamma_{bd} = \frac{\partial x^k}{\partial s^b} \frac{\partial x^l}{\partial s^d} \delta_{kl}. \qquad \square $$

We have also $\gamma = \delta_{ij} \Theta^{ki} \Theta^{lj} \theta_k \otimes \theta_l$. $\gamma$ is well defined as metric of $\mathfrak{M}$ because it is independent of $x$ (of the probe).

**Property 4** $\gamma: \text{Der}(\mathfrak{X})^2 \to \text{Env}(\mathfrak{X})$ is hermitian and positive (and non-degenerate if $\mathcal{Z}(\mathfrak{X})$ is trivial).

Proof: Let $L_V = v_i L_{X^i}$, $L_W = w_i L_{X^i} \in \text{Der}(\mathfrak{X})$ be two tangent vectors of $\mathfrak{M}$ (with $v_i, w_i \in \mathcal{Z}(\text{Env}(\mathfrak{X}))$ selfadjoint).

$$ \gamma(L_V, L_W) = \delta_{ij} \Theta^{ki} \Theta^{lj} v_k w_l \qquad (64) $$

$$ = - \delta_{ij} [V, X^i] [W, X^j] \qquad (65) $$

$$ = - \delta_{ij} ([X^j, W][X^i, V])^\dagger \qquad (66) $$

$$ = - \delta_{ij} ([W, X^j][V, X^i])^\dagger \qquad (67) $$

$$ = \gamma(L_W, L_V)^\dagger \qquad (68) $$

In particular $\gamma(L_V, L_V)$ is selfadjoint and has then a real spectrum. Let $\psi \in \mathcal{H}$.

$$ \langle \psi | \gamma(L_V, L_V) | \psi \rangle = - \delta_{ij} \langle \psi | [V, X^i] [V, X^j] \psi \rangle \qquad (69) $$

$$ = - \delta_{ij} \langle [X^i, V] \psi | [V, X^j] \psi \rangle \qquad (70) $$

$$ = \delta_{ij} \langle [V, X^i] \psi | [V, X^j] \psi \rangle \qquad (71) $$

$$ = \sum_i \| [V, X^i] \psi \|^2 \qquad (72) $$

$$ \geq 0 \qquad (73) $$
Moreover $\langle \psi|\gamma(L_V, L_V)|\psi \rangle = 0$ for all $\psi$ if and only if $V = 0$ except if $V$ commutes with all $X^i$. If the center of $\mathfrak{X}$ is trivial ($\mathcal{Z}(\mathfrak{X}) = \text{Lin}(\text{id})$), this cannot arise. $\square$

At the semi-classical limit where $\Theta^{ij} = -i[X^i, X^j] \sim \vartheta^{ij}$ defines a Poisson structure of $\mathbb{R}^3$ ($\{f, g\} = \vartheta^{ij} \partial_i f \partial_j g, \forall f, g \in \mathbb{R}^3$), $\gamma$ defines an effective metric $G^{kl} = \delta_{ij} \vartheta^{ki} \vartheta^{lj}$ which plays the role in quantum gravity matrix models of the metric of the emergent curved spacetime [32, 10] (the spacetime is flat and noncommutative at the microscopic scale but it emerges a curved commutative spacetime at the macroscopic scale with the semi-classical limit). In this meaning, $\gamma$ can be then viewed as the quantum metric of a quantum spacetime.

Since $\gamma_{kl} = \delta_{ij}\Theta_k^i\Theta_l^j$ (with respect to the representation with $\theta_i$), we can set $E_k^i = \Theta_k^i = [X_k, X^i]$ as triads of $\mathfrak{M}$. The analogue of the Weitzenböck connection [13] of $\mathfrak{M}$ is then

$$ \hat{\Gamma}_{jk}^{i} = L_{X^k}(E_j^i) = i[X_k, [X^i, X_j]] \qquad (74) $$

which defines an analogue of the Weitzenböck torsion:

$$ \hat{T}_{ij}^k = L_{X^j}(E_i^k) - L_{X^i}(E_j^k) \qquad (75) $$

$$ = i[X_j, [X^k, X_i]] - i[X_i, [X^k, X_j]] \qquad (76) $$

$$ = i[X^k, [X_i, X_j]] \qquad (77) $$

(where we have used the Jacobi identity).

Remark : $E_i^j$ induces the following dual diads on $M_\Lambda$:

$$ \hat{e}_i^a = \omega_x([\pi_x]^a_j E_i^j) \qquad (78) $$

$$ = -i\gamma^{ab} \frac{\partial x^j}{\partial s^b} \langle \Lambda(x) | [X_i, X_j] | \Lambda(x) \rangle \qquad (79) $$

the triads being by construction $e_a^i = \frac{\partial x^i}{\partial s^a}$.

In BFFS matrix model, $M_\Lambda$ is the classical space closest as possible to the physics of the quantum space $\mathfrak{M}$. $\mathfrak{M}$ is the quantisation of a flat space (as we can see it with its metric $\gamma = \delta_{ij}dX^i \otimes dX^j$). In this kind of models, the gravitational effects emerge at the macroscopic level from the noncommutativity at the microscopic level (with the thermodynamical limit where the number of strings tends to infinity with constant density). At the microscopic level, the gravitational effects emerge at the quasicoherent picture (which is the classical geometry closest to the quantum geometry) [12]. These gravitational effects are induced, in accordance with general relativity, by the curvature of $M_\Lambda$ with its metric $\gamma_{ab}$.

For the applications to quantum information, $M_\Lambda$ is the constrained control parameter manifold of the qubit. With adiabatic control in the whole of $\mathbb{R}^3$, the adiabatic transport induces in addition to the geometric phase a dynamical phase which, in contrast with the geometric phase, cannot be fixed only by the control design (the shape of paths $\mathcal{C}$ in $M_\Lambda$ [30, 31]). In strong adiabatic transport, these dynamical phases modify the interferences, and in weak adiabatic transport, these operator valued dynamical phases modify the reached density matrix as a kind of effective Hamiltonian evolution in addition to the geometric effects [8, 23]. To have a pure geometric control, the adiabatic control must be constrained onto $M_\Lambda$. The geometry
of this manifold is then characteristic of the controllability of the qubit in presence of the environment.

## 4.2. Use of the noncommutative metric

To use in practice the metric $\gamma$ we need to define the equivalents of paths on $\mathfrak{M}$. Due to the non-separability of the quasicoherent states and of Heisenberg uncertainties concerning the coordinate observables, the usual notion of path cannot be obviously generalised. To solve this problem, we introduce (in accordance with the Perelomov coherent state theory [11]) the notion of displacement operator.

### 4.2.1. The displacement operator

**Definition 7 (Displacement operator)** Let $M_\Lambda$ be the eigenmanifold of a fuzzy space $\mathfrak{M}$, supposed weakly non-degenerate. Let $x, y \in M_\Lambda$ be two distinct points. We call displacement operator $\mathbf{D}(y, x) \in e^{i\mathfrak{X}} \subset \text{Env}(\mathfrak{X})$ from $x$ to $y$, the invertible operator, if it exists, such that

$$ \exists u_{yx} \in U(2)_{\text{a.s.}}, \quad u_{yx} \otimes \mathbf{D}(y, x) |\Lambda(x)\rangle\rangle = |\Lambda(y)\rangle\rangle \quad (80) $$

and then

$$ \omega_y = \omega_x \circ \text{Ad}_{\mathbf{D}(y,x)} \quad (81) $$

If $\mathbf{D}(y, x)$ exists, $x$ and $y$ are said linkable.

Up to a local orientation and local phase changes, $\mathbf{D}(y, x)$ induces the move from the pseudo-point $\omega_x$ to the pseudo-point $\omega_y$. A similar definition of the displacement operator exists for the Perelomov coherent states (by definition the coherent states are the orbit of the vacuum state by the action of the group of displacement operators [11, 27]).

In the context of quantum information theory, the displacement operator models a “control jump”, i.e. a local operation quantum protocol on the qubit ($u_{yx}$) and on its environment ($\mathbf{D}(y, x)$) permitting to abruptly change the control parameters from $x$ to $y$.

The displacement operator associated with $|\hat{\Lambda}\rangle\rangle$ after an external gauge change eq. (49) is related to the one associated with $|\Lambda\rangle\rangle$ by

$$ \hat{\mathbf{D}}(y_2, y_1) = \mathbf{D}(J^{-1}y_2, J^{-1}y_1) \quad (82) $$

$$ = \mathbf{D}(y_2, y_1)\mathbf{G}_J(y_2, y_1) \quad (83) $$

with $\mathbf{G}_J(y_2, y_1) = \mathbf{D}(y_2, y_1)^{-1}\mathbf{D}(J^{-1}y_2, J^{-1}y_1) \in \text{Env}(\mathfrak{X})$.

## Example 2: Fuzzy surface plots

The Perelomov coherent states of the harmonic oscillator algebra are defined by [11]

$$ \mathcal{D}(\alpha)|0\rangle = |\alpha\rangle \quad \text{with} \quad \mathcal{D}(\alpha) = e^{\alpha a^{+} - \bar{\alpha} a} = e^{-|\alpha|^2/2}e^{\alpha a^{+}}e^{-\bar{\alpha} a} \quad (84) $$

and since $\mathcal{D}(\alpha)a\mathcal{D}(\alpha)^{\dagger} = a - \alpha$, we have also $|n\rangle_{\alpha} = \mathcal{D}(\alpha)|n\rangle$. Moreover we have

$$ \mathcal{D}(\alpha)\mathcal{D}(\beta) = e^{i\text{Im}(\alpha\beta)}\mathcal{D}(\alpha + \beta) \quad (85) $$

and then

$$ \mathcal{D}(\beta - \alpha)|\alpha\rangle = e^{-i\text{Im}(\alpha\beta)}|\beta\rangle \quad (86) $$
This defines a displacement operator for the fuzzy spaces of the fuzzy plane family by

$$ \mathbf{D}(\beta, \alpha) = \mathcal{D}(\beta - \alpha) = e^{(\beta - \alpha)a^{+} - (\beta - \bar{\alpha})a} \quad (87) $$

The local gauge change associated with the displacement operator, $u_{\beta\alpha} \otimes \mathbf{D}(\beta, \alpha)$, are

$$ u_{\beta\alpha} = \begin{cases} e^{i\operatorname{Im}(\alpha\bar{\beta})} & \text{for the fuzzy plane} \\ e^{i\operatorname{Im}(\alpha\bar{\beta})} \begin{pmatrix} 1 & 0 \\ \epsilon(\alpha - \beta) & 1 \\ 0 & 1 \end{pmatrix} & \text{for the fuzzy elliptic paraboloid} \\ e^{i\operatorname{Im}(\alpha\bar{\beta})} \begin{pmatrix} 1 \\ \epsilon(\bar{\alpha} - \bar{\beta}) \\ 1 \end{pmatrix} & \text{for the fuzzy hyperbolic paraboloid} \end{cases} \quad (88) $$

The fuzzy hyperboloid is totally unlinkable and no displacement operator exists. Nevertheless, the fuzzy hyperboloid is “paralinkable” and it is possible to generalise the discussion for it, see Appendix B.

**Property 5** If $(\mathcal{X}, \mathcal{H})$ is a selfadjoint irreducible representation of a Lie algebra $\mathfrak{g}$ (involving that $\mathcal{X}$ is stable by the commutator $[\bullet, \bullet]$), and if $|\Lambda(x)\rangle\rangle$ and $|\Lambda(y)\rangle\rangle$ are separable, then it exists a displacement operator $\mathbf{D}(y, x)$.

*Proof:* By an abusive notation we denote also by $\iota$ the induced unitary irreducible representation of the Lie group $\mathcal{G}$ of $\mathfrak{g}$ onto $\mathcal{H}$.

**Lemma 1** Let $\iota$ be an unitary irreducible representation of the Lie group $\mathcal{G}$ onto $\mathcal{H}$. $\forall \psi, \phi \in \mathcal{H}, \|\psi\| = \|\phi\| = 1, \exists g \in \mathcal{G}$ such that $|\psi\rangle = \iota(g)|\phi\rangle$

By reductio ad absurdum we suppose that $\exists \psi, \phi \in \mathcal{H} \setminus \{0\}$ with $|\psi\rangle \notin \iota(G)\mathrm{Lin}|\phi\rangle$. By construction $\iota(G)\mathrm{Lin}|\phi\rangle$ is stable by $\mathcal{G}$. $\iota$ being irreducible, this implies that $\iota(G)\mathrm{Lin}|\phi\rangle = \{0\}$ or $\iota(G)\mathrm{Lin}|\phi\rangle = \mathcal{H}$. But $\iota(G)\mathrm{Lin}|\phi\rangle = \mathcal{H}$ is in contradiction with $|\psi\rangle \notin \iota(G)\mathrm{Lin}|\phi\rangle$, and $\iota(G)\mathrm{Lin}|\phi\rangle = \{0\}$ implies that $\phi = 0$ in contradiction with $\phi$ normalisable.

$|\Lambda(x)\rangle\rangle$ being separable it can be written as $|\Lambda(x)\rangle\rangle = |O(x)\rangle\rangle \otimes |\Omega(x)\rangle\rangle$ with $|O(x)\rangle\rangle \in \mathbb{C}^2$ and $|\Omega(x)\rangle\rangle \in \mathcal{H}$. Two normalised vectors of $\mathbb{C}^2$ can be related by an unitary transformation: $|O(y)\rangle\rangle = u_{yx}|O(x)\rangle\rangle$ ($\mathbb{C}^2$ is an unitary irreducible representation of $U(2)$). Since $(\mathcal{X}, \mathcal{H})$ is an unitary irreducible representation of $\mathcal{G}$, $\exists g \in \mathcal{G}$ such that $|\Omega(y)\rangle\rangle = \iota(g)|\Omega(x)\rangle\rangle$. Let $(\mathcal{X}^\alpha, \mathrm{id})$ be a set of generators of $\mathfrak{g}$, and let $\xi_\alpha^i \in \mathbb{R}$ be such that $X^i = \xi_\alpha^i \iota(\mathcal{X}^\alpha)$ (and $\iota(\mathrm{id}) = \mathrm{id}$). We have then $\exists \mu_\alpha \in \mathbb{R}$ such that $\iota(g) = e^{i\varphi}e^{i\mu_\alpha\iota(\mathcal{X}^\alpha)} = e^{i\varphi}e^{i\mu_\alpha\xi_\alpha^i X^i} \equiv \mathbf{D}(y, x) \in e^{i\mathcal{X}}$. $\square$

**Example 1: Fuzzy sphere**

The Fuzzy sphere being defined by the $(2j+1)$-dimensional unitary irreducible representation of $\mathfrak{su}(2)$ and its quasicoherent states being separable, it satisfies this property. More precisely let $\mathcal{D}_j(x) = e^{i\theta\bar{m}\cdot\bar{J}}$ with $x = rj(\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta)$ and $\bar{m} = (\sin\varphi, -\cos\varphi, 0)$. The quasicoherent state of the Fuzzy sphere is then

$$ |\zeta\rangle_{1/2} \otimes |\zeta\rangle_j = \mathcal{D}_{1/2}(x) \otimes \mathcal{D}_j(x) |1/2, 1/2\rangle \otimes |j, j\rangle \quad (89) $$

$$ \zeta = e^{i\varphi} \tan \frac{\theta}{2}. \text{ We have [11]} $$

$$ \mathcal{D}_j(x')\mathcal{D}_j(x) = \mathcal{D}_j(x'')e^{i\Phi(x',x)J^3} \quad (90) $$

with $x'' = rj(\sin(\theta + \theta')\cos(\varphi + \varphi'), \sin(\theta + \theta')\sin(\varphi + \varphi'), \cos(\theta + \theta'))$ and $\Phi(x', x)$ is the solid angle of the geodesic triangle (at the surface of the sphere) linking the north pole to $x$ and $x'$. It follows that we have the following displacement operator for the Fuzzy sphere:

$$ u_{x'x} \otimes \mathbf{D}(x', x) = \mathcal{D}_{1/2}(x')e^{-\frac{1}{2}\Phi(\bar{x}, x)\sigma_3}\mathcal{D}_{1/2}(x')^{\dagger}\mathcal{D}_{1/2}(\bar{x}) \\ \otimes \mathcal{D}_j(x')e^{-i\Phi(\bar{x}, x)J^3}\mathcal{D}_j(x')^{\dagger}\mathcal{D}_j(\bar{x}) \quad (91) $$

with $\bar{x} = rj(\sin(\theta' - \theta)\cos(\varphi' - \varphi), \sin(\theta' - \theta)\sin(\varphi' - \varphi), \cos(\theta' - \theta))$.
Note that in general situations, the existence of a displacement operator is not ensured.

We can see the displacement operators as replacing the notion of smooth paths on a classical manifold (due to the quantum inseparability of the pure states, the usual notion of path has no meaning). More precisely, the path category of a classical manifold $\mathcal{PM}$ ($\text{Obj}\mathcal{PM} = M$ and $\text{Morph}\mathcal{PM}$ is the set of directed smooth paths on $M$ with the source and the target maps returning the end points) is replaced by the category $\mathcal{PM}$ with $\text{Obj}\mathcal{PM} = \mathcal{P}(\mathfrak{M})$, $\text{Morph}\mathcal{PM} = \text{InnAutEnv}(\mathfrak{X}) \times \mathcal{P}(\mathfrak{M})$ ($\text{InnAutEnv}(\mathfrak{X})$ denoting the set of inner automorphisms of $\text{Env}(\mathfrak{X})$) with source, target and identity maps defined by: $s(a, \omega) = \omega$, $t(a, \omega) = \omega \circ a$, $\text{id}_\omega = (\text{id}, \omega)$ and arrow composition defined by $(a', \omega \circ a) \circ (a, \omega) = (a \circ a', \omega)$.

Suppose that we proceed to a measurement with the probe at $x$, permitting to prepare $\mathfrak{M}$ in the quasicoherent state $|\Lambda(x)\rangle\rangle$. After that, we abruptly proceed to a second measurement with the probe at $y \neq x$ (in contrast with classical geometry, we cannot assume in quantum physics a smooth continuous measurement from $x$ to $y$). This second measurement has a probability $|\langle\langle\Lambda(y)|\Lambda(x)\rangle\rangle|^2 \neq 0$ to project $\mathfrak{M}$ in $|\Lambda(y)\rangle\rangle$. In this case, the evolution induced by the two measurements is defined by $u_{yx} \otimes \mathbf{D}(y, x)$. We can then well view $\mathbf{D}(y, x)$ as a “quantum path” from $\omega_x$ to $\omega_y$.

Moreover, the group of the automorphisms of the $C^*$-algebra $\text{Env}(\mathfrak{X})$ plays the role of the diffeomorphism group: $\text{Diff}_{n.c.}\mathfrak{M} \sim \text{AutEnv}(\mathfrak{X})$. This suggests a relation between the displacement operators and the diffeomorphisms of $M_\Lambda$. Let $\mathcal{M}_\Lambda$ be the category such that $\text{Obj}(\mathcal{M}_\Lambda) = M_\Lambda$, $\text{Morph}(\mathcal{M}_\Lambda) = \text{Diff}_{M_\Lambda} \times M_\Lambda$, $s(\varphi, x) = x$, $t(\varphi, x) = \varphi(x)$, $\text{id}_x = (\text{id}, x)$ and $(\varphi', \varphi(x)) \circ (\varphi, x) = (\varphi' \circ \varphi, x)$. We call $\mathcal{M}_\Lambda$ the eigen category of $\mathfrak{M}$. $\mathcal{M}_\Lambda$ and $\mathcal{PM}$ are related by the functor $\omega$:

$$ \left\{ \begin{array}{l} \omega(x) = \omega_x \\ \omega(\varphi, x) = (\text{Ad}_{\mathbf{D}(\varphi(x),x)}, \omega_x) \end{array} \right. \qquad (92) $$

The categorical aspects related to the noncommutative geometry are studied in details Appendix D, especially Appendix D.3 where we compare the categorical and the noncommutative geometries.

