Polar duality is a well-known concept from convex geometry and analysis. In the present paper we study a symplectically covariant versions of polar duality, having in mind their applications to quantum harmonic analysis. It makes use of the standard symplectic form on phase space and allows a precise study of the covariance matrix of a density operator.

The concept of polar dual set in convex geometry corresponds to the concept of dual space in linear algebra. Given a convex body X in the Euclidean space $Rn$ its polar dual is the set X of all $p∈(Rn)*$ such that ⟨p, x⟩ ≤ ; here is a positive constant, usually taken to be one in the standard literature (we use for flexibility a parameter-dependent definition; in quantum mechanics would be Planck’s constant h divided by 2π; in harmonic analysis one would take = 1/2π while the standard choice in the theory of partial differential equations is = 1). We will most of the time identify the dual space $(Rn)*$ with $Rn$ itself, in which case the polar dual of X is identified, using the standard Euclidean structure (x, p) ⟼ p⋅x with the set
$Xℏ={p∈Rn:supx∈X(p⋅x)≤ℏ}.$
(1)
We will study a variants of polar duality on the symplectic space $(T*Rn,σ)$ where σ is the standard symplectic form $∑j=1ndpj∧dxj$; we will identify the cotangent bundle $T*Rn$ with $R2n≡Rxn×Rpn$. This variant is what we call “symplectic polar duality”: if $Ω⊂R2n$ is a convex body we define its symplectic polar dual by
$Ωℏ,σ={z′∈R2n:supz∈Ωσ(z,z′)≤ℏ};$
(2)
clearly Ω,σ = J) where J is the standard symplectic automorphism J(x, p) = (px). The interest of this notion comes (among other things) from the fact that it has the symplectic covariance property S,σ) = [S(Ω)],σ for every S ∈ Sp(n).

As we will see in the course of this paper, symplectic polar duality is closely related to difficult questions of positivity for trace class operators, and allows to express “quantization conditions” in an elegant and concise geometric way. On the other hand, Lagrangian polar duality allows a geometric redefinition of the notion of quantum state; these states are usually viewed as “wavefunctions” in physics; in our approach they appear as geometric objects defined in terms of convex products $Xℓ×(Xℓ)ℓ′ℏ$ whose factors are supported by transversal Lagrangian planes, and their functional aspects appear only as subsidiary through the use of the John ellipsoid (maximum volume ellipsoid).

The applications of concepts of convex geometry and analysis outside their original area is not new, see for instance Milman34 who applies such methods to probability theory; also see the treatise3 by Aubrun and Szarek.

Let us describe some highlights of this work, emphasizing what we hold for the most important results (our choice being of course somewhat subjective, and highly depending on the authors’ tastes).

The covariance matrix of a physical state (be it classical, or quantum) is a statistical object whose importance in information theory is crucial; it encodes the statistical properties of the state and its study is, as we will see, greatly facilitated by the polar duality approach. The central result is, no doubt, Theorem 17 who gives two criteria for what we call “quantum admissibility” of a phase space ellipsoid (the definition of this notion of admissibility is closely related to the uncertainty principle of quantum mechanics, an is rigorously defined in Definition 14). The first criterion say that an ellipsoid Ω is admissible if and only if it contains its symplectic polar dual Ω,σ; the second is of a more subtle nature; it shows that it is sufficient (and necessary) for admissibility that Ω,σF ⊂ Ω ∩ F for every symplectic subspace of $(R2n,σ)$. This is a tomographic condition reminiscent of an old result by Narcowich35 concerning covariance and information ellipsoid in quantum mechanics. This analogy is made even more convincing in Theorem 27 where we give a dynamical description of quantum admissibility of covariance and information ellipsoids using the techniques we develop; in particular the role of the so fruitful notion of symplectic capacity is highlighted (symplectic capacities are strongly related to Gromov’s famous non-squeezing theorem). We take this opportunity to give a new functional-analytical characterization of those (classical, or quantum) state for which the covariance matrix is well-defined. This is done in terms of a class of modulation spaces.

Notation 1.

The standard symplectic form σ is written in matrix form as σ(z, z′) = Jz⋅zwhere $J=0n×nIn×n−In×n0n×n$ is the standard symplectic matrix. The standard symplectic group Sp(n) is the group of automorphisms S of $T*Rn≡R2n$ such that S*σ = σ; in matrix notation: SSp(n) if and only if STJS = J (or, equivalently, SJST = J), ST the transpose of S). The unitary representation of the double cover of Sp(n) (the metaplectic group) is denoted by πMp: Mp(n) ⟶ Sp(n). The Lagrangian Grassmannian of $(R2n,σ)$ is denoted Lag(n), thus Lag(n) if and only if is a linear subspace of $R2n$ with dim  = n and σ| = 0. The elements of Lag(n) will be called Lagrangian planes.

For detailed treatments of the topics of convex geometry and analysis used in this article we refer to the treatise3 by Aubrun and. Szarek and to Vershynin’s online lecture notes.40 For a comprehensive study of convex geometry with applications to optimization theory see Boyd et al.8

#### 1. Definition

Let X and Y be convex sets in $Rn$; then,
$(X∪Y)ℏ=Xℏ∩Yℏ , (X∩Y)ℏ=Xℏ∪Yℏ̃$
(3)
$X⊂Y⟹Yℏ⊂Xℏ , X closed ⟹(Xℏ)ℏ=X$
(4)
$A∈GL(n,R)⟹(AX)ℏ=(AT)−1Xℏ$
(5)
(in the second formula (3) $Xℏ∪Yℏ̃$ is the convex hull of XY). If $A=AT∈GL(n,R)$ is positive definite then
${x∈Rxn:Ax⋅x≤ℏ}ℏ={p∈Rpn:A−1p⋅p⋅≤ℏ}$
(6)
hence, in particular,
$BXn(ℏ)ℏ=BPn(ℏ) , BPn(ℏ)ℏ=BXn(ℏ)$
(7)
where $BXn(ℏ)$ (resp. $BPn(ℏ)$) is the ball in $Rxn$ (resp. $Rpn$) with radius $ℏ$ and centered at 0.

Remark 2.

The properties of polar duality is less transparent for convex bodies not centered at the origin and requires the use of the so-called Santaló point.37

#### 2. Projections and intersections

Polar duality exchanges the projection and the intersection operations.3 While this result seems to be well-known it seems difficult to find a detailed proof in the literature, so we prove this important result, following.40 For this we need the following elementary lemma:

Lemma 3.

