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.

## I. INTRODUCTION

*X*in the Euclidean space $Rn$ its polar dual is the set

*X*

^{ℏ}of all $p\u2208(Rn)*$ such that ⟨

*p*,

*x*⟩ ≤

*ℏ*; here

*ℏ*is a positive constant, usually taken to be one in the standard literature (we use for flexibility a parameter-dependent definition; in quantum mechanics

*ℏ*would be Planck’s constant

*h*divided by 2

*π*; in harmonic analysis one would take

*ℏ*= 1/2

*π*while the standard choice in the theory of partial differential equations is

*ℏ*= 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

*σ*is the standard symplectic form $\u2211j=1ndpj\u2227dxj$; we will identify the cotangent bundle $T*Rn$ with $R2n\u2261Rxn\xd7Rpn$. This variant is what we call “symplectic polar duality”: if $\Omega \u2282R2n$ is a convex body we define its symplectic polar dual by

^{ℏ,σ}=

*J*(Ω

^{ℏ}) where

*J*is the standard symplectic automorphism

*J*(

*x*,

*p*) = (

*p*−

*x*). The interest of this notion comes (among other things) from the fact that it has the symplectic covariance property

*S*(Ω

^{ℏ,σ}) = [

*S*(Ω)]

^{ℏ,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 $X\u2113\xd7(X\u2113)\u2113\u2032\u210f$ 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 Milman^{34} who applies such methods to probability theory; also see the treatise^{3} 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^{ℏ,σ}; the second is of a more subtle nature; it shows that it is sufficient (and necessary) for admissibility that Ω^{ℏ,σ} ∩ *F* ⊂ Ω ∩ *F* for every symplectic subspace of $(R2n,\sigma )$. This is a tomographic condition reminiscent of an old result by Narcowich^{35} 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.

*The standard symplectic form* *σ* *is written in matrix form as* *σ*(*z*, *z*′) = *Jz⋅z*′ *where* $J=0n\xd7nIn\xd7n\u2212In\xd7n0n\xd7n$ *is the standard symplectic matrix. The standard symplectic group* *Sp*(*n*) *is the group of automorphisms* *S* *of* $T*Rn\u2261R2n$ *such that* *S***σ* = *σ**; in matrix notation:* *S* ∈ *Sp*(*n*) *if and only if* *STJS* = *J* *(or, equivalently,* *SJS*^{T} = *J**),* *S*^{T} *the transpose of* *S**). The unitary representation of the double cover of* *Sp*(*n*) *(the metaplectic group) is denoted by* *π*^{Mp}: *Mp*(*n*) ⟶ *Sp*(*n*). *The Lagrangian Grassmannian of* $(R2n,\sigma )$ *is denoted* *Lag*(*n*)*, thus* *ℓ* ∈ *Lag*(*n*) *if and only if* *ℓ* *is a linear subspace of* $R2n$ *with* dim *ℓ* = *n* *and* *σ*|*ℓ* = 0*. The elements of* *Lag*(*n*) *will be called Lagrangian planes.*

## II. SYMPLECTIC POLAR DUALITY AND THE COVARIANCE MATRIX

### A. A short review of usual polar duality

#### 1. Definition

*X*and

*Y*be convex sets in $Rn$; then,

*X*

^{ℏ}∪

*Y*

^{ℏ}). If $A=AT\u2208GL(n,R)$ is positive definite then

*resp*. $BPn(\u210f)$) is the ball in $Rxn$ (

*resp*. $Rpn$) with radius $\u210f$ and centered at 0.

*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:

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

We have $X=A\u22121/2(BXn(\u210f))$ and $P=B\u22121/2(BPn(\u210f))$ and the inclusion *X*^{ℏ} ⊂ *P* is thus equivalent to the inequality *A*^{1/2} ≤ *B*^{−1/2} in the Löwner ordering, that is, to *A* ≤ *B*^{−1} with equality if and only if *X*^{ℏ} = *P*.■

Polar duality exchanges the operations of intersection and orthogonal projection:

*Let*$X\u2282Rxn$

*be a convex body containing*0

*in its interior and*

*F*

*a linear subspace of*$Rxn$

*; we have*

*where*Π

_{F}

*us the orthogonal projection in*$Rxn$

*onto*

*F*

*. [In*$(\Pi FX)\u210f$

*and*(

*X*∩

*F*)

