Rigidity of quantum steering and 1sDI verifiable quantum computation - PowerPoint PPT Presentation

Rigidity of quantum steering and 1sDI verifiable quantum computation [arXiv:1512.07401] Alexandru Gheorghiu, Petros Wallden, Elham Kashefi 8 June 2016 QPL 2016, Glasgow I V N E U R S E I H T Y T O H F G R E U D B I N Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 1 / 17

Nonlocal correlations � p ( a , b | x , y ) � = p ( a | x , λ ) p ( b | y , λ ) p ( λ ) λ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 2 / 17

Nonlocal correlations � p ( a , b | x , y ) � = p ( a | x , λ ) p ( b | y , λ ) p ( λ ) λ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 Tsirelson’s theorem (1980) √ S = 2 2 is the maximum that can be achieved by QM. E.g. by √ having Alice and Bob share | φ + � = ( | 00 � + | 11 � ) / 2 and measure: √ √ A 0 = X , A 1 = Z , B 0 = ( X + Z ) / 2, B 1 = ( X − Z ) / 2 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 2 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. √ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 2 − ǫ ρ AB is the shared state of Alice and Bob Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. √ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 2 − ǫ ρ AB is the shared state of Alice and Bob There exists a local isometry Φ = Φ A ⊗ Φ B Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. √ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 2 − ǫ ρ AB is the shared state of Alice and Bob There exists a local isometry Φ = Φ A ⊗ Φ B Φ( ρ AB ) ≈ | φ + � ⊗ | φ + � ⊗ ... ⊗ | φ + � ⊗ | junk � Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. √ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 2 − ǫ ρ AB is the shared state of Alice and Bob There exists a local isometry Φ = Φ A ⊗ Φ B Φ( ρ AB ) ≈ | φ + � ⊗ | φ + � ⊗ ... ⊗ | φ + � ⊗ | junk � Φ( A 0 ) ≈ X Φ( A 1 ) ≈ Z √ √ Φ( B 0 ) ≈ ( X + Z ) / 2 Φ( B 1 ) ≈ ( X − Z ) / 2 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Rigidity Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. √ S = | � A 0 B 0 � + � A 0 B 1 � + � A 1 B 0 � − � A 1 B 1 � | ≥ 2 2 − ǫ ρ AB is the shared state of Alice and Bob There exists a local isometry Φ = Φ A ⊗ Φ B Φ( ρ AB ) ≈ | φ + � ⊗ | φ + � ⊗ ... ⊗ | φ + � ⊗ | junk � Φ( A 0 ) ≈ X Φ( A 1 ) ≈ Z √ √ Φ( B 0 ) ≈ ( X + Z ) / 2 Φ( B 1 ) ≈ ( X − Z ) / 2 Saturating nonlocal correlations determines state and strategy! Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

Steering correlations � p ( a , b | x , y ) � = Tr ( ρ AB ( λ )( E a | x ⊗ I )) p ( b | y , λ ) p ( λ ) λ √ S = | � A 0 B 0 � + � A 1 B 1 � | ≥ 2 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

Steering correlations � p ( a , b | x , y ) � = Tr ( ρ AB ( λ )( E a | x ⊗ I )) p ( b | y , λ ) p ( λ ) λ √ S = | � A 0 B 0 � + � A 1 B 1 � | ≥ 2 Theorem S = 2 is the maximum that can be achieved. E.g. by having Alice √ and Bob share | φ + � = ( | 00 � + | 11 � ) / 2 and measure: A 0 = X , A 1 = Z , B 0 = X , B 1 = Z Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

Steering correlations � p ( a , b | x , y ) � = Tr ( ρ AB ( λ )( E a | x ⊗ I )) p ( b | y , λ ) p ( λ ) λ √ S = | � A 0 B 0 � + � A 1 B 1 � | ≥ 2 Theorem S = 2 is the maximum that can be achieved. E.g. by having Alice √ and Bob share | φ + � = ( | 00 � + | 11 � ) / 2 and measure: A 0 = X , A 1 = Z , B 0 = X , B 1 = Z Our main result: Converse is also true! Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A 0 and A 1 (e.g. A 0 = X , A 1 = Z ) Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A 0 and A 1 (e.g. A 0 = X , A 1 = Z ) Bob is untrusted. Measures B ′ 0 and B ′ 1 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A 0 and A 1 (e.g. A 0 = X , A 1 = Z ) Bob is untrusted. Measures B ′ 0 and B ′ 1 Observables have 2 outcomes ± 1 and are also unitary Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A 0 and A 1 (e.g. A 0 = X , A 1 = Z ) Bob is untrusted. Measures B ′ 0 and B ′ 1 Observables have 2 outcomes ± 1 and are also unitary Shared state ρ AB , prepared by Bob (untrusted) Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Assumptions Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A 0 and A 1 (e.g. A 0 = X , A 1 = Z ) Bob is untrusted. Measures B ′ 0 and B ′ 1 Observables have 2 outcomes ± 1 and are also unitary Shared state ρ AB , prepared by Bob (untrusted) In each round Alice and Bob measure the same state | ψ � ( i.i.d. ) Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

Self-testing i.i.d. states Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

Self-testing i.i.d. states A 0 B ′ A 1 B ′ � � � � | + | ≥ 2 − ǫ (1) 0 1 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

Self-testing i.i.d. states A 0 B ′ A 1 B ′ � � � � | + | ≥ 2 − ǫ (1) 0 1 I.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ Φ B such that, for all M A ∈ { I , A 0 , A 1 } , N ′ B ∈ { I , B ′ 0 , B ′ 1 } : B | ψ � ) − | junk � M A N B | φ + � || ≤ O ( √ ǫ ) || Φ( M A N ′ Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

Self-testing i.i.d. states A 0 B ′ A 1 B ′ � � � � | + | ≥ 2 − ǫ (1) 0 1 I.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ Φ B such that, for all M A ∈ { I , A 0 , A 1 } , N ′ B ∈ { I , B ′ 0 , B ′ 1 } : B | ψ � ) − | junk � M A N B | φ + � || ≤ O ( √ ǫ ) || Φ( M A N ′ Cannot do better than O ( √ ǫ )! Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

Removing i.i.d. Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

Removing i.i.d. A 0 B ′ A 1 B ′ � � � � | + | ≥ 2 − ǫ (1) 0 1 Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

Removing i.i.d. A 0 B ′ A 1 B ′ � � � � | + | ≥ 2 − ǫ (1) 0 1 Non-i.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ Φ B such that, for E AB ′ having the role of M A , N ′ B from before, we have for a randomly chosen ρ i : || Φ( E AB ′ ( ρ i )) − E AB ( | φ + � � φ + | ) || ≤ O ( ǫ 1 / 6 ) Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

State and strategy determination Suppose we do K rounds of measurement to certify one Bell state. Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

State and strategy determination Suppose we do K rounds of measurement to certify one Bell state. Do NK rounds of measurement certify N states? Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

State and strategy determination Suppose we do K rounds of measurement to certify one Bell state. Do NK rounds of measurement certify N states? Not implicitly, because of overlap/adaptiveness! Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17