Completely positive semidefinite matrices: conic approximations and matrix factorization ranks Monique Laurent FOCM 2017, Barcelona
Objective ◮ New matrix cone CS n + : completely positive semidefinite matrices Noncommutative analogue of CP n : completely positive matrices ◮ Motivation: conic optimization approach for quantum information ◮ quantum graph coloring ◮ quantum correlations ◮ (Noncommutative) polynomial optimization: common approach for (quantum) graph coloring and for matrix factorization ranks: ◮ symmetric rks: cpsd-rank ( A ) for A ∈ CS n + , cp-rank ( A ) for A ∈ CP n ◮ asymmetric analogues: psd-rank ( A ), rank + ( A ) for A nonnegative ◮ Based on joint works with Sabine Burgdorf, Sander Gribling, David de Laat, Teresa Piovesan
Completely positive semidefinite matrices
Completely positive semidefinite matrices ◮ A matrix A ∈ S n is completely positive semidefinite (cpsd) if A has a Gram factorization by positive semidefinite matrices X 1 ., . . . , X n ∈ S d + of arbitrary size d ≥ 1: A ij = � X i , X j � ( = Tr( X i X j ) ) ∀ i , j ∈ [ n ] The smallest such d is cpsd-rank ( A ) [back to it later] The cpsd matrices form a convex cone � the completely positive semidefinite cone CS n + ◮ If X i are diagonal psd matrices (equivalently, replace X i by nonnegative vectors x i ∈ R d + ), then A is completely positive � the completely positive cone CP n The smallest such d is cp-rank ( A ) [back to it later] ◮ Clearly: CP n ⊆ CS n + ⊆ cl ( CS n + ∩ R n × n =: DNN n + ) ⊆ S n + Is the cone CS n + closed ?
Strict inclusions CP n ⊆ CS n + ⊆ DNN n ◮ CP n = CS n + = DNN n if n ≤ 4; but strict inclusions if n ≥ 5 + \ CP 5 for [Fawzi-Gouveia-Parrilo-Robinson-Thomas’15] A ∈ CS 5 ◮ 1 a b b a a 1 a b b � 2 π � � 4 π � with a = cos 2 , b = cos 2 A = b a 1 a b 5 5 b b a 1 a a b b a 1 √ A ∈ CS 5 A � 0: + because √ ⇒ A = Gram( u 1 u T 1 , . . . , u 5 u T A = Gram( u 1 , . . . , u 5 ) = 5 ) 4 2 0 0 2 2 4 2 0 0 ∈ DNN 5 \ CS 5 [L-Piovesan 2015] A = 0 2 4 3 0 ◮ + 0 0 3 4 2 2 0 0 2 4 because A is supported by a cycle: A ∈ CS n ⇒ A ∈ CP n + ⇐
On the closure cl ( CS n + ) 4 2 0 0 2 2 4 2 0 0 �∈ cl ( CS 5 Moreover, A = 0 2 4 3 0 + ) ! 0 0 3 4 2 2 0 0 2 4 Because [Frenkel-Weiner 2014] show that A does not have a Gram representation by positive elements in any C ∗ -algebra A with trace ... ... while [Burgdorf-L-Piovesan 2015] construct a C ∗ -algebra with trace M U such that cl ( CS n + ) consists of all matrices A having a Gram factorization by positive elements in M U (using tracial ultraproducts of matrix algebras) New cone CS n + C ∗ : all matrices having a Gram representation by positive elements in some C ∗ -algebra with trace . Then A �∈ CS n + C ∗ , CS n + C ∗ is closed, and CS n + ⊆ cl ( CS n + ) ⊆ CS n + C ∗ � DNN n Equality cl ( CS n + ) = CS n + C ∗ under Connes’ embedding conjecture
