Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs Laura Manˇcinska1 David E. Roberson2 1QMATH, University of Copenhagen 2Technical University of Denmark QIP 2021 February 4, 2021 2 / 23 Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Line of attack: Focus on a well-behaved class of games • Nonlocal games provide a general framework for studying • entanglement Line of attack: Focus on a well-behaved class of games • Nonlocal games provide a general framework for studying • entanglement Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Nonlocal games provide a general framework for studying • entanglement Problem: Entanglement-assisted strategies for arbitrary • nonlocal games are hard to analyze Line of attack: Focus on a well-behaved class of games • This talk PART I Quantum isomorphism and different ways to think about it: Nonlocal games • Matrix formulations • Homomorphism counts • PART II Elements of the proof: Intertwiners of quantum groups • Bi-labeled graphs • Homomorphism matrices • 4 / 23 A map f : V(G) V(H) is an isomorphism from G to H if ! f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0 If such a map exists, we say that G and H are isomorphic and write G =∼ H. Matrix formulation: PAGPy = AH for some permutation matrix P Graph isomorphism ∼= 5 / 23 If such a map exists, we say that G and H are isomorphic and write G =∼ H. Matrix formulation: PAGPy = AH for some permutation matrix P Graph isomorphism ∼= A map f : V(G) V(H) is an isomorphism from G to H if ! f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0 5 / 23 Matrix formulation: PAGPy = AH for some permutation matrix P Graph isomorphism ∼= A map f : V(G) V(H) is an isomorphism from G to H if ! f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0 If such a map exists, we say that G and H are isomorphic and write G =∼ H. 5 / 23 Graph isomorphism ∼= A map f : V(G) V(H) is an isomorphism from G to H if ! f is a bijection and • g ∼ g if and only if f(g) ∼ f(g ). • 0 0 If such a map exists, we say that G and H are isomorphic and write G =∼ H. Matrix formulation: PAGPy = AH for some permutation matrix P 5 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0) No communication during game • Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. Intuition: Alice and Bob want to convince a referee that G =∼ H. R A B 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game 6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0) No communication during game • Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. R A B 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. 6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0) No communication during game • Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R g g0 A B 6 / 23 To win players must reply h, h • 0 such that rel(h, h0) = rel(g, g0) No communication during game • Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R g g0 A B h h0 6 / 23 No communication during game • Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0) A B h h0 6 / 23 Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0) No communication during game • A B h h0 6 / 23 Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0) No communication during game • A B h h0 Fact. G =∼ H Classical players can win the game with certainty , 6 / 23 1We work in the commuting rather than the tensor-product model. (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0) No communication during game • A B h h0 Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 6 / 23 (G, H)-Isomorphism Game Intuition: Alice and Bob want to convince a referee that G =∼ H. R To win players must reply h, h0 g g0 • such that rel(h, h0) = rel(g, g0) No communication during game • A B h h0 Fact. G =∼ H Classical players can win the game with certainty , Def. (Quantum isomorphism) ∼ 1 We say that G =qc H if quantum players can win the game with certainty. 1We work in the commuting rather than the tensor-product model. 6 / 23 Eg = fEgh B(H): h V(H)g where 2 2 Egh 0, Egh = I. h P is a unit vector in a Hilbert space H Upon receiving g, Alice performs a local • measurement Eg to get h V(H) 2 Bob measures with F 0 • g E and F 0 0 commute • gh g h Alice and Bob share a quantum state • The probability that players respond with h, h0 on questions g, g0 is p(h, h0jg, g0) = , EghFg h h 0 0 i Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game R g g0 A h h0 B ψ 7 / 23 is a unit vector in a Hilbert space H Eg = fEgh B(H): h V(H)g where 2 2 Egh 0, Egh = I. h P E and F 0 0 commute • gh g h The probability that players respond with h, h0 on questions g, g0 is p(h, h0jg, g0) = , EghFg h h 0 0 i Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game Alice and Bob share a quantum state R • g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) 2 A h h0 B ψ Bob measures with F 0 • g 7 / 23 Eg = fEgh B(H): h V(H)g where 2 2 Egh 0, Egh = I. h P E and F 0 0 commute • gh g h The probability that players respond with h, h0 on questions g, g0 is p(h, h0jg, g0) = , EghFg h h 0 0 i Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game Alice and Bob share a quantum state • R is a unit vector in a Hilbert space H g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) 2 A h h0 B ψ Bob measures with F 0 • g 7 / 23 E and F 0 0 commute • gh g h The probability that players respond with h, h0 on questions g, g0 is p(h, h0jg, g0) = , EghFg h h 0 0 i Quantum commuting strategies ∼ G =qc H := Quantum players can win the (G, H)-isomorphism game Alice and Bob share a quantum state • R is a unit vector in a Hilbert space H g g0 Upon receiving g, Alice performs a local • measurement Eg to get h V(H) 2 A B Eg = fEgh B(H): h V(H)g where h h0 2 2 Egh 0, h Egh = I.