aa r X i v : . [ qu a n t - ph ] S e p Homology of Generic Stabilizer States
Klaus Wirthm¨uller
Fachbereich Mathematik der Technischen Universit¨at Kaiserslautern,Postfach 3049, 67653 Kaiserslautern, Germany [email protected]
Abstract.
This work is concerned with multi-party stabilizer states inthe sense of quantum information theory. We investigate the homologicalinvariants for states of which each party holds a large equal number N of quantum bits. We show that in many cases there is a generic expectedvalue of the invariants : for large N it is approximated with arbitrarilyhigh probability if a stabilizer state is chosen at random. The resultsuggests that typical entanglement of stabilizer states involves but thesets comprising just over one half of the parties.Our main tool is the Bruhat decomposition from the theory of finiteChevalley groups. Entangled multipartite states are central to quantum information theory. As areasonably comprehensive understanding of such states has turned out to belargely illusory, interest has been focused on partial aspects, in particular onspecial classes of states that are easier to study. Among such classes that ofstabilizer states has attracted attention due to the fact that it includes highlyentangled states but, on the other hand, lends itself to a much simpler mathe-matical treatment than is possible for general quantum states.The question addressed in this note has two roots. In [12] a series of invariantsof stabilizer states has been introduced in an attempt to classify qualitativeaspects of entanglement. The nature and construction of these invariants, whichrely on ideas from algebraic geometry, suggest an uneven distribution of theirvalues : while a few special states are expected to have equally special invariants,the large majority is thought to have a common generic value of them.This view is supported by evidence from the first non-trivial invariant H ( | ψ i ).Its meaning is quite explicit since it just measures the number of all-party GHZstates that may be extracted from | ψ i [1,12]. Here the other root of our notecomes in : the work [10] includes a result according to which stabilizer states fromwhich a significant number of GHZ states can be extracted become increasinglyscarce as the number of qubits per party increases. This may be restated say-ing that the suitably normalised generic value of H ( | ψ i ) tends to zero in thissituation (where each party holds the same large number of qubits).Stabilizer states were first described in [4] in the context of error-correctingquantum codes, and the formalism has been repeatedly presented since [2,5,6,7,9,12]. The main point of interest here is that pure stabilizer states in an l -qubit Hilbertspace H are parametrised by the Lagrangian subspaces L of a 2 l -dimensionalsymplectic vector space G over the field F . More precisely to each Lagrangianthere correspond 2 l distinct but equivalent stabilizer states. Thus probabilitiesinvolving properties of stabilizer states may be expressed using the finite set LGr ( G ) of all such Lagrangians as a sample space with Laplace probabilitymeasure P . For our investigation of asymptotics all dimensions in this set-upwill have to be scaled by a large common factor N , since each of the l partieswill be given not one but a large number N of qubits. The homological invariantsof these stabilizer states are then expected to grow of order N too because theybehave additively with respect to direct sums [12].To geometers LGr ( G ) is well-known as a Grassmannian, which naturallycarries the structure of a projective algebraic variety over the field F . The clas-sification of stabilizer states by their homological invariants yields a partitionof this variety into strata which in turn are quasi-projective varieties. Whilestandard arguments based on the notions of dimension, multiplicity, and moregenerally the Hilbert polynomial do give asymptotic results concerning the num-ber of points in such varieties they seem to be less of a suitable tool here sincethe kind of asymptotics is not the right one. The reason is that the context forcesus to work over a fixed finite field — the prime field F being the most importantone — and precludes the possibility of passing to finite extensions, let alone thealgebraic closure. Nevertheless in the search for generic properties of states wefound that the notion of dimension has some predictive value at least.From yet another point of view the variety LGr ( G ) is a homogeneous spaceof the symplectic group Sp(2 l, F ), which is a Chevalley group of type C l overthe field F . This fact makes notions from group theory available which includea method to count the