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Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities
- Phys. Rev. Lett. 117, 070402 (2016)
Analytic and nearly optimal self-testing bounds for the - - PowerPoint PPT Presentation
Analytic and nearly optimal self-testing bounds for the - - PowerPoint PPT Presentation
Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities Phys. Rev. Lett. 117 , 070402 (2016) [ arXiv:1604.08176 ] Jed Kaniewski QMATH, Department of Mathematical Sciences University of
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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What is self-testing?
Bell scenario x a y b Pr[a, b|x, y]
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What is self-testing?
Bell scenario x a y b Pr[a, b|x, y] Def.: Pr[a, b|x, y] is local if Pr[a, b|x, y] =
- λ
p(λ) p(a|x, λ) p(b|y, λ). Otherwise = ⇒ nonlocal or it violates (some) Bell inequality
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What is self-testing?
Obs.: Separable states give local statistics (for all measurements) ρAB =
- λ
pλαλ ⊗ βλ, Pr[a, b|x, y] = tr
- (P x
a ⊗ Qy b)ρAB
- =
- λ
pλ · tr(P x
a αλ)
- p(a|x,λ)
· tr(Qy
bβλ)
- p(b|y,λ)
.
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What is self-testing?
ρAB is separable = ⇒ statistics are local Pr[a, b|x, y] is nonlocal = ⇒ ρAB is entangled
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What is self-testing?
ρAB is separable = ⇒ statistics are local Pr[a, b|x, y] is nonlocal = ⇒ ρAB is entangled smart! anything more specific?
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What is self-testing?
ρAB is separable = ⇒ statistics are local Pr[a, b|x, y] is nonlocal = ⇒ ρAB is entangled smart! anything more specific? sure! let me google it for you...
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What is self-testing?
Given Pr[a, b|x, y] = tr
- (P x
a ⊗ Qy b)ρAB
- deduce properties of ρAB, {P x
a }, {Qy b}
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What is self-testing?
Given Pr[a, b|x, y] = tr
- (P x
a ⊗ Qy b)ρAB
- deduce properties of ρAB, {P x
a }, {Qy b}
We don’t assume that ρAB is pure and it’s important! (ask me if you want to know more)
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What is self-testing?
Given Pr[a, b|x, y] = tr
- (P x
a ⊗ Qy b)ρAB
- deduce properties of ρAB, {P x
a }, {Qy b}
We don’t assume that ρAB is pure and it’s important! (ask me if you want to know more)
- ften only promised some Bell violation
- abxy
cxy
ab Pr[a, b|x, y] = β
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What is self-testing?
Example: the CHSH inequality [Popescu, Rohrlich ’92] βCHSH :=
- abxy
(−1)a+b+xy Pr[a, b|x, y] for a, b, x, y ∈ {0, 1} βCHSH = 2 √ 2 (max) = ⇒ ρAB ≃ ΦAB for |ΦAB =
1 √ 2(|00 + |11).
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What is self-testing?
Example: the CHSH inequality [Popescu, Rohrlich ’92] βCHSH :=
- abxy
(−1)a+b+xy Pr[a, b|x, y] for a, b, x, y ∈ {0, 1} βCHSH = 2 √ 2 (max) = ⇒ ρAB ≃ ΦAB for |ΦAB =
1 √ 2(|00 + |11).
ρAB = ΦAB Inherent limitations
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What is self-testing?
Example: the CHSH inequality [Popescu, Rohrlich ’92] βCHSH :=
- abxy
(−1)a+b+xy Pr[a, b|x, y] for a, b, x, y ∈ {0, 1} βCHSH = 2 √ 2 (max) = ⇒ ρAB ≃ ΦAB for |ΦAB =
1 √ 2(|00 + |11).
ρAB = ΦAB Inherent limitations
- cannot see auxiliary systems (measurements act trivially)
⊗ τA′B′
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What is self-testing?
Example: the CHSH inequality [Popescu, Rohrlich ’92] βCHSH :=
- abxy
(−1)a+b+xy Pr[a, b|x, y] for a, b, x, y ∈ {0, 1} βCHSH = 2 √ 2 (max) = ⇒ ρAB ≃ ΦAB for |ΦAB =
1 √ 2(|00 + |11).
ρAB = ΦAB Inherent limitations
- cannot see auxiliary systems (measurements act trivially)
⊗ τA′B′
- cannot see local unitaries
U( )U † for U = UAA′ ⊗ UBB′
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What is self-testing?
Example: the CHSH inequality [Popescu, Rohrlich ’92] βCHSH :=
- abxy
(−1)a+b+xy Pr[a, b|x, y] for a, b, x, y ∈ {0, 1} βCHSH = 2 √ 2 (max) = ⇒ ρAB ≃ ΦAB for |ΦAB =
1 √ 2(|00 + |11).
