Freeness and the Transpose

Transcription

1 Freeness and the Transpose Jamie Mingo (Queen s University) (joint work with Mihai Popa and Roland Speicher) ICM Satellite Conference on Operator Algebras and Applications Cheongpung, August 8, 04 / 6

2 Kesten s Law I UN is a N N Haar distributed random unitary matrix, distribution of U + U converges to arcsine law. 0.5 U0 + U0 sampled 5000 times Ut + U sampled U0 + U times, (curve = arcsine ) Ut U00 + U U00 sampled 000 times / 6

3 Complex Wigner Example X = (xij ), xij = (aij + bij )/, aij and bij N independent non-gaussian (in this case a sum of independent centred uniform distributions on [, ] with variance ) I X X00 + X )t X00 + (X00 The dotted line shows the free additive convolution of semi-circle with Marchenko-Pastur (c = ). This is strictly a complex phenomena as with the real Wigner we would have t. XN = XN 3 / 6

4 Partial Transpose, Aubrun (00) Motivated by Quantum Information Theory, we investigate the following question: what does the spectrum of A Γ look like? As we will see, the partial transposition dramatically changes the spectrum: the empirical eigenvalue distribution of A Γ is no longer close to a Marchenko-Pastur distribution, but to a shifted semicircular distribution! This is our main theorem. 4 / 6

5 Wishart Random Matrices (Gaussian case) Suppose G,..., G d are d p random matrices where G i = (g (i) jk ) jk and g (i) jk are complex Gaussian random variables with mean 0 and (complex) variance, i.e. E( g (i) jk ) =. Moreover suppose that the random variables {g (i) jk } i,j,k are independent. W = p G. G d ) ( G G d = (G i G j ) ij is a d d d d Wishart matrix. We write W = (W ij ) ij as d d block matrix with each entry the d d matrix G i G j. 5 / 6

6 Partial Transposes G i a d p matrix W ij = d d G i G j, a d d matrix, W = (W ij ) ij is a d d block matrix with entries W ij W T = (W T ji ) ij is the full transpose W Γ = (W ji ) ij is the left partial transpose W Γ = (W T ij ) ij is the right partial transpose we assume that d d p c and 0 < c < eigenvalue distributions of W and W T converge to Marchenko-Pastur with parameter c Γ eigenvalues of W and W Γ converge to a shifted semi-circular with mean c and variance c (Aubrun) thm: (m and popa) the family {W, W T, W Γ, W } is asymptotically free Γ 6 / 6

7 moments of partial transpose (W Γ ) ij = W ji Tr((W Γ ) n ) = = i,...,i n Tr((W Γ ) i i (W Γ ) in i ) Tr((W) i i (W) i i n ) i,...,i n = (W (ii) ) j j (W (iin) ) jn j i,...,i n j,...,j n = (G i G i ) j j (G i G i n ) jn j i,...,i n j,...,j n = (d d ) n g i j t g (i ) j t g i jn t n g (i n) j t n i,...,i n j,...,j n t,...,t n E(tr((W Γ ) n )) = ( d #(σ)+#(γσ ) (n+) d #(σ)+#(γσ) (n+) p ) #(σ) d d σ S n 7 / 6

8 Euler s formula #(σ) denotes the number of cycles in the cycle decomposition of σ let γ be the permutation in S n with the one cycle (,,..., n) given a permutation σ S n we always have #(σ) + #(γσ ) n + with equality only if σ is non-crossing an easy argument shows that only when σ = σ can both σ and σ be non-crossing #(σ) + (#(γσ ) + #(γσ)) (n + ) with equality only if σ is non-crossing and only has blocks of size or 8 / 6

9 asymptotic freeness of W, W T, W Γ, and W Γ E(tr((W Γ ) n )) = ( d #(σ)+#(γσ ) (n+) d #(σ)+#(γσ) (n+) p ) #(σ) d d σ S n thus lim d,d E(tr((WΓ ) n )) = c #(σ) = κ π σ NC, (n) π NC(n) hence κ = c, κ = c, and for r 3, κ r = 0, thus the limiting distribution is semi-circular (Aubrun s thm.) a more involved calculation along similar lines shows that mixed cumulants vanish, thus be a theorem of Nica and Speicher, W, W T, W, and W Γ are asymptotically free Γ 9 / 6

10 going Gauss free if X,..., X s are independent Gaussian random variables then E(X i X ik X j X jk ) = {σ S k i σ = j} in the non-gaussian case we replace this formula with graph sums 0 / 6

