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Quantum Expanders and Growth of Operator Spaces Quantum expanders Quantum Expanders and Growth of Operator Spaces Gilles Pisier Texas A&M University and Université Paris VI INI, Cambridge (UK) September 24, 2013 Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders In operator space (and C∗ algebra) theory the notion of “Exact Operator Space" (cf. Kirchberg....) became quite important in the last 20 years. The spaces that are not exact exhibit a certain “explosion" of the growth of their matricial approximations. This is what we intend to present today We focus on examples that are not “exact" So this work belongs to the non-exact domain...! Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Our investigation is also motivated by the non-compactness/non-separability of the metric space of n-dim operator spaces (joint work with Junge, GAFA 1994) for the following distance −1 dcb(E; F) = inffkukcbku kcb j u : E ! Fg Quantum expanders and Random Matrices yield several refinements of these facts Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders The term “Quantum Expander" was introduced by Hastings and by Ben-Aroya and Ta-Shma in 2007 to designate a (N) (N) (N) (N) sequence fU j N ≥ 1g of n-tuples U = (U1 ; ··· ; Un ) of N × N unitary matrices such that there is an " > 0 satisfying the following “spectral gap" condition: 8N 8x 2 MN with tr(x) = 0 Xn (N) (N)∗ k U xU k2 ≤ n(1 − ")kxk2; (1) 1 j j where k:k2 denotes the Hilbert-Schmidt norm on MN . Suffices to have this for infinitely many N’s i.e. N 2 fN(1) < N(2) < N(3) < · · · g 9 Computer science motivation An n-tuple U(N) satisfying (1) will be called an "-quantum expander. Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Notation: MN : N × N (complex) matrices with operator norm N S2 : same MN but equipped with Hilbert-Schmidt norm Canonically: N N N S2 = `2 ⊗2 `2 Convention: 8aj ; bj 2 MN X N N N N aj ⊗ bj : `2 ⊗2 `2 ! `2 ⊗2 `2 is identified with the operator X ∗ N N x 7! aj xbj : S2 ! S2 Thus X X k a ⊗ b k = sup fk a xb∗k g j j kxk2≤1 j j 2 Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders The "-quantum expander condition can be rewritten Xn (N) (N) U ⊗ U (1 − P) ≤ n(1 − ") j=1 j j j P where P is the orthogonal projection onto CI (I = ej ⊗ e¯j ) we will write also Xn (N) (N) [ U ⊗ U ]j ? ≤ n(1 − ") j=1 j j I Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders In analogy with the classical expanders , one seeks to exhibit (and hopefully to construct explicitly ) sequences (Nm) fU j m ≥ 1g of n-tuples of Nm × Nm unitary matrices that are "-quantum expanders n X (Nm) (Nm) U ⊗ U (1 − P) ≤ n(1 − ") j=1 j j with Nm ! 1 while n and " > 0 remain fixed Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Crucial inspiration: a result due to Hastings 2007 Theorem (Hastings) If we equip U(N)n with its normalized Haar measure P, then for each n and " > 0 p n X lim Pfu = (uj ) 2 U(N) j k[ uj ⊗uj ]jI? k ≤ 2 n − 1+"ng = 1: N!1 j p Best possible : This result fails if 2 n − 1 is replaced by any smaller number. However, our paper includes a quicker proof of a less sharp (butp more general) estimatep that suffices for our needs (where 2 n − 1 is replaced by 4C n, C being a numerical constant). Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Quantum expanders = non-commutative version of classical expanders When G is a finite group generated by S = ft1; ··· ; tng the associated Cayley graph G(G; S) is said to have a spectral gap if the left regular representation λG satisfies X k λG(tj )jI? k < n(1 − ") (2) where I denotes the constant function 1 on G. Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders A sequence of Cayley graphs G(G(m); S(m)) constitutes an expander in the usual sense if Xn k λG(tj )j ? k < n(1 − ") j=1 I is satisfied with " > 0 and n fixed while jG(m)j ! 