points of LGr ( G ) via the Bruhat decomposition intoSchubert cells [3,11] — indeed this would work over an arbitrary finite field andit determines the Weil zeta function of a Grassmannian. It also is the main toolused in this paper as the relevant strata of LGr ( G ) are related to the Schubertcells.The investigation becomes somewhat simpler if the Lagrange Grassmannian LGr ( G ) is replaced by the ordinary one, Gr ( G, k ), which parametrises all k -dimensional linear subspaces of the l -dimensional vector space G over F . Fromthe group theoretic point of view we would then be dealing with SL( l, F ), aChevalley group of type A l − rather than C l . In the context of stabilizer statesthis situation has its own interest as it arises with CSS states : the CSS statewith parameter L ∈ Gr ( G, k ) corresponds to the Lagrangian L ⊕ L ⊥ ⊂ G ⊕ G in LGr ( G ⊕ G ) where G ⊕ G is equipped with the standard symplectic form. Wetherefore treat this case first before proceeding to the Lagrangian one.In Section 2 we state qualitative versions of our results as Theorems 1 and 2.The first refers to the non-Lagrangian case of Nk -dimensional subspaces L ⊂ F Nl just mentioned. It roughly states that with arbitrarily high probability, for a large number N of qubits held by each party, nearly all homology of L is concentratedin the single group H l − k +1 L , and its normalised rank is close to a predeterminednumber that can be calculated from l and k . Similarly Theorem 2 states thatfor large N , with arbitrarily high probability the homology of a Lagrangian L ⊂ F Nl is nearly all concentrated in H l +1 L , and its normalised rank is closeto a predetermined number.The proofs are compiled in Section 3, and include much more precise quan-titative asymptotic results. We will work over a fixed finite field F throughout. While only in the case ofthe prime fields F p there is a direct correspondence between Lagrangians andstabilizer states (in a space of qu- p -its), we require no such restriction and allowthat F has an arbitrary number q = p e of elements. We also fix an integer l ≥ N ∈ N will be variable ; from the point of view of stabilizer states over F we would be dealing with l parties controlling N qudits each.The homology H j ( | ψ i ) of the CSS state | ψ i is H j ( L ⊕ L ⊥ ) by definition,and since homology is additive by [12] Lemma 2 we need only study H j L for anarbitrary linear subspace L ⊂ F Nl . For the same reason we assume the latter ofdimension Nk with fixed k ∈ N , so that dim H j L will be of order N too. We thuslet G k ( N ) := Gr ( F Nl , N k ) denote the Grassmannian variety of Nk -dimensionallinear subspaces of the standard vector space F Nl , and consider G k ( N ) as thesample space of a probability space with Laplacian probability measure P .Recall [12] that the construction of the homology groups H j L starts from achain complex C L δ / / C L δ / / · · · δ j − / / C j L δ j / / C j +1 L δ j +1 / / · · · (1)where the chain group C j L is spanned by all vectors of L that are local to somesubset of j parties. The group H j L then is the j -th cohomology group of thatcomplex, that is H j L = ker δ j / image δ j − for each j > Theorem 1
Let L ∈ G k ( N ) be a sample.1. If j < l − k then lim N →∞ P ( C j L = 0) = 0 .2. If j < l − k or j > l − k +2 then lim N →∞ P ( H j L = 0) = 0 .Now let any real ǫ > be given. Then3. lim N →∞ P ( N dim C l − k L ≥ ǫ ) = 0 , and4. lim N →∞ P ( N dim H l − k L + N dim H l − k +2 L ≥ ǫ ) = 0 .Finally lim N →∞ P (cid:0) N dim H j L ≥ ǫ (cid:1) = 0 if j = l − k +1 , and lim N →∞ P (cid:0)(cid:12)(cid:12) N dim H l − k +1 L − χ k (cid:12)(cid:12) ≥ ǫ (cid:1) = 0 with χ = 0 and χ k = (cid:18) l − k − (cid:19) for k > .Proof. Statements 1 and 3 will be immediate corollaries to the more detailedresults of Proposition 7. Furthermore C j L = 0 trivially implies H j L = 0, andby the duality theorem [12] Theorem 5 the group H j L is the dual of H l − j +2 L ⊥ :thus statement 2 follows from the first, and statement 4 from the third. Finallyfor each j -dimensional coordinate subspace of F l the intersection of the corre-sponding coordinate subspace of F Nl with a generic L has dimension ( j + k − l ) N if j + k ≥ l , again by Proposition 7. Therefore P lj = l − k +1 ( − j (cid:0) lj (cid:1) ( j + k − l ) isthe expected normalised Euler number of the chain complex (1), and a fortiorithat of its cohomology. Using standard identities for binomial coefficients [8] thealternating sum is evaluated to ( − l − k +1 χ k .We now turn to general pure stabilizer states where L ⊂ F Nl may be anarbitrary Lagrangian subspace of F Nl . We let L ( N ) := LGr ( N l ) be the Grass-mannian variety of these subspaces.