For the sake of the simplicity, we treat only the case of totally linkable fuzzy spaces. This assumption can be considered as too restrictive, but it is pertinent since it implies that the magnitude of the quantum entanglement be constant on $M_\Lambda$ (and so corresponds to fuzzy spaces homogeneous from the point of view of the quantum information properties). In Appendix B we generalise the concept of displacement operator in order to treat fuzzy spaces without this assumption (but with a weaker assumption of homogeneity consisting to have a single class of entanglement (entangled or separable) on the whole of $M_\Lambda$ but not necessary with a constant magnitude of the entanglement).

Linkability is a noncommutative equivalent of the classical notion of connectivity. A classical manifold $M_\Lambda$ is connected if $\forall x, y \in M_\Lambda$, it exists a smooth continuous path from $x$ to $y$. A fuzzy space $\mathfrak{M}$ is totally linkable if $\forall x, y \in M_\Lambda$ (eigenmanifold of $\mathfrak{M}$), it exists a “quantum path” $\mathbf{D}(y, x)$ from $\omega_x$ to $\omega_y$. But note that $\mathfrak{M}$ linkable $\Leftrightarrow M_\Lambda$ connected as we will see in the next with examples.
### 4.2.2. The linking vector observable
By analogy with the commutative case, we can define the length of a path $(a, \omega_P) \in \text{InnAutEnv}(\mathcal{X}) \times \mathcal{P}(\mathcal{X})$ if $a = e^{v_i L_{X^i}} = \text{Ad}_{e^{-v_i v_i} X^i}$ by

$$ \ell(\text{Ad}_{e^{-v_i v_i}}, \omega_P) = \omega_P(\sqrt{\gamma(L_V, L_V)}) \quad (93) $$

$$ = \text{tr} \left( P \sqrt{-\delta_{ij}[V, X^i][V, X^j]} \right) \quad (94) $$

$V \in \text{Der}(\mathcal{X})$ being the tangent vector at $(\text{Ad}_{e^{-v_i v_i}}, \omega_P)$, the formula is a noncommutative equivalent to $\ell(\mathcal{C}) = \int_{\mathcal{C}} \sqrt{\gamma(\partial_s, \partial_s)} ds$ with $\partial_s = \frac{\partial x^i}{\partial s} \partial_i \in T_{x(s)} \mathcal{C}$.

If $\mathbf{D}(y, x) = e^{i\mu_i(y,x)X^i} \in e^{i\mathcal{X}}$, $\gamma$ induces the following distance between two quasicoherent states:

$$ \text{dist}_{\gamma}(\omega_x, \omega_y) = \ell(\text{Ad}_{\mathbf{D}(y,x)}, \omega_x) \quad (95) $$

$$ = \langle\langle \Lambda(x) | \sqrt{-\delta_{ij}[\mu_k X^k, X^i][\mu_l X^l, X^j]} | \Lambda(x) \rangle\rangle \quad (96) $$

To interpret $\gamma$, consider two linkable infinitely close points $\vec{x}$ and $\vec{y} = \vec{x} + \delta\vec{x}$. The probe is moved from $x$ to $y$, measurements being performed at these two end points but not between them. In contrast with classical geometry where we can assume a continuously performed measurement along a smooth path, in a quantum context, due to the Born projection rule, this cannot be possible. We must then only consider two end points where measurements are performed, the “path” between them being not physically defined (since no measure occurs). This is the reason for which the equivalent of a smooth path in $\mathfrak{M}$ is an operator depending only on two end points (or is a succession of this kind of operators). The distance between the two points in $\mathbb{R}^3$ is then $\|\delta\vec{x}\|$ and along $M_\Lambda$ is $\text{dist}_\gamma(x, y) = \gamma_{ab}(x)\delta x^a \delta x^b$. But $\vec{x} = \omega_x(\vec{X}) = \langle\langle \Lambda(x) | \vec{X} | \Lambda(x) \rangle\rangle$ is only the mean value of the starting point, which is subject to standard deviation $\Delta_x X^i = \sqrt{\omega_x([X^i]^2) - [\omega_x(X^i)]^2}$. In the reality the measurement of the starting point obeys to a random process, and the coordinates of the starting point are random variables described by $\vec{X}$ for a probability law described by $\omega_x$. In the same manner, the coordinates of the final point are random variables described also by $\vec{X}$ with the probability law $\omega_y$. Since $|\Lambda(y)\rangle\rangle = u_{yx} \otimes \mathbf{D}(y, x)|\Lambda(x)\rangle\rangle$ and $\omega_y(\vec{X}) = \omega_x(\mathbf{D}(y, x)^\dagger \vec{X} \mathbf{D}(y, x))$ we can state that the coordinates of the final point are random variables described by $\mathbf{D}(y, x)^\dagger \vec{X} \mathbf{D}(y, x)$ with the probability law $\omega_x$. The vector linking the two end points is then a random variable of probability law $\omega_x$ described by the following observable:

$$ \delta\vec{\ell}_{yx} = \mathbf{D}(y, x)^\dagger \vec{X} \mathbf{D}(y, x) - \vec{X} \quad (97) $$

$$ = -i[\delta\Pi_x, \vec{X}] + \mathcal{O}(\|\delta\vec{x}\|^2) \quad (98) $$

where we have set $\mathbf{D}(y, x) = \mathbf{D}(x + \delta x, x) = e^{i\delta\Pi_x}$. And so, the distance between the two measured end points is a random variable of probability low $\omega_x$ described by the following observable:

$$ |\delta\ell_{yx}| = \sqrt{(\delta\vec{\ell}_{yx})^2} \quad (99) $$

$$ = \sqrt{-\delta_{ij}[\delta\Pi_x, X^i][\delta\Pi_x, X^j]} \quad (100) $$

$$ = \sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})} \quad (101) $$

$\omega_x(|\delta\ell_{yx}|) = \text{dist}_{\gamma}(\omega_x, \omega_y)$ is the mean value of the distance observable.
We can note that $\mathbf{D}(y, x)^{\dagger} \mathcal{D}_{x} \mathbf{D}(y, x) - \mathcal{D}_{x} = \sigma_{i} \otimes \delta \ell_{yx}^{i}$. It follows that

$$ (\delta \vec{\ell}_{yx})^2 = \frac{1}{2} \mathrm{tr}_{\mathbb{C}^2} (\mathbf{D}(y, x)^{\dagger} \mathcal{D}_{x} \mathbf{D}(y, x) - \mathcal{D}_{x})^2 \quad (102) $$

Let $\Delta \mathcal{D}_{yx} = \mathbf{D}(y, x)^{\dagger} \mathcal{D}_{x} \mathbf{D}(y, x) - \mathcal{D}_{x}$ be the energy gap observable between $x$ and $y$:

$$ \omega_x(\Delta \mathcal{D}_{x+\delta x, x}) = \omega_{x+\delta x}(\mathcal{D}_{x}) - \omega_x(\mathcal{D}_{x}) = \omega_{x+\delta x}(\mathcal{D}_{x}) \quad (103) $$

we have

$$ \mathrm{dist}_{\gamma}(\omega_x, \omega_{x+\delta x}) = \omega_x \left( \sqrt{\frac{1}{2} \mathrm{tr}_{\mathbb{C}^2} (\Delta \mathcal{D}_{x+\delta x, x}^2)} \right) \quad (104) $$

$\frac{1}{2} \mathrm{tr}_{\mathbb{C}^2}(\Delta \mathcal{D}_{x+\delta x, x}^2)$ is the average square energy gap observable for the microcanonical distribution $\frac{1}{2}$id of the spin degree of freedom. Since $\mathrm{tr}_{\mathbb{C}^2} \Delta \mathcal{D}_{x+\delta x, x} = 0$, $\sqrt{\frac{1}{2} \mathrm{tr}_{\mathbb{C}^2}(\Delta \mathcal{D}_{x+\delta x, x}^2)}$ is the uncertainty of the energy gap for the microcanonical distribution, and so $\mathrm{dist}_{\gamma}(\omega_x, \omega_{x+\delta x})$ is the mean value of the energy gap “isotropic” uncertainty (“isotropic” since the uncertainty is computed for the equiprobable direction state distribution). A discrete path $\mathcal{C} = \{x_0, x_1, \dots, x_N\}$ on $M_{\Lambda}$ such that $\omega_{x_n}(|\delta \ell_{x_{n+1}, x_n}|)$ is a small constant value minimises then the quantity $\sum_{n=0}^{N-1} \omega_{x_n} \left( \sqrt{\frac{1}{2} \mathrm{tr}_{\mathbb{C}^2}(\Delta \mathcal{D}_{x_{n+1}, x_n}^2)} \right)$.

### Example 3: “Fuzzy” circle

The definition of the noncommutative metric can be extended to fuzzy spaces with disconnected eigenmanifold $M_{\Lambda}$. We can consider for example the case of the perturbed fuzzy circle with $Z = \sum_{n=0}^{N-1} |n\rangle\langle n+1|$ and $X^3 = \epsilon \hat{N}$ with $\hat{N} = \sum_{n=0}^{N-1} n|n\rangle\langle n|$ and $\epsilon \simeq 0$. By applying the perturbation theory for a separable weakly degenerate quasicoherent state (Appendix A.2), we have for the vertical deformation:

$$ \delta x^3(q) = \epsilon \langle\langle \Lambda_{\psi}(e^{i \frac{2\pi q}{N}}) | \hat{N} | \Lambda_{\psi}(e^{i \frac{2\pi q}{N}}) \rangle\rangle \quad (105) $$

$$ = \frac{\epsilon}{N} \sum_{n=0}^{N-1} n \quad (106) $$

$$ = \epsilon \frac{N-1}{2} \quad (107) $$

The perturbed fuzzy circle is just translated in the $x^3$-direction. Since $|\Lambda_{\psi}(e^{i \frac{2\pi q}{N}})\rangle = |\psi\rangle \otimes \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{i \frac{2\pi q}{N} n} |n\rangle$ we have (at the zero order of perturbation)

$$ |\Lambda_{\psi}(e^{i \frac{2\pi(q+p)}{N}})\rangle = \mathbf{D}^p |\Lambda_{\psi}(e^{i \frac{2\pi q}{N}})\rangle \quad (108) $$

with

$$ \mathbf{D} = \sum_{n=0}^{N-1} e^{i \frac{2\pi}{N} n} |n\rangle \langle n| = e^{i \frac{2\pi}{N} \hat{N}} \quad (109) $$

The points labelled $q$ and $q+p$ are then linkable ($\forall p \in \mathbb{N}$), with a displacement operator $\mathbf{D}(q+p, q) = \mathbf{D}^p$. $\delta\Pi = \frac{2\pi}{N}\hat{N}$ and the linking vector observable is

$$ \delta \vec{\ell}_{q+1,q} = -i \frac{2\pi}{N} [\hat{N}, \vec{X}] \quad (110) $$

$$ = -i \frac{2\pi}{N} \left( \frac{Z^\dagger - Z}{2} \partial_1 - \frac{Z + Z^\dagger}{2i} \partial_2 \right) \quad (111) $$
since $[\hat{N}, Z] = -Z$ and $[\hat{N}, Z^\dagger] = Z^\dagger$. It follows that

$$ \mathrm{dist}_{\gamma}(\omega_{q+1}, \omega_q) = \sqrt{\gamma(L_{\delta\Pi}, L_{\delta\Pi})} = \frac{2\pi}{N} \qquad (112) $$

The noncommutative distance between the points labelled by $q$ and $q+1$ is then the distance along the unit circle between the two points. While $M_\Lambda$ is disconnected, it is totally linkable permitting to compute the noncommutative distance (in contrast with the fuzzy hyperboloid for which $M_\Lambda$ is connected permitting to compute $\gamma$ but is totally unlinkable).

The linking vector observable can also be interpreted as a noncommutative gauge field onto $\mathfrak{M}$ as we can see this in the following section.

### 4.2.3. The noncommutative gauge potential

The displacement operator defines a noncommutative gauge potential on $\mathfrak{M}$, $\forall D \in \mathrm{Env}(\mathfrak{X})$:

$$ A_D = \iota D^\dagger dD \in \Omega_{\mathrm{Der}}^1(\mathfrak{X}) \qquad (113) $$

which is directly associated with the displacement by

$$ \omega_x(A_D(y,x)(L_{X^i})) = \langle\langle \Lambda(x) | (D(y,x)^\dagger X^i D(y,x) - X^i) | \Lambda(x) \rangle\rangle \qquad (114) $$

$$ = y^i - x^i \qquad (115) $$

$\omega_x(A_D(y,x)(L_{X^i}))$ is then the distance between $x$ and $y$ in the $i$-direction of $\mathbb{R}^3$. In other words, the linking vector observable previously introduced eq.(97) is

$$ \delta \vec{\ell}_{yx} = A_D(y,x)(L_{\vec{x}}) \qquad (116) $$

and so

$$ A_D(x+\delta x, x) = d\delta\Pi_x + \mathcal{O}(\|\delta\vec{x}\|^2) \qquad (117) $$

where $y = x + \delta x$ and $D(y, x) = e^{i\delta\Pi_x}$. Since by construction $dD = -\iota A_D$, we can see $D(y, x)$ as the noncommutative path-ordered exponential of $A_D$ along the noncommutative path $(\mathrm{Ad}_{D(y,x)}, \omega_x): \forall \varphi \in \mathrm{Diff} M_\Lambda$ and $\forall x \in M_\Lambda$:

$$ D(\varphi(x), x) \equiv \mathbb{P} e^{-i \int_{\omega(\varphi, x)}^{\varphi} A_D} \qquad (118) $$

$D(\varphi(x), x)$ can be then viewed as the noncommutative geometric phase accumulated along the arrow $(\varphi, x) \in \mathrm{Morph}(\mathcal{M}_\Lambda)$.

Under an external gauge change $J \in SO(3)$, the noncommutative gauge potential becomes

$$ \hat{A}_D = G_J^\dagger A_D G_J + \iota G_J^\dagger dG_J \qquad (119) $$

Indeed $D(J^{-1}x_2, J^{-1}x_1) = D(x_2, x_1)G_J(x_2, x_1) \Rightarrow dD(J^{-1}x_2, J^{-1}x_1) = (dD(x_2, x_1))G_J(x_2, x_1) + D(x_2, x_1)dG_J(x_2, x_1)$. Then $\iota \hat{D}^\dagger d\hat{D} = \iota G_J^{-1}(D^{-1}dD)G_J + \iota G_J^{-1}dG_J$, proving the formula for $\hat{A}_D$.

**Property 6** The noncommutative curvature is zero.
Proof: $\forall X, Y \in \mathcal{X}$,

$$ \mathbf{F}_{\mathbf{D}}(L_X, L_Y) = \mathbf{d}\mathbf{A}_{\mathbf{D}}(L_X, L_Y) - \iota[\mathbf{A}_{\mathbf{D}}(L_X), \mathbf{A}_{\mathbf{D}}(L_Y)] \quad (120) $$

$$ = [X, \mathbf{A}_{\mathbf{D}}(L_Y)] - [Y, \mathbf{A}_{\mathbf{D}}(L_X)] - \mathbf{A}_{\mathbf{D}}(L_{[X,Y]}) \\ - \iota[\mathbf{A}_{\mathbf{D}}(L_X), \mathbf{A}_{\mathbf{D}}(L_Y)] \quad (121) $$

$$ = \iota[X, \mathbf{D}^{\dagger}[Y, \mathbf{D}]] - \iota[Y, \mathbf{D}^{\dagger}[X, \mathbf{D}]] - \iota \mathbf{D}^{\dagger}[[X, Y], \mathbf{D}] \\ + \iota[\mathbf{D}^{\dagger}[X, \mathbf{D}], \mathbf{D}^{\dagger}[Y, \mathbf{D}]] \quad (122) $$

$$ = \iota[X, \mathbf{D}^{\dagger}Y\mathbf{D}] - \iota[X, Y] - \iota[Y, \mathbf{D}^{\dagger}X\mathbf{D}] + \iota[Y, X] \\ - \iota\mathbf{D}^{\dagger}[X, Y]\mathbf{D} + \iota[X, Y] \\ + \iota \underbrace{[\mathbf{D}^{\dagger}X\mathbf{D} - X, \mathbf{D}^{\dagger}Y\mathbf{D} - Y]}_{\mathbf{D}^{\dagger}[X,Y]\mathbf{D} - [\mathbf{D}^{\dagger}X\mathbf{D},Y] - [X,\mathbf{D}^{\dagger}Y\mathbf{D}]+[X,Y]}} \quad (123) $$

$$ = 0 \quad (124) $$

□

### 4.3. About the two distances between quasicoherent states

Consider a path $\mathcal{C}$ on $M_\Lambda$ followed by the probe and suppose that along this path measurements are performed on a sequence of infinitely close points $x_0, x_1, ..., x_N$. We can make an analogy between the quantum random processes with classical stochastic processes by considering that the position of n-th point of the sequence is a classical random variable of gaussian law with mean value $\bar{x}_n$ and standard deviation $\Delta_{x_n} \bar{X}$ (this is just an analogy, the quantum process cannot be identified as this because of the noncommutativity of the observables $X^i$ implying that the result of measurements depends on the ordering of the different measures, and so the measurement of the three coordinates is a counterfactual definiteness). In this analogy, the observed trajectory is a Brownian motion. By repeating the experiment, we have then a beam of random paths of average path $\mathcal{C}$. But the length of the average path is not the average length of the random paths. In this analogy, $\ell(\mathcal{C}) = \int_{\mathcal{C}} \sqrt{\gamma(\partial_s, \partial_s)} ds$ is the length of the average path whereas $\ell((\omega_{x_n})_n) = \sum_{n=0}^{N-1} |\delta\ell_{x_{n+1}x_n}| = \sum_{n=0}^{N-1} \sqrt{\gamma(L_{\delta\Pi_{x_n}}, L_{\delta\Pi_{x_n}})}$ is the average length of the random paths. The classical metric $\gamma$ (which results from $d\ell^2 = d\bar{x}_n^2$) is then the square length measure onto the average manifold $M_\Lambda$ whereas the quantum metric $\gamma$ is the average square length measure onto “quantum fluctuations” of the fuzzy space. $\text{dist}_\gamma(x + \delta x, x) = \sqrt{\gamma_{ab}(x)\delta x^a \delta x^b}$ is the distance between the average positions of the states $\omega_x$ and $\omega_{x+\delta x}$ (which are the normal states closest to the notion of point in $\mathfrak{M}$), whereas $\text{dist}_\gamma(\omega_{x+\delta x}, \omega_x) = \omega_x(|\delta\ell_{x+\delta x, x}|)$ is the mean value in the state $\omega_x$ of the distance observable between the two probe positions $x$ and $x + \delta x$.