SDP outer approximations of CS n + Assume A ∈ CS n A = (Tr( X i X j )) for some X 1 , . . . , X n ∈ S d + : + Define the trace evaluation at X = ( X 1 , . . . , X n ): L : R � x 1 , . . . , x n � → R p �→ L ( p ) = Tr( p ( X 1 , . . . , X n )) (1) L is tracial : L ( pq ) = L ( qp ) ∀ p , q ∈ R � x � L ( p ∗ ) = L ( p ) ∀ p ∈ R � x � (2) L is symmetric : L ( p ∗ p ) ≥ 0 (3) L is positive : ∀ p ∈ R � x � L ( p ∗ x i p ) ≥ 0 (4) localizing constraint: ∀ p ∈ R � x � (5) A = ( L ( x i x j )) F t = matrices A ∈ S n for which there exists L ∈ R � x � ∗ 2 t satisfying (1)-(5) � CS n CS n + ⊆ cl ( CS n + ) ⊆ CS n + ⊆ F t +1 ⊆ F t , + C ∗ ⊆ F t t ≥ 1 F t is the solution set of a semidefinite program: (3) M t ( L ) = ( L ( u ∗ v )) u , v ∈� x � t � 0, (4) ( L ( u ∗ x i v )) u , v ∈� x � t − 1 � 0 Noncommutative analogue of outer approximations of CP n [Nie’14]
Quantum graph coloring
Classical coloring number χ ( G ) = minimum number of colors needed for a proper coloring of V ( G ) ∃ x i χ (G) = min k ∈ N s.t. u ∈ { 0 , 1 } for u ∈ V(G), i ∈ [k] i ∈ [k] x i � ∀ u ∈ V(G) u = 1 x i u x i v = 0 ∀ i ∈ [ k ] ∀ uv ∈ E(G) x i u x j u = 0 ∀ i � = j ∈ [ k ], ∀ u ∈ V(G)
Quantum coloring number ∃ x i χ (G) = min k ∈ N s.t. u ∈ { 0 , 1 } for u ∈ V(G), i ∈ [k] i ∈ [k] x i � ∀ u ∈ V(G) u = 1 x i u x i v = 0 ∀ i ∈ [ k ] , ∀ uv ∈ E(G) x i u x j u = 0 ∀ i � = j ∈ [ k ] , ∀ u ∈ V(G) ∃ d ∈ N ∃ X i χ q (G) = min k ∈ N s.t. u ∈ S d + for u ∈ V(G), i ∈ [k] i ∈ [k] X i � u = I ∀ u ∈ V(G) X i u X i v = 0 ∀ i ∈ [ k ] , ∀ uv ∈ E(G) X i u X j ∀ i � = j ∈ [ k ] , ∀ u ∈ V(G) u = 0 χ q ( G ) ≤ χ ( G ) [Cameron, Newman, Montanaro, Severini, Winter: On the quantum chromatic number of a graph, Electronic J. Combinatorics, 2007]
Motivation: non-local coloring game Two players: Alice and Bob, want to convince a referee that they can color a given graph G = ( V , E ) with k colors Agree on strategy before the start, no communication during the game ◮ The referee chooses a pair of vertices ( u , v ) ∈ V 2 with prob. π ( u , v ) ◮ The referee sends vertex u to Alice and vertex v to Bob ◮ Alice answers color i ∈ [ k ], Bob answers color j ∈ [ k ], using some strategy they have chosen before the start of the game � i = j if u = v ◮ Alice & Bob win the game when i � = j if uv ∈ E When using a classical strategy , the minimum number of colors needed to always win the game is the classical coloring number χ ( G )
