Quantum Mechanics As a Sandwich Theory Simone Severini University - PowerPoint PPT Presentation

Quantum Mechanics As a Sandwich Theory Simone Severini University College London with Adán Cabello (Sevilla), Runyao Duan (UTS), Laura Mancinska (IQC/NUS), Giannicola Scarpa (CWI), Andreas Winter (ICREA)

Quantum Mechanics As a Sandwich Theory Simone Severini University College London with Adán Cabello (Sevilla) , Runyao Duan (UTS), Laura Mancinska (IQC), Giannicola Scarpa (CWI), Andreas Winter (ICREA)

Overview Combinatorial optimization techniques, like semidef. programming (see the Parillo-Lasserre SDP hierarchy) are traditionally important in quantum theory. We propose a framework based on optimization on graphs to unify - non-contextuality, - non-locality (via non-local games), - and certain information transmission tasks that use quantum resources.

Plan 1. Introduction: non-contextuality 2. Results: a general framework to study non contextuality; non-contextuality and non-locality 3. Open problems: theoretical; applied; complexity perspective 4. Information theory: zero-error capacities [if there is time] Plan:

1. Introduction: non-contextuality for 1. classical theories 2. non-signaling theories 3. quantum theory 1. Introduction: non-contextuality

Question 1 Question 2 Question 5 Question 4 Question 3 1. Introduction: non-contextuality

Question 1 context : Question 2 a set of mutually compatible questions Question 3 1. Introduction: non-contextuality

Question 1: compatibility : Is the colour red? Answer: Yes Question 2: What is the suit? Answer: Diamonds 1. Introduction: non-contextuality

non-contextuality: answers do not depend on contexts 1. Introduction: non-contextuality

Context 1 Question 2: Question 1: Is Y true? Is X true? Answer: Answer: Yes Yes non-contextuality: answers do not depend on contexts 1. Introduction: non-contextuality

Context 2 Question 2: Is Y true? Answer: Yes non-contextuality: Question 3: answers Is Z true? do not depend Answer: on contexts No 1. Introduction: non-contextuality

Context 2 Context 1 Question 2: Question 1: Is Y true? Is X true? Answer: Answer: Yes Yes non-contextuality: Question 3: answers Is Z true? do not depend Answer: on contexts No 1. Introduction: non-contextuality

Context 2 Context 1 Question 2: Question 1: Is Y true? Is X true? Answer: Answer: Yes Yes non-contextuality: Question 3: answers Is Z true? do not depend Answer: on contexts No 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 4 Question 5 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 1. Introduction: non-contextuality

compatibility structure Question 1 Question 2 Question 3 Question 5 Question 4 1. Introduction: non-contextuality

exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 0/1 0/1 0/1 Question 5 Question 4 Answer: Answer: 0/1 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 1 0/1 0/1 Question 5 Question 4 Answer: Answer: 0/1 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 1 0 0/1 Question 5 Question 4 Answer: Answer: 0/1 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 1 0 0/1 Question 5 Question 4 Answer: Answer: 0 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 the answers 1 have expectation ≤ 2 if non-contextual and exclusive 1. Introduction: non-contextuality

classical theories give expectation ≤ 2 1. Introduction: non-contextuality

non-contextuality: probabilities of answers do not depend on contexts 1. Introduction: non-contextuality

Context 1 Question 2: Question 1: Is Y true? Is X true? Pr[answer Yes] = b Pr[answer Yes] = a non-contextuality: probabilities of answers do not depend on contexts 1. Introduction: non-contextuality

Context 2 Question 2: Is Y true? Pr[answer Yes] = b non-contextuality: Question 3: probabilities of answers Is Z true? do not depend on contexts Pr[answer Yes] = c 1. Introduction: non-contextuality

Context 2 Context 1 Question 2: Question 1: Is Y true? Is X true? Pr[answer Yes] = b Pr[answer Yes] = a non-contextuality: Question 3: probabilities of answers Is Z true? do not depend on contexts Pr[answer Yes] = c 1. Introduction: non-contextuality