In another viewpoint, suppose that $\delta\Pi_x$ commutes with two coordinate observables but not with $X^1$. In this case, $\sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})} = -\iota[\delta\Pi_x, X^1]$ and then we have the following Heisenberg uncertainty relation $\Delta_x X^1 \Delta_x \delta\Pi_x \ge \frac{1}{2}|\omega_x(\sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})})| = \frac{1}{2}\text{dist}_\gamma(\omega_{x+\delta x}, \omega_x)$. $L_{\delta\Pi_x}$ being the generator of the displacement operator it can be viewed as an infinitesimal translation operator, and so as a linear momentum observable (conjugated to $X^1$ here) in the fuzzy space. $\text{dist}_\gamma(\omega_{x+\delta x}, \omega_x)$ is then position-momentum uncertainty product in the fuzzy space (we recall that the quasi-coherent states minimising the quantum uncertainties). But in the fuzzy space, the position observables are not independent $[X^i, X^j] \ne 0$, and so in general a linear
momentum observable $\delta\Pi_x$ does not commute with any position observable. So, $\sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})}$ is a kind of average commutator of the linear momentum observable with the position observables. We could then define $\Delta_x X_x \equiv \frac{\sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})}}{2\Delta_x V_x}$ as the quantum uncertainty of the location in the direction defined by $L_{\delta\Pi_x}$ (even if we cannot define a position observable $X_x$ conjugated to $L_{\delta\Pi_x}$: in general $\not{X}_x$ such that $[\delta\Pi_x, X_x]^2 = \delta_{ij}[\delta\Pi_x, X^i][\delta\Pi_x, X^j]$).

We can also relate the double definition of the metric with the category structure of $\mathcal{M}_\Lambda$. For $x, x + dx \in \text{Obj}(\mathcal{M}_\Lambda) = M_\Lambda$, $\text{dist}_\gamma(x, x + dx) = \sqrt{\gamma_{ab}(x)}dx^a dx^b$ is the distance between two objects. For $(\varphi, x) \in \text{Morph}(\mathcal{M}_\Lambda) = \text{Diff}(M_\Lambda) \times M_\Lambda$ with $\varphi(x) - x \equiv \delta x$ (infinitesimal diffeomorphism), $\text{dist}_\gamma(\omega_x, \omega_{\varphi(x)}) = \omega_x(\sqrt{\gamma(L_{\delta\Pi_x}, L_{\delta\Pi_x})})$ is the length of the arrow $\omega(\varphi, x) = (\text{Ad}_{\mathcal{D}(\varphi(x), x)}, \omega_x)$, whereas $\text{dist}_\gamma(x, \varphi(x))$ is the distance between its source and target. Appendix D.3 summarises the relations between the different metrics.

## Example 2: Fuzzy surface plots

From the displacement operator we find that the linear momentum observable is

$$ \delta\Pi_{\alpha} = -i(\delta\alpha a^{+} - \delta\bar{\alpha}a) \qquad (125) $$

In physical models, we can see $\delta\alpha$ as the complex representation of an electric field, if we interpret $a$ and $a^+$ as creation/annihilation operator of photons, the linear momentum observable $\delta\Pi_\alpha$ can be identified with the second quantised electric field operator.

$$ [\delta\Pi_{\alpha}, X^1] = i\text{Re}(\delta\alpha) \qquad (126) $$

$$ [\delta\Pi_{\alpha}, X^2] = i\text{Im}(\delta\alpha) \qquad (127) $$

It follows that for the fuzzy plane $\gamma(L_{\delta\Pi_\alpha}, L_{\delta\Pi_\alpha}) = |\delta\alpha|^2$ and then

$$ \text{dist}_{\gamma}(\alpha, \alpha + \delta\alpha) = \text{dist}_{\gamma}(\omega_{\alpha}, \omega_{\alpha + \delta\alpha}) = |\delta\alpha| \qquad (128) $$

For the fuzzy plane, the length of the average path is equal to the average length of the random paths.

For the fuzzy elliptic paraboloid, we have

$$ [\delta\Pi_{\alpha}, X^3] = -i\epsilon[\delta\alpha a^{+} - \delta\bar{\alpha}a, a^{+}a] \qquad (129) $$

$$ = i\epsilon(\delta\alpha a^{+} + \delta\bar{\alpha}a) \qquad (130) $$

It follows that $\gamma(L_{\delta\Pi_\alpha}, L_{\delta\Pi_\alpha}) = |\delta\alpha|^2 + \epsilon^2(\delta\alpha a^{+} + \delta\bar{\alpha}a)^2$ and so

$$ \sqrt{\gamma(L_{\delta\Pi_\alpha}, L_{\delta\Pi_\alpha})} = |\delta\alpha| + \frac{\epsilon^2}{2} |\delta\alpha| \left( \frac{\delta\alpha}{\delta\bar{\alpha}}(a^+)^2 + \frac{\delta\bar{\alpha}}{\delta\alpha}a^2 + 2a^+a + 1 \right) + \mathcal{O}(\epsilon^3) \qquad (131) $$

and then

$$ \text{dist}_{\gamma}(\omega_{\alpha}, \omega_{\alpha + \delta\alpha}) = |\delta\alpha| \left( 1 + \epsilon^2 \left( \text{Re} \left( \alpha^2 \frac{\delta\bar{\alpha}}{\delta\alpha} \right) + |\alpha|^2 + 1/2 \right) \right) + \mathcal{O}(\epsilon^3) \qquad (132) $$

whereas

$$ \text{dist}_{\gamma}(\alpha, \alpha + \delta\alpha) = (1 + 4\epsilon^2 \text{Re}(\alpha)^2)\delta\text{Re}(\alpha)^2 + (1 + 4\epsilon^2 \text{Im}(\alpha)^2)\delta\text{Im}(\alpha)^2 + 8\epsilon^2 \text{Re}(\alpha)\text{Im}(\alpha)\delta\text{Re}(\alpha)\delta\text{Im}(\alpha) \qquad (133) $$

To help the comparison, we can consider the case $\alpha = x \in \mathbb{R}$ and $\delta\alpha = \delta x \in \mathbb{R}$, for which

$$ \text{dist}_{\gamma}(\omega_x, \omega_{x+\delta x}) = \delta x(1 + 2\epsilon^2 x^2 + \epsilon^2/2) + \mathcal{O}(\epsilon^3) \qquad (134) $$

$$ \text{dist}_{\gamma}(x, x + \delta x) = \delta x(1 + 2\epsilon^2 x^2) \qquad (135) $$
<table><caption>**Table 2.** Behaviours of the energy with respect to the different paths for $\Delta t$ a fixed small value. $\Psi(t)$ being defined eq.(15)</caption><thead><tr><th></th><th>energy mean value</th><th>energy uncertainty</th></tr></thead><tbody><tr><td>strong adiabatic transport along any path</td><td>⟨⟨D<sub>x(t)</sub>⟩⟩<sub>Ψ(t)</sub> = 0</td><td>Δ<sub>Ψ(t)</sub>D<sub>x(t)</sub><sup>2</sup> = 0</td></tr><tr><td>strong adiabatic transport along a minimising geodesic path</td><td>⟨⟨D<sub>x(t)</sub>⟩⟩<sub>Ψ(t+Δt)</sub> = 0</td><td>Δ<sub>Ψ(t+Δt)</sub>D<sub>x(t)</sub><sup>2</sup> minimised</td></tr><tr><td>strong adiabatic transport along a path such that ω<sub>x(t)</sub>(|δℓ<sub>x(t+Δt),x(t)</sub>|) is a small constant</td><td>⟨⟨D<sub>x(t)</sub>⟩⟩<sub>Ψ(t+Δt)</sub> = 0</td><td>⟨⟨√(1/2) tr<sub>C2</sub> ΔD<sub>x(t+Δt),x(t)</sub><sup>2</sup>⟩⟩<sub>Ψ(t)</sub> minimised</td></tr></tbody></table>

As expected, the average length of the random paths is larger than the length of the average path.

For the fuzzy hyperbolic paraboloid, we have the same thing:

$$[\delta\Pi_{\alpha}, X^3] = -\frac{\ell}{2}\epsilon[\delta\alpha a^+ - \delta\bar{\alpha}a, a^2 + (a^+)^2] \quad (136)$$

$$= \iota\epsilon(\delta\alpha a + \delta\bar{\alpha}a^+) \quad (137)$$

and then

$$\sqrt{\gamma(L_{\delta\Pi_{\alpha}}, L_{\delta\Pi_{\alpha}})} = |\delta\alpha| + \frac{\epsilon^2}{2} |\delta\alpha| \left( \frac{\delta\alpha}{\delta\bar{\alpha}} a^2 + \frac{\delta\bar{\alpha}}{\delta\alpha} (a^+)^2 + 2a^+ a + 1 \right) + \mathcal{O}(\epsilon^3) \quad (138)$$

Finally we have (by taken into account the contribution of the entanglement of $|\Lambda(\alpha)\rangle\rangle$):

$$\mathrm{dist}_{\gamma}(\omega_{\alpha}, \omega_{\alpha+\delta\alpha}) = |\delta\alpha| \left( 1 + \epsilon^2 \left( \mathrm{Re} \left( \alpha^2 \frac{\delta\alpha}{\delta\bar{\alpha}} \right) + 2|\alpha|^2 + 3/4 \right) \right) + \mathcal{O}(\epsilon^3) \quad (139)$$

whereas

$$\mathrm{dist}_{\gamma}(\alpha, \alpha + \delta\alpha) = (1 + 4\epsilon^2 \mathrm{Re}(\alpha)^2)\delta\mathrm{Re}(\alpha)^2 + (1 + 4\epsilon^2 \mathrm{Im}(\alpha)^2)\delta\mathrm{Im}(\alpha)^2 \\ - 8\epsilon^2 \mathrm{Re}(\alpha)\mathrm{Im}(\alpha)\delta\mathrm{Re}(\alpha)\delta\mathrm{Im}(\alpha) \quad (140)$$

Anew for $\alpha = x \in \mathbb{R}$ and $\delta\alpha = \delta x \in \mathbb{R}$ we have

$$\mathrm{dist}_{\gamma}(\omega_x, \omega_{x+\delta x}) = \delta x(1 + 3\epsilon^2 x^2 + \frac{3}{4}\epsilon^2) + \mathcal{O}(\epsilon^3) \quad (141)$$

$$\mathrm{dist}_{\gamma}(x, x + \delta x) = \delta x(1 + 2\epsilon^2 x^2) \quad (142)$$

or for $\alpha = (1 + i)x$ and $\delta\alpha(1 + i)\delta x$,

$$\mathrm{dist}_{\gamma}(\omega_{(1+i)x}, \omega_{(1+i)(x+\delta x)}) = \sqrt{2}\delta x(1 + 2\epsilon^2 x^2 + \frac{3}{4}\epsilon^2) + \mathcal{O}(\epsilon^3) \quad (143)$$

$$\mathrm{dist}_{\gamma}((1 + i)x, (1 + i)(x + \delta x)) = \sqrt{2}\delta x \quad (144)$$

Table 2 summarises the interpretations concerning the energy of the paths chosen on $M_{\Lambda}$.

### 4.4. The quantum geometric tensor

To characterise the geometry of quantum state spaces, Provost and Valle introduced the concept of quantum geometric tensor $Q_{ab} = \gamma_{ab} + i\mathcal{S}_{ab}$ in [33] (see also [30, 34]) which is defined such that $Q_{ab}$ is invariant by phase change, the symmetric part $\gamma_{ab}$ is
the component of a relevant metric of the manifold of quantum states, and the anti-symmetric part $\mathcal{S}_{ab}$ is the component of a relevant symplectic form of the manifold of quantum states. For example, in adiabatic approximations where the manifold is the set of parameters of a continuous eigenvector, or when the manifold is the projected Hilbert space of all quantum states, $Q_{ab} = \langle \partial_a \psi | (1 - |\psi\rangle\langle\psi|) |\partial_b \psi\rangle$ (with $\|\psi\| = 1$). $\gamma$ is the Fubini-Study metric (see [30]) measuring the distance between two quantum states by $|\langle\psi_1|\psi_2\rangle|^2$ and $\mathcal{S}$ is the Berry curvature measuring the non-adiabatic factors.

In accordance with the definition of the metric of $M_\Lambda$, the quantum geometric tensor of the quasicoherent geometry is defined by

$$Q_{ab} = \langle\langle \partial_a \Lambda | \mathcal{D}_x^2 | \partial_b \Lambda \rangle\rangle \qquad (145)$$

$$= \gamma_{ab} + i \mathcal{S}_{ab} \qquad (146)$$

where $\gamma_{ab}$ is the metric of $M_\Lambda$ and where

$$\mathcal{S}_{ab} = \varepsilon_{ij}{}^k \langle\langle \Lambda | \sigma_k | \Lambda \rangle\rangle \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} \qquad (147)$$

$$= \frac{1}{2} \varepsilon_{ij}{}^k \langle\langle \Lambda | \sigma_k | \Lambda \rangle\rangle \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b} \qquad (148)$$

The symplectic 2-form is then

$$\mathcal{S} = \varepsilon_{ij}{}^k \langle\langle \Lambda | \sigma_k | \Lambda \rangle\rangle \frac{\partial x^i}{\partial s^1} \frac{\partial x^j}{\partial s^2} ds^1 \wedge ds^2 \qquad (149)$$

$$= \vec{n}_x \cdot (\vec{t}_{x1} \times \vec{t}_{x2}) ds^1 \wedge ds^2 \qquad (150)$$

$$= \|\vec{n}_x\| ds^1 \wedge ds^2 \qquad (151)$$

where $\vec{t}_{xa} = \frac{\partial x^i}{\partial s^a} \partial_i$ are two orthonormal tangent vectors of $M_\Lambda$ at $x$ (with $\{s^1, s^2\}$ local curvilinear coordinates on $M_\Lambda$) and so $\vec{t}_{x1} \times \vec{t}_{x2}$ is the unitary normal vector at $x$. $\mathcal{S}$ inherits then its interpretation from $\|\vec{n}_x\|$. It measures the quantum purity of the quasicoherent geometry at $x$ (inverse measure of the entanglement of the quasicoherent state). For this reason, we call $\mathcal{S}$ the purity form of $M_\Lambda$. Note that in accordance with this interpretation, $\mathcal{S}$ can be degenerate at $x$ ($\mathcal{S}(x) = 0$) if $|\Lambda(x)\rangle\rangle$ is maximally entangled.

To finish, it is necessary to verify that $Q_{ab}$ is independent of the gauge choice:

**Property 7** $Q_{ab}$ is invariant by internal gauge changes.

*Proof:* Firstly, let $|\tilde{\Lambda}\rangle = h|\Lambda\rangle$ with $h \in U(1)$. We have then $|\partial_a \tilde{\Lambda}\rangle = \frac{\partial h}{\partial s^a} |\Lambda\rangle + h|\partial_a \Lambda\rangle$ and then $|\tilde{D}_x|\partial_a \tilde{\Lambda}\rangle = h|\tilde{D}_x|\partial_a \tilde{\Lambda}\rangle$. It follows that $\tilde{Q}_{ab} = \langle\langle \partial_a \tilde{\Lambda} | \tilde{D}_x | \partial_b \tilde{\Lambda} \rangle\rangle = Q_{ab}$.

Now let $|\tilde{\Lambda}\rangle = u|\Lambda\rangle$ with $u \in U_x \subset SU(2)_{a.s.}$. $|\tilde{D}_x u|\Lambda\rangle = 0 \Rightarrow |\tilde{D}_x|\partial_a \tilde{\Lambda}\rangle = \sigma_i |\tilde{\Lambda}\rangle \frac{\partial x^i}{\partial s^a}$. It follows that $\tilde{Q}_{ab} = \gamma_{ab} + i\varepsilon_{ij}{}^k \langle\langle \Lambda | u^\dagger \sigma_k u | \Lambda \rangle\rangle \frac{\partial x^i}{\partial s^a} \frac{\partial x^j}{\partial s^b}$.

* If $\mathfrak{M}$ is strongly non-degenerate at $x$, then $u = \text{id}$ and $\tilde{Q}_{ab} = Q_{ab}$.

* If $\mathfrak{M}$ is not strongly non-degenerate and presents at least two independent non-trivial normal vectors at $x$, then we can set $\tilde{\mathcal{S}} = \mathcal{S} = 0$ since $\dim M_\Lambda \le 1$ at $x$ (and then $ds^1 \wedge ds^2 = 0$).

* In the other cases, $\langle\langle \Lambda | u^\dagger \tilde{\sigma} u | \Lambda \rangle\rangle = \langle\langle \Lambda | \tilde{\sigma} | \Lambda \rangle\rangle$ since $u$ belongs to the isotropy group of $\rho_\Lambda$ (or the all purity normal vectors are zero), and then $\tilde{Q}_{ab} = Q_{ab}$.
<figure><img src="image_6.png" /></figure>

We can remark that $\omega_x^*\pi_x^*\mathcal{S} = \frac{1}{2}\epsilon_{ij}{}^k\langle\langle\Lambda|\sigma_k|\Lambda\rangle\rangle dX^i \wedge dX^j$ (with $dX^i \wedge dX^j = dX^i \otimes dX^j - dX^j \otimes dX^i$). We can then set the spin noncommutative 2-form $\mathcal{S} = \frac{1}{2}\epsilon_{ij}{}^k\sigma_k \otimes dX^i \wedge dX^j$ to replace in the noncommutative geometry of $\mathfrak{M}$ the symplectic form (the dimension being odd no symplectic form can be defined); and since $\gamma = \delta_{ij}dX^i \otimes dX^j$, we set as quantum noncommutative geometric tensor:

$$ \mathcal{Q}_{ij} = \delta_{ij} + i\epsilon_{ij}{}^k \sigma_k \qquad (152) $$

which satisfies

$$ \mathcal{Q}_{ij} \otimes (X^i - x^i)(X^j - x^j) = \|\vec{X} - \vec{x}\|^2 + i\epsilon_{ij}{}^k \sigma_k \otimes X^i X^j \qquad (153) $$

$$ = \mathcal{D}_x^2 \qquad (154) $$

$$ = d\vec{\ell}_x^2 \qquad (155) $$

The meaning of $\mathcal{S}$ can be understood by considering two displacement operators $\mathbf{D}(y, x) = e^{i\delta\Pi_x^1}$ and $\mathbf{D}(z, x) = e^{i\delta\Pi_x^2}$ with $y = x + \delta x^1$ and $z = x + \delta x^2$ ($\delta x^1 \neq \delta x^2$).

$$ \mathcal{S}(L_{\delta\Pi_x^1}, L_{\delta\Pi_x^2}) = -\epsilon_{ij}{}^k \sigma_k \otimes [\delta\Pi_x^1, X^i][\delta\Pi_x^2, X^j] \qquad (156) $$

$$ = \epsilon_{ij}{}^k \sigma_k \otimes \delta\vec{\ell}_{yx}^i \delta\vec{\ell}_{zx}^j \qquad (157) $$

$$ = \vec{\sigma} \odot \delta\vec{\ell}_{yx} \times \delta\vec{\ell}_{zx} \qquad (158) $$

$\delta\vec{\ell}_{yx}$ and $\delta\vec{\ell}_{zx}$ being two linking vector observables at $M_\Lambda$ and $\vec{\sigma}$ being the normal vector observable at $M_\Lambda$, $\mathcal{S}(L_{\delta\Pi_x^1}, L_{\delta\Pi_x^2})$ is the volume observable of the noncommutative parallelepiped defined by $(\vec{\sigma}, \delta\vec{\ell}_{yx}, \delta\vec{\ell}_{zx})$. We can then interpret $\mathcal{S}$ as the non-commutative volume form of $\mathfrak{M}$. The quantum noncommutative geometric tensor $\mathcal{Q}_{ij}$ provides then the bivector metric $\gamma$ and the volume form $\mathcal{S}$ of the fuzzy space $\mathfrak{M}$ (and also by contraction with $X^i - x^i$ the square length observable metric $d\vec{\ell}_x^2$).