Let X = {x: Ax⋅x ≤ 1} and P = {p: Bp⋅p ≤ 1} (A, B positive definite and symmetric) be two ellipsoids. We have XP if and only if AB−1, and X = P if and only if AB = In×n.

Proof.

We have $X=A−1/2(BXn(ℏ))$ and $P=B−1/2(BPn(ℏ))$ and the inclusion XP is thus equivalent to the inequality A1/2B−1/2 in the Löwner ordering, that is, to AB−1 with equality if and only if X = P.■

Polar duality exchanges the operations of intersection and orthogonal projection:

Proposition 4.
Let $X⊂Rxn$ be a convex body containing 0 in its interior and F a linear subspace of $Rxn$; we have
$(ΠFX)ℏ=Xℏ∩F , (X∩F)ℏ=ΠF(Xℏ)$
(8)
where ΠF us the orthogonal projection in $Rxn$ onto F. [In $(ΠFX)ℏ$ and (XF) the polar duals are taken inside the subspace F equipped with the induced inner product.]

Proof.
It suffices to prove the first formula (8) since the second follows by duality:
$X∩F=(Xℏ)ℏ∩F=(ΠFXℏ)ℏ$
and hence (XF) = ΠF(X). Let us next show that ΠF(X) ⊂ (XF). For pX we have, for every xXF,
$x⋅ΠFp=ΠFx⋅p=x⋅p≤ℏ$
hence ΠFp ∈ (XF). To prove the inclusion ΠF(X) ⊃ (XF) we note that it is sufficient, by the anti-monotonicity and reflexivity properties of polar duality, to prove that $(ΠF(Xℏ))ℏ⊂X∩F$. Let $x∈(ΠF(Xℏ))ℏ$; we have x⋅ΠFp for every pX. Since xF (because the dual of a subset of F is taken inside F) we also have
$ℏ≥x⋅ΠFp=ΠFx⋅p=x⋅p$
from which follows that $x∈(Xℏ)ℏ=X$, which shows that xXF. This concludes the proof.■

#### 3. John and Löwner ellipsoids

A fundamental tool in convex geometry and Banach space geometry is the John ellipsoid of a convex body $Ω⊂R2n$. It is4,8 the (unique) ellipsoid ΩJohn of maximal volume contained in Ω; similarly the (unique) minimum enclosing ellipsoid is the Löwner ellipsoid $ΩLöwner$. We note that both the John and the Löwner ellipsoids transform covariantly under linear (and affine) transforms: if $L∈GL(n,R)$ then
$(L(X))John=L(XJohn), (L(X))Löwner=L(XJohn).$
(9)

Remark 5.

One can define the John ellipsoid in any finite-dimensional normed space, regardless of whether the space is Euclidean or not. The definition of the John ellipsoid in a normed space is the same as the one given earlier.

Polar duality interchanges the John and Löwner ellipsoids; we have the following duality relations hold for convex symmetric bodies:3
$(XJohn)ℏ=(Xℏ)Löwner, (XLöwner)ℏ=(Xℏ)John.$
(10)

#### 4. Blaschke–Santaló inequality and Mahler volume

Let X be an origin symmetric convex body in $Rxn$. By definition, the Mahler volume (or volume product) of X is the product
$υ(X)=Voln(X)Voln(Xℏ)$
(11)
where Voln is the usual Euclidean volume on $Rxn$. The Mahler volume is a dimensionless quantity because of its rescaling invariance: we have υ(λX) = υ(X) for all λ > 0. More generally, the Mahler volume is invariant under linear automorphisms of $Rxn$: if $L∈GL(n,R)$ we have
$υ(LX)=Voln(LX)Voln(LT)−1Xℏ$
(12)
$=Voln(X)Voln(Xℏ)=v(X).$
(13)
A remarkable property of polar duality is the Blaschke–Santaló inequality:6 assume again that X is a centrally symmetric body; then
$υ(X).≤Voln(Bn(ℏ))2=(πℏ)nΓn2+12$
(14)
and equality is attained if and only if $X⊂Rxn$ is an ellipsoid centered at the origin (see Ref. 5 for a proof using Fourier analysis). It is conjectured (the “Mahler conjecture”32) that one has the lower bound
$υ(X)≥(4ℏ)nn!$
(15)
with equality only when X is the hypercube C = [−1, 1]n. Bourgain and Milman7 have shown the existence, for every $n∈N$, of a constant Cn > 0 such that
$Voln(X)Voln(Xℏ)≥Cnℏn/n!$
(16)
and more recently Kuperberg31 has shown that one can choose Cn = (π/4)n.

Remark 6.
In view of the invariance property (12) this is equivalent to saying that the minimum is attained by any n-parallelepiped
$X=[−2σx1x1,2σx1x1]×⋅⋅⋅×[−2σxnxn,2σxnxn].$
(17)
This is related to the covariances of the tensor product ψ = ϕ1 ⊗⋅⋅⋅⊗ ϕn of standard one-dimensional Gaussians $ϕj(x)=(πℏ)−1/4e−xj2/2ℏ$; the function ψ is a minimal uncertainty quantum state in the sense that it reduces the Heisenberg inequalities to equalities. This observation might lead to aquantum proofof the Mahler conjecture.

Here is an elementary application of polar duality which highlights the role it plays in questions related to the uncertainty principle of quantum mechanics. We are following the presentation we gave in Ref. 20.

Hardy’s uncertainty principle27 in its original (one dimensional) formulation says that if the moduli of $ψ∈L1(R)∩L2(R)$ and of its Fourier transform, here defined by
$ψ̂(p)=Fψ(p)=12πℏ∫−∞∞e−iℏpxψ(x)dx$
are different from zero and satisfy estimates
$|ψ(x)|≤CAe−a2ℏx2 , |ψ̂(p)|≤CBe−b2ℏp2$
(CA, CB > 0, a, b > 0), then we must have ab ≤ 1 and if ab = 1 we have $ψ(x)=Ce−a2ℏx2$ for some complex constant C. We have proven a multidimensional version of this result (see Refs. 21 and, 23, Chapter 10). Let $ψ,ψ̂∈L1(Rn)∩L2(Rn)$, ψ ≠ 0 where
$ψ̂(p)=Fψ(p)=1(2πℏ)n/2∫Rne−iℏp⋅xψ(x)dx.$
Then:

Proposition 7.
Let $A,B∈Sym(n,R)$ be positive definite and ψ as above. Assume that there exist a constants CA, CB > 0 such that
$|ψ(x)|≤CAe−12ℏAx⋅x and |Fψ(p)|≤CBe−12ℏBp⋅p.$
(18)

(i) The eigenvalues λj, 1 ≤ jn, of AB are $≤1$; (ii) If λj = 1 for all j, then $ψ(x)=ke−12ℏAx2$ for some $k∈C$.

