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C OMPLETENESS OF H ARDY N ON - LOCALITY : C ONSEQUENCES & A PPLICATIONS Shane Mansfield QPL 2014 Overview Theorem* For all ( 2 , k , 2 ) and ( 2 , 2 , l ) scenarios, Hardy non-locality ( ) Logical non-locality Consequences &

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SLIDE 1

COMPLETENESS OF HARDY NON-LOCALITY:

CONSEQUENCES & APPLICATIONS Shane Mansfield

QPL 2014

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SLIDE 2

Overview

Theorem*

For all (2,k,2) and (2,2,l) scenarios, Hardy non-locality ( ) Logical non-locality

Consequences & Applications

  • 1. Hardy subsumes other paradoxes
  • 2. Complexity results for logical non-locality
  • 3. Bell states are anomalous
  • 4. Hardy non-locality can be realised with certainty

*S Mansfield, T Fritz - Foundations of Physics, 2012

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SLIDE 3

Non-locality

measurement device mA

  • A

measurement device mB

  • B

preparation p

Bell-CHSH Inequality:

  • E(mA,mB)+E(mA,m0

B)+E(m0 A,mB)E(m0 A,m0 B)

  •  2
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SLIDE 4

Logical Non-locality

A more intuitive approach to non-locality

  • Probabilities

! Truth values (possibilities)

  • Inequalities

! Logical deductions Logical NL > NL Examples:

  • Hardy, GHZ, KS, etc.
  • Hardy’s argument is considered to be the simplest
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SLIDE 5

Hardy’s Non-locality Paradox

Bob Alice " # G W " 1 # G W

  • Outcome (",") is possible
  • If A measures spin and B

measures colour, or vice versa, the outcomes (",W) or (W,") are never obtained

  • When spin " is recorded, the
  • ther subsystem must have

colour G

  • Since (",") is possible, then

(G,G) must be possible

  • Contradiction!
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SLIDE 6

Generalisations of Hardy Non-locality

Measurements have up to l outcomes

1

···

l

  • 1 ···om2
  • m2+1 ···ol

1

1 ··· . . .

l

  • 1

. . .

  • m1

··· . . . ... . . . ···

  • m1+1

. . .

  • l

. . .

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SLIDE 7

Generalisations of Hardy Non-locality

k measurement settings per party

1 ... ...

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SLIDE 8

Generalisations of Hardy Non-locality

n > 2 parties

Figure : The n = 3 Hardy paradox. Blue $ truth value ‘1’, red $ ‘0’

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SLIDE 9

Completeness of Hardy Non-locality

Hardy non-locality can be defined for all (n,k,l) scenarios.

  • n parties
  • k measurement settings per party
  • l outcomes to each measurement

Theorem*

For all (2,k,2) and (2,2,l) scenarios, Hardy non-locality ( ) Logical non-locality

*S Mansfield, T Fritz - Foundations of Physics, 2012

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SLIDE 10

Hardy Subsumes Other Paradoxes

The Chen et al. paradox* occurs if at least one starred entry is non-zero. Relevant entries are either above or below the diagonal.

* ··· * ··· ... . . . ... . . . * ··· ... . . . . . . ... ···

*JL Chen, A Cabello, ZP Xu, HY Su, C Wu, LC Kwek - Physical Review A, 2013

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SLIDE 11

Hardy Subsumes Other Paradoxes

The Chen et al. paradox* occurs if at least one starred entry is non-zero. Relevant entries are either above or below the diagonal.

1 ··· * ··· ... . . . ... . . . * ··· ... . . . . . . ... ···

*JL Chen, A Cabello, ZP Xu, HY Su, C Wu, LC Kwek - Physical Review A, 2013

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SLIDE 12

Hardy Subsumes Other Paradoxes

The Chen et al. paradox* occurs if at least one starred entry is non-zero. Relevant entries are either above or below the diagonal.

1 ··· . . .

*JL Chen, A Cabello, ZP Xu, HY Su, C Wu, LC Kwek - Physical Review A, 2013

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SLIDE 13

Hardy Subsumes Other Paradoxes

The Chen et al. paradox* occurs if at least one starred entry is non-zero. Relevant entries are either above or below the diagonal.

1 ··· . . .

*JL Chen, A Cabello, ZP Xu, HY Su, C Wu, LC Kwek - Physical Review A, 2013

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SLIDE 14

Complexity of Logical Non-locality

Hardy non-locality ( ) Logical non-locality So, in relevant scenarios, one has only to search for Hardy paradoxes

Proposition

Polynomial algorithms can be given for deciding logical non-locality in (2,2,l) and (2,k,2) scenarios.

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SLIDE 15

Bell States are Anomalous

Are all entangled states logically non-local? Logically Non-local

  • Hardy: all non-maximally entangled 2-qubit states
  • Abramsky, Constantin & Ying: all entangled n-qubit states
  • GHZ, Cabello: Many maximally entangled n > 2 qubit states

Exception!

  • Bell States (maximally entangled 2-qubit states)

1 p 2 (|00i+|11i), etc.

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SLIDE 16

Bell States Are Anomalous: Proof (Sketch)

  • Need only consider

1 p 2 (|00i+|11i), etc.