## 5. Adiabatic regimes and geodesics

The quasicoherent geometry is related to the eigenvector $|\Lambda(x)\rangle\rangle$ of $\mathcal{D}_x$. We can then consider for slow moves of the probe $t \mapsto x(t)$ onto $M_\Lambda$ the adiabatic regimes supported by $|\Lambda\rangle\rangle$. This could permit to define the classical spacetime closest to $\mathfrak{M}$, in the meaning that the space part is related to quasicoherent states (states minimising the Heisenberg uncertainties) and the time part is related to the adiabatic approximation (quantum dynamics closest to a semi-classical approximation).

### 5.1. The strong adiabatic regime

#### 5.1.1. Berry gauge potential and Berry curvature
The inner product $\langle\langle\bullet|\bullet\rangle\rangle$ defines a natural gauge potential on $M_\Lambda$:

$$ A = -i\langle\langle\Lambda|d|\Lambda\rangle\rangle \in \Omega^1(M_\Lambda, \mathbb{R}) \qquad (159) $$

$A$ is the generator of the geometric phase accumulated during a strong adiabatic transport of $|\Lambda\rangle\rangle$ along a path $\mathcal{C}$ on $M_\Lambda$ [30, 31]. If $|\Lambda\rangle\rangle$ is strongly nondegenerate along the path $\mathcal{C}$ linking $x$ and $y$, by using the standard adiabatic theorem [29] (strong adiabatic approximation), we have

$$ |\Psi(T)\rangle\rangle = e^{-i\int_{\mathcal{C}} A} |\Lambda(y)\rangle\rangle + \mathcal{O}(1/T) \quad \text{if } |\Psi(0)\rangle\rangle = |\Lambda(x)\rangle\rangle \qquad (160) $$
T being the transport duration along $\mathcal{C}$ and $|\Psi\rangle\rangle$ being the solution of the Schrödinger equation eq.(13). The condition of parallel transport associated with the strong adiabatic approximation is just the usual one introduced by Simon [30]:

$$ \langle\langle\Psi|\dot{\Psi}\rangle\rangle = 0 \qquad (161) $$

**Property 8** Under an inner gauge transformation $(h, u) \in U(1) \times U_x \subset U(1) \times SU(2)_{a.s.}$, the gauge potential $A$ becomes:

$$ \tilde{A} = A - i d \ln h - i \omega_x (u^{-1} du) \qquad (162) $$

Under a local external gauge transformation $J \in SO(3)_{M_\Lambda}$, the gauge potential $A$ becomes:

$$ \hat{A} = J^* A + i \omega_{J^{-1}x} (du_J u_J^{-1}) \qquad (163) $$

where $u_J \in SU(2)$ is the transformations associated with $J$ viewed as a linear map of $\mathbb{R}^3$: $(Jy)^i = J^i_j y^j$. $J^* : \Omega^1\mathbb{R}^3 \to \Omega^1\mathbb{R}^3$ is defined by $(J^*k)(x) = k_a(J^{-1}x)((J^{-1})^i_j dx^j - (J^{-1}dJJ^{-1})^i_j x^j)$, $\forall k \in \Omega^1\mathbb{R}^3$.

*Proof:* $|\tilde{\Lambda}\rangle\rangle = hu|\Lambda\rangle\rangle \Rightarrow d|\tilde{\Lambda}\rangle\rangle = hud|\Lambda\rangle\rangle + dh \times u|\Lambda\rangle\rangle + du \times h|\Lambda\rangle\rangle$. So $-i|d\tilde{\Lambda}\rangle\rangle\langle\langle\tilde{\Lambda}| = u(-i|d\Lambda\rangle\rangle\langle\langle\Lambda|)u^{-1} - i\tilde{h}^{-1}dh \times u|\Lambda\rangle\rangle\langle\langle\Lambda|u^{-1} - i\tilde{du}u^{-1} \times u|\Lambda\rangle\rangle\langle\langle\Lambda|u^{-1}$. By taking the total trace of this last equation we find the formula for $\tilde{A}$.

We recall that we have $|\hat{\Lambda}(x)\rangle\rangle = u_J^{-1}|\Lambda(J^{-1}x)\rangle\rangle$. Then $-i|d\hat{\Lambda}\rangle\rangle\langle\langle\hat{\Lambda}| = i\tilde{u}_J^{-1}du_J u_J^{-1}|\Lambda\rangle\rangle\langle\langle\Lambda|u_J - i\tilde{u}_J^{-1}J^*|d\Lambda\rangle\rangle\langle\langle\Lambda|u_J$ since by construction $df(J^{-1}x) = (J^*df)(x)$ for any function $f$. By taking the total trace of this last equation we find the formula for $\hat{A}$ $\square$

The Berry gauge potential defines the Berry curvature $F = dA \in \Omega^2(M_\Lambda, \mathbb{R})$ [30, 31]. In accordance with usual interpretations of the Berry phases, the gauge fields can be assimilated to virtual electromagnetic fields, as we can see this on the example of the fuzzy surface plots:

**Example 2: Fuzzy surface plots**

By using the following derivative rules:

$$ \frac{\partial}{\partial \alpha} |n\rangle_{\alpha} = \frac{\bar{\alpha}}{2} |n\rangle_{\alpha} + \sqrt{n+1} |n+1\rangle_{\alpha} \qquad (164) $$

$$ \frac{\partial}{\partial \bar{\alpha}} |n\rangle_{\alpha} = - \frac{\alpha}{2} |n\rangle_{\alpha} - \sqrt{n} |n-1\rangle_{\alpha} \qquad (165) $$

($\partial_{\bar{\alpha}}|0\rangle_{\alpha} = -\frac{\alpha}{2}|n\rangle_{\alpha}$), we have for the fuzzy plane and the fuzzy paraboloids, the following Berry potential:

$$ A = -i \langle\langle \Lambda | d | \Lambda \rangle\rangle \qquad (166) $$

$$ = -i \frac{\bar{\alpha} d\alpha - \alpha d\bar{\alpha}}{2} + \mathcal{O}(\epsilon^2) \qquad (167) $$

$$ = x^1 dx^2 - x^2 dx^1 + \mathcal{O}(\epsilon^2) \qquad (168) $$

$$ = r^2 d\theta + \mathcal{O}(\epsilon^2) \qquad (169) $$

$$ = r^2 d\theta + \mathcal{O}(\epsilon^2) \qquad (170) $$

with $r = |\alpha| = \sqrt{(x^1)^2 + (x^2)^2}$ and $\theta = \arg \alpha = \arctan \frac{x^2}{x^1}$. The rest $\mathcal{O}(\epsilon^2)$ is exactly 0 for the fuzzy plane. The associated Berry curvature is then

$$ F = dA \qquad (171) $$

$$ = i d\alpha \wedge d\bar{\alpha} + \mathcal{O}(\epsilon^2) \qquad (172) $$

$$ = 2dx^1 \wedge dx^2 + \mathcal{O}(\epsilon^2) \qquad (173) $$

$$ = 2rdr \wedge d\theta + \mathcal{O}(\epsilon^2) \qquad (174) $$
Following the usual interpretation of the Berry curvature, $A \in \Omega^1(M_\Lambda, \mathbb{R})$ and $F \in \Omega^2(M_\Lambda, \mathbb{R})$ are similar to a classical magnetic potential and to a classical magnetic field of a magnetic monopole at $\alpha = 0$, the magnetic field being normal to $M_\Lambda$ (even for the curved cases, this is due to the perturbative analysis at first order). Since $D(\beta, \alpha) = e^{(\beta-\alpha)a^{+}-(\bar{\beta}-\bar{\alpha})a}$ for the three models, we have

$$ \delta\Pi_{\alpha} = -\iota(\delta\alpha a^{+} - \delta\bar{\alpha}a) \qquad (175) $$

$$ = 2(\delta x^2 X^1 - \delta x^1 X^2) \qquad (176) $$

It follows that the noncommutative gauge potential is

$$ A_{D(\alpha+d\alpha,\alpha)} = d\delta\Pi_{\alpha} + \mathcal{O}(|\delta\alpha|^2) \qquad (177) $$

$$ = -\iota(\delta\alpha da^{+} - \delta\bar{\alpha}da) + \mathcal{O}(|\delta\alpha|^2) \qquad (178) $$

$$ = 2(\delta x^2 dX^1 - \delta x^1 dX^2) + \mathcal{O}(\|\delta\vec{x}\|^2) \qquad (179) $$

Since $\delta\alpha$ can be assimilated to the complex representation of an electric field, and if we interpret $a$ and $a^+$ as creation/annihilation operators of photons, we can see $A_{D(\alpha+d\alpha,\alpha)} \in \Omega_{\text{Der}}^1(\mathcal{X})$ as similar to a second quantised electric field in $\mathfrak{M}$. The zero noncommutative curvature $F_{\mathfrak{D}} = 0$ is the noncommutative equivalent to the classical zero curl of the electric field (in the stationary regime). The electric field like appears as a quantised field whereas the magnetic field like appears as a classical field.

The interpretation of the gauge fields as virtual electromagnetic fields in $M_\Lambda/\mathfrak{M}$ is just a useful analogy.

### 5.1.2. The foliated eigen spacetime of $\mathfrak{M}$
By invoking the adiabatic approximation in the previous section, we have considered the Schrödinger equation $i|\dot{\Psi}\rangle = \mathcal{D}_{x(t)}|\Psi\rangle$. The time $t$ here is the time corresponding to slow variations of the probe $x$ onto $M_\Lambda$. It is then the time of the clock of the observer which studies the fuzzy space $\mathfrak{M}$ by measurements on the probe. In BFSS matrix models, $t$ is the proper time of the classical probe D0-brane, or in other words, the proper time of an observer comoving with the test particle revealing the gravitational effects. In quantum information theory, $t$ is the control time which runs during the realisation of a logical gate. It is then natural to think the emergent classical geometry as a spacetime $M_\Lambda = \bigsqcup_{t \in \mathbb{R}} (t, M_\Lambda^{(t)})$ defined as a time foliation of leafs $M_\Lambda^{(t)}$ diffeomorphic to $M_\Lambda$. There is an infinite number of different foliations of $M_\Lambda$ and there is no reason to think that the pertinent foliation is the trivial one (with null shift vector). We argue that the shift vector consistent with the geometry of $\mathfrak{M}$ is $A^a = \gamma^{ab}A_b$ where $A \in \Omega^1(M_\Lambda, \mathbb{R})$ is the $U(1)$-gauge potential (providing the geometric interpretation of $A$). Indeed, suppose that the leaf $M_\Lambda^{(t)}$ is associated with $|\Lambda(x)\rangle$, then $M_\Lambda^{(t+dt)}$ is associated with $e^{-iA_a s^a dt}|\Lambda(x)\rangle$ by applying the strong adiabatic assumption. Starting at $|\Lambda(x)\rangle$ at $t$, we arrive at $e^{-iA_a s^a dt}|\Lambda(x)\rangle$ at $t+dt$. $A$ (by definition of a gauge potential on a principal bundle) defines then the shift between the two states. Since the quasicoherent states are the quantum states closest to the notion of points of a classical manifold, $A$ is then equivalent to the shift between the two pseudo-points. This corresponds to the definition of a shift vector of a foliated classical manifold [35]: let $x$ be the point on $M_\Lambda^{(t)}$ of curvilinear coordinates $s$, $x'$ be the point at the intersection of the normal vector to $M_\Lambda^{(t)}$ in $M_\Lambda$ at $x$ and $M_\Lambda^{(t+dt)}$, and $x''$ be the point of $M_\Lambda^{(t+dt)}$ of curvilinear coordinates $s$; then the shift vector of the foliation is $\overrightarrow{x'x''} = \vec{A}dt$. This choice of shift vector is moreover consistent with the interpretation in the adiabatic picture of $\mathcal{D}_x$ as the Dirac operator of a
fermionic string in the BFSS model [12].

The eigen spacetime $\mathcal{M}_\Lambda$ of $\mathfrak{M}$ is then endowed with the following metric:

$$d\tau^2 = dt^2 - (\gamma^{ac}A_c dt + ds^a)(\gamma^{bd}A_d dt + ds^b)\gamma_{ab} \quad (180)$$

(the laps function being supposed equal to 1). In BFSS matrix model, $\tau$ is the proper time of the fermionic string end moving onto the noncommutative D2-brane defined by $\mathfrak{M}$.

Note that $d\tau^2 = dt^2 - \omega_{s+ds}(\mathcal{D}_{x(s^a-\gamma^{ab}A_b dt)}^2)$ (by writing that $\mathcal{D}_{x(s^a-\gamma^{ab}A_b dt)} = \mathcal{D}_x + \gamma^{ab}A_b \frac{\partial x^i}{\partial s^b}\sigma_i dt + \mathcal{O}(dt^2)$). $\sqrt{\omega_{s+ds}(\mathcal{D}_{x(s^a-\gamma^{ab}A_b dt)}^2)}$ is the energy uncertainty with a double “slipping”, the measurement is realised with a misalignment $ds$ onto the trajectory of the shifted point $x(s^a - \gamma^{ab}A_b dt)$ corresponding to the leaf defined with a time lag $dt$. For quantum information theory, it is then $\sqrt{dt^2 - d\tau^2} = \sqrt{\omega_{s+ds}(\mathcal{D}_{x(s^a-\gamma^{ab}A_b dt)}^2)}$ which is physically meaningful (and not directly $d\tau$).

**Property 9** Let $h \in U(1)$ be an abelian inner gauge change. $d\tau^2$ is invariant under the gauge transformation $\tilde{A} = A - i\partial\ln h$ if and only if this one goes with $\varphi_t \in \text{Diff}\mathcal{M}_\Lambda$ such that $\varphi_t(x(s)) = x(\tilde{s})$ with $\tilde{s} = f(s, t)$ ($f \in C^\infty(\mathbb{R}^2 \times \mathbb{R}, \mathbb{R}^2)$) satisfying the following equation

$$\partial_0 f^{\tilde{a}} = i \gamma^{ac} \frac{\partial f^{\tilde{a}}}{\partial s^a} \partial_c \ln h \quad (181)$$

$\partial_0$ standing for derivative with respect to $t$.

*Proof:*

$$d\tilde{\tau}^2 = dt^2 - (\tilde{\gamma}^{\tilde{a}\tilde{c}} \tilde{A}_{\tilde{c}} dt + d\tilde{s}^{\tilde{a}})(\Box^{\tilde{b}})\tilde{\gamma}_{\tilde{a}\tilde{b}} \quad (182)$$

$$= dt^2 - (\tilde{\gamma}^{\tilde{a}\tilde{c}}(A_{\tilde{c}} - i\partial_{\tilde{c}} \ln h)dt + d\tilde{s}^{\tilde{a}})(\Box^{\tilde{b}})\tilde{\gamma}_{\tilde{a}\tilde{b}} \quad (183)$$

where $\Box^{\tilde{b}}$ means the “same expression with $\tilde{b}$ in place of $\tilde{a}$”. $\tilde{s}^{\tilde{a}} = f^{\tilde{a}}(s, t) \Rightarrow d\tilde{s}^{\tilde{a}} = \frac{\partial f^{\tilde{a}}}{\partial s^{\tilde{a}}} ds^{\tilde{a}} + \partial_0 f^{\tilde{a}} dt$.

$$d\tilde{\tau}^2 = dt^2 - \left( \gamma^{ac} \frac{\partial f^{\tilde{a}}}{\partial s^a} \frac{\partial f^{\tilde{c}}}{\partial s^c} (A_{\tilde{c}} - i \partial_{\tilde{c}} \ln h) dt + \frac{\partial f^{\tilde{a}}}{\partial s^a} ds^a + \partial_0 f^{\tilde{a}} dt \right) (\Box^{\tilde{b}}) \tilde{\gamma}_{\tilde{a}\tilde{b}} \quad (184)$$

$$= dt^2 - \left( \frac{\partial f^{\tilde{a}}}{ds^a} (\gamma^{ac} A_c dt + ds^a) + \underbrace{(\partial_0 f^{\tilde{a}} - i \gamma^{ac} \frac{\partial f^{\tilde{a}}}{\partial s^a} \partial_c \ln h)}_{\equiv 0} \right) (\Box^{\tilde{b}}) \tilde{\gamma}_{\tilde{a}\tilde{b}} \quad (185)$$

$$= dt^2 - (\gamma^{ac} A_c dt + ds^a)(\Box^{\tilde{b}})\gamma_{ab} \quad (186)$$

$$= d\tau^2 \quad (187)$$

□

Diff$\mathcal{M}_\Lambda$ must be restricted to diffeomorphisms for which it exists $h$ such that eq.(181) holds. This includes Diff$\mathcal{M}_\Lambda$ (time independent diffeomorphisms) which are associated with $h = 1$. Moreover the $U(1)$-gauge changes must be restricted to the ones for which eq.(181) has solutions.

The couple $(h, \varphi_t) \in U(1) \times \text{Diff}\mathcal{M}_\Lambda$ constitutes a change of foliation of $\mathcal{M}_\Lambda$ (the leaf at time $t$ is modified by $\varphi_t$ and the shift vector is modified by $-i\partial\ln h$). The
U(1)-symmetry in the quantum space ℳ is then associated with the foliation changes of its eigen spacetime.

### Example 4: Fuzzy plane and fuzzy Painlevé-Gullstrand spacetime

For the fuzzy plane, the spacetime metric associated with the quasicoherent state $|\Lambda(\alpha)\rangle = |0\rangle \otimes |\alpha\rangle$ is

$$d\tau^2 = dt^2 - (A^a dt + ds^a)(A^b dt + ds^b)\delta_{ab} \qquad (188)$$

$$= dt^2 - (A^r dt + dr)^2 - r^2 (A^\theta dt + d\theta)^2 \qquad (189)$$

in polar coordinates $(\alpha = r e^{i\theta})$, with $A_r = 0$ and $A_\theta = r^2$. So $d\tau^2 = dt^2 - dr^2 - r^2(dt + d\theta)^2$. This is the metric of a flat spacetime with the effect of A which will be discussed in the next section. But consider the following inner gauge change

$$|\tilde{\Lambda}(\alpha)\rangle\rangle = h(\alpha)|\Lambda(\alpha)\rangle\rangle = e^{2i\sqrt{r_S|\alpha|}}|\Lambda(\alpha)\rangle\rangle \qquad (190)$$

$$\tilde{A} = A - id \ln h = A + \sqrt{\frac{r_S}{r}} dr \qquad (191)$$

with $r_S > 0$ a constant parameter. With this gauge choice, the metric becomes:

$$d\tilde{\tau}^2 = dt^2 - (dr + \sqrt{\frac{r_S}{r}} dt)^2 - r^2(dt + d\theta)^2 \qquad (192)$$

In place of the metric a flat spacetime with the effect of the Berry potential, we have the metric of a Painlevé-Gullstrand spacetime (reduced of one dimension) with the effect of A. So the metric of a Schwarzschild black hole of Schwarzchild radius $r_S$ in the Painlevé-Gullstrand coordinates which are the coordinates such that $t$ is the proper time of a free-falling observer who starts from far away with zero velocity. This in accordance to the fact that, in quantum gravity matrix models, fuzzy spaces are the quantisation of the space from the point of view of an ideal Galilean observer for which $t$ is its classical clock (and so the proper time of this free-falling observer). The link between the plane and the Painlevé-Gullstrand spacetime is classically well-known, the space slices of the Painlevé-Gullstrand spacetime are flat. At the quantum level, the difference between the flat spacetime and the Painlevé-Gullstrand black-hole spacetime is just the gauge choice for $|\Lambda\rangle$. But note that the two gauge choices are not physically equivalent because the gauge change $h$ is realised without its associated diffeomorphism. Let $\tilde{r} = f(r, t)$ be the generator of the diffeomorphism associated with $h$:

$$\frac{\partial f}{\partial t} = i\gamma^{rr} \frac{\partial f}{\partial r} \frac{\partial \ln h}{\partial r} \iff \frac{\partial f}{\partial t} = -\sqrt{\frac{r_S}{r}} \frac{\partial f}{\partial r} \qquad (193)$$

$$\tilde{r} = f(r, t) = \left( r^{3/2} - \frac{3}{2} \sqrt{r_S} t \right)^{2/3} \qquad (194)$$

(with the initial condition $\tilde{r} = r$ at $t = 0$). With this change of foliation in addition to the gauge change $h$, $d\tau^2$ is invariant and the flat spacetime remains the flat spacetime with just another coordinate system.

