An operator approach to Bell inequalities Daniel Alsina, Torun, 17 - PowerPoint PPT Presentation

An operator approach to Bell inequalities Daniel Alsina, Torun, 17 th June 2017 Based on D. Alsina, A. Cervera, D. Goyeneche, J. I. Latorre and K. Zyczkowski, Operational approach to Bell inequalities: application to qutrits , Phys. Rev. A 94 , 032102 (2016). arXiv:1606.01991 [quant-ph]

Summary ● Basics of entanglement ● Bell inequalities ● Bell operator ● Bell operator for qutrits ● Mapping states to Bell operators ● Experiments ● Conclusions

Basics of entanglement ● Entanglement: quantum correlations between particles Intuitive conditions for entanglement : 1) Result of a measurement on A is somewhat (possibly totally) uncertain 2) A measurement on B will give more (possibly total) information about A. Paradigmatic example: Bell/EPR state | ψ⟩= 1 √ 2 ( | 0 A 0 B ⟩+ | 1 A 1 B ⟩) We don't know the result of a measure on A (could be 0 or 1) but a measure on B will give us the key to A (it will be 0 if B has been 0, and 1 if B has been 1)

Basics of entanglement | ψ⟩= 1 √ 2 ( | 0 A 0 B ⟩+ | 1 A 1 B ⟩) A measures 0, B also measures 0 A measures 1, B also measures 1 This suggests there is some information traveling between A and B to “tell” the other particle which state should it collapse to.

Basics of entanglement A B Creation of 2 entangled | ψ⟩= 1 particles √ 2 ( | 0 A 0 B ⟩+ | 1 A 1 B ⟩) Separation B B B B A A A A ρ A ρ B I am 0! Measurement and B B B B A A A A collapse of A ρ B | ψ A ⟩= | 0 A ⟩ | ψ A ⟩= | 0 A ⟩ I am 0! Information traveling B B B B A A A A ρ B | ψ A ⟩= | 0 A ⟩ | ψ A ⟩= | 0 A ⟩ B B B B A A A A Collapse of B | ψ A ⟩= | 0 A ⟩ | ψ A ⟩= | 0 A ⟩ | ψ B ⟩= | 0 B ⟩

Basics of entanglement Spacelike separation between A and B? Possibility 1: B will collapse independently of A (entanglement is lost at some point) Possibility 2: Results of collapses in A and B match anyway (entanglement is a non-local property) Possibility 2 is what really happens!

Basics of entanglement Spooky action at a distance... Letter Einstein to Born (1947) 2 new possibilities: 1- Hidden variables: QM is incomplete and there are new variables that determine the outcome of any experiment with certainty. EPR (1935), De Broglie-Bohm (1927-1952) 2- QM is intrinsically non-local and we have to live with it. Is there an experiment to differentiate between those two?

Bell inequalities ● The answer is YES: Bell inequalities Constraints of local realism 1 + E ( bc )⩾ | E ( ab ) − E ( ac ) | a,b,c = +1,-1 E(x): Expected value Original Bell inequality (1964) " If [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says ”. (John Bell, 1987)

Bell inequalities ● CHSH Inequality (Clauser et al., 1969) a ( b + b' )+ a' ( b − b' )⩽ 2 stand for 4 different variables with a ,a' ,b,b' values {+1,-1}. A observes randomly a or a', B observes randomly b or b' In QM applied to a spin 1/2 system, a is to be interpreted as σ · ⃗ ⃗ a σ= { σ x , σ y , σ z } where is the Pauli vector. ⃗ a b a' A B b'

Bell inequalities | E ( ab )+ E ( a' b )+ E ( ab' ) − E ( a' b' ) | ⩽ 2 If we take the singlet state: | ψ⟩= 1 √ 2 ( | 0 A 1 B ⟩ −| 1 A 0 B ⟩) we have the simple form: a , ⃗ E ( ab )= − cos (⃗ b ) taking a = 90 º ,b = 45 º ,a' = 0 º ,b' = 135 º |− √ 2 2 − √ 2 2 − √ 2 2 − √ 2 2 | = 2 √ 2 > 2 Violation of the Bell inequality! Quantum limit of CHSH

Bell inequalities CHSH inequality S = a ( b + b' )+ a' ( b − b' ) S=4 Non- signaling S=2 Quantum mechanics S=2.82 Local realism

Bell inequalities We had to guess first which state would violate | ψ⟩= 1 the BI: √ 2 ( | 0 A 1 B ⟩ −| 1 A 0 B ⟩) Compute the expected value (which can be more complicated for other states): a , ⃗ E ( ab )= − cos (⃗ b ) and then optimize the directions: a = 90 º ,b = 45 º ,a' = 0 º ,b' = 135 º Useful for experiments where we have a concrete state (up to uncertainties) but not from an analytical point of view.

Bell operator Let's look for a more deductive way to find the classical and quantum limits of a BI and the states that saturate the quantum limit . B = ab + ab' + a' b − a' b' Bell operator The maximal eigenvalue of C gives the maximum violation of the BI, and the corresponding eigenvector will be the state responsible for it. We only have to maximize over the directions now.

Bell operator We can still do better with a little trick: computing B² 2 = 4 I a ⊗ I b − [ a,a' ][ b,b' ] ([ x , y ]= xy − yx ) B From this expression we can obtain a lot of information: 2 = 4 - Classically, all commutators are 0, so B clas - In QM, It is thus easy to see that in [σ i , σ j ]= 2 i ϵ ijk σ k . order to maximize the commutators, it's enough to impose that a and a' are perpendicular ( the same for b and b' ). Each commutator will then give a maximum value of 2, and: 2 ⩽ 8 B quant

Bell operator But mathematically: Av i =λ i v i → A 2 v i = A (λ i v i )=λ i Av i =λ i 2 v i so we readily deduce that ⟨⟦ B ⟧⟩ LR = 2 ⟨⟦ B ⟧⟩ QM = 2 √ 2 and the states responsible for the maximum quantum violation of B will be the same as those of B², so we got all information almost for free!