ρAB = ΦAB Inherent limitations
- cannot see auxiliary systems (measurements act trivially)
⊗ τA′B′
- cannot see local unitaries
U( )U † for U = UAA′ ⊗ UBB′ Necessary... but also sufficient!
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
? ρAB state certification
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
? ρAB state certification ? P x
a , Qy b(ρAB)
measurement certification
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
? ρAB state certification ? P x
a , Qy b(ρAB)
measurement certification
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
? ρAB state certification ? P x
a , Qy b(ρAB)
measurement certification Which states can be certified?
Ψ?
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What is self-testing?
- abxy cxy
ab Pr[a, b|x, y] = β
? ρAB state certification ? P x
a , Qy b(ρAB)
measurement certification Which states can be certified?
Ψ?
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What is self-testing?
What is experimentally-relevant? The CHSH inequality: βC = 2 and βQ = 2 √ 2
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What is self-testing?
What is experimentally-relevant? The CHSH inequality: βC = 2 and βQ = 2 √ 2 Non-trivial bounds for... [Bardyn et al. ’09]: β ≥ 1 + √ 2 ≈ 2.41 [McKague et al. ’12]: β ≥ βQ − 2 · 10−5
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What is self-testing?
What is experimentally-relevant? The CHSH inequality: βC = 2 and βQ = 2 √ 2 Non-trivial bounds for... [Bardyn et al. ’09]: β ≥ 1 + √ 2 ≈ 2.41 [McKague et al. ’12]: β ≥ βQ − 2 · 10−5
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What is self-testing?
What is experimentally-relevant? The CHSH inequality: βC = 2 and βQ = 2 √ 2 Non-trivial bounds for... [Bardyn et al. ’09]: β ≥ 1 + √ 2 ≈ 2.41 [McKague et al. ’12]: β ≥ βQ − 2 · 10−5 The loophole-free Bell experiment from Delft β = 2.4 ± 0.2
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What is self-testing?
What is experimentally-relevant? The CHSH inequality: βC = 2 and βQ = 2 √ 2 Non-trivial bounds for... [Bardyn et al. ’09]: β ≥ 1 + √ 2 ≈ 2.41 [McKague et al. ’12]: β ≥ βQ − 2 · 10−5 The loophole-free Bell experiment from Delft β = 2.4 ± 0.2 4 orders of magnitude off!
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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Previous results
Self-testable the singlet [McKague et al. ’12] graph states [McKague ’14] high-dimensional maximally entangled state [Slofstra ’11, Yang, Navascués ’13, McKague ’16, Salavrakos et al. ’16 + . . . ] non-maximally entangled states of 2 qubits [Bamps, Pironio ’15] Only for almost perfect statistics (ε ≈ 10−4).
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Previous results
Self-testable the singlet [McKague et al. ’12] graph states [McKague ’14] high-dimensional maximally entangled state [Slofstra ’11, Yang, Navascués ’13, McKague ’16, Salavrakos et al. ’16 + . . . ] non-maximally entangled states of 2 qubits [Bamps, Pironio ’15] Only for almost perfect statistics (ε ≈ 10−4). Experimentally-relevant robustness a single analytic result for the singlet-CHSH case [Bardyn et
- al. ’09]
swap trick: a numerical method, versatile but computationally expensive (so far up to 4 qubits or 2 qutrits) [Yang et al. ’14, Bancal et al. ’15]
[see arXiv:1604.08176 for references]
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New findings
New approach for analytic self-testing bounds improvement for the CHSH and Mermin3 Mermin3 is actually tight (!) self-testing problem operator inequalities
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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Self-testing from operator inequalities
Extractability of ΨA′B′ from ρAB Ξ(ρAB → ΨA′B′) := maxΛA,ΛB F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
local extraction channels fidelity
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Self-testing from operator inequalities
Extractability of ΨA′B′ from ρAB Ξ(ρAB → ΨA′B′) := maxΛA,ΛB F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
local extraction channels fidelity Obs1: Ξ(ρAB → ΨA′B′) = 1 ⇐ ⇒ ρAB = V (ΨA′B′ ⊗ σA′′B′′)V † for V = VA′A′′→A ⊗ VB′B′′→B
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Self-testing from operator inequalities
Extractability of ΨA′B′ from ρAB Ξ(ρAB → ΨA′B′) := maxΛA,ΛB F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
local extraction channels fidelity Obs1: Ξ(ρAB → ΨA′B′) = 1 ⇐ ⇒ ρAB = V (ΨA′B′ ⊗ σA′′B′′)V † for V = VA′A′′→A ⊗ VB′B′′→B Obs2: Ξ(ρAB → ΨA′B′) ∈ [λ2
max, 1]
largest Schmidt coefficient
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Self-testing from operator inequalities
Idea: measurement operators extraction channels! Analytical bound of [Bardyn et al.] in 2 steps [1] Solve the problem for 2 qbits (local measurements determine a local unitary correction) [2] Use Jordan’s lemma to argue that it holds in arbitrary dimension
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Self-testing from operator inequalities
Idea: measurement operators extraction channels! Analytical bound of [Bardyn et al.] in 2 steps [1] Solve the problem for 2 qbits (local measurements determine a local unitary correction) [2] Use Jordan’s lemma to argue that it holds in arbitrary dimension
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 β 0.25 0.50 0.75 1.00 [C. E. Bardyn, T. C. H. Liew, S. Massar,
- M. McKague, and V. Scarani.