It turns out that this result can easily be restated in terms of polar duality:

Corollary 8.

The Hardy estimates are satisfied by a non-zero function $ψ∈L2(Rn)$ if and only if the ellipsoids X = {x: Ax⋅x} and P = {p: Bx⋅x} satisfy XP with equality X = P if and only $ψ(x)=ke−12ℏAx⋅x$ for some $k∈C$.

Proof.

This immediately follows from Lemma 3.■

We will apply the results above to sub-Gaussian estimates of the Wigner function in Sec. III C.

In what follows we identify the dual of the symplectic space $(R2n,σ)$ with itself.

#### 1. Symplectic covariance

Let $J=0n×nIn×n−In×n0n×n$ be the standard symplectic matrix. Using the matrix formulation σ(z, z′) = Jz⋅z′ of the symplectic form it is straightforward to verify It is straightforward to verify that the symplectic polar dual
$Ωℏ,σ={z′∈R2n:supz∈Ωσ(z,z′)≤ℏ}$
(19)
of a convex body $Ω⊂R2n$ is related to the ordinary polar dual Ω by the formula
$Ωℏ,σ=(JΩ)ℏ=J(Ωℏ).$
(20)
It follows that:

Lemma 9.
Consider the phase space ellipsoid
$ΩM={z:Mz⋅z≤r2}$
where M is symmetric and positive definite and r > 0. Then
$ΩMℏ,σ={z:−JM−1Jz⋅z≤(ℏ/r)2}.$
(21)

Proof.

The ordinary -polar dual $ΩMℏ$ of ΩM is defined by M−1zz ≤ (/r)2. In view of (20) $ΩMℏ,σ=J(ΩM)$ hence the result.

Formula (20) can easily be generalized to yield the following important symplectic covariance result:■

Proposition 10.
Let Ω be a symmetric convex body and SSp(n). (i) we have
$(S(Ω))ℏ,σ=S(Ωℏ,σ).$
(22)
(ii) More generally for SSp(n) and F a linear subspace of $R2n$ we have
$S(Ω∩F)ℏ,σ=(SΩ∩SF)ℏ,σ.$
(23)

Proof.
(i) Using successively (20), the scaling property (5) in dimension 2n, and again (20), we have
$S(Ωℏ,σ)=SJ(Ωℏ)=J(ST)−1(Ωℏ)=J(S(Ω))ℏ=(S(Ω))ℏ,σ.$
(ii) Formula (23) follows from formula (22) since Ω ∩ F is convex and symmetric.■

#### 2. Quantum blobs

Symplectic balls with radius $ℏ$ are the only fixed ellipsoids for ordinary polar duality. They play an important role in various formulations of the uncertainty principle of quantum mechanics14,18,24 where they represent minimum uncertainty units; this motivates the following definition:

Definition 11.

A quantum blob QS(z0) is a symplectic ball with radius $ℏ$: $QS(z0)=S(B2n(z0,ℏ))$ for some SSp(n). When z0 = 0 we write QS = QS(0).

We will see later (Lemma 16) that a characteristic property of quantum blobs is that their orthogonal projections on symplectic planes can never become smaller than πℏ.

Proposition 12.

Let Ω be a centered ellipsoid in $(R2n,σ)$. We have Ω = Ω,σ if and only if Ω is a quantum blob, i.e. if Ω = QS for some SSp(n).

Proof.
That $QSℏ,σ=QS$ is clear in view of (22):
$QSℏ,σ=S(B2n(ℏ))ℏ,σ=S(B2n(ℏ))=QS.$
Suppose conversely that the ellipsoid Ω is defined by Mzz; then its symplectic polar dual Ω,σ is defined by −JM−1Jz⋅z (Lemma 9) and we have Ω = Ω,σ if and only if M = −JM−1J. This condition is trivially equivalent to MJM = J which implies M ∈ Sp(n) hence $Ω=S(B2n(ℏ))$ with S = M−1/2 ∈ Sp(n).■

Proposition 4 relating orthogonal projections and intersections generalizes as follows to the case of symplectic polar duality:

Proposition 13.
Let $Ω⊂R2n$ be a centrally symmetric convex body and F a linear subspace of $R2n$. We have
$(ΠFΩ)ℏ,σ=Ωℏ,σ∩(JF) and (Ω∩F)ℏ,σ=ΠJF(Ωℏ,σ).$
(24)

Proof.
Since Ω is symmetric we have J(Ω ∩ F) = −J(Ω ∩ F) hence the kernel of the projector −JΠFJ is F and its range is Jℓ so that −JΠFJ = ΠJF. This proves the first equality (24). We have, by definition, (Ω ∩ F),σ = J(Ω ∩ F). In view of formula (8) we have (Ω ∩ F) = ΠF) and hence (Ω ∩ F),σ = JΠF). Thus
$(ΠFΩ)ℏ,σ=−JΠF(Ωℏ)=(−JΠFJ)(Ωℏ,σ);$
which is the second equality (24). It immediately follows from Proposition 4 noting that
$(ΠFΩ)ℏ,σ=J(Ωℏ∩F)=Ωℏ,σ∩JF.$
We finally note that the Blaschke–Santaló inequality (14) becomes in this context
$Vol2n(Ω)Vol2n(Ωℏ,σ).≤Vol2n(B2n(ℏ))2$
(25)
with equality if and only the convex set X is an ellipsoid. This follows from (14) noting that
$Vol2n(Ωℏ,σ)=Vol2n(JΩℏ)=Vol2n(Ωℏ).$

#### 1. Definition and a necessary and sufficient condition

The following definition will be motivated below:

Definition 14.

Let $ΩM⊂R2n$ be the ellipsoid {z: Mz⋅ z} (M = Mt > 0). We will say that ΩM is quantum admissible if it contains a quantum blob $QS=S(B2n(ℏ))$, SSp(n).

