Polar Duality and the Reconstruction of Quantum Covariance Matrices from Partial Data

We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work. We apply our constructions to Gaussian quantum states which leads to a non-trivial generalization of Pauli's reconstruction problem and we state a simple tomographic characterization of such states.


Abstract
We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work.We apply our constructions to

Introduction
The covariance matrix is a fundamental concept in both classical and quantum mechanics, serving distinct purposes in each domain.In classical mechanics, the covariance matrix is employed to characterize statistical relationships and correlations between different variables within a system [5].In quantum mechanics, the covariance matrix holds particular significance in the context of quantum correlations [6].According to Born's rule, the quantum covariance matrix encapsulates all available statistical information about a quantum state.Moreover, covariance matrices serve as a powerful tool for detecting entanglement, playing a key role in identifying and analyzing quantum entangled states [7,33].
Notably, the quantum covariance matrix fully characterizes Gaussian states, their Wigner distributions are parametrized by their covariance matrices and their center.This correspondence is central to the discussion in the present paper to address the problem of the determination of covariance matrices from partial data.This problem is not new and has been studied by many authors, see for instance Řeháček et al. [30].We have initiated such a study from a wider point of view in [15,20,18] using the notion of polar duality from convex geometry; the present work considerably extends and synthesizes these preliminary works.More precisely, let Σ be an arbitrary real positive definite 2n × 2n matrix; such a matrix can always be viewed as the covariance of the multivariate Gaussian distribution (we are using the notation z = (x, p) ∈ R n x × R n p ); this distribution qualifies as a the Wigner distribution of a quantum state provided that we impose a constraint on Σ; the latter is usually chosen to be [3,10,22,28]: where J is the standard symplectic matrix.We have shown [11,12,22] that this condition is equivalent to The covariance ellipsoid Ω Σ contains a quantum blob ( (a quantum blob is a symplectic ball with radius √ ); in [11,22] we also formulated these conditions using the topological notion of symplectic capacity [9] which is closed related to Gromov's symplectic non-squeezing theorem.On the operator level the conditions (2) and (3) guarantees that the trace class operator with Weyl symbol (2π ) n ρ is positive semidefinite and has trace one, and thus qualifies as a density operator [19].
The main results of this paper are • Theorem 12 which states that a generalized Gaussian (and hence a covariance matrix) can be reconstructed from the knowledge of only two marginal distributions of along a pair of transverse Lagrangian planes; it provides a generalization of the solution of Pauli's reconstruction problem; • Theorem 13 provides a geometric interpretation of the previous results; identifies Gaussian states with quantum blobs viewed as John ellipsoids of a convex set constructed using the notion of Lagrangian polar duality, which can be viewed as a geometric variant of the uncertainty principle: • Theorem 16 uses the notion of symplectic polar duality Ω −→ Ω ,ω we introduces in [21,17].We prove that in order to prove that a phase space ellipsoid Ω is the covariance ellipsoid of a Gaussian quantum state it suffices that Ω ,ω ∩ ℓ ⊂ Ω ∩ ℓ for one Lagrangian subspace ℓ (and hence for all).
Notation and terminology We will denote by ω the standard symplectic form on We denote by Sym ++ (m, R) the convex set of all symmetric positive definite real m × m matrices.
The group of all automorphisms of the symplectic space (R 2n , ω) leaving the standard symplectic form invariant is called the (standard) symplectic group and is denoted by Sp(n).The properties of Sp(n) and of its double covering, the metaplectic group Mp(n) are summarized in the Appendix A.
A linear subspace of (R 2n , ω) with dimension n is on which ω vanishes identically is called a Lagrangian subspace (or plane); the set of all Lagrangian subspaces of (R 2n , ω) will be denoted by Lag(n), and is called the Lagrangian Grassmannian of (R 2n , ω).
Let B 2n ( √ ) be the ball with center 0 and radius √ in R 2n (equipped with the usual Euclidean norm).The image S(B 2n ( √ )) of that ball by S ∈ Sp(n) is called a quantum blob.

The RSUP for Gaussians
The Robertson-Schrödinger uncertainty principle (RSUP) is, as opposed to the elementary Heisenberg inequalities, a fundamental concept in quantum mechanics that describe the trade-off between the uncertainties in the measurements of two non-commuting observables, such as position and momentum.These inequalities are typically expressed in terms of standard deviations or (co)variances.

Multivariate Correlated Gaussians
We denote by φ 0 the standard Gaussian (called "fiducial state" by Littlejohn [24]) and by W φ 0 its Wigner function: here Consider now the set Gauss 0 (n) of all centered generalized Gaussians: we have ψ XY ∈ Gauss 0 (n) if and only if where X and Y are symmetric real n × n matrices such that X > 0 and γ ∈ R is a real constant.This function is normalized to unity: ||ψ γ X,Y || L 2 = 1 and its Wigner transform is given by [10,14] where G is the symmetric and symplectic matrix (The result seems to back to Bastiaans, see [24].)Observe that G = S T S where An almost trivial, but fundamental, observation is that every ψ γ X,Y ∈ Gauss 0 (n) can be obtained from the standard Gaussian using elementary metaplectic transformations, namely In fact: Proposition 1 The metaplectic group Mp(n) acts transitively on Gaussians ψ X,Y = ψ 0 X,Y up to a unimodular factor: Proof.To see this it is sufficient, in view of formula (9) to show that if S ∈ Mp(n) then Sφ 0 = i µ ψ X,Y for some µ ∈ R and ψ X,Y ∈ Gauss 0 (n).In view of the pre-Iwasawa factorization (Appendix A, formula (76)) we have S = π Mp ( S) = V P M L R and hence We claim that It follows that Rφ 0 = i µ φ 0 for some µ ∈ R. In fact we have, by the symplectic covariance of the Wigner transform, the second equality in view of the rotational invariance of φ 0 ; it follows that hence Sφ 0 ∈ Gauss 0 (n) as claimed (this result can also be obtained using Fourier integral operators, at the cost of much more complicated calculations involving Cayley transforms (see e.g.[24]).There remains to show that Sψ γ X,Y ∈ Gauss 0 (n) for all ψ γ X,Y ∈ Gauss 0 (n).Every S ∈ Mp(n) can be written S = V P M L,m R (m ∈ {0, 2}) corresponding to the pre-Iwasaw factorization of S = π Mp ( S).For S ′ ∈ Mp(n); we have We claim that Rφ 0 = i µ φ 0 for some µ ∈ R. We have the second equality in view of the rotational invariance of φ 0 ; hence