^{ℏ}

*the polar duals are taken inside the subspace*

*F*

*equipped with the induced inner product.]*

*X*∩

*F*)

^{ℏ}= Π

_{F}(

*X*

^{ℏ}). Let us next show that Π

_{F}(

*X*

^{ℏ}) ⊂ (

*X*∩

*F*)

^{ℏ}. For

*p*∈

*X*

^{ℏ}we have, for every

*x*∈

*X*∩

*F*,

_{F}

*p*∈ (

*X*∩

*F*)

^{ℏ}. To prove the inclusion Π

_{F}(

*X*

^{ℏ}) ⊃ (

*X*∩

*F*)

^{ℏ}we note that it is sufficient, by the anti-monotonicity and reflexivity properties of polar duality, to prove that $(\Pi F(X\u210f))\u210f\u2282X\u2229F$. Let $x\u2208(\Pi F(X\u210f))\u210f$; we have

*x*⋅Π

_{F}

*p*≤

*ℏ*for every

*p*∈

*X*

^{ℏ}. Since

*x*∈

*F*(because the dual of a subset of

*F*is taken inside

*F*) we also have

*x*∈

*X*∩

*F*. This concludes the proof.■

#### 3. John and Löwner ellipsoids

^{4,8}the (unique) ellipsoid Ω

_{John}of maximal volume contained in Ω; similarly the (unique) minimum enclosing ellipsoid is the Löwner ellipsoid $\Omega Lo\u0308wner$. We note that both the John and the Löwner ellipsoids transform covariantly under linear (and affine) transforms: if $L\u2208GL(n,R)$ then

*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.*

^{3}

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

*X*be an origin symmetric convex body in $Rxn$. By definition, the Mahler volume (or volume product) of

*X*is the product

_{n}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 $L\u2208GL(n,R)$ we have

^{6}assume again that

*X*is a centrally symmetric body; then

^{32}) that one has the lower bound

*X*is the hypercube

*C*= [−1, 1]

^{n}. Bourgain and Milman

^{7}have shown the existence, for every $n\u2208N$, of a constant

*C*

_{n}> 0 such that

^{31}has shown that one can choose

*C*

_{n}= (

*π*/4)

^{n}.

*In view of the invariance property (12) this is equivalent to saying that the minimum is attained by any*

*n*

*-parallelepiped*

*This is related to the covariances of the tensor product*

*ψ*=

*ϕ*

_{1}⊗⋅⋅⋅⊗

*ϕ*

_{n}

*of standard one-dimensional Gaussians*$\varphi j(x)=(\pi \u210f)\u22121/4e\u2212xj2/2\u210f$

*; 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 a*“

*quantum proof*”

*of the Mahler conjecture.*

### B. Example: Hardy’s uncertainty principle

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.

^{27}in its original (one dimensional) formulation says that if the moduli of $\psi \u2208L1(R)\u2229L2(R)$ and of its Fourier transform, here defined by

*C*

_{A},

*C*

_{B}> 0,

*a*,

*b*> 0), then we must have

*ab*≤ 1 and if

*ab*= 1 we have $\psi (x)=Ce\u2212a2\u210fx2$ for some complex constant

*C*. We have proven a multidimensional version of this result (see Refs. 21 and, 23, Chapter 10). Let $\psi ,\psi \u0302\u2208L1(Rn)\u2229L2(Rn)$,

*ψ*≠ 0 where

*Let*$A,B\u2208Sym(n,R)$

*be positive definite and*

*ψ*

*as above. Assume that there exist a constants*

*C*

_{A},

*C*

_{B}> 0

*such that*

*(i) The eigenvalues* *λ*_{j}*,* 1 ≤ *j* ≤ *n**, of* *AB* *are* $\u22641$*; (ii) If* *λ*_{j} = 1 *for all* *j**, then* $\psi (x)=ke\u221212\u210fAx2$ *for some* $k\u2208C$.