We are going to prove, using symplectic polarity, two simple but important necessary and sufficient conditions for an ellipsoid to be quantum admissible. We first recall the Williamson symplectic diagonalization result41 (see 16 and 28 for “modern” proofs). For every M = MT > 0, there exists S ∈ Sp(n) such that
$M=STDS , D=Λσ0n×n0n×nΛσ$
(26)
where $Λσ=diag(λ1σ,…,λnσ)$ the $λjσ$ being the symplectic eigenvalues of M (i.e. the moduli of the eigenvalues of JMM1/2JM1/2). It is usual to rank the symplectic eigenvalues in non-increasing order
$λmaxσ=λ1σ≥λ2σ≥⋅⋅⋅≥λnσ=λminσ.$
Note that the symplectic spectrum of M−1 is $((λ1σ)−1,…,(λnσ)−1)$. It is usual to call the factorization (26) the “Williamson normal form of M.”

Remark 15.

The diagonalizing symplectic matrix S in (26) is not unique; see Son et al.39 for a detailed analysis of the set of diagonalizing symplectic matrices.

Recall from Proposition 12 that the equality $ΩMℏ,σ=ΩM$ occurs if and only if $ΩM=S(B2n(ℏ))$ for some S ∈ Sp(n), i.e. if and only if Ω is a “quantum blob.” Below we state and prove a general criterion for admissibility which we glorify it by giving it the status of a theorem. Let us first introduce some preparatory material:

• A two-dimensional subspace F of $(R2n,σ)$ is called a symplectic plane if the restriction σ|F of symplectic form σ is non-degenerate; equivalently F has a basis {e1, e2} such that σ(e1, e2) = 1. In particular every plane Fj of conjugate coordinates xj, pj is symplectic; and for every symplectic plane F there exists Sj ∈ Sp(n) such that F = Sj(Fj).

• We will use Gromov’s symplectic non-squeezing theorem;26 it says (in its simplest form) that no symplectomorphism f ∈ Symp(n) of $(R2n,σ)$ can send a ball B2n(R) into a cylinder $Zj2n(r):xj2+pj2≤r2$ if r < R [we are denoting by Symp(n) the group of all symplectomorphisms28,36,42 of $(R2n,σ)$].

We will also need the following immediate consequence of Gromov’s theorem:

Lemma 16.
Let F be a symplectic plane in $(R2n,σ)$ and fSymp(n). The area of the orthogonal projection ΠF of f[B2n(z0, r)] on F satisfies
$Area(ΠFf(B2n(z0,r)))≥πr2.$
(27)

Proof.
It is sufficient to suppose that z0 = 0 since areas are translation-invariant. Assume that $ΠFj(S(B2n(r)))=πR2$ with R < r. Then $fB2n(r)$ must be contained in the cylinder Zj(R), but this contradicts Gromov’s non-squeezing theorem. (ii) Assume that R < r. Then, by (27),
$πr2≤Area(ΠFS(B2n(R)))

Let us now state and prove our theorem:

Theorem 17.
The ellipsoid ΩM is quantum admissible if and only if the two following equivalent conditions are satisfied: (i) We have the inclusion
$ΩMℏ,σ⊂ΩM.$
(28)
(ii) We have the inequality
$Area(ΩMℏ,σ∩F)≤πℏ$
(29)
for every symplectic plane F in $(R2n,σ)$.

Proof.
(i) Suppose that ΩM is quantum admissible; then there exists S ∈ Sp(n) such that $QS=S(B2n(ℏ))⊂ΩM$. By the anti-monotonicity of symplectic polar duality this implies that we have
$ΩMℏ,σ⊂QSℏ,σ=QS⊂ΩM,$
which proves the necessity of the condition. Suppose conversely that $ΩMℏ,σ⊂ΩM$. We have
$ΩMℏ,σ={z∈R2n:(−JM−1J)z⋅z≤ℏ}$
(30)
hence the inclusion $ΩMℏ,σ⊂ΩM$ implies that M ≤ (−JM−1J) ($≤$ stands here for the Löwner ordering of matrices). Performing a symplectic diagonalization (26) of M and using the relations JS−1 = STJ, $(ST)−1J=JS$ this is equivalent to
$M=STDS≤ST(−JD−1J)S$
that is to D ≤ −JD−1J. This implies that we have $Λσ≤(Λσ)−1$ and hence $λjσ≤1$ for 1 ≤ jn; thus DI and M = STDSSTS. The inclusion $S(B2n(ℏ))⊂ΩM$ follows and we are done. (ii) Suppose that ΩM is admissible and let ΠF be the orthogonal projection in $R2n$ on F. By Proposition 13 we have
$ΩMℏ,σ∩F=(ΠJFΩM)ℏ,σ.$
Since ΩM is an ellipsoid the symplectic version (25) of the Blaschke–Santaló inequality becomes the equality
$Area(ΠJFΩM)ℏ,σArea(ΠJFΩM)=(πℏ)2$
(31)
that is
$Area(ΩMℏ,σ∩F)Area(ΠJFΩM)=(πℏ)2.$
The inequality (29) follows: since ΩM is admissible, it contains a quantum blob $(B2n(ℏ))$ hence Area(ΠJFΩM) ≥ πℏ in view of Lemma 16. Assume conversely that $Area(ΩMℏ,σ∩F)≤πℏ$ for every symplectic plane F; by (31) we must then have Area(ΠJFΩM) ≥ πℏ for every F. Let us show that this implies that ΩM must be admissible. Since admissibility is preserved by symplectic conjugation we may assume, using a Williamson diagonalization (26), that M is of the diagonal type $Λσ0n×n0n×nΛσ$ where $Λσ=diag(λ1σ,…,λnσ)$ the $λjσ$ being the symplectic eigenvalues of M. The ellipsoid ΩM is thus given by
$λ1σ(x12+p12)+⋅⋅⋅+λnσ(xn2+pn2)≤ℏ.$
Let us intersect ΩM with the symplectic plane F1 (the plane of coordinates x1, p1). It is the ellipse $x12+p12≤ℏ/λ1σ$ which has area $πℏ/λ1σ$. Now, ΩM is admissible if and only if $λ1σ=λmaxσ≤1$ which is equivalent to the condition Area(ΩMF) ≥ π, hat is to $Area(ΩMℏ,σ∩F)≤πℏ$ again in view of the Blaschke–Santaló equality (31).■

Condition (29) is truly remarkable; it shows that given an ellipsoid and its symplectic polar dual the datum of a sequence of two-dimensional conditions suffices to decide whether the ellipsoid is admissible or not. This “tomographic” property is related to a condition using the Poincaré invariant given by Narcowich35 we will briefly discuss in our study of covariance and information ellipsoids in Sec. V B.

