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) = inf{kukcbku kcb | u : E → F} 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 {U | N ≥ 1} of n-tuples U = (U1 , ··· , Un ) of N × N unitary matrices such that there is an ε > 0 satisfying the following “spectral gap" condition: ∀N ∀x ∈ MN with tr(x) = 0
Xn (N) (N)∗ k U xU k2 ≤ n(1 − ε)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 ∈ {N(1) < N(2) < N(3) < ···} ∃ Computer science motivation An n-tuple U(N) satisfying (1) will be called an ε-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: ∀aj , bj ∈ 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 {k a xb∗k } 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 ε-quantum expander condition can be rewritten
Xn (N) (N) U ⊗ U (1 − P) ≤ n(1 − ε) j=1 j j
| P where P is the orthogonal projection onto CI (I = ej ⊗ e¯j ) we will write also
Xn (N) (N) [ U ⊗ U ]| ⊥ ≤ n(1 − ε) 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) {U | m ≥ 1} of n-tuples of Nm × Nm unitary matrices that are ε-quantum expanders
n X (Nm) (Nm) U ⊗ U (1 − P) ≤ n(1 − ε) j=1 j j
with Nm → ∞ while n and ε > 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 ε > 0 √ n X lim P{u = (uj ) ∈ U(N) | k[ uj ⊗uj ]|I⊥ k ≤ 2 n − 1+εn} = 1. N→∞ j √ 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 (but√ more general) estimate√ 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 = {t1, ··· , tn} 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 )|I⊥ k < n(1 − ε) (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 )| ⊥ k < n(1 − ε) j=1 I
is satisfied with ε > 0 and n fixed while |G(m)| → ∞. 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 = {t1, ··· , tn} Assume X k λG(tj )|I⊥ k < n(1 − ε) Then for any non trivial irreducible representation π of G
Uj = π(tj )
is an ε-quantum expander i.e. X k Uj ⊗ Uj (1 − P)k ≤ n(1 − ε)
[π ⊗ π¯] ⊥ ⊂ λ ⊥ Proof: I G|I
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 − ε) (“ε-separated")
[π ⊗ σ¯] ⊂ λ ⊥ Proof: G|I One can show (de la Harpe-Robertson-Valette 1993) that in the presence of an n-element ε-expander in G (n, ε > 0 fixed)
0 0 02 ∀N card{π | dπ ≤ N } ≤ exp (cεnN ) 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 ∀t = (tj ) ∈ 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) |<(hu, viH )| ≤ 1 − δ and hence √ 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) √ √ 2 2 |T | ≤ (1 + 2/ 2δ)2nN = enN 2/δ
2 ⇒ |T | ≤ exp cδnN
where cδ > 0 is a constant depending ONLY on δ.
−−−−−−−−−−−−−−−−−−−−−−−−−−
This yields the HRV bound:
0 0 02 ∀N card{π | dπ ≤ N } ≤ exp (cεnN )
Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Main result
Theorem ∃δ > 0 ∃β > 0 such that for each 0 < ε < 1, for all sufficiently large integers n (n ≥ n0(ε)) for all N ≥ 1 there is a δ-separated family {u(t) | t ∈ T } ⊂ U(N)n formed of ε-quantum expanders such that
|T | ≥ exp βnN2.