Theorem 2
Let L ∈ L ( N ) be a sample.1. If j < l/ then lim N →∞ P ( C j L = 0) = 0 .2. If j < l/ or j > l/ then lim N →∞ P ( H j L = 0) = 0 .Let any ǫ > be given and assume that l is even. Then3. lim N →∞ P ( N dim C l/ L ≥ ǫ ) = 0 , and4. lim N →∞ P ( N dim H l/ L + N dim H l/ L ≥ ǫ ) = 0 .Under the same assumptions let χ = 2 χ l/ = 2 (cid:18) l − l/ − (cid:19) be the absolute valueof the expected normalised Euler number of L . Then5. lim N →∞ P (cid:0) N dim H j L ≥ ǫ (cid:1) = 0 if j = l/ , and lim N →∞ P (cid:16)(cid:12)(cid:12) N dim H l/ L − χ (cid:12)(cid:12) ≥ ǫ (cid:17) = 0 . Proof.
This follows from Proposition 12 in the same way as Theorem 1 followsfrom Proposition 7.
Remarks.
We do not know whether an analogue of statement 5 holds for odddimension l . — As mentioned in the introduction the rank of H ( | ψ i ) is thenumber of all-party GHZ states that may be extracted from | ψ i . By statement l > N , with arbitrarilyhigh probability. In fact in the binary case q = 2 the more detailed results ofProposition 12 imply the bound of (1 + log 2) 2 − ( l − N for the probability thata GHZ state may be extracted from | ψ i . In the case of l = 4 the number ofextractable all-party GHZ states is still likely to be arbitrarily small compared to N , by the third statement of Theorem 2. Again Proposition 12 explicitly boundsthe probability of the event that Nǫ or more GHZ states may be extracted from | ψ i . In the binary case the bound is proportional to 2 − N ǫ and thus improvesthat given in [10] for this particular question, which is exponential in N ratherthan N .Theorem 2 may be read as a statement about the typical entanglement ofan l -party stabilizer state : all such entanglement comes from the entanglementof the ( l/ L, L ⊥ ⊂ F l ofcomplementary dimension. Remarkably, in the symmetric case dim L = l/ X and Z generators. We begin by recalling the well-known partition of the Grassmannian Gr ( l, k )into Schubert cells. The latter are indexed by right congruence classes of per-mutations σ ∈ Sym l / (Sym k × Sym l − k ). More precisely, given any σ ∈ Sym l the corresponding Schubert cell C σ comprises those subspaces of F l which arespanned by the columns of some unipotent upper triangular matrix u ∈ GL( l, F ),with column index in { σ , σ , . . . , σk } . Explicitly, if σ is chosen in its congruenceclass such that σ < σ < · · · < σk then a subspace belongs to C σ if and only ifit is the image subspace of some matrix ∗ ∗ · · · · · · ∗ ... ... ... ∗ ∗ · · · · · · ∗ · · · · · · ∗ · · · · · · ∗ ... ... ∗ · · · · · · ∗ · · · · ∗ . . . ... ∗ ∈ Mat ( l × k, F ) where the special rows are those with indices σ , . . . , σk , and the entries at thestarred positions are arbitrary. It follows that C σ is an affine space of dimension d σ = (cid:12)(cid:12)(cid:8) ( a, b ) ∈ { , . . . , k }×{ k +1 , . . . , l } (cid:12)(cid:12) σa > σb (cid:9)(cid:12)(cid:12) . From this fact the number of points G lk := | Gr ( l, k ) | is easily computed : Proposition 3 G lk = Q li = l − k +1 ( q i − Q ki =1 ( q i − .Proof. Clearly G l = G ll = 1. For 0 < k < l the set σ { , . . . , k } may or may notcontain l , and each case corresponds to a summand in the recurrence relation G lk = q l − k G l − ,k − + G l − ,k . The formula now follows by induction.We let 0 = F ⊂ F ⊂ · · · ⊂ F l − ⊂ F l be the standard flag formed by thecoordinate subspaces with vanishing last components. Then for any L ∈ C σ wehave dim L ∩ F j = k − (cid:12)(cid:12)(cid:8) a ∈ { , . . . , k } (cid:12)(cid:12) σa > j (cid:9)(cid:12)(cid:12) . (2)In particular the collection of these dimensions is an alternative way to charac-terise the cell C σ .If j + k ≤ l then the intersection L ∩ F j generically is zero, and we determine F lkj := (cid:12)(cid:12)(cid:8) L ∈ Gr ( l, k ) (cid:12)(cid:12) L ∩ F j = 0 (cid:9)(cid:12)(cid:12) (3)in this case. Proposition 4
For j + k ≤ l one has F lkj = q kj · Q l − ji = l − k − j +1 ( q i − Q ki =1 ( q i − .Proof. From (2) we know that L ∩ F j = 0 occurs if and only if L belongs to acell C σ such that σ sends { , . . . , k } into { j + 1 , . . . , l } . For 0 < k < l the set σ { , . . . , k } may or may not contain l : this gives the recurrence relation F lkj = q l − k F l − ,k − ,j + F l − ,k,j , which together with F l j = 1 and F lk = G lk allows to prove the formula byinduction. For j + k ≥ l the dimension of L ∩ F j is at least (and generically equal to) j + k − l , and we extend (3) to this case putting F lkj := (cid:12)(cid:12)(cid:8) L ∈ Gr ( l, k ) (cid:12)(cid:12) dim L ∩ F j = j + k − l (cid:9)(cid:12)(cid:12) ( j + k ≥ l ) . The analogue of Proposition 4 is
Proposition 5
For j + k ≥ l one has F lkj = q ( l − j )( l − k ) · Q ji = j + k − l +1 ( q i − Q l − ki =1 ( q i − .Proof. Passing to orthogonal complements we see thatdim L ∩ F j = j + k − l if and only if L ⊥ ∩ ( F j ) ⊥ = 0 . Still in the case j + k ≥ l we now let more generally s be an integer with j + k − l ≤ s ≤ min { j, k } , and calculate the cardinality H lkjs := (cid:12)(cid:12)(cid:8) L ∈ Gr ( l, k ) (cid:12)(cid:12) dim L ∩ F j = s (cid:9)(cid:12)(cid:12) . Proposition 6 H lkjs = G js · F l − s,k − s,j − s for ≤ j + k − l ≤ s ≤ min { j, k } .Proof. The first factor counts the possible intersections L ′ := L ∩ F j , and thesecond the possibilities to realise a fixed L ′ : to this end consider the factor spaces L/L ′ ∈ Gr ( L/L ′ , k − s ) with L/L ′ ∩ F j /L ′ = 0.We are now ready to study the asymptotic behaviour of the entities intro-duced so far. From the point of view of algebraic geometry the spaces L ∈ Gr ( l, k ) that have a non-generic, that is larger than expected intersection withsome F j are points of a proper algebraic subvariety of Gr ( l, k ). This implies thatthe proportion F lkj : G lk approaches 1 when the ground field F q is replacedby a large finite extension field. While in our context the ground field is fixed,and rather the dimensions l , k , and j are scaled by a large common factor N the proportion in question still behaves by and large the same way. The preciseresult is this : Proposition 7
Let l , k , and j be fixed.1. If j + k = l then ≤ − F Nl,Nk,Nj G Nl,Nk ≤ | log(1 − q − ) | − q − · q −| l − k − j | N for all N ∈ N .