The multivariate RSUP
A mixed Gaussian quantum state centered at z 0 is a density operator ρ on L 2 (R n ) whose Wigner distribution ρ is of the type where Σ is the covariance (or noise) matrix; the condition ρ ≥ 0 is equivalent to saying that the eigenvalues of the Hermitian matrix Σ+ i 2 J are ≥ 0 [3, 10], which we write for short as the Löwner ordering: The state ρ is pure if and only if det Σ = 2 2n (see below).The condition (12) is equivalent to the following three statements: The symplectic capacity of the covariance ellipsoid is at least π .
Recall [10] that the symplectic eigenvalues of a positive definite matrix Σ are the numbers λ ω j > 0, (0 ≤ j ≤ n) such that the ±iλ ω j are the eigenvalues of JΣ (which are the same as those of the antisymmetric matrix Σ 1/2 JΣ 1/2 ).The terminology "quantum blob" to denote a symplectic ball with radius √ was introduced in [12,11]).For the notion of symplectic capacity and its applications to the uncertainty principle see [11].The proof of the first claim is well-known and widespread in the literature, see for instance [3,12,22].It is easily proven using Williamson's symplectic diagonalization theorem [10] which says that for every Σ > 0 there exists S ∈ Sp(n) such that where The proof of the second claim easily follows from a geometric argument; see [11,22].The case where λ ω j = 1 2 for all j is of particular interest; in this case we have Σ = 2 S T S (and hence det Σ = 2 2n ) and so that where S ∈ Mp(n) covers S and T (z 0 ) is the Heisenberg-Weyl displacement operator [10]: From these considerations follows that the Wigner distribution of the Gaussian state ρ is the Wigner function of a multivariate Gaussian T (z 0 )ψ X,Y centered at z 0 .We will write the covariance matrix in block-form with Σ XX = (σ x j x k ) 1≤j,k≤n , Σ P P = (σ p j p k ) 1≤j,k≤n , Σ P P = (σ x j p k ) 1≤j,k≤n .
Proposition 5 Let Σ be the covariance matrix of the Gaussian state ρ; (i) This state is pure, that is there exists ψ X,Y such that ρ = ψ X,Y , if and only if (ii) We have (Σ XP ) 2 ≥ 0, i.e. the eigenvalues of (Σ XP ) 2 are ≥ 0 hence the generalizes Heisenberg inequality Σ XX Σ P P ≥ 2 4 I n×n holds.
Proof.Suppose that ρ is a pure state; then Σ = 2 S T S and the conditions ( 15)-( 16) are just a restatement of the relations (72) in APPENDIX A, taking into account the fact that S T S > 0. To prove that (Σ XP ) 2 ≥ 0 we note that since Σ XX Σ XP = Σ P X Σ XX we have Σ XP = Σ −1 XX Σ P X Σ XX hence Σ XP and Σ P X have the same eigenvalues; since Σ P X = Σ T XP these eigenvalues must be real, hence those of (Σ XP ) 2 are ≥ 0.
In particular, when n = 1 one recovers the usual saturated Robertson-Schrödinger uncertainty principle satisfied by all pure one-dimensional Gaussian states.
Note that when Σ is diagonal, i.e. the state is a tensor product of onedimensional functions one recovers the well-known fact that the Heisenberg uncertainty inequalities are saturated (reduce to equalities): σ x j xj σ p j p j = 2 /4 for all j = 1, ..., n.

Orthogonal projections of the covariance ellipsoid
We begin with a general result.For M ∈ Sym ++ (2n, R) we define the phase space ellipsoid Writing M in block-matrix form M = M XX M XP M P X M P P where the blocks are n × n matrices the condition M ∈ Sym ++ (2n, R) ensures us [34] that M XX > 0, M P P > 0, and M P X = M T XP .Using classical formulas for the inversion of block matrices [31] the inverse of M is where M/M P P and M/M XX are the Schur complements: The following results is well-known (see for instance [13] and the references therein): Lemma 6 The orthogonal projections Π ℓ X Ω and P = Π ℓ P Ω on the coordinate subspaces ℓ X = R n x × 0 and ℓ P = 0 × R n p of the ellipsoid Ω are the ellipsoids Consider now a Gaussian state (11) with covariance matrix Σ = Σ XX Σ XP Σ P X Σ P P ; by definition the covariance (or Wigner) ellipsoid of this state is Setting M = 1 2 Σ −1 the results above yield the inverse of the covariance matrix Σ: Formula ( 24) immediately follows from (18) using the relations Proposition 7 The orthogonal projections on the canonical coordinate subspaces ℓ X and ℓ P of Ω Σ are Proof.The projection formulas ( 27) and ( 28) are a consequence of the corresponding formulas ( 21) and (22).We next note the following remarkable fact showing that the notion of polar duality is related to the uncertainty principle (see APPENDIX C for a short review of polar duality): Proposition 8 (i) The ellipsoids X = Π ℓ X Ω Σ and P = Π ℓ P Ω Σ are such that X ⊂ P where is the -polar dual of X. (ii) We have the equality X = P if and only if ) and P = B n P ( √ ) (the centered balls with radius √ in ℓ X and ℓ P , respectively) Proof.(i) Recall [17,21] that if.
with A ∈ Sym ++ (n, R), then its -polarX ℏ is the ellipsoid It follows that if (with B ∈ Sym ++ (n, R)) the inclusion X ⊂ P holds if and only if AB ≥ I n×n (i.e. the eigenvalues of AB are all ≥ 1); this is equivalent to BA ≥ I n×n .Since we have here A = 1 2 Σ −1 XX and B = 1 2 Σ −1 P P (formulas ( 27) and ( 21)) the inclusion X ⊂ P will hold if and only if 1 where 2 .The orthogonal projections Ω X and Ω P of Ω on the x and p coordinate axes are the intervals Let Ω X be the polar dual of Ω X : it is the set of all numbers p such that px ≤ for − √ 2σ xx ≤ x ≤ √ 2σ xx and is thus the interval Since σ xx σ pp ≥ 1 2 we have the inclusion which reduces to the equality Ω X = Ω P if and only if the Heisenberg inequality is saturated, i.e.
3 Pauli's Problem and its Generalizations