Recall21 that the cross-Wigner function of a pair $(ψ,ϕ)∈L2(Rn)×L2(Rn)$ it is defined by
$W(ψ,ϕ)(z)=12πℏn∫Rne−iℏp⋅yψ(x+12y)ϕ(x−12y)̄dy.$
(32)
W(ψ, ϕ) is a continuous function satisfying the estimate
$|W(ψ,ϕ)(z)|≤2πℏn‖ψ‖L2‖ϕ‖L2.$
(33)
When ψ = ϕ the function W(ψ, ψ) = is the usual Wigner function
$Wψ(z)=12πℏn∫Rne−iℏp⋅yψ(x+12y)ψ(x−12y)̄dy.$
(34)
The Wigner functions of general Gaussian functions is well-known;16,21 if
$ψA,B(x)=1πℏn/4(detA)1/4e−12ℏA+iBx⋅x$
(35)
where $A,B∈Sym(n,R)$, A > 0. Then16,21
$WψA,B(z)=1πℏne−1hGz⋅z$
(36)
where G ∈ Sp(n):
$G=STS∈Sp(n) , S=A1/20n×nA−1/2BA−1/2.$
(37)
Explicitly
$G=A+BA−1BYBA−1A−1BA−1.$
(38)
Notice that, conversely, if $ψ∈L2(Rn)$ is such that $W(z)=(πℏ)−ne−1hMz⋅z$ for some M = MT ∈ Sp(n), M > 0, then ψ = eψA,B ($χ∈R$) where X and Y are determined by (38) and χ| = 1.

Sub-Gaussian estimates for the Wigner function refer to bounds on the magnitude of the Wigner function that ensure that it does not fluctuate too much. A function is sub-Gaussian if its tails decay faster than any Gaussian distribution. Using Proposition 7 one proves21,23 that:

Corollary 18.

Let $ψ∈L2(Rn)$, ψ ≠ 0, and assume that there exists C > 0 such that $Wψ(z)≤Ce−1ℏMz⋅z$ where M = MT > 0. Then the symplectic eigenvalues $λ1σ≥λ2σ≥⋅⋅⋅≥λnσ$ of M are all $≤1$. When $λ1σ=λ2σ=⋅⋅⋅=λnσ=1$ then the function ψ is a generalized Gaussian (35).

It follows from this result that the Wigner function can never have compact support: assume that there exists R > 0 such that (z) = 0 for |z| > R. Then, for every a > 0 there exists a constant C(a) > 0 such that $Wψ(z)≤C(a)e−aℏ|z|2$ for all $z∈R2n$. Choosing a large enough this contradicts the statement in Corollary 18 because as soon as a > 1 the symplectic eigenvalues of M = aIn×n are all equal to a.

Corollary 18 can be elegantly reformulated in terms of symplectic polar duality:

Proposition 19.

The Wigner function of $ψ∈L2(Rn)$ satisfies a sub-Gaussian estimate $Wψ(z)≤Ce−1ℏMz⋅z$ if and only if the ellipsoid ΩM = {z: Mz z} is admissible: $ΩMℏ,σ⊂ΩM$, that is, ΩM contains a quantum blob S(B2n()), SSp(n).

Proof.

In view of Williamson’s diagonalization theorem 26 there exists S ∈ Sp(n) such that M = STDS where $D=Λσ0n×n0n×nΛσ$ hence SM) = ΩD. Since S,σ) = [S(Ω)],σ (Proposition 10) it is sufficient to prove the result for M = D. Since $ΩDℏ,σ=Ω−JD−1J$ we thus have to show that the sub-Gaussian estimate is satisfied if and only if $Ω−JD−1J⊂ΩD$. This is equivalent to D ≤ −JD−1J (in the Löwner ordering) that is to $Λσ≤(Λσ)−1$ which is possible if and only the symplectic eigenvalues $λjσ$ are all $≤1$. When the $λjσ$ are all equal to one we have D = I2n × 2n., that is M = STS so that $Wψ(z)=Ce−1ℏMz⋅z$ for some constant C.■

#### 1. Definition using the Wigner function

Let $Ls1(R2n)$ is the weighted L1-space defined by
$Ls1(R2n)={ρ:R2n⟶C:⟨z⟩sρ∈L1(R2n)}$
(39)
where $⟨z⟩=(1+|z|2)1/2$. In the definition below we are following our presentation of modulation spaces given in Ref. 19; see Refs. 13, 25, and 29 for a definitions used in time-frequency analysis [they are based on the short-time Fourier transform (Gabor transform)]. We denote by $S(Rn)$ the Schwartz space of test functions decreasing rapidly to zero at infinity, together with their derivatives.

Definition 20.

The modulation space $Ms1(Rn)$ $(s∈R)$ consists of all $ψ∈L2(Rn)$ such that $W(ψ,ϕ)∈Ls1R2n$ for every $ϕ∈S(Rn)$. When s = 0 the space $M01(Rn)=S0(Rn)$ is called the Feichtinger algebra.

It turns out that it suffices to check that condition $W(ψ,ϕ)∈Ls1R2n$ holds for one function ϕ ≠ 0 (hereafter called “window” for it then it holds for all $ϕ∈S(Rn)$.The mappings $ψ⟼‖ψ‖ϕ,Ms1$ defined by
$‖ψ‖ϕ,Ms1=‖W(ψ,ϕ)‖Ls1=∫R2n|W(ψ,ϕ)(z)|zsdz$
form a family of equivalent norms, and the topology on $Ms1(Rn)$ thus defined makes it into a Banach space. We have the chain of inclusions
$S(Rn)⊂Ms1(Rn)⊂Ls1(Rn)∩F(Ls1(Rn))$
where F is the Fourier transform; it follows by Riemann–Lebesgue that in particular
$Ms1(Rn)⊂Ls1(Rn)∩C0(Rn).$
Observe that
$Ms1(Rn)⊂Ms′1(Rn)⟺s≥s′$
and one proves25 that
$⋂s≥0Ms1(Rn)=S(Rn).$

An essential property of the spaces $Ms1(Rn)$ is their metaplectic invariance for s ≥ 0. Recall that the metaplectic group Mp(n) is the unitary representation in $L2(Rn)$ of the double cover Sp2(n) of the symplectic group Sp(n) (see for instance Ref. 16 for a detailed study of the metaplectic representation). We will denote πMp the covering projection Mp(n) ⟶ Sp(n); it is a two-to-one epimorphism.