2. If j + k = l let any real ǫ > be given and put S N := (cid:12)(cid:12)(cid:8) L ∈ Gr ( N l, N k ) (cid:12)(cid:12) dim L ∩ F Nj ≥ N ǫ (cid:9)(cid:12)(cid:12) . Then S N G Nl,Nk ≤ e qq − | log(1 − q − ) | · ( q − ( q − − q · q − N ǫ for all N ∈ N .Proof. We first assume that j + k < l and put m = l − k − j . The expressionsfor F lkj and G lk from Propositions 3 and 4 have the same degree as rationalfunctions in q , and the relevant quotient is Q N := F Nl,Nk,Nj G Nl,Nk = Q Nl − Nji = Nl − Nk − Nj +1 (1 − q − i ) Q Nki =1 (1 − q − i ) · Q Nki =1 (1 − q − i ) Q Nli = Nl − Nk +1 (1 − q − i )= Q Nl − Nji = Nm +1 (1 − q − i ) Q Nli = Nl − Nk +1 (1 − q − i ) . Giving away the denominator we abbreviate u = − q · log(1 − q − ), and using m > Q N ≥ Nl − Nj X i = Nm +1 log(1 − q − i ) ≥ − u · Nl − Nj X i = Nm +1 q − i ≥ − u · ∞ X i = Nm +1 q − i = − u · q − Nm − − q − (4)for all N . We conclude1 − Q N ≤ − exp (cid:0) − uq − q − mN (cid:1) ≤ uq − q − mN and need only substitute the value of u .The case of j + k > l is similar, and there remains that of j + k = l . HereProposition 6 supplies the quotient H Nl,Nk,Nj,σ H Nl,Nk,Nj,σ − = q σ − Nl − · ( q Nk − σ +1 − q Nj − σ +1 − q σ − for each σ >
0. In view of H Nl,Nk,Nj, = F Nl,Nk,Nj we obtain the estimate H Nl,Nk,Nj,s F Nl,Nk,Nj ≤ q s ( q − ( q − · · · ( q s − and, summing up, min { Nj,Nk } X σ = s H Nl,Nk,Nj,σ F Nl,Nk,Nj ≤ q s ( q − ( q − · · · ( q s − · ∞ X i =0 q i ( q s +1 − i = q s ( q − ( q − · · · ( q s − · ( q s +1 − ( q s +1 − − q for all s >
0. In the resulting estimate X σ ≥ s H Nl,Nk,Nj,σ G Nl,Nk ≤ q s ( q − ( q − · · · ( q s − · ( q − ( q − − q = 1(1 − q − ) (1 − q − ) · · · (1 − q − s ) · ( q − ( q − − q · q − s the product (1 − q − ) · · · · · (1 − q − s ) is bounded below by e − u/ ( q − as in (4),so that X σ ≥ s H Nl,Nk,Nj,σ G Nl,Nk ≤ e u/ ( q − · ( q − ( q − − q · q − s . It only remains to substitute s := ⌈ N ǫ ⌉ . Remark.
Statement 1 of the proposition definitely does not extend to the case j + k = l , for the quotient F Nl,Nk,Nj G Nl,Nk = Nk Y i =1 − q − i − q − Nj − i clearly cannot converge to one for N → ∞ if j is positive.We now turn to the symplectic vector space F l and let LGr (2 l ) denote theGrassmannian of its Lagrangian subspaces. As mentioned in the introduction LGr (2 l ) is a homogeneous space of the symplectic group Sp(2 l, F ), which is aChevalley group of type C l .Writing e a for the a -th standard column vector in R l the vectors e a − e b for a < b , and e a + e b for a ≤ b form a system of positive roots for Sp(2 l, F ). The corresponding Borel subgroup B consists of all matrices g gs g t ) − ∈ Mat (2 l × l, F ) with g ∈ B ′ and s ∈ Sym ( l, F ) (5) where B ′ ⊂ GL( l, F ) is the ordinary Borel group of upper triangular matrices,and Sym ( l, F ) the space of symmetric matrices of that size. The elements of theWeyl group W = {± } l Sym l act as permutations σ of the set {± , . . . , ± l } withthe property that σ ( − a ) = − σa for all a .The Schubert cells of LGr (2 l ) are indexed by the right congruence classes in W/ Sym l ; they are in one-to-one correspondence with the normalised permuta-tions σ — those which satisfy the condition σ < · · · < σl and thus are shortest in their congruence classes (recall that σ acts on the set {± , . . . , ± l } of signed integers). The cells are simply characterised by the set (cid:8) a ∈ σ { , . . . , l } (cid:12)(cid:12) a < (cid:9) , and the length d σ = (cid:12)(cid:12)(cid:8) ( a, b ) ∈ { , . . . , l } (cid:12)(cid:12) a ≤ b and σa + σb < (cid:9)(cid:12)(cid:12) (6)is the dimension of the Schubert cell C σ labelled by the class of σ . Of course C σ may be described explicitly, as follows. Define n by σn < < σ ( n +1) and put p = l − n . Then a subspace of F l belongs to C σ if and only if it is the imagesubspace of a (unique) matrix ∗ ∗ · · · · · · ∗ ◦ · · · · · · · · · ◦ · · · · · · · · · · · · · · · ∗ ∗ · · · ∗ ◦ · · · · · · · · · ◦ · · · · · · · · · · · · ∗ ◦ · · · · · · · · · ◦ . . . ... ... ... ∗ ◦ · · · · · · · · · ◦ · · · · · · · · · ◦ · · · · · · · · · ◦ ∗ ... . . . ∗ · · · · ∗ · · · · · · ∗ · · · · · · ∗ · · · · · · ∗ ∗ ∈ Mat (2 l × ( p + n ) , F ) (7)where steps occur at the row indices l +1 − σb with σb > l +1 −| σa | with σa < , and where the unspecified entries are subject to the following conditions. Allstarred entries in either the upper left hand or the lower right hand submatrixmay be chosen arbitrarily, and the other half are then determined by the con-dition that these submatrices span mutually orthogonal subspaces of F l . The circled entries are free within a symmetry condition dependent on the starredones as in (5).It is well-known how to count Lagrangians : Proposition 8
There are exactly L l = Q li =1 ( q i + 1) Lagrangian subspaces of F l , and more generally the number of k -dimensional isotropic subspaces is L lk = Q li = l − k +1 ( q i − Q ki =1 ( q i − for k ≤ l .Proof. For l > σ { , . . . , l } contains either − l or l , so (6) gives therecursive formula L l = q l L l − + L l − for L l .For general k ≤ l we make two observations : given an isotropic subspace K ⊂ F l of dimension k the Lagrangians of F l that contain K are in one-to-onecorrespondence with Lagrangian subspaces of K ⊥ /K , while on the other handgiven a Lagrangian L ⊂ F l every subspace of L is isotropic. Thus counting pairs( L, K ) we obtain the identity L l − k · L lk = L l · G lk , and the result follows using Proposition 3.For a given integer j with 0 ≤ j ≤ l we let E j ⊕ F j ⊂ F l ⊕ F l denote the sum of the subspaces spanned by the first j coordinates in F l ⊕ ⊕ F l . In order to compute K lj := (cid:12)(cid:12)(cid:8) L ∈ LGr (2 l ) (cid:12)(cid:12) L ∩ E j = 0 (cid:9)(cid:12)(cid:12) we read off from (7) thatdim L ∩ E j = (cid:12)(cid:12)(cid:8) b ∈ { , . . . , l } (cid:12)(cid:12) σb > l − j (cid:9)(cid:12)(cid:12) , so that K lj counts the cases with σb ≤ l − j for all b , or equivalently σa = a − − l for a = 1 , . . . , j. (8)In particular assuming l and j positive, the condition can only be met if − l rather than l belongs to σ { , . . . , l } , so we have K lj = q l K l − ,j − and conclude : Proposition 9 K lj = q j (2 l − j +1) / · l − j Y i =1 ( q i + 1) . We now compute J lj := (cid:12)(cid:12)(cid:8) L ∈ LGr (2 l ) (cid:12)(cid:12) L ∩ ( E j + F j ) = 0 (cid:9)(cid:12)(cid:12) . Proposition 10
Assume j ≤ l and let σ be a normalised permutation with σb ≤ l − j for all b . The proportion (cid:12)(cid:12)(cid:8) L ∈ C σ (cid:12)(cid:12) L ∩ ( E j + F j ) = 0 (cid:9)(cid:12)(cid:12) : (cid:12)(cid:12)(cid:8) L ∈ C σ (cid:12)(cid:12) L ∩ E j = 0 (cid:9)(cid:12)(cid:12) is the same for all such σ and equal to Q l − ji = l − j +1 (1 − q − i ) . In particular J lj K lj = l − j Y i = l − j +1 (1 − q − i ) . Proof.