Paulis' reconstruction problem
The Pauli reconstruction problem is a particular case of a larger class of phase retrieval problems; see Grohs and Liehr [27] for recent advances on this difficult topic.Pauli asked in [29] whether the probability densities |ψ(x)| 2 and | ψ(p)| 2 of a normed function ψ ∈ L 2 (R) uniquely determine ψ.
The answer is in general negative: consider the correlated Gaussian which has Fourier transform the covariance σ xp can take any of the two values 2 ) 1/2 so the Pauli problem has two possible solutions ψ ± ("Pauli partners": see Corbett's [4] review of Pauli's problem).The same argument works for multivariate Gaussians (5 To find the covariance matrix Σ XP one then uses the Robertson-Schrödinger formula (15) I n×n which has multiple solutions in Σ XP .Another way of seeing Pauli's problem is to use the Wigner formalism; recall that the Wigner transform of ψ ∈ L 2 (R n ) is defined by hold [14].Since W ψ * (x, p) = W ψ(x, −p) the functions ψ and ψ * have the same marginals, which is reflected by the relations (38) which are satisfied by both ψ and its complex conjugate.This idea extends to more general phase retrieval problems, see [27].

Lagrangian frames
We will henceforth call Lagrangian frame a pair (ℓ, ℓ ′ ) of transverse Lagrangian subspaces When ℓ and ℓ ′ are coordinate Lagrangian planes ℓ X and ℓ P we will call it the canonical Lagrangian frame.We denote by F Lag (n) The following property will be essential for our constructions to come (See [10]): Lemma 10 The symplectic group Sp(n) acts transitively on F Lag (n).In particular, every Lagrangian frame (ℓ, ℓ ′ ) ∈ F Lag (n) can be obtained from the canonical frame (ℓ X , ℓ P ) by a symplectic transformation.