11 Graph Sums (with Roland Speicher) G = (V, E) is a directed graph, e E runs from s(e) to t(e) T : E M N (C), e T e = (t (e) ij ) is an oriented assignment of matrices to edges (flip edge = transpose matrix) G(T) = i:v [N] e E i t ij j t (e) i t(e) i s(e) where the sum runs over all functions i : V [N] = {,, 3,..., N}; i k = i(k) Examples i T i T j G(T) = Tr(T) = i t ii G(T) = i,j t ij / 6

12 More Examples i G(T) = i,j,k,l k T T j T 3 l t () ij t () jk t(3) jl i j T T T 3 k G(T) = t () ij t () jk t(3) ki i,j,k / 6

13 the question G(T) = i:v [N] e E t (e) i t(e) i s(e) given G = (V, E) find r(g) such that for all N and all T : E M N (C) we have G(T) N r(g) e E T e where T is the operator norm of T M N (C) note that r(g) answer: r(g) depends on the number of leaves of a tree (or union of trees) associated with G 3 / 6

14 some notation (so we can say what r(g) is) let G = (V, E) be a directed graph a cutting edge is one whose removal disconnects the connected component which contains it a subgraph is two-edge connected if it cannot be disconnected by removing a single edge a two-edge connected component is a two-edge connected subgraph not properly contained in another two-edge connected subgraph F(G) is the graph whose vertices are the two-edge connected components of G and whose edges are the cutting edges of G F(G) is a union of trees (= a forest) a leaf of a tree (or forest) is a vertex with only one incident edge 4 / 6

15 The Theorem let (G = (V, E) be a directed graph with F(G) the forest of two-edge connected components if G is connected r(g) = max{, l/} where l is the number of leaves of F(G) in general, r(g) is the sum of the contributions of each connected component i T T j T T T 3 k l r = 3/ r = i j T 3 k 5 / 6

16 Graphs from partitions Let Γ n be the graph with vertices {,,..., n, n} and an edge from k to k. We put the matrix D k on the edge from k to k D (3) 6 D D (, 4, 6) 5 On the left we have Γ 6. We let π be the partition of [6] with blocks {(, 4, 6)()(3)(5)}. The graph on the right is G π. We have G π (D, D, D 3 ) = i,j,k,l d() ij d () jk d(3) () D D D 3 jl and r(g π ) = 3/. (5) 6 / 6

17 Back to Wishart Matrices (Wigner case) Suppose G,..., G d are d p random matrices where G i = (g (i) jk ) jk and g (i) jk are complex random variables with mean 0 and (complex) variance, i.e. E( g (i) jk ) = and E((g (i) jk ) ) = 0. Moreover suppose that the random variables {g (i) jk } i,j,k are independent. A, A 4,..., A n are d d d d constant matrices A, A 3,..., A n are p p constant matrices W k = p G. G d ( A k G G d ) = (G i A k G j ) ij is a d d d d Wishart matrix. We write W k = (W (k ) ij ) ij as d d block matrix with each entry the d d matrix G i A k G j. 7 / 6

18 given ɛ,..., ɛ n and η,..., η n in {, } we consider ɛ a permutation on the set [±n] = {,,,,..., n, n} given by ɛ(k) = ɛ k k = i ±,...,i ±n Tr((W (ɛ,η) Tr(W (ɛ,η ) A W (ɛ n,η n ) n A n ) ) i i (A ) i i (W (ɛ n,η n ) n ) in i n (A ) i n i ) = Tr((W ) (η) j j (A ) j j (W n ) (η n) j n j n (A ) j n j ) j ±,...,j ±n = ((W ) (η) j j ) r r (A ) (j j ) r r ((W n ) (η n) j n j n ) rn r n (A n ) (j nj ) r n r j r ±,...,r ±n = (W ) (jj ) s s (A ) (j j ) s s (W n ) (j nj n ) s n s n (A n ) (j nj ) s n s s ±,...,s ±n j ( ) where j(k) = i(ɛ(k)), ( ) where s(k) = r(η(k)) 8 / 6

19 Continuing to expand we get So = p n j,st = p n Tr(W (ɛ,η ) A W (ɛ n,η n ) n A n ) g (j ) s t a () t t g (j ) s t a (j j ),t s g (j n) j,st Tr(W (ɛ,η ) A W (ɛ n,η n ) n A n ) s n t n a (n ) t n t n g (j n) s n t n a (j nj ) n,t n s E(g (j ) s t g (j ) s t g (j n) s n t n g (j n) s n t n ) a () t t (j nj ) n,t n s Then we write the mixed moment E(g (j ) s t g (j n) s n t n ) as a sum of cumulants E(g (j ) s t g (j ) s t g (j n) s n t n g (j n) s n t n ) = k π (g (j ) s t, g (j ) s t,..., g (j n) s n t n, g (j n) s n t n ) π P(n) 9 / 6