Clearly, the fuzzyification (of a 2D slice) of the Schwarzschild black hole is not the same for the Schwarzschild coordinates and for the Painlevé-Gullstrand coordinates:
<figure><img src="image_7.png" /></figure>

with $g(t', r) = t' - r_S(-2\sqrt{r/r_S} + \ln \frac{\sqrt{r/r_S+1}}{\sqrt{r/r_S-1}})$. This is due to the fact that the only possible changes of coordinates in $\mathfrak{M}$ are the external gauge changes $\tilde{X}^i = J^i_j X^j$ to have a transformation inner to $\mathfrak{X}$.

**5.1.3. Geodesics** The geodesics for $\gamma$ ($\ddot{s}^a + \check{\Gamma}_{bc}^a \dot{s}^b \dot{s}^s = 0$, with $\check{\Gamma}_{bc}^a = e_i^a \frac{\partial^2 x^i}{\partial s^b \partial s^c}$), i.e. the geodesics minimising the length on $M_\Lambda$ measured with $\gamma$, are the path minimising the cumulated energy uncertainty since $d\ell_s^2 = \omega_{x(s+ds)}(d\ell_{x(s)}^2)$ is the instantaneous energy uncertainty when the measure presents a small misalignment $ds$ with respect to the probe. The strongly adiabatic dynamics following minimising geodesics on $M_\Lambda$ is then such that the energy uncertainty due to the slipping is minimal.

Now we want consider the geodesics of the eigen spacetime $M_\Lambda$. The tetrads of $M_\Lambda$ are

$$e_a^i = \frac{\partial x^i}{\partial s^a} \qquad (195)$$

$$e_0^i = \gamma^{ab} A_a \frac{\partial x^i}{\partial s^b} \qquad (196)$$

$$e_a^0 = 0 \qquad (197)$$

$$e_0^0 = 1 \qquad (198)$$

(the inverse tetrads being $e_i^a = \gamma^{ad} \delta_{ij} \frac{\partial x^i}{\partial s^d}$, $e_i^0 = 0$, $e_0^0 = 1$ and $e_0^a = -\gamma^{ab} A_b$). This defines the following Christoffel symbols:

$$\Gamma_{\beta\gamma}^{\alpha} = 0 \qquad (199)$$

$$\Gamma_{b0}^{a} = e_{i}^{a} \partial_{b} e_{0}^{i} \qquad (200)$$

$$\Gamma_{bc}^{0} = 0 \qquad (201)$$

$$\Gamma_{bc}^{a} = e_{i}^{a} \partial_{b} e_{c}^{i} = \check{\Gamma}_{bc}^{a} \qquad (202)$$

The geometry of $M_\Lambda$ is not torsion free due to $\Gamma_{b0}^a \neq \Gamma_{0b}^a = 0$. Let $\check{\Gamma}_{bc}^a = \frac{\partial^2 x^i}{\partial s^b \partial s^c} \delta_{ij} \gamma^{ad} \frac{\partial x^j}{\partial s^d}$ and $\check{\Gamma}_{b0}^0 = \check{\Gamma}_{b0}^a = \check{\Gamma}_{bc}^0 = \check{\Gamma}_{00}^0 = 0$ be the Christoffel symbols induced by $dt^2 - \gamma_{ab} ds^a ds^b$ (the geometry without the geometric phase generator effects). The contorsion of $M_\Lambda$ is $\kappa_{b0}^a = \Gamma_{b0}^a - \check{\Gamma}_{b0}^a = e_i^a \partial_b e_0^i$. The autoparallel geodesics are then defined by

$$\ddot{s}^{\alpha} + \Gamma_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} = 0 \qquad (203)$$
$$ \Leftrightarrow \ddot{s}^{\alpha} + \check{\Gamma}_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} = -\kappa_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} \qquad (204) $$

We can then say that the probe following an autoparallel geodesic is deviated from the extremal geodesics by a contorsional force. $A = \delta_{ij}e_0^i dx^i$ (with $dx^i = \frac{\partial x^i}{\partial s^a} ds^a$, we recall that $d$ stands for the derivative of $M_\Lambda$) and then $F = \delta_{ij}de_0^i \wedge dx^j$, more precisely $F_{ab} = \delta_{ij}\partial_{[a}e_0^i \frac{\partial x^j}{\partial s^b]}$. But $\gamma_{ac}\kappa_{b0}^c = \delta_{ij}\partial_b e_0^i \frac{\partial x^j}{\partial s^a}$. It follows that:

$$ F_{ab} = \gamma_{[bc}\kappa_{a]0}^{c} \qquad (205) $$

The usual interpretation of the strong adiabatic transport is that this one is equivalent to the transport of a charged particle in $M_\Lambda$ where lives a magnetic field $F$ of potential $A$ [30, 31]. The strong adiabatic transport formula being $|\Psi\rangle = e^{-i\int_F A}|\Lambda\rangle = e^{iS_A}|\Lambda\rangle$ where $S_A = -\int_0^t A_a \cdot \frac{ds^a}{dt}$ is the classical action of the interaction of a particle (of negative unit charge) with a magnetic field, the adiabatic approximation can be seen as a kind of a semi-classical approximation (by analogy with the WKB ansatz for a wave function $\psi = e^{iS}R$). We can then interpret $F$ as a magnetic field in $M_\Lambda$, and so the contorsional force $f_\kappa^a = -\kappa_{b0}^a \dot{s}^b \dot{t}$ is a Laplace-like force acting on the virtual particle representing the probe. The contorsion deviates the geodesics as the magnetic Laplace force deviates the charged particle trajectories.

**Property 10** If $t \mapsto x(s(t)) \in M_\Lambda$ is a geodesic for the strong adiabatic transport on $M_\Lambda$ ($\ddot{s}^a + \check{\Gamma}_{bc}^a \dot{s}^b \dot{s}^c + q\kappa_{b0}^a \dot{s}^b = 0$), then the linear momentum $p = (\dot{x}^i + q\gamma^{ab} \frac{\partial x^i}{\partial s^a} A_b)\partial_i \in T_x M_\Lambda$ is conserved (with $q = \frac{dt}{d\tau}$ a constant).

*Proof:* The geodesic equation for $t$ being $\ddot{t} = 0$, we can set $t = q\tau$.

$$ \ddot{x}^i = \frac{d}{d\tau} \left( \frac{\partial x^i}{\partial s^b} \dot{s}^b \right) = \frac{\partial x^i}{\partial s^b} \ddot{s}^b + \frac{\partial^2 x^i}{\partial s^b \partial s^c} \dot{s}^b \dot{s}^c \qquad (206) $$

It follows that

$$ \delta_{ij}\gamma^{ad}\frac{\partial x^j}{\partial s^d}\ddot{x}^i = \delta_{ij}\gamma^{ad}\frac{\partial x^i}{\partial s^b}\frac{\partial x^j}{\partial s^d}\ddot{s}^b + \delta_{ij}\gamma^{ad}\frac{\partial x^j}{\partial s^d}\frac{\partial^2 x^i}{\partial s^b \partial s^c}\dot{s}^b \dot{s}^c \qquad (207) $$

$$ = \ddot{s}^a + \check{\Gamma}_{bc}^a \dot{s}^b \dot{s}^c \qquad (208) $$

We have then

$$ \ddot{s}^a + \check{\Gamma}_{bc}^a \dot{s}^b \dot{s}^c + \kappa_{b0}^a \dot{s}^b \dot{s}^c q = 0 \qquad (209) $$

$$ \Leftrightarrow \delta_{ij}\gamma^{ad}\frac{\partial x^i}{\partial s^d}\left(\ddot{x}^i + \frac{\partial}{\partial s^b}\left(\gamma^{ce}A_c\frac{\partial x^i}{\partial s^e}\right)\dot{s}^b q\right) = 0 \qquad (210) $$

$$ \Leftrightarrow \pi_{xi} \frac{d}{d\tau} \left( \dot{x}^i + q\gamma^{ab} A_a \frac{\partial x^i}{\partial s^b} \right) = 0 \qquad (211) $$

Let $p^i = \dot{x}^i + q\gamma^{ab}A_a \frac{\partial x^i}{\partial s^b}$, we have $-p^i\sigma_i = \frac{d\check{D}_x}{dt} + q\gamma^{ab}A_a \frac{\partial \check{D}_x}{\partial s^b}$. But $\check{D}_x|\Lambda\rangle = 0 \Rightarrow (\partial_b \check{D}_x)|\Lambda\rangle = \check{D}_x\partial_b|\Lambda\rangle = 0$ and then $\langle\Lambda|(\partial_b \check{D}_x)|\Lambda\rangle = 0$. It follows that $p^i\langle\Lambda|\sigma_i|\Lambda\rangle = 0 \Leftrightarrow \vec{p} \cdot \vec{n}_x = 0$, and so $p^i\partial_i \in T_x M_\Lambda$, inducing than $\pi_x p = p$ ($\pi_x$ being the orthogonal projection onto $T_x M_\Lambda$). $\square$

The conservation of $p^i = \dot{x}^i + qA^i$ is in accordance with the interpretation of the contorsional force as a Laplace-like force. It is equivalent to the conservation of $\frac{d\check{D}_x}{d\tau} + q\gamma^{ab}A_a \frac{\partial \check{D}_x}{\partial s^b}$.
## 5.2. The weak adiabatic regime

### 5.2.1. Lorentz connection
The inner product $\langle \bullet | \bullet \rangle_*$ defines another natural gauge potential on $M_\Lambda$:

$$ \mathfrak{A}\rho_{\Lambda} = -i\langle\Lambda|d|\Lambda\rangle_* = -i \operatorname{tr}_{\mathcal{H}} |d\Lambda\rangle\langle\Lambda| \qquad (212) $$

$\rho_{\Lambda} = \operatorname{tr}_{\mathcal{H}} |\Lambda\rangle\langle\Lambda|$ being the $C^*$-norm of the quasicoherent state, and $d$ denoting the differential of $M_{\Lambda}$. Note that $\operatorname{tr}(\rho_{\Lambda}\mathfrak{A}) = A$, $A$ is the statistical mean value of $\mathfrak{A}$ in the mixed state $\rho_{\Lambda}$. We can then see $\mathfrak{A}$ as a gauge potential random variable depending on the statistical distribution of quantum states of $\mathbb{C}^2$ defined by the density matrix $\rho_{\Lambda}$.

$\mathfrak{A}$ is in general not self-adjoint but $\operatorname{tr}(\rho_{\Lambda}(\mathfrak{A}^{\dagger} - \mathfrak{A})) = 0$, $\mathfrak{A}$ is then almost surely self-adjoint for the probability law defined by $\rho_{\Lambda}$. We write then $\mathfrak{A} \in \Omega^1(M_{\Lambda}, \mathfrak{u}(2)_{\text{a.s.}})$ where $\mathfrak{u}(2)$ denotes the set of self-adjoint operators of $\mathbb{C}^2$.

$\mathfrak{A}$ is the generator of the geometric phases accumulated during a weak adiabatic transport of $|\Lambda\rangle$ along a path $\mathcal{C}$ on $M_{\Lambda}$, i.e. a transport adiabatic with respect to the quantum degree of freedom associated with $\mathcal{H}$ but not with the one associated with $\mathbb{C}^2$. If $|\Lambda\rangle$ is weakly nondegenerate along the path $\mathcal{C}$ we have (see [23] and appendix A in [12])

$$ \begin{gathered} |\Psi(T)\rangle\rangle = e^{-i \int_0^T \lambda_0(t) dt} \mathbb{P} e^{-i \int_{\mathfrak{e}} \mathfrak{A}} |\Lambda(y)\rangle\rangle + \mathcal{O}(\max(\epsilon, 1/T)) \quad (213) \\ \text{if } |\Psi(0)\rangle\rangle = |\Lambda(x)\rangle\rangle \end{gathered} $$

where $T$ is the total duration and $\epsilon$ is a perturbative magnitude of the transitions from the quasicoherent state to the other eigenstates induced by $(\mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}})^{-1} \mathbb{D}_{x} \mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}}$. $\mathbb{P}e$ denotes the path anti-ordered exponential $(\frac{d}{ds} \mathbb{P}e^{-i \int \mathfrak{A}} = -i \mathbb{P}e^{-i \int \mathfrak{A}} \mathfrak{A}_a \dot{s}^a$ for $s$ a curvilinear coordinate along $\mathcal{C}$, $\dot{s}^a \equiv \frac{ds^a}{ds}$). The dynamical phase $\lambda_0(t) = \langle\Lambda(x(t))| (\mathbb{P}e^{-i \int_0^t \mathfrak{A}_a \dot{s}^a dt})^{-1} \mathbb{D}_{x(t)} \mathbb{P}e^{-i \int_0^t \mathfrak{A}_a \dot{s}^a dt} |\Lambda(x(t))\rangle\rangle$ is zero if $|\Lambda\rangle$ is separable [12]. The condition of parallel transport associated with the weak adiabatic approximation introduced in [8] can be rewritten as

$$ \langle \Psi | \dot{\Psi} \rangle_* = 0 \qquad (214) $$

**Property 11** $\mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}} \in U(2)_{\text{a.s.}}$.

*Proof:* Let $u \equiv \mathbb{P}e^{-i \int_{\mathfrak{e}} \mathfrak{A}}$.

$$ \frac{d}{ds}(u\rho_{\Lambda}u^{\dagger}) = \dot{u}\rho_{\Lambda}u^{\dagger} + u\dot{\rho}_{\Lambda}u^{\dagger} + u\rho_{\Lambda}\dot{u}^{\dagger} \qquad (215) $$

$$ = u(-i\mathfrak{A}_a\dot{s}^a\rho_{\Lambda} + \dot{\rho}_{\Lambda} + i\rho_{\Lambda}\mathfrak{A}_a^{\dagger}\dot{s}^a)u^{\dagger} \qquad (216) $$

But

$$ \dot{\rho}_{\Lambda} = \operatorname{tr}_{\mathcal{H}} |\dot{\Lambda}\rangle\langle\Lambda| + \operatorname{tr}_{\mathcal{H}} |\Lambda\rangle\langle\dot{\Lambda}| \qquad (217) $$

$$ = i\mathfrak{A}_a\dot{s}^a\rho_{\Lambda} - i\rho_{\Lambda}\mathfrak{A}_a^{\dagger}\dot{s}^a \qquad (218) $$

and then

$$ \frac{d}{ds}(u\rho_{\Lambda}u^{\dagger}) = 0 \Rightarrow u(s)\rho_{\Lambda}(s)u^{\dagger}(s) = \rho_{\Lambda}(0) \qquad (219) $$

$$ \Rightarrow \operatorname{tr}(\rho_{\Lambda}(s)u^{\dagger}(s)u(s)) = 1 \qquad (220) $$

$$ \Rightarrow u(s) \in U(2)_{\text{a.s.}} \qquad (221) $$
<figure><img src="image_8.png" /></figure>

**Property 12** Under an inner gauge transformation $(h, u) \in U(1) \times U_x \subset U(1) \times SU(2)_{a.s.}$, the gauge potential becomes:

$$ \tilde{\mathfrak{A}} = u\mathfrak{A}u^{-1} - i\mathrm{d}\ln h - i\mathrm{d}uu^{-1} \qquad (222) $$

Under a local external gauge transformation $J \in SO(3)_{M_\Lambda}$, the gauge potential becomes:

$$ \hat{\mathfrak{A}} = u_J^{-1} J^* \mathfrak{A} u_J + i u_J^{-1} du_J \qquad (223) $$

where $u_J \in SU(2)$ is the transformations associated with $J$ viewed as a linear map of $\mathbb{R}^3$. $(Jy)^i = J^i_j y^j$. $J^*$: $\Omega^1\mathbb{R}^3 \to \Omega^1\mathbb{R}^3$ is defined by $(J^*k)(x) = k_a(J^{-1}x)((J^{-1})^i_j dx^j - (J^{-1}dJJ^{-1})^i_j x^j)$, $\forall k \in \Omega^1\mathbb{R}^3$.

*Proof:* $|\tilde{\Lambda}\rangle = hu|\Lambda\rangle \Rightarrow d|\tilde{\Lambda}\rangle = hud|\Lambda\rangle + dh \times u|\Lambda\rangle + du \times h|\Lambda\rangle$. So $-i|d\tilde{\Lambda}\rangle\langle\tilde{\Lambda}| = u(-i|d\Lambda\rangle\langle\Lambda|)u^{-1} - i\tilde{h}^{-1}dh \times u|\Lambda\rangle\langle\Lambda|u^{-1} - i\tilde{du}^{-1} \times u|\Lambda\rangle\langle\Lambda|u^{-1}$. By taking the partial trace over $\mathcal{H}$ we find the formula for $\tilde{\mathfrak{A}}$ multiplied on the right by $\tilde{\rho}_\Lambda = u\rho_\Lambda u^{-1} = \mathrm{tr}_\mathcal{H} |\tilde{\Lambda}\rangle\langle\tilde{\Lambda}|$. We recall that we have $|\hat{\Lambda}(x)\rangle = u_J^{-1}|\Lambda(J^{-1}x)\rangle$. Then $-i|d\hat{\Lambda}\rangle\langle\hat{\Lambda}| = iu_J^{-1}du_J u_J^{-1}|\Lambda\rangle\langle\Lambda|u_J - iu_J^{-1}J^*|d\Lambda\rangle\langle\Lambda|u_J$ since by construction $df(J^{-1}x) = (J^*df)(x)$ for any function $f$. By taking the partial trace on $\mathcal{H}$, we find the formula for $\hat{\mathfrak{A}}$ multiplied on right by $\hat{\rho}_\Lambda = u_J^{-1}\rho_\Lambda u_J$. $\square$

$\mathfrak{A}$ defines the adiabatic fake curvature $\mathfrak{F} = d\mathfrak{A} - i\mathfrak{A} \wedge \mathfrak{A} \in \Omega^2(M_\Lambda, u(2)_{a.s.})$ [8].