Gaussian reconstruction by partial tomography
We are going to generalize the reconstruction procedure for Gaussians using the notion of Lagrangian frame introduced above.For this we need to define the integral of a real integrable function ρ on phase space along an affine subspace ℓ(z) = ℓ+z where ℓ is a Lagrangian subspace of (R 2n , ω).Assuming [8,10] that ℓ is represented by the system of equations Ax+Bp = 0 with A T B (and B T A) symmetric, and A T A + B T B = I n×n (and where z = (x, p) ∈ R 2n is arbitrary.Taking into account the fact that A T A + B T B = I n×n we then define the generalized line integral Applying this definition to the case where ρ = W ψ with ψ ∈ L 1 (R n )∩L 2 (R n ) and choosing ℓ = ℓ P we have B = 0 and A = I n×n so that, in view of the marginal properties (44), and, similarly, choosing ℓ = ℓ P , and We'll generalize these relations below, but we first prove that for fixed z = (x, p) the integral (46) is independent of the choice of parametrization (45).
To see this, we note that the Lagrangian subspace ℓ is the image of ℓ X = {(u, 0) : u ∈ R n } by the symplectic rotation ( [8,10]; Appendix A) the matrices A ′ and B ′ satisfying relations similar to those of A and B, and let R ′ be the corresponding symplectic rotation.The product leading to the same value of the integral 46) since d(Hu) = du.
The following result relates the integral (46) to the notion of marginal value of the Wigner transform: where Proof.The Lagrangian subspace ℓ is the image of the momentum space ℓ P by the symplectic rotation hence (47) in view of the first marginal property (44).Formula (47) is closely related to the definition of the symplectic Radon transform as given in our paper [16].The result below shows that, however, we do not need the full power of the theory of inverse Radon transform to reconstruct Gaussians: Theorem 12 Let (ℓ, ℓ ′ ) ∈ F Lag (n) be a Lagrangian frame.The Wigner transform W ψ X,Y (and hence the Gaussian ψ X,Y itself, up to a unimodular factor) is uniquely determined by the knowledge of the integrals Proof.It is similar to that of Proposition 11 above.In view of Lemma 10 the symplectic group acts transitively on F Lag (n) so we can find S ∈ Sp(n) such tat (ℓ, ℓ ′ ) = S(ℓ X , ℓ P ).Setting S = A B C D let S ∈ Mp(n) be one of the two metaplectic operators covering S. Let ℓ(Sz) = Sℓ X + Sz and ℓ ′ (Sz) = Sℓ P + Sz; we have, using the covariance relation that is, in view of the first marginal formula (44), Similarly, using the second formula (44), These values allow the determination of S −1 ψ X,Y and, hence, of ψ X,Y .

Geometric Interpretation
We now consider the following situation: performing a large number of measurements on the coordinates x 1 , ..., x k , p k+1 , ..., p n (with 1 ≤ k < n) we identify this cloud of measurements with an ellipsoid X ℓ carried by the Lagrangian subspace ℓ with coordinates x 1 , ..., x k , p k+1 , ..., p n and centered at the origin.We now ask whether to this ellipsoid we can in some way associate the covariance ellipsoid Ω Σ of some pure Gaussian state with covariance matrix Σ.In view of the discussion above Ω Σ has to be a quantum blob, i.e. the image of the phase space ball B 2n ( √ ) by some S ∈ Sp(n).The answer is given by the following result, which actually holds for any Lagrangian subspace (not necessarily a coordinate subspace).It introduces a notion of polar duality between Lagrangian subspaces.(The basics of the notion of polar duality for convex sets are shortly reviewed in Appendix C.) of M and using the relations that is to D ≤ −JD −1 J.In the notation in (13) this implies that we have Λ ω ≤ (Λ ω ) −1 and hence λ ω j ≤ 1 for 1 ≤ j ≤ n; thus D ≤ I and

Here is an example:
Example 15 Consider again the covariance ellipse The condition 2 so that Ω Σ is indeed a quantum blob.

A tomographic result
We are going to prove a stronger statement, which can be seen as a "tomographic" result since it involves the intersection of the covariance ellipsoid with a (Lagrangian) subspace.Let us begin with a simple example in the case n = 1.
Proof.(i) The necessity of the condition (60) is trivial (Proposition 14).Let us prove that this condition is also sufficient.Setting as usual M = 1 2 Σ −1 and we have Performing a symplectic diagonalization ( 13) of M we get where Ω D −1 /2 and its dual are explicitly given by where we have used the identity Now, the condition , and Ω D −1 /2 contains a quantum blob in view of Proposition 14..We have thus proven our result in the case where Σ = D −1 /2 and ℓ = ℓ X .
For the general case we take ℓ = S −1 ℓ X where S is a Williamson diagonalizing matrix for Σ; in view of (63) we have . It now suffices to apply Proposition 14.To prove (ii) it is sufficient to note that the equality Since we have in this case M = S T 0 S 0 in view of (13), the proof in the general case can be completed as above.
Example 17 With the notation of the previous examples, we have 2 .We have and is equivalent to the inequality where and hence the inclusion (66) holds if and only if k ≤ 0, that is, if and only if D ≥ 2 /4 which is the Robertson-Schrödinger inequality ensuring us that Ω Σ contains a quantum blob (and is itself a quantum blob when D = 2 /4).

