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

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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

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Rigidity of quantum steering and 1sDI verifiable quantum computation

[arXiv:1512.07401] Alexandru Gheorghiu, Petros Wallden, Elham Kashefi 8 June 2016 QPL 2016, Glasgow

T H E U N I V E R S I T Y O F E D I N B U R G H Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 1 / 17

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Nonlocal correlations

p(a, b|x, y) =

  • λ

p(a|x, λ)p(b|y, λ)p(λ) S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 2 / 17

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Nonlocal correlations

p(a, b|x, y) =

  • λ

p(a|x, λ)p(b|y, λ)p(λ) S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 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: A0 = X, A1 = Z, B0 = (X + Z)/ √ 2, B1 = (X − Z)/ √ 2

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 2 / 17

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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

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Rigidity

Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2 √ 2 − ǫ ρAB is the shared state of Alice and Bob

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

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Rigidity

Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2 √ 2 − ǫ ρAB is the shared state of Alice and Bob There exists a local isometry Φ = ΦA ⊗ ΦB

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

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Rigidity

Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2 √ 2 − ǫ ρAB is the shared state of Alice and Bob There exists a local isometry Φ = ΦA ⊗ ΦB Φ(ρAB) ≈ |φ+ ⊗ |φ+ ⊗ ... ⊗ |φ+ ⊗ |junk

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

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Rigidity

Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2 √ 2 − ǫ ρAB is the shared state of Alice and Bob There exists a local isometry Φ = ΦA ⊗ ΦB Φ(ρAB) ≈ |φ+ ⊗ |φ+ ⊗ ... ⊗ |φ+ ⊗ |junk Φ(A0) ≈ X Φ(A1) ≈ Z Φ(B0) ≈ (X + Z)/ √ 2 Φ(B1) ≈ (X − Z)/ √ 2

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

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Rigidity

Reichardt Unger Vazirani [RUV] (2012) Robust converse of Tsirelson’s theorem is also true. S = | A0B0 + A0B1 + A1B0 − A1B1 | ≥ 2 √ 2 − ǫ ρAB is the shared state of Alice and Bob There exists a local isometry Φ = ΦA ⊗ ΦB Φ(ρAB) ≈ |φ+ ⊗ |φ+ ⊗ ... ⊗ |φ+ ⊗ |junk Φ(A0) ≈ X Φ(A1) ≈ Z Φ(B0) ≈ (X + Z)/ √ 2 Φ(B1) ≈ (X − Z)/ √ 2 Saturating nonlocal correlations determines state and strategy!

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 3 / 17

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Steering correlations

p(a, b|x, y) =

  • λ

Tr(ρAB(λ)(Ea|x ⊗ I))p(b|y, λ)p(λ) S = | A0B0 + A1B1 | ≥ √ 2

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

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Steering correlations

p(a, b|x, y) =

  • λ

Tr(ρAB(λ)(Ea|x ⊗ I))p(b|y, λ)p(λ) S = | A0B0 + A1B1 | ≥ √ 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: A0 = X, A1 = Z, B0 = X, B1 = Z

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

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Steering correlations

p(a, b|x, y) =

  • λ

Tr(ρAB(λ)(Ea|x ⊗ I))p(b|y, λ)p(λ) S = | A0B0 + A1B1 | ≥ √ 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: A0 = X, A1 = Z, B0 = X, B1 = Z Our main result: Converse is also true!

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 4 / 17

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A0 and A1 (e.g. A0 = X, A1 = Z)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A0 and A1 (e.g. A0 = X, A1 = Z) Bob is untrusted. Measures B′

0 and B′ 1

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A0 and A1 (e.g. A0 = X, A1 = 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

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A0 and A1 (e.g. A0 = X, A1 = Z) Bob is untrusted. Measures B′

0 and B′ 1

Observables have 2 outcomes ±1 and are also unitary Shared state ρAB, prepared by Bob (untrusted)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 5 / 17

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Assumptions

Quantum mechanics is true/correct (no supra-quantum correlations) Alice is trusted to measure anticommuting A0 and A1 (e.g. A0 = X, A1 = Z) Bob is untrusted. Measures B′

0 and B′ 1

Observables have 2 outcomes ±1 and are also unitary Shared state ρAB, 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

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Self-testing i.i.d. states

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

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Self-testing i.i.d. states

|

  • A0B′
  • +
  • A1B′

1

  • | ≥ 2 − ǫ

(1)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

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Self-testing i.i.d. states

|

  • A0B′
  • +
  • A1B′

1

  • | ≥ 2 − ǫ

(1) I.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ ΦB such that, for all MA ∈ {I, A0, A1}, N′

B ∈ {I, B′ 0, B′ 1}:

||Φ(MAN′

B |ψ) − |junk MANB |φ+ || ≤ O(√ǫ)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

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Self-testing i.i.d. states

|

  • A0B′
  • +
  • A1B′

1

  • | ≥ 2 − ǫ

(1) I.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ ΦB such that, for all MA ∈ {I, A0, A1}, N′

B ∈ {I, B′ 0, B′ 1}:

||Φ(MAN′

B |ψ) − |junk MANB |φ+ || ≤ O(√ǫ)

Cannot do better than O(√ǫ)!