X ∀s ∈ T k sj ⊗sj (1−P)k ≤ n(1−ε)“ε−quantum expanders”
X ∀s 6= t ∈ 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 ∈ U(N)n s, t being δ − separated means
X ∗ ∗ 2 1/2 2 1/2 ∀x, y ∈ MN |tr( sjxtj y )| ≤ n(1 − δ)(tr(|x| )) (tr(|y| ))
much stronger than just separated in H :
√ X ∗ ks − tkH ≥ 2δ ⇔ Moreover, √ 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" ε-quantum expanders (for ε arbitrarily small) namely 2 > exp cεnN well separated for the H-norm Step 2. Prove that for them essentially √ X 0 k sj ⊗ tj k ≤ n(1 − δ) ⇔<< ks − tkH ≥ 2δ >> n n Recall ∀s = (sj ) ∈ U(N) ∀t = (tj ) ∈ 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 √ ∀N ≥ 1 Ek[ uj ⊗ uj ](I − P)k ≤ C n. j n X C ⇒ P{u ∈ U(N) | k[ uj ⊗ uj ](I − P)k > εn} ≤ √ ε n j < 1/2 if n > n0(ε) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Notation: For all s, t ∈ U(N)n we set d(s, t) = ks − tkH but also (distance of “orbits") 0 d (s, t) = inf{d(UsV , t) | U, V ∈ U(N)} UsV = (Usj V )1≤j≤n Let 0 < ε < 1. Let n n X Sε = {u ∈ U(N) | k uj ⊗ uj (1 − P)k ≤ εn}. 1 Choosing n ≥ n0(ε) we find (for all N) P(Sε) ≥ 1/2. 2 Metric Entropy ⇒ ∃T ⊂ Sε such that |T | ≥ exp βnN and ∀s 6= t ∈ T d 0(s, t) ≥ c > 0 (here β, c > 0 absolute constants) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Step 2 Step 2 follows from the slightly more technical Lemma: ∃ε0 > 0 and ∃0 < δ < 1 such that 0 ∀s, t ∈ Sε0 d (s, t) ≥ c > 0 ⇒ s, t δ − separated. So ∀0 < ε ≤ ε0 the proof is complete Question: Does the main result hold for all 0 < δ < 1 ? Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Applications to Operator Spaces Definition (“Non-commutative Banach spaces") An operator space is a subspace of B(H), i.e. we are given E ⊂ B(H) “Operator space Theory" is now well developed after Ruan’s1987 thesis cf. Effros-Ruan, Blecher-Paulsen, and many more... cf. one book by Effros-Ruan, & one by myself Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Bounded linear maps (Banach spaces) are replaced by Completely bounded linear maps (Operator spaces) Let u : E → F be a linear map between operator spaces. We denote for any given N ≥ 1 uN : MN (E) → MN (F) [aij ] 7→ [u(aij )] X X (in Tensor Product Notation : ak ⊗ xk 7→ ak ⊗ u(xk )) Recall that kukcb = supN≥1 kuN k. Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Isomorphisms (Banach spaces) are replaced by Complete Isomorphisms (Operator spaces) To measure the degree of isomorphism of two spaces we need a "distance" (actually d(E, F) ≤ d(E, G)d(G, F)...“multiplicative distance") Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders For an operator space the norm is replaced by a sequence of norms {k.kMN (E) | N ≥ 1} The ordinary norm on E corresponds to N = 1 We have E ⊂ B(H) MN (E) ⊂ MN (B(H)) with norm induced by MN (B(H)) ' B(H ⊕ · · · ⊕ H) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Growth of Matricial realization Consider E ⊂ B(H) with dim(E) < ∞ Classical Hahn-Banach: ∀a ∈ E kak = sup |f (a)| f ∈BE∗ ∗ ∗ ∗ Matricial analogue: (Roger Smith Lemma) Fix N ≥ 1 ( ∀a = [aij ] ∈ MN (E) k[aij ]kMN (E) = sup k[u(aij )]kMN (MN ) u:E→MN ,kukcb≤1 Natural question: How many u’s are needed ? Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders How many u’s are needed ? More precisely Fix δ > 0. We seek to minimize k = card(T ) over all finite sets T ⊂ {u : E → MN | kukcb ≤ 1} such that ∀a = [aij ] ∈ MN (E)(1 − δ)k[aij ]kMN (E) ≤ sup k[u(aij )]kMN (MN ) u∈T Or with C = (1 − δ)−1 ∀a = [aij ] ∈ MN (E) k[aij ]kMN (E) ≤ C sup k[u(aij )]kMN (MN ) u∈T Smallest k denoted by kE (N, C) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Again kE (N, C) is the smallest k such that ∀a = [aij ] ∈ MN (E) ∃T ⊂ BCB(E,MN ) with |T | = k such that 1/Ck[aij ]kMN (E) ≤ sup k[u(aij )]kMN (MN ) ≤ k[aij ]kMN (E) u∈T Smallest k denoted by kE (N, C) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Minimal case: Exactness Definition E will be called 1-exact if ∀ε > 0 ∃K such that E ⊂ MK (1+ε)−completely isomorphically If E is 1-exact then: for all N large enough, a single u suffices (assuming say C > 1 + ε): ∀N ≥ K kE (N, C) = 1 1 − exact ⇒ No Growth Classical Banach space (commutative) analogue: K every E is 1-exact (E ⊂ `∞) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Opposite Extreme: arbitrary E Using metric entropy in dimension nN2 2 (note dim(CB(E, MN )) = nN ), we find: Proposition 2 ∃T with card(T ) ≤ exp cδnN such that ∀a = [aij ] ∈ MN (E)(1 − δ)k[aij ]kMN (E) ≤ sup k[u(aij )]kMN (MN ) u∈T −1 2 i.e. with C = (1 − δ) we have kE (N, C) ≤ exp cδnN Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Goal: A class of examples of E for which this is essentially best possible, i.e. for any such T for all N large enough 0 2 card(T ) ≥ exp cδnN For these spaces E, there is C > 1 and c > 0 so that 2 ∀N large enough kE (N, C) ≥ exp cnN Such spaces are “extremely not exact" For short, in this talk, let us say that they have “exponential growth" This includes ∗ E1 = span[U1, ··· , Un] ⊂ C (IFn) and E2 = OHn n n These are Operator space analogues of `1 and `2 Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders n Operator space analogue of `1 E1 = span[U1, ··· , Un] ⊂ B(H) (H = `2) Can be realised in the (maximal/full) C∗-algebra of the free group IFn Pn Pn N = 1 (ordinary norm) k x U k = |x | = kxk n j=1 j j j=1 j `1 We will now describe the norm on MN (E) Fix N > 1. Then n n X X ∀aj ∈ MN aj ⊗ Uj = sup aj ⊗ uj n j=1 (uj )∈U(N) j=1 MN (E1) MN (MN ) Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders n Operator space analogue of `2 E2 = span[T1, ··· , Tn] ⊂ B(H) (H = `2) Pn Pn 2 1/2 N = 1 (ordinary norm) k x T k = ( |x | ) = kxk n j=1 j j j=1 j `2 We will now describe the norm on MN (E) Fix N > 1. Then 1/2 Xn Xn 1/2 ∀aj ∈ MN aj ⊗ Tj = aj ⊗ aj = ka.a¯k j=1 1 MN (E2) also Xn ¯ n ¯ aj ⊗ Tj = sup{ka.bk | b ∈ M kb.bk ≤ 1}. j=1 N MN (E2) ¯ def X a.b = aj ⊗ bj Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Main result implies Corollary Both E1 and E2 (and a few more...) have exponential growth. Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Application: Metric entropy for Matricial (“Quantized") Banach-Mazur distance Assuming E ' F, we set (recall uN : MN (E) → MN (F)) −1 dN (E, F) = inf{kuN kk(u )N k} where the inf runs over all the isomorphisms u : E → F. We set dN (E, F) = ∞ if E, F are not isomorphic. Also, if E, F are completely isomorphic, we set −1 dcb(E, F) = inf{kukcbku kcb} where the inf runs over all the complete isomorphisms u : E → F. If E, F finite dim by compactness: dcb(E, F) = sup dN (E, F) N≥1 Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Let OSn denote the set of all operator spaces (identified up to complete isometry) Let N(OSn, dN , r) denote the minimal cardinal of “an r-net" by “an r-net" we mean a finite set T ⊂ OSn such that ∀E ∈ OSn ∃F ∈ T such that dN (E, F) ≤ r Theorem For some r > 1, there are positive constants b, c such that, for any n large enough (say n ≥ n0), we have for all N ≥ 1 2 2 exp(exp bnN ) ≤ N(OSn, dN , r) ≤ exp(exp cnN ). Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces Quantum expanders Thank you ! Gilles Pisier Texas A&M University and Université Paris VI Quantum Expanders and Growth of Operator Spaces