**Example 2: Fuzzy surface plots**

A and $F$ are fields onto the eigenmanifold $M_\Lambda$ spanned by the mean values of the coordinate observables $X^i$. These fields onto the mean value of $\mathfrak{M}$ are also mean values of the quantum observables $\mathfrak{A}$ and $\mathfrak{F}$ in the eigen density matrix $\rho_\Lambda$ ($A = \mathrm{tr}(\rho_\Lambda\mathfrak{A})$ and $F = \mathrm{tr}(\rho_\Lambda\mathfrak{F})$). These nonabelian gauge fields are observables for the spin degree of freedom. By definition $\mathfrak{A}\rho_\Lambda = -i\mathrm{tr}_\mathcal{H}|d\Lambda\rangle\langle\Lambda|$, and so for the fuzzy plane we have:

$$ \mathfrak{A} = A \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \qquad (224) $$

where $\rho_\Lambda = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = |0\rangle\langle0|$ is the projection onto the first state of the canonical basis of $\mathbb{C}^2$ ($\rho_\Lambda$ is a pure state). $A = -i\frac{\bar{\alpha}d\alpha - \alpha d\bar{\alpha}}{2}$. Note that $\mathfrak{A}$ is not single defined, since $\mathfrak{A}' = \mathfrak{A} + \xi$ is also solution of eq. (212) if $|0\rangle \in \mathrm{ker} \xi$. We can then think $\xi$ as a gauge change of a third kind (the two first ones being the internal and external gauge changes). But by definition $\xi$ is almost surely zero, $\mathrm{tr}(\rho_\Lambda\xi) = 0$ and plays no fundamental role. We can then set $\xi = 0$ without lost of generality. The interpretation of the observable $\mathfrak{A}$ is obvious, the magnetic potential is $A$ for the spin state up $|0\rangle$ (eigenvector of $\mathfrak{A}$) and the magnetic potential is set to zero for the spin state down $|1\rangle$, since $|0\rangle$ is the only one spin state appearing in the quasicoherent state of the fuzzy plane. For the fuzzy elliptic paraboloid we have

$$ \mathfrak{A}\rho_{\Lambda} = A \begin{pmatrix} 1 & -\epsilon\bar{\alpha} \\ -\epsilon\alpha & 0 \end{pmatrix} + i\epsilon d\alpha \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} + \mathcal{O}(\epsilon^2) \qquad (225) $$

where the pure eigenstate $\rho_\Lambda = \begin{pmatrix} 1 & -\epsilon\bar{\alpha} \\ -\epsilon\alpha & 0 \end{pmatrix} + \mathcal{O}(\epsilon^2) = |O_\alpha\rangle\langle O_\alpha|$ is the projection onto $|O_\alpha\rangle = |0\rangle - \epsilon\alpha|1\rangle$. It follows that we can set

$$ \mathfrak{A} = A \begin{pmatrix} 1 & -\epsilon\bar{\alpha} \\ -\epsilon\alpha & 0 \end{pmatrix} - i\frac{dc}{\alpha} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} + \mathcal{O}(\epsilon^2) \qquad (226) $$
Note that the second member is almost surely zero at the perturbative first order
tr(ρ<sub>Λ</sub>|1⟩⟨1|) = O(ε<sup>2</sup>). For the fuzzy hyperbolic paraboloid we have

$$
\mathfrak{A}\rho_{\Lambda} = A \begin{pmatrix} 1 & -\epsilon\alpha \\ -\epsilon\bar{\alpha} & 0 \end{pmatrix} + i\frac{\epsilon}{2} \begin{pmatrix} 0 & d\alpha \\ d\bar{\alpha} & 0 \end{pmatrix} + \mathcal{O}(\epsilon^2) \quad (227)
$$

with $\rho_\Lambda = |O_\alpha\rangle\langle O_\alpha|$ the projection onto $|O_\alpha\rangle = |0\rangle - \epsilon\bar{\alpha}|1\rangle$. It follows that we can set

$$
\mathfrak{A} = A \begin{pmatrix} 1 & -\epsilon\alpha \\ -\epsilon\bar{\alpha} & 0 \end{pmatrix} - \frac{i}{2} \begin{pmatrix} \frac{d\alpha}{\alpha} & \frac{d\alpha}{\epsilon|\alpha|^2} \\ 0 & \frac{d\alpha}{\bar{\alpha}} \end{pmatrix} + \mathcal{O}(\epsilon^2) \quad (228)
$$

A new the second member is almost surely zero.

For the geometric viewpoint, $\mathfrak{A}$ defines a Lorentz connection onto the classical spacetime $\mathcal{M}_\Lambda$:

**Proposition 1** Let $\Omega^{\mu\nu} \in \Omega^1(M_{\Lambda}, \mathbb{R})$ be the 1-form valued antisymmetric tensor defined by

$$
\Omega^{jk} = - \varepsilon^{jk} {}_i \mathrm{Re} \mathrm{tr}(\sigma^i \mathfrak{A}) \qquad (229)
$$

$$
\Omega^{i0} = \mathrm{Im} \operatorname{tr}(\sigma^i \mathfrak{A}) \qquad (230)
$$

then $\Omega$ is a Lorentz connection onto $\mathbb{R} \times M_\Lambda$. It follows that $R^{\mu\nu}_{ab}$ is the Riemann
curvature tensor of $\mathbb{R} \times M_\Lambda$ associated with $\Omega$ (a, b standing for indices of local
curvilinear coordinates on $M_\Lambda$), with $R^{jk}_{ab} = -\epsilon^{jk} {}_i \mathrm{Re} \mathrm{tr}(\sigma^i \mathfrak{F}_{ab})$ and $R^{i0}_{ab} =
\mathrm{Im} \mathrm{tr}(\sigma^i \mathfrak{F}_{ab})$. The Ricci curvature tensor $R^\mu_b = R^a_{\mu b} = \gamma^{ac} \delta_{ik} \frac{\partial x^k}{\partial s^c} R^{i\mu}_{ab}$ is then
$R^j_b = -\epsilon^{jk} {}_i \mathrm{Re} \mathrm{tr}(\sigma_k \mathfrak{F}^a_{\ b}) \frac{\partial x^i}{\partial s^a}$ and $R^0_b = \mathrm{Im} \mathrm{tr}(\sigma^i \mathfrak{F}^a_{\ b}) \frac{\partial x^i}{\partial s^a}$.

We can see that Ω is well a Lorentz connection by considering its transformation un-
der frame changes (see Appendix C.1). The definition of the Lorentz connection Ω by
A is consistent with the BFFS Dirac equation ℐ|Ψ⟩ = D̂<sub>x</sub>|Ψ⟩ at the weak adiabatic
limit [12]. We can show that operator valued geometric phases generated by A induces
spin rotations which are the Einstein-de Sitter spin precessions induced by Ω. The
definition of Ω is then consistent with the physical interpretation of M as a string
theory matrix model.

In quantum information theory, Ω permits to represent the qubit logical gate generated by the operator valued geometric phases induced by A by a precession effect induced by Ω. The adiabatic quantum control of the qubit in presence of the environment is then completely translated in the geometry of M<sub>Λ</sub>.

As for usual gauge fields, (A, F, 𝒜, 𝒤) are local data of a connection onto
a complicated fibre bundle which is studied Appendix D.2, the adiabatic regimes
appearing as horizontal lifts in this one (Appendix D.1).

5.2.2. Torsion and geodesics In weak adiabatic transport, the time dependent state is not proportional to |Λ⟩ but is P e⁻¹ ∫<Λ|Λ⟩, so the energy uncertainty is not minimised during the dynamics, and the relevant geodesics are not the ones associated with γ<sub>ab</sub> (minimising geodesics) but the ones associated with the Lorentz connection Ω (autoparallel geodesics). Moreover the energy mean value ⟨Ψ|D̃<sub>x</sub>|Ψ⟩ = ⟨Λ|(P e⁻¹ ∫<Λ>)†D̃<sub>x</sub>P e⁻¹ ∫<Λ>|Λ⟩ is not constant except if |Λ⟩ is separable. The Lorentz connection with the triads define the following Christoffel symbols on M<sub>Λ</sub>:

$$
\Gamma^a_{bc} = e^a_i \partial_b e^i_c + e^a_i \Omega^i_{bj} e^j_c \qquad (231)
$$
<table><caption>**Table 3.** Geometric behaviours with respect to the different dynamical regimes.</caption>
  <tr>
    <td>strong adiabatic transport along any path</td>
    <td>quantum transport minimising the uncertainties Δ<sub>Ψ(t)</sub>X<sup>i</sup> without spin precession</td>
  </tr>
  <tr>
    <td>strong adiabatic transport along a minimising geodesic path</td>
    <td>the length of the average path is minimised</td>
  </tr>
  <tr>
    <td>strong adiabatic transport along a path such that ω<sub>x(t)</sub>(|δl<sub>x(t+Δt),x(t)</sub>|) is small constant</td>
    <td>the average of the path length observable is minimised</td>
  </tr>
  <tr>
    <td>weak adiabatic transport along any path</td>
    <td>quantum transport minimising the uncertainties Δ<sub>Ψ(t)</sub>X<sup>i</sup> with spin precession</td>
  </tr>
  <tr>
    <td>weak adiabatic transport along an autoparallel geodesic path</td>
    <td>the spin precession is autoparallel</td>
  </tr>
</table>

where $e_i^a = \Xi^a_c \hat{e}_i^c = \gamma^{ad} \delta_{ij} \frac{\partial x^j}{\partial s^d}$ where $\hat{e}_i^c$ are the dual triads induced by $E_i^c$ and $\Xi$ the inverse matrix of $(\hat{e}_i^a e_b^i)$. Since these Christoffel symbols do not derive directly from $\gamma_{ab}$, the geometry of $M_\Lambda$ is not torsion free. We can show after some algebra that this torsion is

$$T_{bc}^{a} = \frac{1}{2} \gamma^{ad} \mathrm{Im} \operatorname{tr}(\tau_d[\tau_{[c}, \mathfrak{A}_{b}]]) \quad (232)$$

$$= \gamma^{ad} \epsilon_{ij}{}^k \mathrm{Re} \operatorname{tr}(\sigma_k \mathfrak{A}_{[b}} \frac{\partial x^j}{\partial s^c} \frac{\partial x^i}{\partial s^d}) \quad (233)$$

with $\tau_a \equiv \sigma_i \frac{\partial x^i}{\partial s^a}$. This connection is associated with the covariant derivatives in $\mathfrak{M}$, see Appendix C.2.

If $\mathfrak{A}$ is self-adjoint, a weak adiabatic transport along an autoparallel geodesic provides $\langle\langle \Lambda(x(t))|(\mathbb{P}e^{-i\int_{t_0}^t \mathfrak{A}})^{-1}\sigma_i \mathbb{P}e^{-i\int_{t_0}^t \mathfrak{A}}|\Lambda(x(t))\rangle\rangle \dot{x}^i(0) = 0$ (see Appendix C.2): the rotated (by $\mathbb{P}e^{-i\int_{t_0}^t \mathfrak{A}}$) normal vector remains normal to the initial tangent vector. The adiabatic transport is well autoparallel (in the general case, it is just almost surely autoparallel). Table 3 summarises the physical meaning of the different regimes. In another viewpoint, let $\Delta_0 \not D = \frac{d\not D_x}{dt}|_{t=0}$, $\Delta t$ be the energy uncertainty observable for an initial time uncertainty $\Delta t$. The autoparallel geodesics minimises the mean value of this energy uncertainty at each time (it is zero if $\mathfrak{A}$ is self-adjoint), see Appendix C.2.

The Lorentz connection $\Omega_a^{\mu\nu} = -\epsilon^{0\mu\nu}{}_{i} \mathrm{Re} \operatorname{tr}(\sigma^i \mathfrak{A}_a) + (\delta^{\nu 0} \delta^\mu_i - \delta^{\mu 0} \delta^\nu_i) \mathrm{Im} \operatorname{tr}(\sigma^i \mathfrak{A}_a)$ both with the tetrads eq.(195-198) define the following Christoffel symbols on $M_\Lambda$:

$$\Gamma_{0\beta}^{\alpha} = 0 \quad (234)$$

$$\Gamma_{b0}^{0} = \Omega_{bj}^{0} e_{0}^{j} \quad (235)$$

$$\Gamma_{b0}^{a} = e_{i}^{a} \partial_{b} e_{0}^{i} + e_{0}^{a} \Omega_{bj}^{0} e_{0}^{j} + e_{i}^{a} \Omega_{b0}^{i} e_{0}^{j} \quad (236)$$

$$\Gamma_{bc}^{0} = \Omega_{bj}^{0} e_{c}^{j} \quad (237)$$

$$\Gamma_{bc}^{a} = e_{i}^{a} \partial_{b} e_{c}^{i} + e_{i}^{a} \Omega_{bj}^{i} e_{c}^{j} + e_{0}^{a} \Omega_{bj}^{0} e_{c}^{j} \quad (238)$$

The geometry of $M_\Lambda$ is not torsion free, and we have

$$T_{ab}^{0} = \mathrm{Im} \operatorname{tr}(\sigma_i \mathfrak{A}_{[a}) \frac{\partial x^i}{\partial s^b]) \quad (239)$$

$$T_{a0}^{0} = \mathrm{Im} \operatorname{tr}(\sigma_i \mathfrak{A}_a) \gamma^{bc} A_b \frac{\partial x^i}{\partial s^c} \quad (240)$$

$$T_{b0}^{a} = \gamma^{ac} \frac{\partial x_i}{\partial s^c} (\partial_b e_0^i + \mathrm{Im} \operatorname{tr}(\sigma^i \mathfrak{A}_b) + \varepsilon^i_{jk} \mathrm{Re} \operatorname{tr}(\sigma^k \mathfrak{A}_b) \gamma^{cd} A_c \frac{\partial x^j}{\partial s^d})$$
$$ - \gamma^{ac} \mathrm{Im} \operatorname{tr}(\sigma_j \mathfrak{A}_b) \gamma^{de} A_c A_d \frac{\partial x^j}{\partial s^e} \qquad (241) $$

$$ T_{bc}^{a} = \gamma^{ad} \varepsilon_{ij}{}^{k} \mathrm{Re} \operatorname{tr}(\sigma_k \mathfrak{A}_{[b]} \frac{\partial x^j}{\partial s^c} \frac{\partial x^i}{\partial s^d} - \gamma^{ad} \mathrm{Im} \operatorname{tr}(\sigma_j \mathfrak{A}_{[b]} \frac{\partial x^j}{\partial s^c}) A_d \qquad (242) $$

Let $\check{\Gamma}_{bc}^a = \frac{\partial^2 x^i}{\partial s^b \partial s^c} \delta_{ij} \gamma^{ad} \frac{\partial x^j}{\partial s^d}$ and $\check{\Gamma}_{b0}^0 = \check{\Gamma}_{b0}^a = \check{\Gamma}_{bc}^0 = \check{\Gamma}_{00}^0 = 0$ be the Christoffel symbols induced by $dt^2 - \gamma_{ab}ds^a ds^b$ (the geometry without the geometric phase generator effects), and $K_{\beta\gamma}^a = \Gamma_{\beta\gamma}^a - \check{\Gamma}_{\beta\gamma}^a$ be the contorsion of $M_\Lambda$. This one is

$$ K_{b0}^{0} = \Omega_{bj}^{0} e_{0}^{j} \qquad (243) $$

$$ K_{b0}^{a} = e_{i}^{a} \partial_{b} e_{0}^{i} + e_{0}^{a} \Omega_{bj}^{0} e_{0}^{j} + e_{i}^{a} \Omega_{b0}^{i} + e_{i}^{a} \Omega_{bj}^{i} e_{0}^{j} \qquad (244) $$

$$ K_{bc}^{0} = \Omega_{bj}^{0} e_{c}^{j} \qquad (245) $$

$$ K_{bc}^{a} = e_{i}^{a} \Omega_{bj}^{i} e_{c}^{j} + e_{0}^{a} \Omega_{bj}^{0} e_{c}^{j} \qquad (246) $$

The autoparallel geodesics are defined by

$$ \ddot{s}^{\alpha} + \Gamma_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} = 0 \qquad (247) $$

$$ \Leftrightarrow \ddot{s}^{\alpha} + \check{\Gamma}_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} = -K_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma} \qquad (248) $$

The contorsion can be split in two terms, $\kappa_{b0}^a = e_i^a \partial_b e_0^i$ the part associated with the magnetic field $F$ and $K_{\beta\gamma}^a = e_\mu^a \Omega_{b\nu}^\mu e_\gamma^\nu$. We recall that $\mathbb{P}e^{-i\int_e \mathfrak{A}^{off}} = \mathbb{P}e^{\int_0^t \Omega_a^{ij} \frac{1}{2}[\sigma_i, \sigma_j] s^a dt}$ ($\mathfrak{A}^{off} = \mathfrak{A} - \mathrm{tr}\mathfrak{A}$) is responsible of a spin precession in the weak adiabatic transport:

$$ \rho_{\Psi}(t) = \mathbb{P}e^{\int_{0}^{t} \Omega_{a}^{ij} \frac{1}{2}[\sigma_{i}, \sigma_{j}] \dot{s}^{a} dt} \rho_{\Lambda}(x(t)) \left( \mathbb{P}e^{\int_{0}^{t} \Omega_{a}^{ij} \frac{1}{2}[\sigma_{i}, \sigma_{j}] \dot{s}^{a} dt} \right)^{\dagger} \qquad (249) $$

where $\rho_{\Psi} = \mathrm{tr}_{\mathcal{H}}|\Psi\rangle\langle\Psi|$ is the spin density matrix and $\rho_{\Lambda} = \mathrm{tr}_{\mathcal{H}}|\Lambda\rangle\langle\Lambda|$ is the eigen density matrix. The contorsional force $f_{\kappa}^{\alpha} = -K_{\beta\gamma}^{\alpha} \dot{s}^{\beta} \dot{s}^{\gamma}$ as the same interpretation than in Einstein-Cartan theory [36]: as the spacetime curvature and energy momentum tensor influence each other (Einstein equation), spacetime torsion and spin influence each other (Sciama-Kibble equation [36]). So the geometry of $M_\Lambda$ induces a spin precession, but in return the spin modifies the geometry of $M_\Lambda$ inducing a deviation from the extremal geodesics by the force $f_{\kappa}$.

To summarise, onto $M_\Lambda$ we have four notions of geodesic:

* Geodesics minimising the length of the mean path (GMLM): geodesics for the metric $\omega_{x(s+ds)}(d\ell^2_{x(s)}) = \gamma_{ab}ds^a ds^b$ for which the energy uncertainty with probe misalignment is minimised;

* Geodesics minimising the average length of the quantum paths (GMAL): geodesics for the metric $(\omega_{x(s)}(\sqrt{\gamma(L_{\delta\Pi}, L_{\delta\Pi})})^2 = \tilde{\gamma}_{ab}\delta s^a \delta s^b$ (this ones being applied onto a discrete set of points along the geodesic corresponding to measures onto the quantum system), for which the average energy gap isotropic uncertainty is minimised;

* Strongly adiabatic autoparallel geodesics (SAAG): geodesics including the contorsion $\kappa_{b0}^a$ corresponding to the effect of the Laplace like force induced by the Berry curvature $F$ (viewed as a magnetic field), for which energy uncertainty with probe misalignment and time lag is minimised;

* Weakly adiabatic autoparallel geodesics (WAAG): geodesics including the contorsions $\kappa_{b0}^a$ and $K_{\beta\gamma}^a$ corresponding to the Einstein-Cartan spinning effect, for which the spin precession is autoparallel.
## Example 2: Fuzzy surface plots

For the fuzzy plane, the minimising geodesics are straight lines whereas the autoparallel geodesics are circles (see [12]). Under the effect of the magnetic field $F$ (normal to the plane) the geodesics are curved to induce a cyclic rotation of the charged particle. GMLM and GMAL are the same since the two distances are the same, and SAAG and WAAG are the same since $\mathfrak{A} = A|0\rangle\langle 0|$ (implying than $K_{\beta\gamma}^{\alpha} = 0$).