20 we will have k π (g (j ) s t, g (j ) s t,..., g (j n) s n t n, g (j n) s n t n ) = 0 unless for each block of π all of the matrix entries are the same (or their conjugate); this gives a condition on the indices j, s, t which gives us our graph sum. Thus = p n j,st = p n E(Tr(W (ɛ,η ) A W (ɛ n,η n ) n A n )) π P(n) = p n π P(n) k π (g (j ) s t,,..., g (j n) s n t n )a () t t (j nj ) n,t n s π P(n) k(π) j,st a () t t (j nj ) n,t n s k(π)g π(a,... A n )G π(a,..., A n ) 0 / 6

21 the two graph sums a (n ) t nt n t n t k t t t t n a () t t a (3) t t a (k ) t k t k t k t s n a (j (n )j n) n,s (n ) s n s (n ) s n s s a (j nj) n,s ns a (j (k )j k ) k,s (k ) s k s k s (k ) a (j j),s s s G π(a,... A n ) is the quotient of Γ n by π and G π(a,..., A n ) is the quotient of Γ n by π / 6

22 let r (π) be the exponent of the graph G π, then G π(a,..., A n ) p r (π) A A n let r (π) be the exponent of the graph G π, then G π(a,..., A n ) (d d ) r (π) A A n so letting r(π) = r (π) + r (π) we have p n G π (A,... A n )G π(a,..., A n ) p r (π) n (d d ) r (π) i A i = ( ) d d r (π) p r(π) n p i A i so, for which partitions is r(π) n? thm: r(π) n with equality only if π is a non-crossing pairing. / 6

23 = p n π NC (n) Tr(W (ɛ,η ) A W (ɛ n,η n ) n A n ) ( k(π)g π(a,... A n )G π(a,..., A n ) + O p) From now on we shall assume that all A s are the identity matrix (of appropriate sizes). So for ( ) a () t t a (j j ),s s a (n ) t n t n a (j nj ) n,t n s 0, j, s, t must satisfy some conditions which we describe in terms of the symmetric group. Let γ = (,,..., n) be the permutation with one cycle. Extend γ to a permutation on [±n] = {,,..., n, n} by setting γ( k) = k (for k > 0). The conditions for ( ) are (after some manipulation) j ɛδγ δγδɛ = j s ɛδγ δγδɛ = s t δ = t 3 / 6

24 In order to have k π (g (j ) s t, g (j ) s t,..., g (j n) s n t n, g (j n) s n t n ) = for a non-crossing pairing π we must have j π = j, s π = s, and t π = t. Note that by our assumption that E((g (i) jk ) ) = 0, π must connect a g (j k) s k t k to a g (j l) s l t l. We then combine this with our previous conditions j ɛδγ δγδɛ = j ( ) s ɛδγ δγδɛ = s t δ = t to get = p n π NC (n) E(Tr(W (ɛ,η ) W (ɛ n,η n ) )) ( k π (g (j ) s t, g (j ) s t,..., g (j n) s n t n, g (j n) s n t n ) + O p) j,s,t ( ) 4 / 6

25 Let f ɛ (π) = #(ɛδγ δγδɛ π) ( means the sup of partitions). Then = π NC (n) E(Tr(W (ɛ,η ) W (ɛ n,η n ) )) ( ) fɛ d (π)( ) fη d ( (π)p #(πδ)+ (f ɛ(π)+f η (π)) n +O p p p) Now when π is non-crossing #(πδ) + (f ɛ(π) + f η (π)) = n and is strictly less otherwise. So = π NC (n) E(tr(W (ɛ,η ) W (ɛ n,η n ) )) ( d p ) fɛ (π) ( d p ) fη (π) + O ( p ). 5 / 6

26 Summing up let σ = πδ restricted to [n] = {,,..., n} so we can write = σ NC(n) E(tr(W (ɛ,η ) W (ɛ n,η n ) )) ( ) fɛ d (σ) ( ) fη (σ) ( ) d + O. p p p The conditions j π = j, s π = s, and t π = t means that ɛ and η are constant on cycles of σ if ɛ = η on a cycle of σ then this cycle can only be of length or σ can only connect W s if both ɛ and η agree, thus W, W, W Γ, and W T are asymptotically free for W and W Γ only cycles of length or appear, thus W and W Γ are asymptotically semi-circular since there is no restriction on the size of cycles for W and W T there are asymptotically Marchenko-Pastur (just a sanity check) Γ Γ Γ 6 / 6