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

*The Hardy estimates are satisfied by a non-zero function* $\psi \u2208L2(Rn)$ *if and only if the ellipsoids* *X* = {*x*: *Ax⋅x* ≤ *ℏ*} *and* *P* = {*p*: *Bx⋅x* ≤ *ℏ*} *satisfy* *X*^{ℏ} ⊂ *P* *with equality* *X*^{ℏ} = *P* *if and only* $\psi (x)=ke\u221212\u210fAx\u22c5x$ *for some* $k\u2208C$.

This immediately follows from Lemma 3.■

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

## III. THE SYMPLECTIC CASE

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

### A. Basic properties

#### 1. Symplectic covariance

*σ*(

*z*,

*z*′) =

*Jz⋅z*′ of the symplectic form it is straightforward to verify It is straightforward to verify that the symplectic polar dual

^{ℏ}by the formula

*Consider the phase space ellipsoid*

*where*

*M*

*is symmetric and positive definite and*

*r*> 0

*. Then*

*Let*Ω

*be a symmetric convex body and*

*S*∈

*Sp*(

*n*)

*. (i) we have*

*(ii) More generally for*

*S*∈

*Sp*(

*n*)

*and*

*F*

*a linear subspace of*$R2n$

*we have*

#### 2. Quantum blobs

Symplectic balls with radius $\u210f$ are the only fixed ellipsoids for ordinary polar duality. They play an important role in various formulations of the uncertainty principle of quantum mechanics^{14,18,24} where they represent minimum uncertainty units; this motivates the following definition:

*A quantum blob* *Q*_{S}(*z*_{0}) *is a symplectic ball with radius* $\u210f$*:* $QS(z0)=S(B2n(z0,\u210f))$ *for some* *S* ∈ *Sp*(*n*)*. When* *z*_{0} = 0 *we write* *Q*_{S} = *Q*_{S}(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*.

*Let* Ω *be a centered ellipsoid in* $(R2n,\sigma )$*. We have* Ω = Ω^{ℏ,σ} *if and only if* Ω *is a quantum blob, i.e. if* Ω = *Q*_{S} *for some* *S* ∈ *Sp*(*n*).

*Mz*⋅

*z*≤

*ℏ*; then its symplectic polar dual Ω

^{ℏ,σ}is defined by −

*JM*

^{−1}

*Jz⋅z*≤

*ℏ*(Lemma 9) and we have Ω = Ω

^{ℏ,σ}if and only if

*M*= −

*JM*

^{−1}

*J*. This condition is trivially equivalent to

*MJM*=

*J*which implies

*M*∈ Sp(

*n*) hence $\Omega =S(B2n(\u210f))$ with

*S*=

*M*

^{−1/2}∈ Sp(

*n*).■

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

*Let*$\Omega \u2282R2n$

*be a centrally symmetric convex body and*

*F*

*a linear subspace of*$R2n$

*. We have*

^{ℏ}is symmetric we have

*J*(Ω ∩

*F*)

^{ℏ}= −

*J*(Ω ∩

*F*)

^{ℏ}hence the kernel of the projector −

*J*Π

_{F}

*J*is

*F*and its range is

*Jℓ*so that −

*J*Π

_{F}

*J*= Π

_{JF}. This proves the first equality (24). We have, by definition, (Ω ∩

*F*)

^{ℏ,σ}=

*J*(Ω ∩

*F*)

^{ℏ}. In view of formula (8) we have (Ω ∩

*F*)

^{ℏ}= Π

_{F}(Ω

^{ℏ}) and hence (Ω ∩

*F*)

^{ℏ,σ}=

*J*Π

_{F}(Ω

^{ℏ}). Thus

*X*is an ellipsoid. This follows from (14) noting that

### B. Quantum admissible ellipsoids

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

The following definition will be motivated below:

*Let* $\Omega M\u2282R2n$ *be the ellipsoid* {*z*: *Mz⋅ z* ≤ *ℏ*} *(**M* = *M*^{t} > 0*). We will say that* Ω_{M} *is quantum admissible if it contains a quantum blob* $QS=S(B2n(\u210f))$*,* *S* ∈ *Sp*(*n*).

