operator norms & covering nets Fernando Brandão (UCL/MSR) Aram Harrow (MIT) QIP arXiv:

outline 1.separable states and their complexity 2.approximating the set of separable states 3.approximating general operator norms 4.the simple case of the simplex

entanglement and optimization Definition: ρ is separable (i.e. not entangled) if it can be written as ρ = ∑ i p i |v i v i | ⊗ |w i w i | probability distribution unit vectors Weak membership problem: Given ρ and the promise that ρ ∈ Sep or ρ is far from Sep, determine which is the case. Sep = conv{|vv| ⊗ |ww|} = conv{α ⊗ β} = Optimization: h Sep (M) := max { tr[Mρ] : ρ ∈ Sep }

complexity of h Sep h Sep (M) ± 0.99 ||M|| op at least as hard as planted clique [Brubaker, Vempala ‘09] 3-SAT[log 2 (n) / polyloglog(n)] [H, Montanaro ‘10] h Sep (M) ± 0.99 h Sep (M) at least as hard as small-set expansion [Barak, Brandão, H, Kelner, Steurer, Zhou ‘12] h Sep (M) ± ||M|| op / poly(n) at least as hard as 3-SAT[n] [Gurvits ‘03], [Le Gall, Nakagawa, Nishimura ‘12] Suppose local systems have dimension n. h Sep(√n parties) (M) ± 0.99 ||M|| op at least as hard as 3-SAT[n] [Chen, Drucker ‘10] multipartite case

A hierachy of tests for entanglement separable = ∞-extendable 100-extendable all quantum states (= 1-extendable) 2-extendable Algorithms: Can search/optimize over k-extendable states in time n O(k). Question: How close are k-extendable states to separable states? Definition: ρ AB is k-extendable if there exists an extension with for each i.

SDP hierarchies for h Sep doesn’t match hardness Thm: If M =∑ i A i ⊗ B i with ∑ i |B i | ≤ I, each |A i | ≤ I, then h Sep(n,2) (M) ≤ h k-ext (M) ≤ h Sep(n,2) (M) + c (log(n)/k) 1/2 bipartite [Brandão, Christandl, Yard ’10], [Yang ’06], [Brandão, H ’12], [Li, Winter ‘12] Sep(n,m) = conv{ρ 1 ⊗... ⊗ ρ m : ρ i ∈ D n } SepSym(n,m) = conv{ρ ⊗ m : ρ ∈ D n } [Brandão, H ’12], [Li, Smith ’14] Thm: ε-approx to h SepSym(n,m) (M) in time exp(m 2 log 2 (n)/ε 2 ). ε-approx to h Sep(n,m) (M) in time exp(m 3 log 2 (n)/ε 2 ). multipartite ≈matches Chen-Drucker hardness

proof intuition Measure extended state and get outcomes p(a,b 1,…,b k ). Possible because of 1-LOCC form of M. case 1 p(a,b 1 ) ≈ p(a) ⋅ p(b 1 ) case 2 p(a, b 2 | b 1 ) has less mutual information [Brandão, Christandl, Yard ’10] (secretly in [Yang ’06]) [Brandão, H ’12], [Li, Winter ’12], [Li, Smith ‘14]

suspicious coincidence! Run-time exp(c log 2 (n) / ε 2 ) appears in both Algorithm for M in 1-LOCC Hardness for M in SEP. Why? Can we bridge the gap? Can we find multiplicative approximations, or otherwise use these approaches for small-set expansion?

h Sep for 1-LOCC M M =∑ i ∈ [m] A i ⊗ B i with ∑ i A i ≤ I, each A i ≥ 0, |B i | ≤ I h Sep (M) = max α,β tr[M(α ⊗ β)] = max α,β ∑ i ∈ [m] tr[A i α]tr[B i β] p p i = tr[A i α] ∈S⊆Δm∈S⊆Δm = max p ∈ S max β ∑ i ∈ [m] p i tr[B i β] = max p ∈ S ||∑ i ∈ [m] p i B|| op := max p ∈ S ||p|| B ||x|| B = ||∑ i x i B i || op {A i }BiBi α∈Dnα∈Dn β∈Dnβ∈Dn i density matrices probabilities measurements

nets for h Sep h Sep (M) = max p ∈ S ||p|| B p i = tr[A i α] α∈Dnα∈Dn p∈S⊆Δmp∈S⊆Δm ∑ i p i B i ||x|| B = ||∑ i x i B i || 2->2 net size O(1/ε) n net size m log(n)/ε 2 net size O(1/ε) n