The case of mixed states
Sofar we have been considering pure states.Let us generalize our discussion to more general mixed states.We assume that ρ is what we have called in [19] a "Feichtinger state", i.e. a density operator whose Wigner distribution ρ is regular enough to allow the existence of the covariance matrix ( z = R 2n zρ(z)dz is the mean value vector).In order to represent a quantum state a necessary condition is that [3,28] Σ + i 2 J is semidefinite positive (69) which we write for short as Σ + i 2 J ≥ 0 (this condition equivalent to the uncertainty principle in its strong Robertson-Schrödinger form, ibid.).One shows [28] that (69) implies that the covariance matrix Σ of a quantum state is always definite positive, and, conversely, that (69) is sufficient for Gaussian mixed states: a Gaussian of the type (11) introduced above, tha is is the Wigner distribution of a mixed quantum state if and only if the condition (69) holds.Recall that this contition is equivalent to saying that the covariance ellipsoid Ω Σ : 1 2 Σ −1 z • z ≤ 1 contains a quantum blob, from which follows that the orthogonal projetions of Ω Σ : on any conjugate plane x j , p j has area at least π .
Proof.It is just a restatemnt of Theoren 16.

Discussion and Conclusions
Theorem 13 shows that we can identify any pure Gaussian state with a geometric object, the Cartesian product X ℓ ×(X ℓ ) ℓ ′ .The physical interpretation of this correspondence is the following: once a cloud of position-momentum measurements is made on a given Lagrangian plane ℓ, the latter is approximated by an ellipsoid X ℓ .One then chooses a transversal Lagrangian plane ℓ ′ and one calculates the polar dual (X ℓ ) ℓ ′ of X ℓ on ℓ ′ ; the covariance ellipsoid of the Gaussian state we are looking for is then simply the maximal volume ellipsoid of the convex product X ℓ × (X ℓ ) ℓ ′ , and the latter uniquely determines the state (which is here supposed to be centered at the origin; the general case is trivially obtained using phase space translation or the Heisenberg displacement operator).Theorem 16, on the other hand, shows that one can test whether a covariance ellipsoid is that of a quantum state by intersecting it with a single arbitrary Lagrangian plane.This is typically a tomographic result which might have both theoretical and practical applications.
It would be interesting (and important!) to extend the approach and results of this paper to non-Gaussian states; non-Gaussian features are indispensable in many quantum protocols, especially to reach a quantum computational advantage (see the interesting discussions in Ra et al. [25] and Walschaers [32]).A possible approach could be to generalize the "geometric states" described by Theorem 13 to the case where X ℓ no longer is an ellipsoid, but an arbitrary convex body.The Lagrangian plane ℓ would then be replaced with a Lagrangian submanifold of phase space (i.e. a ndimensional submanifold where the tangent spaces are all Lagrangian).We will come back to these intriguing and potentially fruitful possibilities in future work.
An interesting point raised by one of the Reviewers is the question of what happens for reduced covariance matrices where obtaining purity or von entropy Neumann is possible?These questions will be answered in a forthcoming article (There are some delicate points to elucidate, and we have found they deserve to be discussed in a sequel of this work).
APPENDIX A. The Groups Sp(n), U (n), and Mp(n) For details and proofs see [10].The symplectic group Sp(n) consists of all linear automorphisms S : R 2n −→ R 2n such that ω(Sz, Sz ′ ) = ω(z, z ′ ) for all (z, z ′ ) ∈ R 2n × R 2n .A symplectic basis of (R 2n z , σ) being chosen once for all, we can write this condition in matrix form S T JS = SJS T = J and, writing S ∈ Sp(n) in block-matrix form where the entries A, B, C, D are n × n matrices, these conditions are then easily seen to be equivalent to the two following sets of equivalent conditions: The matrix R is a symplectic rotation: R ∈ Sp(n) ∩ O(2n, R).