^{41}(see 16 and 28 for “modern” proofs). For every

*M*=

*M*

^{T}> 0, there exists

*S*∈ Sp(

*n*) such that

*M*(

*i.e.*the moduli of the eigenvalues of

*JM*∼

*M*

^{1/2}

*JM*

^{1/2}). It is usual to rank the symplectic eigenvalues in non-increasing order

*M*

^{−1}is $((\lambda 1\sigma )\u22121,\u2026,(\lambda n\sigma )\u22121)$. It is usual to call the factorization (26) the “Williamson normal form of

*M*.”

Recall from Proposition 12 that the equality $\Omega M\u210f,\sigma =\Omega M$ occurs if and only if $\Omega M=S(B2n(\u210f))$ 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 $(R2n,\sigma )$ is called a symplectic plane if the restriction*σ*|*F*of symplectic form*σ*is non-degenerate; equivalently*F*has a basis {*e*_{1},*e*_{2}} such that*σ*(*e*_{1},*e*_{2}) = 1. In particular every plane*F*_{j}of conjugate coordinates*x*_{j},*p*_{j}is symplectic; and for every symplectic plane*F*there exists*S*_{j}∈ Sp(*n*) such that*F*=*S*_{j}(*F*_{j}).We will use Gromov’s symplectic non-squeezing theorem;

^{26}it says (in its simplest form) that no symplectomorphism*f*∈ Symp(*n*) of $(R2n,\sigma )$ can send a ball*B*^{2n}(*R*) into a cylinder $Zj2n(r):xj2+pj2\u2264r2$ if*r*<*R*[we are denoting by Symp(*n*) the group of all symplectomorphisms^{28,36,42}of $(R2n,\sigma )$].

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

*Let*

*F*

*be a symplectic plane in*$(R2n,\sigma )$

*and*

*f*∈

*Symp*(

*n*)

*. The area of the orthogonal projection*Π

_{F}

*of*

*f*[

*B*

^{2n}(

*z*

_{0},

*r*)]

*on*

*F*

*satisfies*

*z*

_{0}= 0 since areas are translation-invariant. Assume that $\Pi Fj(S(B2n(r)))=\pi R2$ with

*R*<

*r*. Then $fB2n(r)$ must be contained in the cylinder

*Z*

_{j}(

*R*), but this contradicts Gromov’s non-squeezing theorem. (ii) Assume that

*R*<

*r*. Then, by (27),

Let us now state and prove our theorem:

*The ellipsoid*Ω

_{M}

*is quantum admissible if and only if the two following equivalent conditions are satisfied: (i) We have the inclusion*

*(ii) We have the inequality*

*for every symplectic plane*

*F*

*in*$(R2n,\sigma )$.

_{M}is quantum admissible; then there exists

*S*∈ Sp(

*n*) such that $QS=S(B2n(\u210f))\u2282\Omega M$. By the anti-monotonicity of symplectic polar duality this implies that we have

*M*≤ (−

*JM*

^{−1}

*J*) ($\u2264$ stands here for the Löwner ordering of matrices). Performing a symplectic diagonalization (26) of

*M*and using the relations

*JS*

^{−1}=

*S*

^{T}

*J*, $(ST)\u22121J=JS$ this is equivalent to

*D*≤ −

*JD*

^{−1}

*J*. This implies that we have $\Lambda \sigma \u2264(\Lambda \sigma )\u22121$ and hence $\lambda j\sigma \u22641$ for 1 ≤

*j*≤

*n*; thus

*D*≤

*I*and

*M*=

*S*

^{T}

*DS*≤

*S*

^{T}

*S*. The inclusion $S(B2n(\u210f))\u2282\Omega M$ follows and we are done. (ii) Suppose that Ω

_{M}is admissible and let Π

_{F}be the orthogonal projection in $R2n$ on

*F*. By Proposition 13 we have

_{M}is an ellipsoid the symplectic version (25) of the Blaschke–Santaló inequality becomes the equality

_{M}is admissible, it contains a quantum blob $(B2n(\u210f))$ hence Area(Π

_{JF}Ω

_{M}) ≥

*πℏ*in view of Lemma 16. Assume conversely that $Area(\Omega M\u210f,\sigma \u2229F)\u2264\pi \u210f$ for every symplectic plane