Proposition 21.

The modulation spaces $Ms1(Rn)$, s ≥ 0, are invariant under the action of Mp(n): if $ψ∈Ms1(Rn)$ and $Ŝ∈Mp(n)$ then $Ŝψ∈Ms1(Rn)$.

Proof.
By definition we have $ψ∈Ms1(Rn)$ if and only if $W(ψ,ϕ)∈Ls1R2n$ for every $ϕ∈S(Rn)$; similarly $Ŝψ∈Ms1(Rn)$ if and only $W(Ŝψ,ϕ)∈Ls1R2n$ for every ϕ. Now, in view of the symplectic covariance of the cross-Wigner function16,19
$W(Ŝψ,ϕ)=W(Ŝψ,Ŝ(Ŝ−1ϕ))=W(ψ,Ŝ−1ϕ)◦S−1$
(40)
where $S=πMp(Ŝ)$. Since $Ŝ−1ϕ∈S(Rn)$ there remains to prove that $W(ψ,Ŝ−1ϕ)◦S−1∈Ls1R2n$. Set $ϕ′=Ŝ−1ϕ$; we have det S = 1 hence
$∫R2n|W(ψ,ϕ′)(S−1z)|zsdz=∫R2n|W(ψ,ϕ′)(z)|Szsdz.$
We have $Szs≤CSzs$ for some constant CS > 0, thus
$∫R2n|W(ψ,ϕ′)(S−1z)|zsdz≤CS∫R2n|W(ψ,ϕ′)(z)|zsdz<∞$
and we are done.■

Quantum mechanics is inherently probabilistic, and the behavior of quantum systems is described by mathematical entities known as quantum states. These states are represented by density operators (also called density matrices by physicists). An element $ρ̂$ of the algebra $L1(Rn)$ of trace class operators on $L2(Rn)$ is called a density operator if $ρ̂≥0$ and has trace $Trρ̂=1$; the passiveness implies in particular that $ρ̂$ is a self-adjoint compact operator so that there exists a sequence (αj) with αj ≥ 0, ∑jαj = 1 and an orthonormal stem (ψj), $ψj∈L2(Rn)$ such that we have we have the spectral decomposition
$ρ̂=∑jαjρ̂j , αj≥0 , ∑jαj=1$
(41)
where $ρ̂j$ is the rank one orthogonal projection in $L2(Rn)$ on the ray $Cψj$. It follows that that the Weyl symbol of the operator $ρ̂$ is19,
$(2πℏ)nρ=∑jαjWψj$
(42)
(the function ρ is often called the Wigner distribution of $ρ̂$ in the physical literature). In Refs. 15 and 22 we introduced the notion of Feichtinger state:

Definition 22.

A density operator $ρ̂∈L1(Rn)$ is called a aFeichtinger stateif each $ψj∈Ms1(Rn)$ for some s ≥ 0.

Feichtinger states satisfy the marginal properties: we have $ρ∈L1(R2n)$ and
$∫Rnρ(x,p)dp=∑jαj|ψj(x)|2 , ∫Rnρ(x,p)dx=∑jαj|Fψj(x)|2.$
(43)
The main interest of the notion of Feichtinger state in our context is that they also allow to define rigorously the covariance matrix Σcov of a density operator. The latter is defined –if it exists!– as being the symmetric 2n × 2 × n matrix
$Σcov=∫R2n(z−z̄)(z−z̄)Tρ(z)dz$
(44)
where $z̄=∫R2nzρ(z)dz$ is the average (or mean value) vector. It is convenient to write the covariance matrix in n × n block-matrix form as
$Σcov=ΣXXΣXPΣPXΣPP , ΣPX=ΣXPT$
(45)
where $ΣXX=(σxjxk)1≤j,k,≤n$, $ΣPP=(σpjpk)1≤j,k,≤n$, and $ΣXP=(σxjpk)1≤j,k,≤n$ with
$σxjxk=∫R2nxjxkρ(z)dz$
(46)
and so on. We have the following conjugation result:

Proposition 23.

The covariance matrix Σcov of a Feichtinger state $ρ̂$ with s ≥ 2 is well-defined. If $Ŝ∈Mp(n)$ then conjugate state $Ŝρ̂Ŝ−1$ is also a Feichtinger state with s ≥ 2 and the covariance matrix of $Ŝρ̂Ŝ−1$ is SΣcovST.

Proof.
Since s ≥ 2 we have
$∫R2n|ρ(z)|(1+|z|2)dz<∞.$
(47)
It is no restriction to assume $z̄=0$; setting zα = xα if 1 ≤ αn and zα = pα if n + 1 ≤ α ≤ 2n we have $Σ=(σαβ)1≤α,β≤2n$ where the integrals
$σαβ=∫R2nzαzβρ(z)dz$
are absolutely convergent in view of the trivial inequalities |zαzβ| ≤ 1 + |z|2. Let $Ŝ∈Mp(n)$; it follows from the standard properties of Weyl pseudodifferential calculus16,19,38 that the Weyl symbol of the conjugate $Ŝρ̂Ŝ−1$ is(2πℏ)nρS−1. In view of Proposition 21 $ρ∈Ms1(Rn)$ implies that $ρ◦S∈Ms1(Rn)$ and the result follows from (44) by a simple calculation.■

#### 1. A necessary condition for positivity

Assume from now on that $ρ̂$ is a Feichtinger state with s ≥ 2, guaranteeing the existence of the covariance matrix Σcov. It i a well-known property in harmonic analysis that the positivity condition $ρ̂≥0$ implies that Σ must satisfy the algebraic condition
$Σ+iℏ2J is positive semidefinite$
(48)
which we write for short $Σ+iℏ2J≥0$. (Note that the matrix $Σ+iℏ2J$ is selfadjoint since Σ is symmetric and J* = −J.) This condition implies in particular that Σ is positive definite35,19 and hence invertible. This (highly nontrivial) result can be proven by various methods, one can for instance use the notion of η-positivity due to Kastler30 together with a variant of Bochner’s theorem on the Fourier transform of a probability measure; for a simpler approach using methods from harmonic analysis see our recent paper10 with Cordero and Nicola. The condition (48) actually first appeared as a compact formulation of the uncertainty principle in Arvind et al..2 We have rigorously shown in Refs. 16 and 17 that (48) is equivalent to the Robertson–Schrödinger inequalities
$σxjxjσpjpj≥σxjpj2+14ℏ2 for 1≤j≤n$
(49)
which form the textbook statement of the complete uncertainty principle of quantum mechanics.33 While condition (48) is generally only a necessary condition for the positivity of a trace class operator, it is also sufficient for operators with Gaussian Weyl symbol (2πℏ)nρ where
$ρ(z)=1(2π)ndetΣe−12Σ−1(z−z0)⋅(z−z0)$
(50)
(Σ = ΣT > 0 playing the role of a covariance matrix Σcov; see for instance 15, 21, 25, and 28).

