On Orthogonal Projections of Symplectic balls

We study the orthogonal projections of symplectic balls in $\mathbb{R}^{2n}$ on complex subspaces. In particular we show that these projections are themselves symplectic balls under a certain complexity assumption. Our main result is a refinement of a recent very interesting result of Abbondandolo and Matveyev extending the linear version of Gromov's non-squeezing theorem. We use a conceptually simpler approach where the Schur complement of a matrix plays a central role.

dx n be the standard symplectic form on R 2n ≡ R n × R n ; we call symplectic ball the image of the ball B 2n (z 0 , R) = {z ∈ R 2n : |z − z 0 | ≤ R} by a symplectic automorphism S ∈ Sp(n) (the symplectic group of (R 2n , σ)).As a consequence of Gromov's non-squeezing theorem [7,9] the orthogonal projection of a symplectic ball S(B 2n (z 0 , R)) on any two-dimensional symplectic subspace of (R 2n , σ) has area at least equal to πR2 .Abbondandolo and Matveyev asked in [1] the question whether this result could be generalized to subspaces with higher dimensions.They showed that the orthogonal projection Π V S(B 2n (z 0 , R)) of S(B 2n (z 0 , R)) onto an arbitrary complex symplectic subspace (V, σ |V ) of (R 2n , σ) such that dim V = 2k satisfies (1.1) Vol V Π V S(B 2n (z 0 , R)) ≥ (πR 2 ) k k! where Vol V is the volume element on V. Notice that (πR 2 ) k /k! is the volume of the ball B V (Π V z 0 , R) in V: They moreover proved that equality holds in (1.1) if and only S T V is itself a complex subspace of R 2n .The inequality (1.1) implies the linear version of Gromov's theorem when dim V =2 and conservation of volume by linear symplectomorphisms when V = R 2n .Abbondandolo and and Matveyev proved their results using an ingenuous but complicated argument involving the Wirtinger inequality for 2-forms on Kähler manifolds [3].Results of this type are more subtle and difficult than they might appear at first sight; for instance as Abbondandolo and Matveyev show the inequality (1.1) does not hold when one replaces S by a nonlinear symplectomorphism f .In fact, one can construct examples where Vol V Π V f (B 2n (R)) can become arbitrarily small.They however make an interesting conjecture, to which we will come back at the end of this paper.
1.2.What we will do.We will prove by elementary means a stronger version of (1.1) and of its extension.We will actually prove (Theorem 3.1) that the orthogonal projection of a symplectic ball on a symplectic subspace contains a symplectic ball with the same radius in this subspace, and is itself a symplectic ball when the subspace under consideration is complex.The proof will be done in the particular case where the symplectic space V is of the type R 2n A ⊕0 in which case the symplectic orthogonal V σ is 0⊕R 2n B ; our refinement of (1.1) says that for every S ∈ Sp(n) there exists and z 0,A = Π V z 0 .This will be done using the theory of Schur complements and the notion of symplectic spectrum of a positive definite matrix.Since symplectomorphisms are volume-preserving, (1.3) implies (1.1).It is however a much stronger statement than (1.1) because, given two measurable sets Ω and Ω ′ with the same volume, there does not in general exist a symplectomorphism (let alone a linear one) taking Ω to Ω ′ as soon as the dimension of the symplectic space exceeds two [9].
Note that these results are invariant under phase space translations.We will therefore assume henceforth that z 0 = 0. We finally discuss in section 4 some possible extensions to the non-linear case, pointing out the difficulties.

Preliminaries
In what follows M will be a real 2n × 2n positive definite matrix; we will write M > 0. We denote by J the standard symplectic matrix 0 n I n −I n 0 n .
We have, σ(z, z ′ ) = Jz • z ′ when z = (x, p), z ′ = (x ′ , p ′ ).In this notation the condition S ∈ Sp(n) is equivalent to S T JS = J (or SJS T = J) where S T is the transpose of S.

2.1.
Williamson's symplectic diagonalization.By definition the symplectic spectrum of M is the increasing sequence > 0 where the ±iλ σ j (M ) are the eigenvalues of JM (which are the same as those of the antisymmetric matrix M 1/2 JM 1/2 ).We will use the following property, known in the literature as "Williamson's symplectic diagonalization theorem" [4,9]: there exists S ∈ Sp(n) such that where Λ is the diagonal matrix whose eigenvalues are the numbers λ σ j (M ) (all matrices corresponding here to the standard splitting z = (x, p)).The symplectic spectra of M and M −1 are inverses of each other in the sense that: (2.1) λ σ j (M −1 ) = λ σ n−j (M ) −1 for 1 ≤ j ≤ n.We also have the less obvious property ([4], section 8.3.2) The following simple result characterizing positive semi-definiteness in terms of the symplectic spectrum will be very useful for proving Theorem 3.1: S T S + iJ ≥ 0 and SS T + iJ ≥ 0 since λ σ j (S T S) = λ σ j (SS T ) = 1 for all j (because D = I in view of Williamson's diagonalization result).