*F*; by (31) we must then have Area(Π

_{JF}Ω

_{M}) ≥

*πℏ*for every

*F*. Let us show that this implies that Ω

_{M}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 $\Lambda \sigma 0n\xd7n0n\xd7n\Lambda \sigma $ where $\Lambda \sigma =diag(\lambda 1\sigma ,\u2026,\lambda n\sigma )$ the $\lambda j\sigma $ being the symplectic eigenvalues of

*M*. The ellipsoid Ω

_{M}is thus given by

_{M}with the symplectic plane

*F*

_{1}(the plane of coordinates

*x*

_{1},

*p*

_{1}). It is the ellipse $x12+p12\u2264\u210f/\lambda 1\sigma $ which has area $\pi \u210f/\lambda 1\sigma $. Now, Ω

_{M}is admissible if and only if $\lambda 1\sigma =\lambda max\sigma \u22641$ which is equivalent to the condition Area(Ω

_{M}∩

*F*) ≥

*π*, hat is to $Area(\Omega M\u210f,\sigma \u2229F)\u2264\pi \u210f$ 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 Narcowich^{35} we will briefly discuss in our study of covariance and information ellipsoids in Sec. V B.

### C. Sub-Gaussian estimates for the Wigner function

^{21}that the cross-Wigner function of a pair $(\psi ,\varphi )\u2208L2(Rn)\xd7L2(Rn)$ it is defined by

*W*(

*ψ*,

*ϕ*) is a continuous function satisfying the estimate

*ψ*=

*ϕ*the function

*W*(

*ψ*,

*ψ*) =

*Wψ*is the usual Wigner function

^{16,21}if

*A*> 0. Then

^{16,21}

*G*∈ Sp(

*n*):

*M*=

*M*

^{T}∈ Sp(

*n*),

*M*> 0, then

*ψ*=

*e*

^{iχ}

*ψ*

_{A,B}($\chi \u2208R$) 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 proves^{21,23} that:

*Let* $\psi \u2208L2(Rn)$*,* *ψ* ≠ 0*, and assume that there exists* *C* > 0 *such that* $W\psi (z)\u2264Ce\u22121\u210fMz\u22c5z$ *where* *M* = *M*^{T} > 0*. Then the symplectic eigenvalues* $\lambda 1\sigma \u2265\lambda 2\sigma \u2265\u22c5\u22c5\u22c5\u2265\lambda n\sigma $ *of* *M* *are all* $\u22641$*. When* $\lambda 1\sigma =\lambda 2\sigma =\u22c5\u22c5\u22c5=\lambda n\sigma =1$ *then the function* *ψ* *is a generalized Gaussian (35).*

It follows from this result that the Wigner function *Wψ* can never have compact support: assume that there exists *R* > 0 such that *Wψ*(*z*) = 0 for |*z*| > *R*. Then, for every *a* > 0 there exists a constant *C*(*a*) > 0 such that $W\psi (z)\u2264C(a)e\u2212a\u210f|z|2$ for all $z\u2208R2n$. Choosing *a* large enough this contradicts the statement in Corollary 18 because as soon as *a* > 1 the symplectic eigenvalues of *M* = *aI*_{n×n} are all equal to *a*.

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

*The Wigner function* *Wψ* *of* $\psi \u2208L2(Rn)$ *satisfies a sub-Gaussian estimate* $W\psi (z)\u2264Ce\u22121\u210fMz\u22c5z$ *if and only if the ellipsoid* Ω_{M} = {*z*: *Mz z* ≤ *ℏ*} *is admissible:* $\Omega M\u210f,\sigma \u2282\Omega M$*, that is,* Ω_{M} *contains a quantum blob* *S*(*B*^{2n}(*ℏ*))*,* *S* ∈ *Sp*(*n*).

