Quantum strategy, Quantum correlations and Operator algebras Hun Hee Lee Seoul National University Seoul, Nov. 28th, 2016
Personal encounters with CS/mathematicians Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms, Thomas Vidick, Oded Regev Journal of Operator Theory. Locally decodable codes and the failure of cotype for projective tensor products, Jop Briet, Assaf Naor, Oded Regev Electronic Research Announcements in Mathematical Sciences 19 (2012) 120 130. The list continues, but not directly connected to Crypto.
Some remarkable applications of functional analysis in Quantum informations (Hastings, Collins, ) A counter-example of MOE addtivity conjecture using concentration of measure phenomenon and free probability. (Junge, Palazuelos, ) Unbounded violation of tripartite Bell inequality using operator space theory. (Paulsen, Todorov, Winter, ) Quantum correlations, Quantum chromatic numbers using operator system theory.
Finite input/output games A does not know B s input/output and vice versa, A, B both knows the rules. x I A, y I B : inputs, a O A, b O B : outputs λ(x, y, a, b) {0, 1}: the (winning) rule function.
Graph coloring game G = (V, E): a graph. c-coloring of G def h : V {1,, c} such that x y implies h(x) h(y). I A = I B = V {, O A = O B = {1,, c}, x y a b Winning. x = y a = b
Deterministic strategy for games A{ deterministic strategy means a pair of functions f : I A O A g : I B O B. Fundamental questions: 1. Is there any perfect strategy? 2. If not, what is the winning probability (or the value) of a strategy? (Def) Let (f, g) be a deterministic strategy of a game G. Let π Prob(I A I B ) be a given probability density (in many cases, π is uniform). 1. The pair (f, g) is called a perfect strategy if λ(x, y, f (x), g(y)) = 1 for all x I A, y I B. 2. (Winning) W (G, (f, g)) = π(x, y)λ(x, y, f (x), g(y)). x I A,y I B 3. W = W (G) = sup (f,g) W (G, (f, g)).
Graph coloring game: continued G: c-coloring game on (V, E). perfect deterministic strategy (V, E) has a c-coloring. (Ex) G = K 3 : 2-coloring is not possible. For f = g given by f (1) = f (2) = 1 and f (3) = 2 we have W = W (G, (f, g)) = 7 9.
CHSH (Clauser-Horne-Shimony-Holt ) game G: I A = I B = O A = O B = Z 2 or {0, 1} Winning: a + b = x y (mod 2) For f = g 0 we have W = W (G, (f, g)) = 3 4. This game is coming from CHSH inequality, a well-known Bell s inequality.
Probabilistic (or local) strategy of games A probabilistic (or local) strategy means a function I A I B Prob(O A O B ), (x, y) p(, x, y). In other words, there is a probability space {(Ω, µ) such that f x : Ω O A for each (x, y) we have random variables g y : Ω O B with joint probability p(a, b x, y) = µ(f x = a, g y = b). e.g. A and B use today s market value of x-company and y-company, respectively. W loc = π(x, y)λ(x, y, a, b)p(a, b x, y). x I A,y I B (Def) A perfect probabilistic strategy means λ(x, y, a, b) = 0 p(a, b x, y) = 0 for any x, y, a, b. Probabilistic strategy is no better than deterministic strategy.
Postulates of Quantum Mechanics P1 Every physical system is described by a Hilbert space H (state spce) and each unit vector h H is called a state (vector). P2 Quantum measurements are described by linear operators (M i ) i O acting on H, where O is the set of all outcomes. p i = the probability of outcome being i = M i h 2. 1 = i p i = i M ih, M i h = h, ( i M i M i )h. If we observe the outcome i, then the state changes into M i h M i h {. We set P i = Mi M P i 0, i i, then we get i P We call i = I H. such family (P i ) i O a POVM (positive operator-valued measure).
Quantum (or non-local) strategy of games A quantum (or non-local) strategy means a state h H A H B and POVM s assigned for each (x, y), namely (P x,a ) a OA on H A and (P y,b ) b OB on H B. In this case, we denote p q (a, b x, y) = h, (P x,a P y,b )h. W q = π(x, y)λ(x, y, a, b)p q (a, b x, y). x I A,y I B Alice and Bob share a quantum resource, namely a state h usually taken to be entangled. (Def) A perfect quantum strategy means λ(x, y, a, b) = 0 p q (a, b x, y) = 0 for any x, y, a, b. Quantum strategy is sometimes much better than deterministic strategy.
CHSH game: continued Note W (CHSH) = W loc (CHSH) = 3 4. However, we have W q (CHSH) > 3 4. (Proof) H A = H B = C 2, h = 1 2 ( 00 + 11 ) = 1 2 (e 0 e 0 + e 1 e 1 ), P 0,a = orth. proj. onto [ e a and P 1,a := P 0,1+a, cos Q 0,0 = Q 1,1 = A θ = 2 ] θ cos θ sin θ cos θ sin θ sin 2 and θ Q 0,1 = Q 1,0 = I A θ. Then, this choice gives us the proof!
Correlation matrices Fix I A = I B = n and O A = O B = k. (Def) We define correlation matrices in various settings: C loc = {p(a, b x, y) = µ(f x = a, g y = b), (µ, f, g)}, C q = {p q (a, b x, y) = h, (P x,a Q y,b )h, (h, P, Q)}, C qs = {p qs ( ) = h, (P x,a Q y,b )h, (h, P, Q), -dim}, C qc = {p qc ( ) = h, (P x,a Q y,b ), (h, P, Q), PQ = QP} The above are all subsets of R n2 k 2. C loc C q C qs C qa := C q C qc. Corresponding strategies and winning probabilities are defined. (Strong Tsirelson s conjecture) C q = C qc for all n, k. (False by W. Slofstra) (Connes embedding conjecture) C qa = C qc for all n, k.
Tsirelson s conjecture and Connes embedding conjecture (W. Slofstra) Strong Tsirelson s conjecture is false: for the Binary constraint system game there is a perfect qc-strategy, but no perfect quantum strategy! (Connes conjecture, 76) Any separable type II 1 -factor embedds in an ultrapower of the hyperfinite II 1 -factor (Kirchberg) The above is true iff there is only one C -norm on C (F ) C (F ). The list of equivalent formulations continues and Connes conjecture is one of the most important opne problems in the field of operator algebras.
(Quantum) Chromatic numbers G = (V, E): a graph with V = n, χ(g) = min{c : c-coloring exists}, the chromatic number. (Def) The quantum chromatic number χ q (G) is given by χ q (G) := min{c : perfect strategy C q (n, c)}. (Ex) Ω N = (V, E): Hadamard graph, V = {±1} N so that V = 2 N, (v, w) E v w, (Frankl-Rodl) r > 1 such that χ(ω N ) > r N for large N, (Paulsen-Todorov) χ q (Ω N ) N.