Block-matrix partitions and Schur
We denote by Π A (resp.Π B ) the orthogonal projection R 2n −→ R 2n A (resp.R 2n B ).We choose symplectic bases B A , B B in R 2n A and R 2n B and identify linear mappings R 2n −→ R 2n with their matrices in the symplectic basis B = B A ⊕ B B of R 2n .Such a matrix will be written as Similarly, the standard symplectic matrix J will be split as Since M is positive definite and symmetric the upper-left and lower-right blocks in (2.4) are themselves positive-definite and symmetric: M AA > 0 and M BB > 0. In particular the Schur complements are well defined and invertible [13], and the inverse of the matrix M is given by the formula 2.3.Orthogonal projections of ellipsoids in R 2n .We will also need the following general characterization of the orthogonal projection of an ellipsoid on a subspace: We have

8).
Interchanging A and B the orthogonal projection of Ω on R 2n B is similarly given by (2.10)

Orthogonal Projections of Symplectic Balls
3.1.The main result: statement and proof.Let us now prove the main result.We assume again the matrix M is written in block-form (2.4).To simplify notation we also assume that all balls B 2n (z 0 , R) are centered at the origin and set B 2n (0, R) = B 2n (R).The case of a general ball B 2n (z 0 , R) trivially follows using the translation z −→ z + z 0 .(ii) We have if and only if S = S A ⊕ S B for some S B ∈ Sp(n B ), in which case we also have We are going to show that the symplectic eigenvalues λ σ A j (M/M BB ) are all ≤ 1.The inclusion (3.1) will then follow since we have, in view of Williamson's diagonalization result, for some S A ∈ Sp(n A ) and It follows that: To prove that we indeed have we begin by noting that the symplectic eigenvalues λ σ j (M ) of M = (SS T ) −1 are all trivially equal to one, and hence also those of its inverse M −1 : λ σ j (M −1 ) = 1 for 1 ≤ j ≤ n.In view of Lemma 2.1 the Hermitian matrix M −1 + iJ is positive semidefinite: where which is in turn equivalent to the set of conditions Multiplying the identity (3.12) on the right by (M/M AA )X T and using (3.15), we obtain Since (M/M AA ) > 0, this is possible if and only if X = 0. Finally, from (3.13) we conclude that (M/M AA ) −1 ∈ Sp(n B ).Moreover, since (M/M AA ) −1 is symmetric and positive definite, there exists S B ∈ Sp(n B ), such that (M/M AA ) −1 = S B S T B .Altogether hence S = S A ⊕ S B , which concludes the proof.
We remark that the proof above actually provides the means to calculate explicitly the symplectic automorphisms S A in (3.1).Recapitulating, it is constructed as follows: given S ∈ Sp(n) calculate and then obtain the Schur complement (2.5) The matrix S A is then obtained from (3.14) (observe that S A is only defined up to a symplectic rotation, but this ambiguity is irrelevant since B 2n A (R) is rotationally invariant).
3.2.Discussion and extension.Theorem 3.1 implies de facto the Abbondandolo and Matveyev result (1.1) since formula (3.1) has the immediate consequence that (3.17) Abbondandolo and Matveyev's [1] however prove these relations for projections on a general complex symplectic subspace V of (R 2n , σ) (that is such that JV = V) and they show that the equality in (1.1) holds if and only if the subspace S T V is complex, that is if JS T V =S T V. Let us check that these conditions are satisfied when V = R 2n A ⊕ 0. We first note that

Perspectives
A first natural question that arises is whether Theorem 3.1 can be extended in some way to non-linear symplectic mappings, that is to general symplectomorphisms of (R 2n , σ).The first answer is that there are formidable roadblocks to the passage from the linear to the nonlinear case, as shortly mentioned in the Introduction.For instance, Abbondandolo and Matveyev [1] show, elaborating on ideas of Guth [8], that for every ε > 0 one can find a symplectomorphism f of (R 2n , σ) defined near B 2n (0, 1) such that Vol(Π V f (B 2n (0, 1)) < ε.They however speculate in [1] that their projection result might still hold true when the linear symplectic automorphism S ∈ Sp(n) is replaced with a symplectomorphism f of (R 2n , σ) close to a linear one.It would be interesting to apply our methods to tackle this difficult problem.
Also, Theorem 3.1 could be used to shed some light on packing problems (see the review [12] by Schlenk) which form a notoriously difficult area of symplectic topology.
Given the partitioning of R 2n = R 2n A ⊕ R 2n B it seems natural to expect some connection between orthogonal projections of symplectic balls and the separability/entanglement problem in quantum mechanics [6,10,11].We intend to address this problem in a future work.