In view of Williamson’s diagonalization theorem 26 there exists *S* ∈ Sp(*n*) such that *M* = *S*^{T}*DS* where $D=\Lambda \sigma 0n\xd7n0n\xd7n\Lambda \sigma $ hence *S*(Ω_{M}) = Ω_{D}. Since *S*(Ω^{ℏ,σ}) = [*S*(Ω)]^{ℏ,σ} (Proposition 10) it is sufficient to prove the result for *M* = *D*. Since $\Omega D\u210f,\sigma =\Omega \u2212JD\u22121J$ we thus have to show that the sub-Gaussian estimate is satisfied if and only if $\Omega \u2212JD\u22121J\u2282\Omega D$. This is equivalent to *D* ≤ −*JD*^{−1}*J* (in the Löwner ordering) that is to $\Lambda \sigma \u2264(\Lambda \sigma )\u22121$ which is possible if and only the symplectic eigenvalues $\lambda j\sigma $ are all $\u22641$. When the $\lambda j\sigma $ are all equal to one we have *D* = *I*_{2n × 2n}., that is *M* = *S*^{T}*S* so that $W\psi (z)=Ce\u22121\u210fMz\u22c5z$ for some constant *C*.■

## IV. MODULATION SPACES AND COVARIANCE MATRICES

### A. The modulation spaces $Ms1$

#### 1. Definition using the Wigner function

*L*

^{1}-space defined by

*The modulation space* $Ms1(Rn)$ $(s\u2208R)$ *consists of all* $\psi \u2208L2(Rn)$ *such that* $W(\psi ,\varphi )\u2208Ls1R2n$ *for every* $\varphi \u2208S(Rn)$*. When* *s* = 0 *the space* $M01(Rn)=S0(Rn)$ *is called the Feichtinger algebra.*

*one*function

*ϕ*≠ 0 (hereafter called “window” for it then it holds for

*all*$\varphi \u2208S(Rn)$.The mappings $\psi \u27fc\Vert \psi \Vert \varphi ,Ms1$ defined by

*F*is the Fourier transform; it follows by Riemann–Lebesgue that in particular

^{25}that

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 Sp_{2}(*n*) of the symplectic group Sp(*n*) (see for instance Ref. 16 for a detailed study of the metaplectic representation). We will denote *π*^{Mp} the covering projection Mp(*n*) ⟶ Sp(*n*); it is a two-to-one epimorphism.

*The modulation spaces* $Ms1(Rn)$*,* *s* ≥ 0*, are invariant under the action of* *Mp*(*n*)*: if* $\psi \u2208Ms1(Rn)$ *and* $S\u0302\u2208Mp(n)$ *then* $S\u0302\psi \u2208Ms1(Rn)$.

*ϕ*. Now, in view of the symplectic covariance of the cross-Wigner function

^{16,19}

*S*= 1 hence

*C*

_{S}> 0, thus

### B. Density operators and covariance matrices

*α*

_{j}) with

*α*

_{j}≥ 0, ∑

_{j}

*α*

_{j}= 1 and an orthonormal stem (

*ψ*

_{j}), $\psi j\u2208L2(Rn)$ such that we have we have the spectral decomposition

^{19},

*ρ*is often called the

*Wigner distribution*of $\rho \u0302$ in the physical literature). In Refs. 15 and 22 we introduced the notion of

*Feichtinger state*:

*A density operator* $\rho \u0302\u2208L1(Rn)$ *is called a a* “*Feichtinger state*” *if each* $\psi j\u2208Ms1(Rn)$ *for some* *s* ≥ 0.

_{cov}of a density operator. The latter is defined –if it exists!– as being the symmetric 2

*n*× 2 ×

*n*matrix

*n*×

*n*block-matrix form as

*The covariance matrix* Σ_{cov} *of a Feichtinger state* $\rho \u0302$ *with* *s* ≥ 2 *is well-defined. If* $S\u0302\u2208Mp(n)$ *then conjugate state* $S\u0302\rho \u0302S\u0302\u22121$ *is also a Feichtinger state with* *s* ≥ 2 *and the covariance matrix of* $S\u0302\rho \u0302S\u0302\u22121$ *is* *S*Σ_{cov}*S*^{T}.