Quantum strategy for the coloring game � ◮ ∀ u ∈ V A i Alice has POVM { A i u } i ∈ [ k ] : A i u ∈ H d + , u = I i ∈ [ k ] � ◮ ∀ v ∈ V Bob has POVM { B j B j v ∈ H d B j v } j ∈ [ k ] : + , v = I j ∈ [ k ] ◮ Alice and Bob share an entangled state Ψ ∈ C d ⊗ C d (unit vector) ◮ Probability of answer ( i , j ): p ( i , j | u , v ) := � Ψ , A i u ⊗ B j v Ψ � ◮ Alice and Bob win the game if they never give a wrong answer : p ( i , j | u , v ) = 0 if ( u = v & i � = j ) or ( uv ∈ E & i = j ) ◮ Theorem: [Cameron et al. 2007] The minimum number of colors for which there is a quantum winning strategy is equal to χ q ( G )
Classical and quantum coloring numbers ◮ χ q ( G ) ≤ χ ( G ) ◮ ∃ G for which χ q ( G ) = 3 < χ ( G ) = 4 [Fukawa et al. 2011] ◮ The separation χ q < χ is exponential for Hadamard graphs G n : n = 4 k , with vertices x ∈ { 0 , 1 } n , edges ( x , y ) if d H ( x , y ) = n / 2 χ ( G n ) ≥ (1 + ǫ ) n [Frankl-R¨ odl’87] χ q ( G n ) = n [Avis et al.’06][Mancinska-Roberson’16] ◮ Deciding whether χ q ( G ) ≤ 3 is NP-hard [Ji 2013] ◮ Approach: Model χ q ( G ) as conic optimization problem using the cone of completely positive semidefinite matrices
Conic formulation for quantum graph coloring χ q ( G ) = min k s.t. ∃ X i u � 0 ( u ∈ V , i ∈ [ k ]) satisfying: � i ∈ [ k ] X i u = � j ∈ [ k ] X j v ( � = 0) ( u , v ∈ V ) (Q1) X i u X j X i u X i u = 0 ( i � = j ∈ [ k ] , u ∈ V ) , v = 0 ( i ∈ [ k ] , uv ∈ E ) (Q2) Set A := Gram ( X i u ). Then: X i u X j ⇒ Tr( X i u X j v = 0 ⇐ v ) = 0 = A ui , vj χ q (G) = min k s.t. ∃ A ∈ CS nk Then: satisfying: + � i , j ∈ [ k ] A ui , vj = 1 ( u , v ∈ V ) , (C1) A ui , uj = 0 ( i � = j ∈ [ k ] , u ∈ V ) , A ui , vi = 0 ( i ∈ [ k ] , uv ∈ E ) . (C2) Theorem (L-Piovesan 2015) ◮ Replacing CS + by the cone CP , we get χ ( G ) ◮ Replacing CS + by the cone DNN , get the theta number ϑ + ( G ) ◮ Hence: ϑ + ( G ) ≤ χ q ( G ) [Mancinska-Roberson 2015]
SDP relaxations for coloring If ( X i u ) is solution to χ q ( G ) = k , its normalized trace evaluation satisfies (1) L (1) = 1 (2) L is symmetric, tracial, positive (on Hermitian squares) (3) L = 0 on the ideal generated by 1 − � k i =1 x i x i u x j x i u x i u ( u ∈ V ) , u ( i � = j , u ∈ V ) , v ( uv ∈ E , i ∈ [ k ]) Restricting to the truncated polynomial space R � x � 2 t , get the parameters: t ( G ) = min k such that ∃ L ∈ R � x � ∗ ξ nc 2 t satisfying (1)-(3) t ( G ) = min k such that ∃ L ∈ R [ x ] ∗ ξ c 2 t satisfying (1)-(3) ξ nc ξ c t ( G ) ≤ χ q ( G ) t ( G ) ≤ χ ( G ) ◮ For t = 1 get the theta number: ξ nc 1 ( G ) = ξ c 1 ( G ) = ϑ + ( G ) ◮ ξ c t ( G ) = χ ( G ) ∀ t ≥ n [Gvozdenovi´ c-L 2008] ◮ ξ nc t 0 ( G ) = χ C ∗ ( G ) ≤ χ q ( G ) ∀ t ≥ t 0 [Gribling-de Laat-L 2017] u ∈ A for any C ∗ -algebra A with trace χ C ∗ ( G )= allow solutions X i [Ortiz-Paulsen 2016]
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