For the fuzzy elliptic paraboloid, GMAL are slightly deviated from GMLM because the average length of quantum paths between two measures is slightly larger than the length of the mean path. SAAG are not circles but epicycles around deferents corresponding to circles of the rotational symmetry (the epicyclic moves being induced by the magnetic field $F$ and the curvature of $M_{\Lambda}$). WAAG are very complicated trajectories on $M_{\Lambda}$ difficult to interpret which seem to trap the particle in the neighbourhood of $\alpha = 0$. Fig. 4 presents numerical integrations of geodesic equations for the fuzzy elliptic paraboloid.

# 6. Time-dependent Fuzzy spaces

In the previous sections, we have supposed that $\mathfrak{M}$ was time-independent and that only the probe $x$ was moved. But the physical situations induce that the coordinate observables $X^i$ can be time-dependent. Indeed, in string theory BFSS matrix models, the D2-brane obeys to the following equation of motion (without vacuum fluctuation) [4]:

$$ \ddot{X}^i - [X_j, [X^i, X^j]] = 0 \qquad (250) $$

with the Gauss constraint $[\dot{X}_i, X^i] = 0$. This equation is the noncommutative version of the Einstein equation in the vacuum ($\Box_g g_{\mu\nu} + ... = 0$ in harmonic local coordinates with $\Box_g$ the spacetime Laplace-Beltrami operator – the spacetime D’Alembertian –; $[X_j, [\bullet, X^j]]$ being the noncommutative Laplacian of $\mathfrak{M}$).

Moreover, in quantum information theory, the environmental noise operators satisfy the following Heisenberg equation:

$$ \dot{X}^i = i[\tilde{H}_\mathcal{E}, X^i] \qquad (251) $$

where $H_{\mathcal{E}} \in \mathcal{L}(\mathcal{H})$ is the Hamiltonian of the environment (and $\tilde{H}_{\mathcal{E}} = U_{\mathcal{E}}^{\dagger}H_{\mathcal{E}}U_{\mathcal{E}}$ with $i\dot{U}_{\mathcal{E}} = H_{\mathcal{E}}U_{\mathcal{E}}; \tilde{H}_{\mathcal{E}} = H_{\mathcal{E}}$ if it is time-independent).

If the time-dependence of $X^i$ is fast, we cannot apply directly the adiabatic approximations and it is needed to treat the problem with a Schrödinger-Floquet picture (for the periodic cases) [37] or a Schrödinger-Koopman picture (for the quasi-periodic and aperiodic cases) [38]. These pictures are equivalent to add a fourth coordinate observable $X^0$ (as a quantum time coordinate observable) with a probe space $\mathbb{R}^4$ endowed with the Minkowski metric [12]. This method has been discussed in a previous paper. In this paper we want to focus on the case where $X^i$ depends slowly on the time permitting to use directly the adiabatic approximations.

With a time-dependent fuzzy space, the eigen equation becomes:

$$ \sigma_i \otimes (X^i(t) - x^i)|\Lambda(x, t)\rangle\rangle = 0 \qquad (252) $$

for $x^i \in M_{\Lambda}(t)$, the eigenmanifold being time dependent. By using the perturbation theory Appendix A with $\delta X^i = \dot{X}^i dt$ we have $M_{\Lambda}(t) = \{x(t) \in \mathbb{R}^3\}$ where $t \mapsto x(t)$ are the solutions of the following differential equation:

$$ \dot{x}^i = \langle \Omega_{x(t),t} | \dot{X}^i | \Omega_{x(t),t} \rangle \qquad (253) $$
<figure><img src="image_9.png" /><figcaption>**Figure 4.** In the parameter space spanned by $\alpha$, the geodesics for the fuzzy plane and the fuzzy elliptic paraboloid (and these ones onto $M_\Lambda$). GMLM is a geodesic minimising the length of the mean path (associated with $\gamma$); GMAL is geodesic minimising the average length of quantum paths between two measurements (this one is then discrete, and is associated with $\text{dist}_\gamma$); SAAG is a strongly adiabatic autoparallel geodesic (including the effect of the magnetic field $F$); and WAAG is a weakly adiabatic autoparallel geodesic (including the effects of $F$ and of the torsion associated with $\Omega$). In all cases the initial conditions are $\alpha(0) = 1 - 3i$ and $\dot{\alpha}(0) = i$, with $\epsilon = 0.1$.</figcaption></figure>
if $|\Lambda(x,t)\rangle\rangle = |O_{x,t}\rangle\rangle \otimes |\Omega_{x,t}\rangle$ is separable, or else

$$ \dot{x}^i = \langle\langle \Lambda(x(t),t) | \sigma_j \otimes \dot{X}^j | \Lambda(x(t),t) \rangle\rangle \frac{p_{x(t)}^i}{\vec{n}_{x(t)} \cdot \vec{p}_{x(t)}} \quad (254) $$

with $\vec{n}_{x(t)} = \langle\langle \Lambda(x(t),t) | \sigma^i | \Lambda(x(t),t) \rangle\rangle \partial_i \in N_{x(t)} M_\Lambda(t)$ and $\vec{p}_{x(t)} = \frac{\langle\langle \Lambda(x(t),t) | \dot{X}^i | \Lambda(x(t),t) \rangle\rangle}{\|\langle\langle \dot{X} \rangle\rangle(t)\|} \partial_i$.

The set of initial conditions $M_\Lambda(0)$ is such that $\ker(\sigma_i \otimes (X^i(0) - x^i(0))) \neq \{0\}$. Moreover we have $\forall x(t) \in M_\Lambda(t)$:

$$ \left. \frac{\partial}{\partial t} |\Lambda(x, t)\rangle\rangle \right|_{x=x(t)} = - \sum_{n>0} \frac{\langle\langle \lambda_n(x(t),t) | \sigma_i \otimes (\dot{X}^i - \dot{x}^i) | \Lambda(x(t),t) \rangle\rangle}{\lambda_n(x(t),t)} |\lambda_n(x(t),t)\rangle\rangle \quad (255) $$

where $\tilde{D}_x(t)|\lambda_n(x,t)\rangle\rangle = \lambda_n(x,t)|\lambda_n(x,t)\rangle\rangle$ (with $\lambda_0(x,t) = 0$).

Let $(t, s^1, s^1)$ be a local coordinate system onto $M_\Lambda = \bigsqcup_t(t, M_\Lambda(t))$. The metric of the leaf $M_\Lambda(t)$ satisfies

$$ \dot{\gamma}_{ab} = \delta_{ij} \frac{\partial x^i}{s^{(a)}} \frac{\partial \dot{x}^j}{\partial s^{(b)}} \quad (256) $$

The main difference with the previous sections is that now the gauge potentials are forms of $M_\Lambda$

$$ A = -i(\langle\langle \Lambda(x,t) | d | \Lambda(x,t) \rangle\rangle + \langle\langle \Lambda(x,t) | \partial_t | \Lambda(x,t) \rangle\rangle dt) \in \Omega^1(M_\Lambda, \mathbb{R}) \quad (257) $$

$$ \begin{aligned} \mathfrak{A}\rho_\Lambda = -i(\langle\langle \Lambda(x,t) | d | \Lambda(x,t) \rangle\rangle_* &+ \langle\langle \Lambda(x,t) | \partial_t | \Lambda(x,t) \rangle\rangle_* dt) \in \Omega^1(M_\Lambda, \mathfrak{u}(2)_{a.s.}) & (258) \\ \mathfrak{A}\rho_\Lambda = -i(\langle\langle \Lambda(x,t) | d | \Lambda(x,t) \rangle\rangle_* &+ \langle\langle \Lambda(x,t) | \partial_t | \Lambda(x,t) \rangle\rangle_* dt) \in \Omega^1(M_\Lambda, \mathfrak{u}(2)_{a.s.}) & (258) \end{aligned} $$

and we must redefine one tetrad by

$$ e_0^0 = 1 + A_0 \quad (259) $$

The metric of $M_\Lambda$ is then now

$$ d\tau^2 = (1 + A_0)^2 dt^2 - (\gamma^{ac} A_c dt + ds^a)(\gamma^{bd} A_d dt + ds^b)\gamma_{ad} \quad (260) $$

where $A$ and $\gamma$ depends on $s$ and $t$. $\vec{A} = \gamma^{ab} A_b \partial_a$ is still the shift vector of the foliation of $M_\Lambda$, but now $1 + A_0(x, t)$ is its laps function.

The Lorentz connection has now new components $\Omega_0^{\mu\nu} = \varepsilon^{0\mu\nu}_i \mathrm{Re} \operatorname{tr}(\sigma^i \mathfrak{A}_0) + (\delta^{\nu 0} \delta^\mu_i - \delta^{\mu 0} \delta^\nu_i) \mathrm{Im} \operatorname{tr}(\sigma^i \mathfrak{A}_0)$. With the modification of the inverse tetrads $e_0^a = -\frac{\gamma^{ab} A_b}{1+A_0}$ and $(e^{-1})_0^0 = (e_0^0)^{-1} = \frac{1}{1+A_0}$, this induces these new Christoffel symbols:

$$ \Gamma_{b0}^0 = \partial_b \ln e_0^0 + (e_0^0)^{-1} \Omega_{bj}^0 e_0^j \quad (261) $$

$$ \Gamma_{b0}^a = e_0^a \partial_b e_0^0 + e_i^a \partial_b e_0^i + e_0^a \Omega_{bj}^0 e_0^j + e_i^a \Omega_{b0}^i e_0^0 + e_i^a \Omega_{bj}^i e_0^j \quad (262) $$

$$ \Gamma_{bc}^0 = (e_0^0)^{-1} \Omega_{bj}^0 e_c^j \quad (263) $$

$$ \Gamma_{bc}^a = e_i^a \partial_b e_c^i + e_i^a \Omega_{bj}^i e_c^j + e_0^a \Omega_{bj}^0 e_c^j \quad (264) $$

$$ \Gamma_{0c}^a = e_i^a \partial_0 e_c^i + e_0^a \Omega_{0j}^0 e_c^j + e_i^a \Omega_{0j}^i e_c^j \quad (265) $$

$$ \Gamma_{0c}^0 = (e_0^0)^{-1} \Omega_{0j}^0 e_c^j \quad (266) $$

$$ \Gamma_{00}^0 = \partial_0 \ln e_0^0 + (e_0^0)^{-1} \Omega_{0j}^0 e_0^j \quad (267) $$
**Property 13** In the case where $\dot{X}^i = i[\tilde{H}_\mathcal{E}, X^i]$ $\iff$ $X^i(t) = U_\mathcal{E}^\dagger(t)X^i(0)U_\mathcal{E}(t)$, we have:

$$M_{\Lambda}(t) = M_{\Lambda}(0) \qquad (268)$$

$$|\Lambda(x, t)\rangle\rangle = U_{\mathcal{E}}^{\dagger}(t)|\Lambda(x, 0)\rangle\rangle \qquad (269)$$

$$A = A_a(x, 0)ds^a + \langle\langle\Lambda(x, 0)|H_{\mathcal{E}}|\Lambda(x, 0)\rangle\rangle dt \qquad (270)$$

$$\mathfrak{A} = \mathfrak{A}_a(x, 0)ds^a + \mathrm{tr}_{\mathcal{H}}(|\Lambda(x, 0)\rangle\rangle\langle\langle\Lambda(x, 0)|H_{\mathcal{E}})\rho_{\Lambda}(x, 0)^{-1}dt\rangle\rangle(271)$$

$$\mathbf{A}_{\mathbf{D}_t} = U_{\mathcal{E}}^{\dagger}(t)\mathbf{A}_{\mathbf{D}_0}U_{\mathcal{E}}(t) + iU_{\mathcal{E}}^{\dagger}(t)dU_{\mathcal{E}}(t) + i\mathbf{D}_t dU_{\mathcal{E}}^{\dagger}(t)U_{\mathcal{E}}(t)\mathbf{D}(272)$$

($\rho_{\Lambda}^{-1}$ standing for a pseudo-inverse), where $\mathbf{D}_t = U_{\mathcal{E}}^{\dagger}(t)\mathbf{D}_0U_{\mathcal{E}}(t)$. The flow $\varphi^t$ defined by eq.(253,254) is just a time-dependent diffeomorphism of $M_{\Lambda}(0)$ ($\varphi^t \in \mathrm{Diff}\,M_{\Lambda}(0)$).

*Proof:* $\tilde{D}_x(t) = U_\mathcal{E}^\dagger(t)\tilde{D}_x(0)U_\mathcal{E}(t) \Rightarrow \tilde{D}_x(0)U_\mathcal{E}(t)|\Lambda(x,t)\rangle\rangle = 0 \Rightarrow |\Lambda(x,0)\rangle\rangle = U_\mathcal{E}(t)|\Lambda(x,t)\rangle\rangle$. It follows that $\partial_t|\Lambda(x,t)\rangle\rangle = -U_\mathcal{E}^\dagger(t)\dot{U}_\mathcal{E}(t)U_\mathcal{E}^\dagger(t)|\Lambda(x,0)\rangle\rangle = -U_\mathcal{E}^\dagger(t)H_\mathcal{E}|\Lambda(x,0)\rangle\rangle$.

$u_{yx} \otimes \mathbf{D}_0(y, x)|\Lambda(x, 0)\rangle\rangle = |\Lambda(y, 0)\rangle\rangle \Rightarrow u_{yx} \otimes U_\mathcal{E}^\dagger \mathbf{D}_0(y, x)U_\mathcal{E}|\Lambda(x, t)\rangle\rangle = |\Lambda(y, t)\rangle\rangle$. It follows $\mathbf{D}_t(y, x) = U_\mathcal{E}^\dagger \mathbf{D}_0(y, x)U_\mathcal{E}$ and then $d\mathbf{D}_t = dU_\mathcal{E}^\dagger \mathbf{D}_0 U_\mathcal{E} + U_\mathcal{E}^\dagger d\mathbf{D}_0 U_\mathcal{E} + U_\mathcal{E}^\dagger \mathbf{D}_0 dU_\mathcal{E}$. $\square$

In the case where the quasicoherent state is separable, the flow is similar to the Ehrenfest theorem: $\frac{d}{dt}\langle\Omega_{x(t),t}|X^i|\Omega_{x(t),t}\rangle = i\langle\Omega_{x(t),t}|[\tilde{H}_\mathcal{E}, X^i]|\Omega_{x(t),t}\rangle$.

Another simple time-dependent case arises with time-dependent external gauge changes:

**Property 14** In the case where $X^i(t) = J(t)^i_j X^j(0)$ with $J \in \mathrm{SO}(3)$ (independent of $x$), we have:

$$M_{\Lambda}(t) = \{J(t)^i_j x^j(0), x(0) \in M_{\Lambda}(0)\} \qquad (273)$$

$$|\Lambda(x, t)\rangle\rangle = u_J(t)^{-1}|\Lambda(J(t)^{-1}x, 0)\rangle\rangle \qquad (274)$$

$$A = A_i(J^{-1}(t)x, 0)J^{-1}(t)^i_j \frac{\partial x^j}{\partial s^a} ds^a \\ + i\omega_{J^{-1}x}(\dot{u}_J(t)u_J(t)^{-1})dt \qquad (275)$$

$$\mathfrak{A} = u_J(t)^{-1}\mathfrak{A}_i(J^{-1}(t)x, 0)u_J(t)J^{-1}(t)^i_j \frac{\partial x^j}{\partial s^a} ds^a \\ + i u_J(t)^{-1} \dot{u}_J(t) dt \qquad (276)$$

$$\mathbf{A}_{\mathbf{D}(y,x)}(t) = \mathbf{G}_J(y, x, t)^{\dagger} \mathbf{A}_{\mathbf{D}(y,x)}(0) \mathbf{G}_J(y, x, t) \\ + i \mathbf{G}_J(y, x, t)^{\dagger} d\mathbf{G}_J(y, x, t) \qquad (277)$$

where $u_J \in \mathrm{SU}(2)$ is such that $u_J^{-1}\sigma_j u_J = J^i_j\sigma_i$ and $\mathbf{G}_J(y, x, t) = \mathbf{D}(y, x)^{-1}\mathbf{D}(J(t)^{-1}y, J(t)^{-1}x)$.

The flow $\varphi^t$ defined by eq.(253,254) is just the rotation of $M_{\Lambda}(0)$ in $\mathbb{R}^3$ defined by $J(t)$.

*Proof:* We apply simply the properties 8 and 12 concerning the external gauge changes with a time-dependent $J$ in place of a $x$-dependent $J$. $\square$

### Example 2: Fuzzy surface plots

We consider fuzzy surface plots governed by the environmental Hamiltonian $H_\mathcal{E} = \omega a^+ a$ ($\mathcal{X}$ being the set of observables of a harmonic oscillator or of a single-mode boson field, $\omega$ is the quantum of energy of the environment). $U_\mathcal{E}(t) = e^{-i\omega t a^+ a}$ and

$$U_{\mathcal{E}}^{\dagger}(t)aU_{\mathcal{E}}(t) = e^{-i\omega t}a \qquad (278)$$
<figure><img src="image_10.png" /><figcaption>**Figure 5.** The eigenmanifold $M_\Lambda$ of the fuzzy hyperbolic paraboloid with some orbits of the flow $\phi^t$ defined by eq.(254) for a fuzzy space dynamics governed by the Hamiltonian $H_\varepsilon = \omega a^+ a$.</figcaption></figure>

It follows that

$$|\Lambda(\alpha, t)\rangle\rangle = U_{\varepsilon}^{\dagger}(t)|\Lambda(\alpha, 0)\rangle\rangle = |\Lambda(e^{i\omega t}\alpha)\rangle\rangle_0 \quad (279)$$

where $|\Lambda(\alpha)\rangle\rangle_0$ is the quasicoherent state of the time-independent case ($U_{\varepsilon}^{\dagger}(t)|\alpha\rangle = |e^{i\omega t}\alpha\rangle$ for the Perelomov coherent states of the CCR algebra [11, 27]). We have then for the fuzzy plane and the fuzzy paraboloids

$$A = -i \frac{\bar{\alpha} d\alpha - \alpha d\bar{\alpha}}{2} + \omega |\alpha|^2 dt + \mathcal{O}(\epsilon^2) \quad (280)$$

The flow onto $M_\Lambda$ is then $\phi^t(\alpha_0) = e^{-i\omega t}\alpha_0$ which consists to a rotation on the surface of $M_\Lambda$ around $\alpha = 0$ at the frequency $\omega$ as for example fig. 5.