*s*≥ 2 we have

*z*

_{α}=

*x*

_{α}if 1 ≤

*α*≤

*n*and

*z*

_{α}=

*p*

_{α}if

*n*+ 1 ≤

*α*≤ 2

*n*we have $\Sigma =(\sigma \alpha \beta )1\u2264\alpha ,\beta \u22642n$ where the integrals

*z*

_{α}

*z*

_{β}| ≤ 1 + |

*z*|

^{2}. Let $S\u0302\u2208Mp(n)$; it follows from the standard properties of Weyl pseudodifferential calculus

^{16,19,38}that the Weyl symbol of the conjugate $S\u0302\rho \u0302S\u0302\u22121$ is(2

*πℏ*)

^{n}

*ρ*◦

*S*

^{−1}. In view of Proposition 21 $\rho \u2208Ms1(Rn)$ implies that $\rho \u25e6S\u2208Ms1(Rn)$ and the result follows from (44) by a simple calculation.■

#### 1. A necessary condition for positivity

*s*≥ 2, guaranteeing the existence of the covariance matrix Σ

_{cov}. It i a well-known property in harmonic analysis that the positivity condition $\rho \u0302\u22650$ implies that Σ must satisfy the algebraic condition

*J** = −

*J*.) This condition implies in particular that Σ is positive definite

^{35,19}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 Kastler

^{30}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 paper

^{10}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

^{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 (2

*πℏ*)

^{n}

*ρ*where

^{T}> 0 playing the role of a covariance matrix Σ

_{cov}; see for instance 15, 21, 25, and 28).

## V. SYMPLECTIC CAPACITIES AND POLAR DUALITY

### A. The notion of symplectic capacity

Symplectic capacities were defined by Ekeland and Hofer^{11,12} (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.

*symplectic capacities*. A (normalized) symplectic capacity on $(R2n,\sigma )$ associates to every subset $\Omega \u2282R2n$ 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 $\lambda \u2208R$ we have*c*(*λ*Ω) =*λ*^{2}*c*(Ω);SC3

*Symplectic invariance*:*c*[*f*(Ω)] =*c*(Ω) for every*f*∈ Symp(*n*);SC4

*Normalization*: For 1 ≤*j*≤*n*,

*r*based on the

*x*

_{j},

*p*

_{j}plane.

It follows that the symplectic capacity of a quantum blob $QS(z0)=S(B2n(\u210f))$ is *c*(*Q*_{S}(*z*_{0})) = *πℏ*.

*c*

_{max}and

*c*

_{min}, such that

*c*

_{min}≤

*c*≤

*c*

_{max}for every symplectic capacity

*c*, they are defined by

*c*

_{min}and

*c*

_{min}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 $(R2n,\sigma )$ with smooth boundary

*∂*Ω then

*pdx*=

*p*

_{1}

*dx*

_{1}+ ⋅⋅⋅ +

*p*

_{n}

*dx*

_{n}and

*γ*

_{min}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 $c(\Omega +z)=c(\Omega )$ for every $z\u2208R2n$.

#### 1. The case of ellipsoids

*M*. If in particular Ω = Ω

_{M}:

*Mz⋅z*≤

*ℏ*then

_{M}to Williamson normal form

*∂*Ω

_{M}is given by Hamilton’s equations for the Hamiltonian function $H1(x1,p1)=\lambda 1\sigma (x12+p12)$ with the condition

*H*

_{1}(

*x*

_{1},

*p*

_{1}) =

*ℏ*. One verifies that this periodic solution verifies

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

*Let*Ω

_{M}

*be as above and let*$\Omega M\u210f,\sigma $

*be its symplectic polar dual. (i) We have*

*where*$\lambda max\sigma $

*(resp.*$\lambda min\sigma $

*) is the largest (rep. smallest)symplectic eigenvalue of*

*M*.

*(ii) In particular*

*with equality if and only if*$\Omega M=\lambda B2n(\u210f)$

*for some*

*λ*> 0.

*M*which reduces the problem to the case of an ellipsoid with axes contained in the conjugate

*x*

_{j},

*p*

_{j}planes (see Refs. 28 and 16 for details). The symplectic polar dual of Ω

_{M}is

*N*= −

*JM*

^{−1}

*J*: we have

*JN*=

*M*

^{−1}

*J*hence the eigenvalues of

*JN*are those of

*M*

^{−1/2}

*JM*

^{−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 $\Omega M=\lambda B2n(\u210f)$ 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

*S*

^{T}

*S*.■

### B. Symplectic polarity and covariance ellipsoids

#### 1. Covariance and information ellipsoids

_{cov}is the covariance matrix of $\rho \u0302$. The symplectic polar dual of Ω

_{cov}is the ellipsoid

_{cov}. It turns out that Ω

_{cov}and Ω

_{inf o}are Legendre duals of each other.