  • Projective measurements necessarily lead to (2,k,2) scenarios

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

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SLIDE 17

Bell States Are Anomalous: Proof (Sketch)

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

State: 1 p 2 (|00i+|11i), etc. Observables: {A1,A2,B3,B4} Eigenvectors: |0ii = cos θi 2 |0i+eiφi sin θi 2 |1ii = sin θi 2 |0i+eiφi cos θi 2 Outcome probabilities: ⌦ 0j0k|ψ ↵ = 1 p 2 cos θj 2 cos θk 2 +ei

⇣ φj+φk ⌘

sin θj 2 sin θk 2 ! ⌦ 0j1k|ψ ↵ = 1 p 2 cos θj 2 sin θk 2 +ei

⇣ φjφk ⌘

sin θj 2 cos θk 2 ! ⌦ 1j0k|ψ ↵ = 1 p 2 sin θj 2 cos θk 2 +ei

⇣ φjφk ⌘

sin θj 2 cos θk 2 ! ⌦ 1j1k|ψ ↵ = 1 p 2 sin θj 2 sin θk 2 +ei

⇣ φj+φk ⌘

cos θj 2 cos θk 2 !

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SLIDE 18

Bell States Are Anomalous: Proof (Sketch)

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

State: 1 p 2 (|00i+|11i), etc. Observables: {A1,A2,B3,B4} Eigenvectors: |0ii = cos θi 2 |0i+eiφi sin θi 2 |1ii = sin θi 2 |0i+eiφi cos θi 2 Outcome probabilities: p(01 | AB) = p(10 | AB) p(00 | AB) = p(11 | AB)

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SLIDE 19

Bell States Are Anomalous: Proof (Sketch)

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

Symmetries + No-signalling + Hardy Paradox:

1⁄2 1⁄2 1⁄2 1⁄2 1⁄2 1-q⁄2 q⁄2 1⁄2 q⁄2 1-q⁄2

0 < q  1 q = 0: Local q = 1: PR box Outcome probabilities: p(01 | AB) = p(10 | AB) p(00 | AB) = p(11 | AB)

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SLIDE 20

Bell States Are Anomalous: Proof (Sketch)

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

Symmetries + No-signalling + Hardy Paradox:

1⁄2 1⁄2 1⁄2 1⁄2 1⁄2 1-q⁄2 q⁄2 1⁄2 q⁄2 1-q⁄2

0 < q  1 q = 0: Local q = 1: PR box Outcome probabilities: p(01 | AB) = p(10 | AB) p(00 | AB) = p(11 | AB) Observables: A1 = A2 = B3 = B4 = ±X

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SLIDE 21

Bell States Are Anomalous: Proof (Sketch)

Claim

For any observables {A1,A2,B3,B4} there is no Hardy paradox

Symmetries + No-signalling + Hardy Paradox:

1⁄2 1⁄2 1⁄2 1⁄2 1⁄2 1-q⁄2 q⁄2 1⁄2 q⁄2 1-q⁄2

0 < q  1 q = 0: Local q = 1: PR box Outcome probabilities: p(01 | AB) = p(10 | AB) p(00 | AB) = p(11 | AB) Observables: A1 = A2 = B3 = B4 = ±X ) q = 0 Contradiction!

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SLIDE 22

The Paradoxical Probability

Bob Alice " # G W " 0.09 # G W

  • An almost probability free

non-locality proof

  • Experimental motivations for

maximising this probability

  • Considered a measure of the

quality of Hardy non-locality

Model Probability Hardy

5 p 511 2

⇡ 0.09 Hardy Ladder (k ! ∞) 0.5 Ghosh et al. (tripartite) 0.125 Choudhary (non-quantum, NS) 0.5 Chen et al. (l ! ∞) ⇡ 0.4

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SLIDE 23

Probability Free Hardy Non-locality?

  • Recall: Chen et al. sum paradoxical probabilities

* ··· * ··· ... . . . ... . . . * ··· ... . . . . . . ... ···

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SLIDE 24

Probability Free Hardy Non-locality?

  • Recall: Chen et al. sum paradoxical probabilities
  • If we allow this, we can achieve Hardy non-locality with

certainty! Example: the PR box 1 1 1 1 1 1 1 1

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SLIDE 25

Probability Free Hardy Non-locality?

  • Recall: Chen et al. sum paradoxical probabilities
  • If we allow this, we can achieve Hardy non-locality with

certainty! Example: the PR box 1 1 1 1 1 1 1 1

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SLIDE 26

Probability Free Hardy Non-locality?

  • Recall: Chen et al. sum paradoxical probabilities
  • If we allow this, we can achieve Hardy non-locality with

certainty! Example: the PR box 1 1 1 1 1 1 1 1

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SLIDE 27

Hardy Non-locality With Certainty

The GHZ model: Local X & Y measurements on |GHZi = 1 p 2 (|000i+|111i) 000 001 010 011 100 101 110 111 X X X 1 1 1 1 X Y Y 1 1 1 1 Y X X 1 1 1 1 Y Y X 1 1 1 1

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SLIDE 28

Hardy Non-locality With Certainty

The GHZ model: Local X & Y measurements on |GHZi = 1 p 2 (|000i+|111i)

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SLIDE 29

Hardy Non-locality With Certainty

Model Probability Hardy

5 p 511 2

⇡ 0.09 Hardy Ladder (k ! ∞) 0.5 Ghosh et al. (tripartite) 0.125 Choudhary (non-quantum, NS) 0.5 Chen et al. (l ! ∞) ⇡ 0.4 PR box (non-quantum, NS) 1 GHZ 1

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SLIDE 30

Conclusion

Theorem*

For all (2,k,2) and (2,2,l) scenarios, Hardy non-locality ( ) Logical non-locality

Consequences & Applications

  • 1. Hardy subsumes other paradoxes
  • 2. Complexity results for logical non-locality
  • 3. Bell states are anomalous (not logically non-local)
  • 4. Hardy non-locality can be realised with certainty

*S Mansfield, T Fritz - Foundations of Physics, 2012