Bell operator Can the trick be extended to other BI? General BI: BI(n,m,d) ● n: number of parties: a,b,c... ● m: number of settings: a,a',a''… ● d: local dimension: 2,3,4… CHSH is the most simple BI: it's a BI(2,2,2)

Bell operator for qutrits ● Motivations for qutrits Existence of the AME(4,3): Absolutely maximally entangled state of four qutrits ∣ψ〉= 1 3 (∣ 0000 〉+∣ 0112 〉+∣ 0221 〉+∣ 1011 〉+∣ 1120 〉 +∣ 1202 〉+∣ 2022 〉+∣ 2101 〉+∣ 2210 〉) There is no complete characterization of the family of BI(n,2,3) Experimentally qutrits are harder to realize but are more robust against decoherence

Bell operator for qutrits ● Qutrits have values {0,1,2} or {1,0,-1}. The operator basis needs to be expanded to {a,a 2 } (because now , ) The elements of the basis are no longer the 3 Pauli Matrices of SU(2) but the 8 Gell-Mann matrices of SU(3).

Bell operator for qutrits BI(2,2,3): (CGLMP Inequality, 2002) Its Bell operator is: The form of B² is too complicated. But we can just optimize over the directions to find:

Bell operator for qutrits for the directions: And the corresponding state is: Not the maximally entangled state!!

Bell operator for qutrits A smarter alternative is to turn to complex values {1,w,w 2 } where w=Exp(2i Pi /3). The elements of the basis of operators are the generalized Pauli matrices: Shift matrix Clock matrix

Bell operator for qutrits CHSH for qubits

Bell operator for qutrits We call them Multiplets of Optimal Settings (MOS). They have the property that their commutator and anticommutator are nilpotent matrices.

Bell operator for qutrits ● 3 qutrits: ● 4 qutrits:

Bell operator for qutrits ● For both inequalities, Mutually unbiased bases! Apparently two different families for 2qt and 3,4 qt. Is there a generating formula for BI(n,2,3)?

Bell operator for qutrits Nº qutrits 2 3 4 5 6 3 √ 3 √ 3 3 √ 3 9 √ 3 9 √ 3 ⟨⟦ B ⟧ A ⟩ LR − 2 √ 3 − 3 √ 3 − 6 √ 3 − 9 √ 3 − 18 √ 3 m ⟨⟦ B ⟧ A ⟩ LR 3 3 9 9 27 ⟨⟦ B ⟧ H ⟩ LR -3 -6 -9 -18 -27 m ⟨⟦ B ⟧ H ⟩ LR 2.524 5.058 9.766 15.575 32.817 ⟨⟦ B ⟧ x ⟩ QM Ratio 1.457 1.686 1.879 1.731 2.105 Settings MOS MUB MUB Num. MOS Purity 0.347 0.342 1/3 0.351 0.334

Mapping states to Bell operators ● Observation: Maximally entangled state | ψ⟩= 1 √ 2 ( | + A 0 B ⟩+ |− A 1 B ⟩) | ±⟩= 1 √ 2 ( | 0 ⟩± | 1 ⟩) can be expanded into | ψ⟩= 1 √ 2 ( | 0 A 0 B ⟩+ | 0 A 1 B ⟩+ | 1 A 0 B ⟩ −| 1 A 1 B ⟩) | ψ⟩ B | 0 ⟩ A B = ab + ab' + a' b − a' b' a | 0 ⟩ B b CHSH operator | 1 ⟩ A a' | 1 ⟩ B b'

Mapping states to Bell operators ● An equivalent strategy for qutrits would be: | ψ⟩ B Obtaining Bell | 0 ⟩ A a inequalities of | 0 ⟩ B b qutrits and | 1 ⟩ A three settings, a' i.e. BI (n, 3, 3) | 1 ⟩ B b' | 2 ⟩ A a'' | 2 ⟩ B b'' We apply the map to the GHZ of 2 and 4 qutrits and to the AME of 4 qutrits | AME ⟩= 1 3 ( | 0000 ⟩+ | 0112 ⟩+ | 0221 ⟩+ | 1011 ⟩+ | 1120 ⟩ + | 1202 ⟩+ | 2022 ⟩+ | 2101 ⟩+ | 2210 ⟩)

Mapping states to Bell operators Table of results: State 2 (GHZ) 2 4 (GHZ) 4 (AME) 4 9 √ 3 3 √ 3 9 √ 3 3 √ 3 √ 3 ⟨⟦ B ⟧ A ⟩ LR − 3 √ 3 − 9 √ 3 − 9 √ 3 − 6 √ 3 − 2 √ 3 m ⟨⟦ B ⟧ A ⟩ LR 4.5 3 13.5 13.5 9 ⟨⟦ B ⟧ H ⟩ LR -4.5 -3 -27 -27 -9 m ⟨⟦ B ⟧ H ⟩ LR 5.117 2.524 26.025 25.372 9.766 ⟨⟦ B ⟧ x ⟩ QM R 1.137 1.457 1.928 1.879 1.879 Settings MUB MOS Num. MUB and MUB Num. P 1/3 0.347 1/3 1/3 1/3

Experiments BI have been tested experimentally for quite a while now, albeit generally just in the CHSH form with the singlet state. Various experiments in the 70's and 80's (ex. Aspect et al. (1982) found confirmations of CHSH violation Now already quite close to the quantum bound: 10 -3 in Poh et al., (2015)