Symplectic capacities were defined by Ekeland and Hofer11,12 (see 9 and 28 for review of that notion). They are closely related to Gromov’s symplectic non-squeezing theorem shortly discussed above; in fact the existence of a single symplectic capacity is equivalent to Gromov’s theorem.

Gromov’s theorem ensures us of the existence of symplectic capacities. A (normalized) symplectic capacity on $(R2n,σ)$ associates to every subset $Ω⊂R2n$ a number c(Ω) ∈ [0, + ∞] such that the following properties hold (Ref. 28, see Ref. 24 for a review):
• SC1 Monotonicity: If Ω ⊂ Ω′ then c(Ω) ≤ c(Ω′);

• SC2 Conformality: For every $λ∈R$ we have c(λΩ) = λ2c(Ω);

• SC3 Symplectic invariance: c[f(Ω)] = c(Ω) for every f ∈ Symp(n);

• SC4 Normalization: For 1 ≤ jn,

$c(B2n(r))=πr2=c(Zj2n(r))$
(51)
where $Zj2n(r)$ is the cylinder with radius r based on the xj, pj plane.

It follows that the symplectic capacity of a quantum blob $QS(z0)=S(B2n(ℏ))$ is c(QS(z0)) = πℏ.

There exist symplectic capacities, cmax and cmin, such that cminccmax for every symplectic capacityc, they are defined by
$cmax(Ω)=inff∈Symp(n){πr2:f(Ω)⊂Zj2n(r)}$
(52)
$cmin(Ω)=supf∈Symp(n){πr2:f(B2n(r))⊂Ω}.$
(53)
That cmin and cmin indeed are symplectic capacities follows from the axioms (SC1)–(SC4). Note that the conformality and normalization properties (SC2) and (SC4) show that for n > 1 symplectic capacities are not related to volume; they have the dimension of an area,1 or equivalently, that of an action. For instance, the Hofer–Zehnder capacity 28 is characterized by the property that when Ω is a compact convex set in $(R2n,σ)$ with smooth boundary Ω then
$cHZ(Ω)=∫γminpdx$
(54)
where pdx = p1dx1 + ⋅⋅⋅ + pndxn and γmin is the shortest positively oriented Hamiltonian periodic orbit carried by Ω.

One also has the weaker notion of linear (or affine) symplectic capacity, obtained by replacing condition (SC3) with

SC3lin Linear symplectic invariance: c[S(Ω)] = c(Ω) for every S ∈ Sp(n) and $c(Ω+z)=c(Ω)$ for every $z∈R2n$.

The corresponding minimal and maximal linear symplectic capacities $cminlin$ and $cmaxlin$ are then given by
$cminlin(Ω)=supS∈Sp(n){πR2:S(B2n(z,R)),z∈R2n}$
(55)
$cmaxlin(Ω)=inff∈Sp(n){πr2:S(Ω)⊂Zj2n(z,r),z∈R2n}.$
(56)

#### 1. The case of ellipsoids

It turns out that all symplectic capacities (linear as well as non-linear) agree on ellipsoids. Assume that
$ΩM={z∈R2n:Mz⋅z≤r2}$
where $M∈Sym+(2n,R)$, and let $λmaxσ=λ1σ≥λ2σ≥⋅⋅⋅≥λnσ$ be the symplectic eigenvalues of M. If in particular Ω = ΩM: Mz⋅z then
$c(ΩM)=πℏ/λmaxσ.$
(57)
We have in particular
$cHZ(ΩM)=∫γminpdx=πℏ/λmaxσ.$
This is easy to verify using the calculations in the Proof of Theorem 17(ii): reducing ΩM to Williamson normal form
$λ1σ(x12+p12)+⋅⋅⋅+λnσ(xn2+pn2)≤ℏ$
the shortest Hamiltonian orbit carried by ΩM is given by Hamilton’s equations for the Hamiltonian function $H1(x1,p1)=λ1σ(x12+p12)$ with the condition H1(x1, p1) = . One verifies that this periodic solution verifies
$∫γminpdx=πℏ/λ1σ=πℏ/λmaxσ=c(ΩM).$

For the symplectic polar dual ellipsoid we have the following result, which yields a Blaschke–Santaló type inequality for symplectic capacities of ellipsoids:

Proposition 24.
Let ΩM be as above and let $ΩMℏ,σ$ be its symplectic polar dual. (i) We have
$c(ΩM)=πℏ/λmaxσ and c(ΩMℏ,σ)=πℏλminσ$
(58)
where $λmaxσ$ (resp. $λminσ$) is the largest (rep. smallest)symplectic eigenvalue of M. (ii) In particular
$c(ΩM)c(ΩMℏ,σ)≤(πℏ)2$
(59)
with equality if and only if $ΩM=λB2n(ℏ)$ for some λ > 0.