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 6 / 17

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Removing i.i.d.

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

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Removing i.i.d.

|

  • A0B′
  • +
  • A1B′

1

  • | ≥ 2 − ǫ

(1)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

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Removing i.i.d.

|

  • A0B′
  • +
  • A1B′

1

  • | ≥ 2 − ǫ

(1) Non-i.i.d. self-testing theorem If inequality 1 is satisfied, then there exists a local isometry Φ = I ⊗ ΦB such that, for EAB′ having the role of MA, N′

B from

before, we have for a randomly chosen ρi: ||Φ(EAB′(ρi)) − EAB(|φ+ φ+|)|| ≤ O(ǫ1/6)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 7 / 17

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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

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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

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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

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy Sguess = (ρAB, EA, GB) denotes a guessing strategy

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy Sguess = (ρAB, EA, GB) denotes a guessing strategy S′

guess = (|φ+ ⊗ ... ⊗ |φ+ , EA, GB) second guessing strategy

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy Sguess = (ρAB, EA, GB) denotes a guessing strategy S′

guess = (|φ+ ⊗ ... ⊗ |φ+ , EA, GB) second guessing strategy

S is ǫ-structured ↔ observed correlation is greater than 2 − ǫ

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy Sguess = (ρAB, EA, GB) denotes a guessing strategy S′

guess = (|φ+ ⊗ ... ⊗ |φ+ , EA, GB) second guessing strategy

S is ǫ-structured ↔ observed correlation is greater than 2 − ǫ S1 ≈ S2 ↔ ρ1 ≈ ρ2, EA,1 ≈ EA,2, EB,1 ≈ EB,2

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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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! Sreal = (ρAB, EA, E′B) denotes the real strategy Sideal = (|φ+ ⊗ ... ⊗ |φ+ , EA, EB) denotes the ideal strategy Sguess = (ρAB, EA, GB) denotes a guessing strategy S′

guess = (|φ+ ⊗ ... ⊗ |φ+ , EA, GB) second guessing strategy

S is ǫ-structured ↔ observed correlation is greater than 2 − ǫ S1 ≈ S2 ↔ ρ1 ≈ ρ2, EA,1 ≈ EA,2, EB,1 ≈ EB,2 Objective: Sreal ≈ Sideal

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 8 / 17

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State and strategy determination - Proof sketch

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 9 / 17

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State and strategy determination - Proof sketch

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 10 / 17

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State and strategy determination - Proof sketch

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 11 / 17

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State and strategy determination - Proof sketch

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 12 / 17

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State and strategy determination - Proof sketch

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 13 / 17

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Application: verification of quantum computation

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 14 / 17

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Application: verification of quantum computation

Computationally limited, trusted verifier

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 14 / 17

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Application: verification of quantum computation

Computationally limited, trusted verifier Powerful, untrusted quantum server(s)

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 14 / 17

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Application: verification of quantum computation

Computationally limited, trusted verifier Powerful, untrusted quantum server(s) Alice = verifier, Bob = server

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 14 / 17

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Application: verification of quantum computation

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 15 / 17

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Conclusions

Saturating correlations ↔ ideal states and measurements I.i.d. self-testing → Non-i.i.d. self-testing → Rigidity Lower bounded Ω(√ǫ) closeness Tight bounds for non-i.i.d. and rigidity? Most natural application is quantum verification

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 16 / 17

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Related work and references

Presentation based primarily on this work: [Gheorghiu, Kashefi, Wallden, ’15] - arXiv:1512.07401 Related works on self-testing and rigidity: [Hoban, ˇ Supi´ c ’16] - arXiv:1601.01552 [Bancal, Navascu´ es, Scarani, V´ ertesi, Yang ’13] - arXiv:1307.7053 [Reichardt, Unger, Vazirani ’12] - arXiv:1209.0448 [McKague, Yang, Scarani ’12] - arXiv:1203.2976 Related works on verification: [Gheorghiu, Kashefi, Wallden ’15] - arXiv:1502.02571 [Kashefi, Wallden ’15] - arXiv:1510.07408 [McKague ’15] - arXiv:1309.5675 Thank you!

Alexandru Gheorghiu, Petros Wallden, Elham Kashefi [arXiv:1512.07401] 17 / 17