1. Expanders (equivalently expanding graphs) have been extremely useful, especially (in the applied direction) since Margulis 1973 and Lubotzky-Phillips-Sarnak 1988 (“Ramanujan graphs") obtained explicit constructions But there are also useful random ones, cf. Joel Friedman’s work Memoirs AMS 2008. See the recent survey: S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications Bull. Amer. Math. Soc. 43 (2006), 439-561. Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders From classical expanders to quantum ones: G a finite group generated by S = ft1; ··· ; tng Assume X k λG(tj )jI? k < n(1 − ") Then for any non trivial irreducible representation π of G Uj = π(tj ) is an "-quantum expander i.e. X k Uj ⊗ Uj (1 − P)k ≤ n(1 − ") [π ⊗ π¯] ? ⊂ λ ? Proof: I GjI Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Moreover, if σ 6= π is another irreducible representation, let Uj = π(tj ) Vj = σ(tj ) Then X k Uj ⊗ Vj k ≤ n(1 − ") (“"-separated") [π ⊗ σ¯] ⊂ λ ? Proof: GjI One can show (de la Harpe-Robertson-Valette 1993) that in the presence of an n-element "-expander in G (n; " > 0 fixed) 0 0 02 8N cardfπ j dπ ≤ N g ≤ exp (c"nN ) Open problem (Meshulam-Wigderson): Is the growth of this UPPER BOUND optimal ? Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Actually, this UPPER BOUND is rather soft. More generally, Suppose we have a family of n-tuples of N × N unitary matrices T ⊂ U(N)n n n Denote 8t = (tj ) 2 MN simply H = MN equipped with n !1=2 1 X 1 ktk = tr(t∗t ) H n N j j 1 Recall X u; v δ − separated , k uj ⊗ vj k ≤ n(1 − δ) This implies a fortiori (much weaker) j<(hu; viH )j ≤ 1 − δ and hence p kukH = kvkH = 1 and ku − vkH ≥ 2δ Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders n Note H = MN Now since dim(H) = nN2 We must have (e.g. by a simple volume consideration) p p 2 2 jT j ≤ (1 + 2= 2δ)2nN = enN 2/δ 2 ) jT j ≤ exp cδnN where cδ > 0 is a constant depending ONLY on δ. −−−−−−−−−−−−−−−−−−−−−−−−−− This yields the HRV bound: 0 0 02 8N cardfπ j dπ ≤ N g ≤ exp (c"nN ) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Main result Theorem 9δ > 0 9β > 0 such that for each 0 < " < 1, for all sufficiently large integers n (n ≥ n0(")) for all N ≥ 1 there is a δ-separated family fu(t) j t 2 T g ⊂ U(N)n formed of "-quantum expanders such that jT j ≥ exp βnN2: X 8s 2 T k sj ⊗sj (1−P)k ≤ n(1−")\"−quantum expanders" X 8s 6= t 2 T k sj ⊗ tj k ≤ n(1 − δ)\δ − separated" Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Important note : Consider s; t 2 U(N)n s; t being δ − separated means X ∗ ∗ 2 1=2 2 1=2 8x; y 2 MN jtr( sjxtj y )j ≤ n(1 − δ)(tr(jxj )) (tr(jyj )) much stronger than just separated in H : p X ∗ ks − tkH ≥ 2δ , <hs; ti ≤ 1 − δ , < tr( sjtj ) ≤ n(1 − δ)N Moreover, p X X 1=2 k sj ⊗ tj k ≤ n(1 − δ) ) k (sj − tj ) ⊗ (sj − tj )k ≥ 2nδ but not conversely 6( Key: Some converse is valid but for Quantum expanders Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Sketch of proof of main result: Step 1. Prove that there are “very many" "-quantum expanders (for " arbitrarily small) namely 2 > exp c"nN well separated for the H-norm Step 2. Prove that for them essentially p X 0 k sj ⊗ tj k ≤ n(1 − δ) ,<< ks − tkH ≥ 2δ >> n n Recall 8s = (sj ) 2 U(N) 8t = (tj ) 2 U(N) kskH = ktkH = 1 Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Step 1 Instead of Hastings 07 we use : Theorem There is a universal constant C such that: X p 8N ≥ 1 Ek[ uj ⊗ uj ](I − P)k ≤ C n: j n X C ) Pfu 2 U(N) j k[ uj ⊗ uj ](I − P)k > "ng ≤ p " n j < 1=2 if n > n0(") Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Notation: For all s; t 2 U(N)n we set d(s; t) = ks − tkH but also (distance of “orbits") 0 d (s; t) = inffd(UsV ; t) j U; V 2 U(N)g UsV = (Usj V )1≤j≤n Let 0 < " < 1.
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