^{35}Consider in fact the quadratic forms $w(z)=12\Sigma cov\u22121z\u22c5z$ and $w\sigma (z\u2032)=12\Sigma covz\u2032\u22c5z\u2032$. The Legendre transform of

*w*(

*z*) is defined by

*z*is expressed in terms of

*z*′ by solving the equation $z\u2032=\u2202zw(z)=\Sigma cov\u22121z$ hence

*w*′(

*z*′) =

*w*

^{σ}(

*z*′).

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

*The covariance ellipsoid* Ω_{cov} *of a Feichtinger state* $\rho \u0302$ *satisfies the condition* $\Omega cov\u210f,\sigma \u2282\Omega cov$ *and is hence quantum admissible.*

_{M}:

*Mz z*≤

*ℏ*and we have to prove that the positivity condition $\Sigma +i\u210f2J\u22650$ is equivalent to the condition $\Omega M\u210f,\sigma \u2282\Omega M$. In view of Propositions 17 it is sufficient to prove that this conditions hold if and only if Ω

_{M}contains a quantum blob $QS=S(B2n(\u210f))$,

*S*∈ Sp(

*n*). Performing a symplectic diagonalization

*M*=

*S*

^{T}

*DS*(26) of

*M*the condition $\Sigma +i\u210f2J=M\u22121+iJ\u22650$ implies that

*D*

^{−1}+

*iJ*≥ 0, that is

*M*. The eigenvalues of

*D*

^{−1}+

*iJ*are the real numbers $\lambda j=(\lambda j\sigma )\u22121\xb11$ and the condition

*D*

^{−1}+

*iJ*≥ 0 thus implies that we must have $\lambda j\sigma \u22641$ for 1 ≤

*j*≤

*n*. It follows that the ellipsoid Ω

_{D}:

*Dz⋅z*≤

*ℏ*contains the ball $B2n(\u210f)$ and hence Ω

_{M}contains the quantum blob $S(B2n(\u210f))$ where

*S*∈ Sp(

*n*) is the diagonalizing matrix. The result now follows applying Propositions 17.■

*Assume that the Wigner distribution of*$\rho \u0302$

*is a Gaussian (50) with*

*z*

_{0}= 0

*. If*Ω

^{ℏ,σ}= Ω

*it follows from Propositions*17)

*that*Ω

*is a quantum blob*$S(B2n(\u210f))$

*and hence*

*for some*

*S*∈

*Sp*(

*n*)

*. Then*

^{16,19}$\rho =W(S\u0302\u22121\varphi 0\u210f)$

*where*

*and*$S\u0302\u2208Mp(n)$

*has projection*$\pi Mp(S\u0302)=S$.

#### 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 $H\u2208C\u221e(R2n,R)$: *i.e*. *∂*Ω = {*z*: *H*(*z*) = *E*} for some $E\u2208R$. 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:

*The ellipsoid*Ω

*is a quantum covariance ellipsoid*Ω

_{cov}

*(resp. an information ellipsoid*Ω

_{inf o}

*) if and only if the following equivalent conditions are satisfied: (i) We have*

*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*

*(iii) Let*

*F*

*be an arbitrary two-dimensional subspace of*$R2n$

*an let the ellipse*$\gamma F*=\u2202\Omega *\u2229F$

*be positively oriented. We have*

*Nz z*≤

*ℏ*where $N=\u210f24M\u22121$. The symplectic spectrum of

*N*is thus $\u210f24((\lambda n\sigma )\u22121,\u2026,\lambda 1\sigma )\u22121)$ where $(\lambda 1\sigma ,\u2026,\lambda n\sigma )$ 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

*F*is a null space for the symplectic form [i.e.

*F*has a basis {

*e*

_{1},

*e*

_{2}} such that

*σ*(

*e*

_{1},

*e*

_{2}) = 0]. The, by Stokes’s theorem we have

*F*is a symplectic plane. Then, by formula (29) in Theorem 17 we have

## ACKNOWLEDGMENTS

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

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