Context 2 Context 1 Question 2: Question 1: Is Y true? Is X true? Pr[answer Yes] = b Pr[answer Yes] = a non-contextuality: Question 3: probabilities of answers Is Z true? do not depend on contexts Pr[answer Yes] = c 1. Introduction: non-contextuality

exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 0/1 0/1 0/1 Question 5 Question 4 Answer: Answer: 0/1 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: p 2 p 3 p 1 Pr[1] = Pr[1] = Pr[1] = Question 5 Question 4 Answer: Answer: p 5 p 4 Pr[1] = Pr[1] = exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Context 1 Question 1 Question 2 p 1  p 2 = 1 Answer: Answer: p 1 p 2 Pr[1] = Pr[1] = exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: Pr[1] = 1/2 Pr[1] = 1/2 Pr[1] = 1/2 Question 5 Question 4 Answer: Answer: Pr[1] = 1/2 Pr[1] = 1/2 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality 1. Introduction: non-contextuality

non-signaling theories* give expectation ≤ 2.5 *also called “general probabilistic theories” 1. Introduction: non-contextuality 1. Introduction: non-contextuality

axiomatically: classical theories give expectation ≤ 2 non-signaling theories give expectation ≤ 2.5 1. Introduction: non-contextuality 1. Introduction: non-contextuality

axiomatically: classical theories give expectation ≤ 2 quantum theory give expectation ≤ ? non-signaling theories give expectation ≤ 2.5 1. Introduction: non-contextuality 1. Introduction: non-contextuality

∣  〉 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 0/1 0/1 0/1 ∣  〉 Question 5 Question 4 Answer: Answer: 0/1 0/1 exclusiveness: answers to adjacent questions are not both 1 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Question 1 Question 2 Question 3 Answer: Answer: Answer: 2 2 2 Pr[1] = ∣ 〈 ∣ v 1 〉 ∣ ∣ 〈 ∣ v 2 〉 ∣ ∣ 〈 ∣ v 3 〉 ∣ Pr[1] = Pr[1] = ∣  〉 Question 5 Question 4 Answer: Answer: 2 2 ∣ 〈 ∣ v 5 〉 ∣ ∣ 〈 ∣ v 4 〉 ∣ Pr[1] = Pr[1] = extra axiom: expectation: 5 ∣ 〈 ∣ v i 〉 ∣ Born rule ∑ i = 1 2 1. Introduction: non-contextuality 1. Introduction: non-contextuality

Context i Question i Question i + mod(5) Answer: Answer: 2 2 ∣ 〈 ∣ v i  1mod  5  〉 ∣ Pr[1] = ∣ 〈 ∣ v i 〉 ∣ Pr[1] = 〈 v i ∣ v i  1mod  5  〉= 0 compatibility 1. Introduction: non-contextuality

∣  〉 ∈ℝ 3 ∣ v 5 〉 ∣ v 4 〉 2 = 1 ∣ 〈 ∣ v i 〉 ∣  5 ∣ v 1 〉 ∣ v 3 〉 ∣ v 2 〉 1. Introduction: non-contextuality

∣  〉 ∈ℝ 3 ∣ v 5 〉 ∣ v 4 〉 2 = 1 ∣ 〈 ∣ v i 〉 ∣  5 5 ∣ 〈 ∣ v i 〉 ∣ ∣ v 1 〉 ∑ i = 1 2 =  5 ∣ v 3 〉 ∣ v 2 〉 1. Introduction: non-contextuality

∣  〉 ∈ℝ 3 quantum theory gives expectation ≤  5 ∣ v 5 〉 ∣ v 4 〉 2 = 1 ∣ 〈 ∣ v i 〉 ∣  5 5 ∣ 〈 ∣ v i 〉 ∣ ∣ v 1 〉 ∑ i = 1 2 =  5 ∣ v 3 〉 ∣ v 2 〉 1. Introduction: non-contextuality

classical theories* give expectation ≤ 2 quantum theory** gives expectation ≤  5 ≈ 2.23 non-signaling theories* give expectation ≤ 2.5 *Wright (1978); **Klyachko et al. (2008) 1. Introduction: non-contextuality