Fix σ as in the hypothesis and let L ∈ C σ be the Lagrangian with span-ning matrix (7). In view of (8) only the columns with index p +1 , . . . , p + j cancontribute to the intersection L ∩ ( E j + F j ), and the submatrix comprising thesecolumns has the form e ′ e ′′ f ′′ ∈ Mat (2 l × j, F ) (9)with blocks e ′ and the unit matrix 1 of height j , and e ′′ , f ′′ of height l − j .Furthermore, e ′′ has p zero rows (with indices l +1 − σb where n < b ≤ l ) and f ′′ has n − j zero rows (with indices 2 l +1 −| σa | where j < a ≤ n , all indices countedwith respect to the full matrix, on a scale from 1 to 2 l ). The remaining rows of e ′′ and f ′′ are free and together form an ( l − j ) × j matrix g which determineswhether L ∩ ( E j + F j ) = 0 : this occurs if and only if rk g = j .Thus the proportion in question is that of the matrices of full rank amongstall ( l − j ) × j matrices, and therefore equal to | GL( l − j, F ) | q j ( l − j ) · | GL( l − j, F ) | : q j ( l − j ) = q − j (2 l − j ) · q ( l − j ) · Q l − ji =1 (1 − q − i ) q ( l − j ) · Q l − ii =1 (1 − q − i )= Q l − ji = l − j +1 (1 − q − i ) . We extend the definition of J lj to the case of 2 j ≥ l in terms of the moregeneral M ljs := (cid:12)(cid:12)(cid:8) L ∈ LGr (2 l ) (cid:12)(cid:12) dim L ∩ ( E j + F j ) = s (cid:9)(cid:12)(cid:12) by putting J lj = M lj, j − l if 2 j ≥ l . Corollary 11
We have J lj = q j · Q l − ji =1 ( q i − Q l − ji =1 ( q i − if j ≤ l, and more generally M ljs = q ( j − s ) · Q ji = j − s +1 ( q i − · Q l − ji =1 ( q i − Q si =1 ( q i − · Q l + s − ji =1 ( q i − for max { , j − l } ≤ s ≤ j. Proof.
The first formula results from Propositions 9 and 10, and it implies thesecond by the following reasoning.For the Lagrangians L to be counted the intersection L ′ := L ∩ ( E j + F j ) is an s -dimensional isotropic subspace of E j ⊕ F j , and Proposition 8 gives the number L js of possible L ′ . Now fixing L ′ we determine the number of Lagrangians L with L ∩ ( E j + F j ) = L ′ , and for this purpose may assume L ′ = E s . Such L intersect F s trivially, and therefore, under the projection E l ⊕ F l → E l /E s ⊕ F l /F s theycorrespond to Lagrangians L ⊂ E l /E s ⊕ F l /F s with L ∩ ( E j /E s ⊕ F j /F s ) = 0.Thus there are exactly J l − s,j − s such L for any given L ′ , and we obtain theformula M ljs = L js · J l − s,j − s and thereby the result.We finally state the asymptotic properties of the data we have collected. Proposition 12
Let l and j be fixed.1. If j = l then ≤ − J Nl,Nj L Nl ≤ q − + | log(1 − q − ) | − q − · q −| l − j | N for all N ∈ N .2. Assume that j ≤ l and, for any given ǫ > put T N := (cid:12)(cid:12)(cid:8) L ∈ LGr (2 N l ) (cid:12)(cid:12) dim L ∩ ( E Nj ⊕ F Nj ) ≥ N ǫ (cid:9)(cid:12)(cid:12) . Then T N L Nl ≤ e qq − | log(1 − q − ) | · ( q − ( q − − q · q − N ǫ for all N ∈ N .Proof. Quite analogous to that of Proposition 7.
References