APPENDIX B. Lagrangian Subspaces
By definition a Lagrangian coordinate subspace is a n-dimensional subspace ℓ (α,β) of the (R 2n , ω) given by the relations x (α) = 0 and p (β) where α and β form a partition of the set of integers {1, ..., n}.Thus, for instance, x 1 = 0 and p 1 = 0 defines coordinate Lagrangian subspaces in R 4 .The choices α = ∅ and β = ∅ correspond to the canonical coordinate planes ℓ X and ℓ P , respectively.Let ℓ (α,β) be a Lagrangian coordinate subspace; we assume for notational simplicity that α = {1, ..., k}, β = {k + 1, ..., n} (1 ≤ k < n).It is represented by the equation where A and B are the diagonal matrices A = I k×k ⊗ 0 (n−k)×(n−k) and B = 0 k×k ⊗ I (n−k)×(n−k) .A remarkable feature of coordinate Lagrangian subspaces is that the symplectic form ω vanishes identically on them if z, z ′ ∈ ℓ (α,β) then ω(z, z) = 0.This motivates the following definition [8,10]: a ndimensional subspace ℓ of R 2n is called a Lagrangian subspace (or plane) if ω(z, z) = 0 for all z, z ′ ∈ ℓ. (Lagrangian subspaces intervene in many areas of mathematical physics; for instance they are the tangent spaces to the invariant tori of classical mechanics [1,8]).In the case n = 1 Lagrangian planes are just the straight lines through the origin in the phase plane.In the general case they are represented by equations Ax + Bp = 0 where rank(A, B) = n and A T B = B T A [8].It turns out that every Lagrangian subspace can be obtained from any Lagrangian coordinate subspace using a symplectic transformation.This follows from the fact the symplectic group Sp(n) acts transitively on the set Lag(n) of all Lagrangian subspaces of (R 2n , ω) (see [10] for a proof using symplectic bases).In particular: The action U (n) × Lag(n) −→ Lag(n) is transitive where U (n) ⊂ Sp(n) is the group of symplectic rotations (see Appendix A).Let ℓ (α,β) ∈ Lag(n) be a Lagrangian coordinate subspace.It follows that There exists non-unique) S (α,β) , S ′ (α,β) ∈ Sp(n) such that ℓ (α,β) = S (α,β) ℓ X = S ′ (α,β) ℓ P (notice that we can take S ′ (α,β) = S (α,β) J).

4 2 Σ − 1 XX Σ − 1 P 1 4 2 .
P ≥ I n×n or, equivalently Σ XX Σ P P ≥ But this is the generalized Heisenberg inequality of Proposition 5. (ii) We have X = P if and only if AB = I n×n .Example 9 Let us illustrate the result above in the case n = 1.Here Σ XX = σ xx > 0, Σ P P = σ pp > 0, and Σ XP = Σ P X = σ xp and the covariance ellipse is

A
T C, B T D symmetric, and A T D − C T B = I (72) AB T , CD T symmetric, and AD T − BC T = I. the n × n blocks E and F are given by E = (AA T + BB T ) −1/2 A , F = (AA T + BB T ) −1/2 B.
and only if / 2σ pp ≤ 2D/σ pp which is equivalent to D ≥ 1 4 2 .More generally, if ℓ a : p = ax for any a ∈ R then Ω Σ ∩ ℓ a and Ω ,ω Σ ∩ ℓ a are determined by the inequalities