nets from sampling Performance k ≃ log(n)/ε 2, m=poly(n) run-time O(m k ) = exp(log 2 (n)/ε 2 ) Algorithm: Enumerate over k-sparse q check whether ∃ p ∈ S, ||p-q|| B ≤ε if so, compute ||q|| B Lemma: ∀ p ∈ Δ m ∃ q k-sparse (each q i = integer / k) Pf: Sample i 1,..., i k from p. Operator Chernoff says [Ahlswede-Winter ‘00]

operator norms operator norm l 2  l 2 largest singular value. ||X|| 2->2 = ||X|| op l ∞  l 1 MAX-CUT = max{vec(X), a ⊗ b: ||a|| ∞, ||b|| ∞ ≤ 1} l 1  l ∞ max i,j |X i,j | = max{vec(X), a ⊗ b: ||a|| 1, ||b|| 1 ≤ 1} S 1 -> S 1 of X ⊗ id channel distinguishability (cb norm, diamond norm) S 1 -> S p max output p-norm, min output Rènyi-p entropy l 2  l 4 hypercontractivity, small-set expansion S 1  S ∞ h Sep = max{ Choi(X), a ⊗ b : ||a|| S 1, ||b|| S 1 ≤ 1 } Examples

complexity of l 2  l 4 norm UG ≈ SSE ≤ 2->4 = h Sep G = normalized adjacency matrix P λ = largest projector s.t. G ≥ ¸P Theorem: All sets of volume ≤ ± have expansion ≥ 1 - ¸ O(1) iff ||P λ || 2->4 ≤ 1/± O(1) Unique Games (UG): Given a system of linear equations: x i – x j = a ij mod k. Determine whether ≥1-² or ≤² fraction are satisfiable. Small-Set Expansion (SSE): Is the minimum expansion of a set with ≤±n vertices ≥1-² or ≤²?

nets for Banach spaces X:A->B ||X|| A->B = sup ||Xa|| B / ||a|| A operator norm ||X|| A->C->B = min {||Z|| A->C ||Y|| C->B : X=YZ} factorization norm Let A,B be arbitrary. C = l 1 m Only changes are sparsification (cannot assume m≤poly(n)) and operator Chernoff for B. result: estimated in time exp(T 2 (B) 2 log(m)/ε 2 ) T 2 (S 2 ) = 1 T 2 (S ∞ ) T 2 (S 1 ) = Type-2 constant: T 2 (B) is smallest λ such that

applications S 1  S p norms of entanglement-breaking channels N(ρ) = ∑ i tr[A i ρ]B i, where ∑ i A i = I, ||B i || 1 = 1. Can estimate ||N|| 1  p ±εin time n f(p,ε). (uses bounds on T 2 (S p ) from [Ball-Carlen-Lieb ’94] low-rank measurements: h Sep (∑ i A i ⊗ B i )±εfor ∑ i |A i |=1, ||B i || ∞ ≤1, rank B i ≤r in time n O(r/ε 2 ) l 2  l p for even p≥4 in time n O(p/ε 2 ) Multipartite versions of 1-LOCC norm too [cf. Li-Smith ‘14]

lots of coincidences! ε-nets vs. SoS Problemε-netsSoS/info theory max p ∈ Δ p T ApBK ’02, KLP ‘06DF ’80 BK ’02, KLP ‘06 approx NashLMM ’03HNW ‘16 free gamesAIM ‘14Brandão-H ‘13 unique gamesABS ’10BRS ‘11 small-set expansion ABS ’10BBHKSZ ’12 h Sep Shi-Wu ’11 BKS ’13 Brandão-H ‘15 BCY ‘10 Brandão-H ’12 BKS ‘13

simplest version: polynomial optimization over the simplex Δ n = {p ∈ R n : p≥0, ∑ i p i = 1} Given homogenous degree-d poly f(p 1, …, p n ), find max p f(p). NP-complete: given graph G with clique number α, max p p T Ap = 1 – 1/α. [Motzkin-Strauss, ‘65] Approximation algorithms Net: Enumerate over all points in Δ n (k) := Δ n ∩ Z n /k. Hierarchy: min λs.t. (∑ i p i ) k (λ(∑ i p i ) d -f(p)) has all nonnegative coefficients. Thm: Each gives error ≤ (max p f(p)-min p f(p)) exp(d) / k in time n O(k). [de Klerk, Laurent, Parrilo, ’06]

open questions Application to unique games, small-set expansion, etc.Application to unique games, small-set expansion, etc. Tight hardness results, e.g. for h Sep.Tight hardness results, e.g. for h Sep. What about operators with unknown factorization?What about operators with unknown factorization? Explain the coincidences!Explain the coincidences!