Proof.
(i) The formula $c(ΩM)=πr2/λmaxσ$ is easily proven using a symplectic diagonalization (26) of M which reduces the problem to the case of an ellipsoid with axes contained in the conjugate xj, pj planes (see Refs. 28 and 16 for details). The symplectic polar dual of ΩM is
$ΩMℏ,σ={z∈R2n:−JM−1Jz⋅z≤ℏ}$
[formula (21) in Lemma 9]. Set N = −JM−1J: we have JN = M−1J hence the eigenvalues of JN are those of M−1/2JM−1/2 so the symplectic eigenvalues of N are the inverses of those of M; the second formula (58) follows. (ii) Formula (59) is obvious; that we have equality if and only if all $ΩM=λB2n(ℏ)$ follows from the fact that if the symplectic eigenvalues of M are all equal then by Williamson’s theorem M is a scalar multiple of a matrix STS.■

#### 1. Covariance and information ellipsoids

We are going to express condition (48) in a simple geometric way using symplectic polarity. By definition, the covariance ellipsoid of a Feichtinger state $ρ̂$ is
$Ωcov={z∈R2n:12Σcov−1z⋅z≤1}$
(60)
where Σcov is the covariance matrix of $ρ̂$. The symplectic polar dual of Ωcov is the ellipsoid
$Ωcovℏ,σ={z∈R2n:−12JΣJz⋅z≤1}.$
By definition the associated information (or precision) ellipsoid is
$Ωinfo={z∈R2n:12Σcovz⋅z≤1}$
by the equality $Ωcovℏ,σ=J(Ωinfo)$, that is, $Ωinfo=J(Ωcovℏ)$ where $Ωcovℏ$ is the ordinary polar dual of Ωcov. It turns out that Ωcov and Ωinf o are Legendre duals of each other.35 Consider in fact the quadratic forms $w(z)=12Σcov−1z⋅z$ and $wσ(z′)=12Σcovz′⋅z′$. The Legendre transform of w(z) is defined by
$w′(z′)=z⋅z′−w(z)$
where z is expressed in terms of z′ by solving the equation $z′=∂zw(z)=Σcov−1z$ hence w′(z′) = wσ(z′).

We are going to prove that the covariance ellipsoid of a Feichtinger state always is quantum admissible:

Proposition 25.

The covariance ellipsoid Ωcov of a Feichtinger state $ρ̂$ satisfies the condition $Ωcovℏ,σ⊂Ωcov$ and is hence quantum admissible.

Proof.
Setting set $M=ℏ2Σ−1$ the covariance ellipsoid is then defined by ΩM: Mz z and we have to prove that the positivity condition $Σ+iℏ2J≥0$ is equivalent to the condition $ΩMℏ,σ⊂ΩM$. In view of Propositions 17 it is sufficient to prove that this conditions hold if and only if ΩM contains a quantum blob $QS=S(B2n(ℏ))$, S ∈ Sp(n). Performing a symplectic diagonalization M = STDS (26) of M the condition $Σ+iℏ2J=M−1+iJ≥0$ implies that D−1 + iJ ≥ 0, that is
$D−1+iJ=(Λσ)−1iIn×n−iIn×n(Λσ)−1≥0$
where $Λσ=diag(λ1σ,…,λnσ)$ the $λjσ$ being the symplectic eigenvalues of M. The eigenvalues of D−1 + iJ are the real numbers $λj=(λjσ)−1±1$ and the condition D−1 + iJ ≥ 0 thus implies that we must have $λjσ≤1$ for 1 ≤ jn. It follows that the ellipsoid ΩD: Dz⋅z contains the ball $B2n(ℏ)$ and hence ΩM contains the quantum blob $S(B2n(ℏ))$ where S ∈ Sp(n) is the diagonalizing matrix. The result now follows applying Propositions 17.■

Remark 26.
Assume that the Wigner distribution of $ρ̂$ is a Gaussian (50) with z0 = 0. If Ω,σ = Ω it follows from Propositions 17) that Ω is a quantum blob $S(B2n(ℏ))$ and hence
$ρ(z)=1(2π)ne−12Gz⋅z , G=STS$
(61)
for some SSp(n). Then16,19 $ρ=W(Ŝ−1ϕ0ℏ)$ where
$ϕ0ℏ(x)=(πℏ)−ne−|x|2/2ℏ$
and $Ŝ∈Mp(n)$ has projection $πMp(Ŝ)=S$.

#### 2. A dynamical characterization of admissibility

Let Ω be an ellipsoid in $R2n$ with smooth boundary Ω. We assume that Ω is the energy hypersurface of some (quadratic) Hamiltonian function $H∈C∞(R2n,R)$: i.e. Ω = {z: H(z) = E} for some $E∈R$. We ask now when Ω can be viewed as the covariance ellipsoid of a quantum state; the following is in a sense a restatement of Theorem 17, but we give an independent proof here:

Theorem 27.
The ellipsoid Ω is a quantum covariance ellipsoid Ωcov (resp. an information ellipsoid Ωinf o) if and only if the following equivalent conditions are satisfied: (i) We have
$∫γpdx≥c(Ω)≥πℏ$
(62)
for every periodic Hamiltonian orbit γ carried by Ω; if we have equality for the shortest orbit then Ω is a quantum blob. (ii) Let Ω* be the Legendre transform of Ω; we have
$c(Ω*)≤4π/ℏ.$
(63)
(iii) Let F be an arbitrary two-dimensional subspace of $R2n$ an let the ellipse $γF*=∂Ω*∩F$ be positively oriented. We have
$∫γF*pdx≤4πℏ.$
(64)

Proof.
(i) The inequality (62) follows from (54) and (58). (ii) If Ω is defined by $12Σ−1z⋅z≤1$ then Ω* is defined by $12Σz⋅z≤1$. Setting $M=ℏ2Σ−1$ the ellipsoid Ω* is given by Nz z where $N=ℏ24M−1$. The symplectic spectrum of N is thus $ℏ24((λnσ)−1,…,λ1σ)−1)$ where $(λ1σ,…,λnσ)$ is the symplectic spectrum of M (recall our convention to rank symplectic eigenvalues in non-increasing order). It follows that the symplectic capacity of Ω* is
$c(Ω*)=πℏℏ24(λnσ)−1−1≥4πℏ$
the last inequality because Ω is quantum admissible if and only if $λnσ=λmaxσ≤1$. To prove the action inequality (64) we can proceed as follows: suppose first that F is a null space for the symplectic form [i.e. F has a basis {e1, e2} such that σ(e1, e2) = 0]. The, by Stokes’s theorem we have
$∫γF*pdx=∫Ω*∩Fσ=0$
(65)
so that (64) is trivially verified. Assume next that F is a symplectic plane. Then, by formula (29) in Theorem 17 we have
$Area(ΩMℏ,σ∩F)≤πℏ$
(66)
but this is precisely (64) since $M=ℏ2Σ−1$.■

The first author (MdG) has been financed by the Grant Nos. P 33447 N and PAT 2056623 of the Austrian Research Foundation FWF.

The authors have no conflicts to disclose.

Maurice de Gosson: Funding acquisition (equal); Investigation (equal); Project administration (equal). Charlyne de Gosson: Data curation (equal); Project administration (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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