In contrast, if we consider a fuzzy surface plot governed by the D2-brane equation of motion eq.(250) with $X^i(t) = u(t)X_0^i$ for $u(t) \in \mathbb{R}^+$ such that $u(0) = 1$ ($X_0^i$ being the coordinate observables of the time-independent case), we have $\ddot{u} = 0 \Leftrightarrow u(t) = vt + 1$ (for some $v \in \mathbb{R}$). We have then for the fuzzy plane and the fuzzy paraboloids:

$$\sigma_i \otimes (X^i(t) - x^i)|\Lambda(x, t)\rangle\rangle = 0 \quad (281)$$

$$\Leftrightarrow \sigma_i \otimes (u(t)X_0^i - x^i)|\Lambda(x, t)\rangle\rangle = 0 \quad (282)$$

$$\Leftrightarrow \sigma_i \otimes (X_0^i - \frac{x^i}{u(t)})|\Lambda(x, t)\rangle\rangle = 0 \quad (283)$$

$$\Rightarrow \quad \left\{ \begin{array}{l} x^i(t) = u(t)x_0^i \\ |\Lambda(x, t)\rangle\rangle = |\Lambda(\frac{x}{u(t)})\rangle\rangle_0 \end{array} \right. \quad (284)$$

where $|\Lambda(\alpha)\rangle\rangle_0$ is the quasi-coherent state of the time-independent case. $M_\Lambda(t)$ grows with an expansion factor $u(t) = vt + 1$. We have then

$$A = -i \frac{\bar{\alpha} d\alpha - \alpha d\bar{\alpha}}{2u(t)^2} + \mathcal{O}(\epsilon^2) \quad (285)$$

$$\text{Now we consider the time-dependent SO(3) transformation } J = \begin{pmatrix} \cos(\omega t) & -\sin(\omega t) & 0 \\ \sin(\omega t) & \cos(\omega t) & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

corresponding to a rotation around the third axis at angular velocity $\omega$. We have then
$u_J = e^{\frac{1}{2}\omega t\sigma_3}$. Now $M_\Lambda(t)$ is just the rotation around the third axis of the eigenmanifold of the time-independent case and

$$|\Lambda(\alpha, t)\rangle\rangle = e^{-\frac{1}{2}\omega t\sigma_3} |\Lambda(e^{-i\omega t}\alpha)\rangle\rangle_0 \qquad (286)$$

and

$$A = -i \frac{\bar{\alpha} d\alpha - \alpha d\bar{\alpha}}{2} + \omega \left( \frac{1}{2} \mathcal{O}(\epsilon^2) \right) \left( \frac{1}{2} \mathcal{O}(\epsilon^2) \right) \left( \frac{1}{2} \mathcal{O}(\epsilon^2) \right) \qquad (287)$$

Note the difference between the dynamics induced by $J$ and $H_\mathcal{E}$. The two ones induces a rotation around 0 in the base complex plane, but $J$ as rotation in $\mathbb{R}^3$ around the third axis does not affect the z-perturbation in contrast with $H_\mathcal{E}$. $J$ induces a rotation of $\mathfrak{M}/M_\Lambda$ in $\mathbb{R}^3$ as a solid whereas $H_\mathcal{E}$ induces a rotation inside $\mathfrak{M}/M_\Lambda$ as a fluid traversed by a current of flow $\varphi^t$.

# 7. Conclusion

A fuzzy space $\mathfrak{M}$ can be endowed with two quantum metrics $d\ell^2_x$ (square length observable) and $\gamma$ (noncommutative symmetric quantum bivector). The square root of the quantum average of the first one ($\sqrt{\gamma} = \sqrt{\omega_{x(s+ds)}(d\ell^2_{x(s)})}$) measures the length of mean path onto $\mathfrak{M}$ (the average being defined with respect to the quantum fluctuations of the paths induced by the Heisenberg uncertainties). The quantum average of the square root of the second one ($\text{dist}_\gamma = \omega_x(\sqrt{\gamma})$) measures the mean length of the fluctuating quantum paths. The two associated (minimising) geodesics correspond to minimise the energy uncertainty (for the first one, for the quasicoherent state with a misalignment of the probe; for the second one, for the gap between two quasicoherent states with imposing the microcanonical state as spin state).

Gauge theories onto $\mathfrak{M}$ induces torsions. The Berry phase generator $A$ of the strong adiabatic regime is associated with a contorsional force similar to a Laplace force generated by the magnetic field $F = dA$. The operator valued geometric phase generator $\mathfrak{A}$ of the weak adiabatic regime is associated with a contorsional force generating spin precessions as in Einstein-Cartan gravity theory. The two associated (autoparallel) geodesics are in accordance with these interpretations.

The presence of two metrics is related with the two possible evolutions: smooth moves of the probe and transition from $|\Lambda(x)\rangle\rangle$ to $|\Lambda(y)\rangle\rangle$ after a jump of the probe and induced by the projection after a measurement (which can be modelled by the quantum path $D(y, x)$). If the language of the noncommutative geometry is the more natural to describe fuzzy spaces, these ones are also related to category structures as studied in details Appendix D because of this double evolution.

For the sake of the simplicity we have considered only 3D fuzzy spaces in this paper. However the discussion can be generalised to higher dimensions, with possible interesting applications:

## 7.1. Higher dimension

We can consider a ND fuzzy space $(\mathfrak{X}, \mathbb{C}^N \otimes \mathcal{H}, \mathcal{D}_x)$ ($x \in \mathbb{R}^N$) with (for the Euclidean case) $\mathfrak{X} = \text{Lin}_\mathbb{R}(X^1, ..., X^N, \text{id})$ and $\mathcal{D}_x = \Sigma_i \otimes (X^i - x^i)$ such that $\text{tr}(\Sigma_i \Sigma_j) = 2\delta_{ij}$, or (for the Lorentzian case) $\mathfrak{X} = \text{Lin}_\mathbb{R}(X^0, ..., X^{N-1}, \text{id})$ and $\mathcal{D}_x = \gamma_\mu \otimes (X^\mu - x^\mu)$ such that $\text{tr}(\gamma_\mu \gamma_\nu) = 2\eta_{\mu\nu}$ (Minkowski metric). In these cases $\dim M_\Lambda = N - 1$. We have the following applications:
* $N = 3$ and $\Sigma_i = \sigma_i$ (Pauli matrices): string theory BFFS matrix models with orbifoldisation and control of a qubit submitted to environmental noises.

* $N = 4$ and $\gamma_\mu$ the Dirac matrices: string theory IKKT matrix models with orbifoldisation.

* $N = 9$ or $N = 10$ and $\gamma_\mu$ generalisations of Dirac matrices: string theory BFSS or IKKT matrix models without orbifoldisation.

* $N = 8$ and $\Sigma_i$ the Gell-Mann matrices: control of a qutrit submitted to environmental noises.

* $N = 9$ and $\Sigma_{(ij)} = \frac{1}{\sqrt{2}}\sigma_i \otimes \sigma_j$: control of two qubits submitted to environmental noises.

By “orbifoldisation” we mean a dimension reduction as explained in ref. [7].

It is interesting to note that the fuzzy space models permit to picture an analogy between the two domains of application:
<table><caption>7.2. Analogy quantum gravity - quantum information</caption><thead><tr><th></th><th>Noncommutative geometry</th><th>String theory</th><th>Quantum information</th></tr></thead><tbody><tr><td>C<sup>2</sup></td><td>orientation state space</td><td>fermionic string spin state space</td><td>qubit state space</td></tr><tr><td>σ</td><td>orientation observables</td><td>fermionic string spin observables</td><td>qubit observables</td></tr><tr><td>H</td><td>quantum coordinate state space</td><td>state space of the fermionic string end</td><td>environment state space</td></tr><tr><td>X̄</td><td>quantum coordinate observables</td><td>field operators of D2-brane</td><td>environmental noise observables</td></tr><tr><td>Env(Χ)</td><td>set of quantum spatial observables</td><td>set of D2-brane observables</td><td>set of environment observables</td></tr><tr><td>Der(Χ)</td><td>quantum tangent vectors</td><td>set of D2-brane superoperators</td><td>set of environment superoperators</td></tr><tr><td>M × R</td><td>quantum space-time</td><td>D2-brane</td><td>environment</td></tr><tr><td>x̄</td><td>probe (test particle)</td><td>D0-brane</td><td>qubit control</td></tr><tr><td>R<sup>3</sup></td><td>class. space of a Galilean observer</td><td>embedding space of strings</td><td>control parametric space</td></tr><tr><td>t ↦→ x(t)</td><td>probe move</td><td>D0-brane dynamics</td><td>quantum computation</td></tr><tr><td>|Λ⟩</td><td>quasicoherent state of the geometry</td><td>ferm. string state of zero displ. energy</td><td>bipartite state of zero int. energy</td></tr><tr><td>M<sub>Λ</sub></td><td>classical space slice closest to M</td><td>D0-brane locations of zero displ. energy</td><td>control manifold of zero int. energy</td></tr><tr><td>D̄<sub>x</sub></td><td>square length observable</td><td>fermionic string Dirac operator</td><td>interaction picture Hamiltonian</td></tr><tr><td>dℓ<sup>2</sup><sub>x</sub> = D̄<sup>2</sup><sub>x</sub></td><td>quantum metric</td><td>inducing space metric at thermo. limit</td><td></td></tr><tr><td>γ</td><td>length of the average path</td><td>displ. ener. uncert. meas. misalign.</td><td>energy uncert. with meas. misalign.</td></tr><tr><td>√γ<sub>ab</sub>ds<sup>a</sup>ds<sup>b</sup></td><td>average length of quantum paths</td><td>average displ. ener. gap isotr. uncert.</td><td>average energy gap isotropic uncert.</td></tr><tr><td>dist<sub>γ</sub>(ω<sub>x+δx</sub>, ω<sub>x</sub>)</td><td>probe proper time</td><td>D0-brane proper time</td><td>ener. uncert. misalign. + time lag</td></tr><tr><td>dτ</td><td></td><td></td><td></td></tr><tr><td>√dτ<sup>2</sup> - dτ<sup>2</sup></td><td></td><td></td><td></td></tr><tr><td>A</td><td>shift vector</td><td>U(1)-gauge potential of the D0-brane</td><td>Berry phase induced by the control</td></tr><tr><td>e<sup>-i∫A</sup></td><td>spacetime Lorentz connection</td><td>fermionic string Lorentz connection</td><td>logical gate induced by the control</td></tr><tr><td>Λ</td><td>linking vector observable</td><td>quantised gauge field of the D2-brane</td><td>control jump</td></tr><tr><td>Pe<sup>-i∫Λ</sup> →</td><td></td><td></td><td></td></tr><tr><td>A<sub>D</sub>(L<sub>X̄</sub>)</td><td></td><td></td><td></td></tr><tr><td>D(y, x)</td><td></td><td></td><td></td></tr></tbody></table>
The noncommutative manifold can be seen as a quantised extended solid body or as a direct noncanonical quantisation of spacetime. The string theory model consists to a fermionic string linking a bosonic D2-brane and a D0-brane. The thermodynamical limit consists to a number of bosonic strings tending to infinity with constant density (semi-classical limit to the macroscopic scale). The displacement energy is the “energy tension” of the fermionic string. The two branes are submitted to gauge fields (electro-magnetic interaction). The quantum information model consists to a qubit controlled by classical fields and interacting with a large environment. For this case, the energy corresponds to the total energy interaction (qubit-control + qubit-environment). The variation of the control fields is assimilated to a quantum computation (by assimilating the resulting evolution operator to a logical gate).

The measure misalignment means that the measurement is performed with a small location error; the time lag means that the measurement slightly lags behind the dynamics.

This analogy between quantum gravity (described by noncommutative geometry or string theory) and quantum information theory can be related to other puzzling arguments (analogy between the classifications of qubit entanglements and charged string black holes [39], ER=EPR conjecture [40], AdS/CFT correspondance [41], string gauge structures arising in semi-classical light-matter interaction [37]) suggesting that the “spacetime Reality” is like a quantum computer (the “It for qubit” proposal [42]).

### 7.3. Future research directions

An important ingredient of the present study is the displacement operator $D(y, x)$. But a lot of the fuzzy spaces are not linkable (in the meaning of this paper). In Appendix B we have generalised to “paralinkable” fuzzy spaces for which it exists a generalised displacement operator. But no argument permits to state that all fuzzy spaces are totally paralinkable (especially, it is not the case if the quasicoherent state is separable at some points but is entangled at some others). It needs then to generalise the present study to unlikable fuzzy spaces. Moreover, the quantum local orientation (the spin degree of freedom) playing an important role, it could be interesting to study the possibility of defining non-orientable fuzzy spaces (at least fuzzy spaces having non-orientable eigenmanifolds).

## Appendix A. Perturbation theory of fuzzy spaces

Let $\mathfrak{M} = (\mathcal{X}, \mathbb{C}^2 \otimes \mathcal{H}, \mathcal{D}_x)$ be a fuzzy space, $(\delta X^i)$ be a set of perturbations of the coordinate observables, $\mathfrak{M} = (\hat{\mathcal{X}}, \mathbb{C}^2 \otimes \mathcal{H}, \hat{\mathcal{D}}_x)$ be the perturbed fuzzy space: $\hat{X}^i = X^i + \delta X^i$. In the whole of this appendix, “$\simeq$” stands for equal up to negligible element of magnitude $O(\|\delta_{ij}\delta X^i \delta X^j\|)$, where $\|\cdot\|$ is the operator norm induced by $\langle\cdot|\cdot\rangle$.

The perturbed quasicoherent state is solution of

$$ (\sigma_i \otimes (X^i - x^i) + \sigma_i \otimes (\delta X^i - \delta x^i))|\hat{\Lambda}\rangle = 0 \quad (A.1) $$

with $x \in M_\Lambda$ (eigen manifold of $\mathfrak{M}$) and $x + \delta x \in \hat{M}_\Lambda$ (eigen manifold of $\mathfrak{M}$).
## Appendix A.1. M strongly non-degenerate at x

The perturbation vector $\langle\delta\vec{X}\rangle = \langle\hat{\Lambda}|\delta\vec{X}|\hat{\Lambda}\rangle \simeq \langle\Lambda|\delta\vec{X}|\Lambda\rangle$ defines the perturbation direction $\vec{p}_x = \frac{\langle\delta\vec{X}\rangle}{\|\langle\delta\vec{X}\rangle\|}$. We set then $\delta x^i = \|\delta\vec{x}\|p_x^i$. At the first order of the perturbation theory, the eigenvalue $\hat{\lambda}_0$ issued from the zero eigenvalue of $\mathcal{D}_x$ is

$$ \hat{\lambda}_0(x, \delta x) \approx \langle\Lambda(x)|\sigma_i \otimes (\delta X^i - \delta x^i)|\Lambda(x)\rangle \quad (A.2) $$

The condition that $x + \delta x \in \hat{M}_{\Lambda}$ implies that:

$$ \hat{\lambda}_0 \approx 0 \Rightarrow n_i(x)\delta x^i = \langle\Lambda(x)|\sigma_i \otimes \delta X^i|\Lambda(x)\rangle \quad (A.3) $$

$$ \Rightarrow \|\delta\vec{x}\|\vec{p}_x \cdot \vec{n}_x = \langle\Lambda(x)|\sigma_i \otimes \delta X^i|\Lambda(x)\rangle \quad (A.4) $$

$$ \Rightarrow \delta\vec{x} = \frac{\langle\Lambda(x)|\sigma_i \otimes \delta X^i|\Lambda(x)\rangle}{\vec{p}_x \cdot \vec{n}_x} \vec{p}_x \quad (A.5) $$

where $\vec{n}_x = \langle\Lambda(x)|\vec{\sigma}|\Lambda(x)\rangle$ is the normal vector to $M_{\Lambda}$ at $x$.

If the quasicoherent state is separable, $|\Lambda(x)\rangle = |O_x\rangle \otimes |\Omega_x\rangle$, then $\langle\delta\vec{X}\rangle = \langle\Omega_x|\delta\vec{X}|\Omega_x\rangle$, $\langle\Lambda(x)|\sigma_i \otimes \delta X^i|\Lambda(x)\rangle = \langle O_x|\vec{\sigma}|O_x\rangle \cdot \langle\Omega_x|\delta\vec{X}|\Omega_x\rangle$ and $\vec{n}_x = \langle O_x|\vec{\sigma}|O_x\rangle$. It follows that $\delta\vec{x} = \langle\Omega_x|\delta\vec{X}|\Omega_x\rangle$.

The metric of $\hat{M}_{\Lambda}$ is

$$ \hat{\gamma}_{ab} \approx \gamma_{ab} + \delta_{ij} \left( \frac{\partial x^i}{\partial s^a} \frac{\partial \delta x^j}{\partial s^b} + \frac{\partial x^i}{\partial s^b} \frac{\partial \delta x^j}{\partial s^a} + \frac{\partial \delta x^i}{\partial s^a} \frac{\partial \delta x^j}{\partial s^b} \right) \quad (A.6) $$

The perturbed quasicoherent state is then

$$ |\hat{\Lambda}(x+\delta x)\rangle \approx |\Lambda(x)\rangle - \sum_{n>0} \frac{\langle\lambda_n(x)|\sigma_i \otimes (\delta X^i - \delta x^i)|\Lambda(x)\rangle}{\lambda_n(x)} |\lambda_n(x)\rangle \quad (A.7) $$

where $\mathcal{D}_x|\lambda_n(x)\rangle = \lambda_n(x)|\lambda_n(x)\rangle$, $\forall x \in M_{\Lambda}$ ($\lambda_0 = 0$).

## Appendix A.2. M only weakly non-degenerate at x

We consider the case of a strictly weakly non-degenerate separable quasicoherent state. In this case, all state $|O_{\Theta}\rangle \otimes |\Omega_x\rangle$ is quasicoherent state with $|O_{\Theta}\rangle = \cos\theta|0\rangle + e^{i\varphi}\sin\theta|1\rangle$ ($\Theta = (\theta, \varphi)$, $|0\rangle, |1\rangle$) is the canonical basis of $\mathbb{C}^2$). Let $|I_{\Theta}\rangle = -e^{-i\varphi}\sin\theta|0\rangle + \cos\theta|1\rangle$ be an orthogonal vector to $|O_{\Theta}\rangle$. The perturbation direction does not depend on the choice of $\Theta$ since $\langle\Lambda(x, \Theta)|\delta\vec{X}|\Lambda(x, \Theta)\rangle = \langle\Omega_x|\delta\vec{X}|\Omega_x\rangle$. The eigenvalue matrix is then

$$ [\hat{\lambda}_0] = \begin{pmatrix} \langle O_{\Theta} | \sigma_i | O_{\Theta} \rangle & \langle O_{\Theta} | \sigma_i | I_{\Theta} \rangle \\ \langle I_{\Theta} | \sigma_i | O_{\Theta} \rangle & \langle I_{\Theta} | \sigma_i | I_{\Theta} \rangle \end{pmatrix} \langle \Omega_x | (\delta X^i - p_x^i \|\delta\vec{x}\|) | \Omega_x \rangle \quad (A.8) $$

$[\hat{\lambda}_0] = 0$ if $\delta\vec{x} = \langle\Omega_x|\delta\vec{X}|\Omega_x\rangle$ (the perturbed strictly weakly degenerate quasicoherent state is still strictly weakly degenerate). The perturbed quasicoherent states are then

$$ |\hat{\Lambda}(x, \Theta)\rangle \approx |O_{\Theta}\rangle \otimes |\Omega_x\rangle - \sum_{n>0} \frac{\langle\lambda_n(x)|\sigma_i \otimes (\delta X^i - \delta x^i)|O_{\Theta}\rangle \otimes |\Omega_x\rangle}{\lambda_n(x)} |\lambda_n(x)\rangle \quad (A.9) $$

For an entangled quasicoherent state, we have several perturbation directions $\vec{p}_{x,u} = \langle\Lambda(x)|u^\dagger u\delta\vec{X}|\Lambda(x)\rangle$ where $u \in \text{Lin}(\text{id}, u_1, u_2, u_3)$ with $u_i$ such that $\text{tr}(\rho_{\Lambda}u_i) = 0$