- Phys. Rev. A, 80(6), 2009. arXiv:0907.2170]
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Self-testing from operator inequalities
Refined approach: assume ΛA := ΛA({P x
a })
and ΛB := ΛB({Qy
b}).
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Self-testing from operator inequalities
Refined approach: assume ΛA := ΛA({P x
a })
and ΛB := ΛB({Qy
b}).
for ΨA′B′ pure F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
= (ΛA ⊗ ΛB)(ρAB), ΨA′B′ = ρAB, (Λ†
A ⊗ Λ† B)(ΨA′B′) = tr(KρAB)
for K = (Λ†
A ⊗ Λ† B)(ΨA′B′)
important: K depends only on ΨA′B′, {P x
a }, {Qy b}, not on ρAB!
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Self-testing from operator inequalities
Refined approach: assume ΛA := ΛA({P x
a })
and ΛB := ΛB({Qy
b}).
for ΨA′B′ pure F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
= (ΛA ⊗ ΛB)(ρAB), ΨA′B′ = ρAB, (Λ†
A ⊗ Λ† B)(ΨA′B′) = tr(KρAB)
for K = (Λ†
A ⊗ Λ† B)(ΨA′B′)
important: K depends only on ΨA′B′, {P x
a }, {Qy b}, not on ρAB!
... just like the Bell operator W =
- abxy
cxy
abP x a ⊗ Qy b.
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Self-testing from operator inequalities
Forget the input state ρAB! Want s, µ ∈ R such that K ≥ sW + µ1 holds for all possible measurement operators
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Self-testing from operator inequalities
Forget the input state ρAB! Want s, µ ∈ R such that K ≥ sW + µ1 holds for all possible measurement operators Challenging!... but if works then tr(KρAB) ≥ s tr(WρAB) + µ tr(ρAB) equivalent to F
- (ΛA ⊗ ΛB)(ρAB), ΨA′B′
≥ sβ + µ precisely a (linear) self-testing statement!
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Self-testing from operator inequalities
Main technical challenge: find channels and s, µ such that K ≥ sW + µ1 holds for all possible measurement operators
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Self-testing from operator inequalities
Main technical challenge: find channels and s, µ such that K ≥ sW + µ1 holds for all possible measurement operators Jordan’s lemma: any two binary, projective measurements can be simultaneously block-diagonalised into 2 × 2 blocks (at most) each block parametrised by an angle a ∈ [0, π/2] (up to unitary) this becomes tractable: 1-parameter per party
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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CHSH self-testing: proof in 4 steps
1 Extraction channels: angle-dependent dephasing
full partial none partial full
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CHSH self-testing: proof in 4 steps
1 Extraction channels: angle-dependent dephasing
full partial none partial full
2 Find suitable s, µ (numerics): s = (4 + 5
√ 2)/16 and µ = −(1 + 2 √ 2)/4
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CHSH self-testing: proof in 4 steps
1 Extraction channels: angle-dependent dephasing
full partial none partial full
2 Find suitable s, µ (numerics): s = (4 + 5
√ 2)/16 and µ = −(1 + 2 √ 2)/4
3 Prove
K(a, b) ≥ sW(a, b) + µ1 for all a, b ∈ [0, π/2] (2-parameter family of 4 × 4 matrices)
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CHSH self-testing: proof in 4 steps
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 β 0.25 0.50 0.75 1.00 Bardyn et al. [BLM+09] Bancal et al. [BNS+15] current work
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CHSH self-testing: proof in 4 steps
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 β 0.25 0.50 0.75 1.00 Bardyn et al. [BLM+09] Bancal et al. [BNS+15] current work
non-trivial for β > (16 + 14 √ 2)/17 ≈ 2.11 ?
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Mermin3 self-testing: proof in 4 steps
1 Same extraction channels 2 Find suitable s, µ (numerics): s = (2 +
√ 2)/8 and µ = −1/ √ 2
3 Prove
K(a, b, c) ≥ sW(a, b, c) + µ1 for all a, b, c ∈ [0, π/2] (3-parameter family of 8 × 8 matrices)
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Mermin3 self-testing: proof in 4 steps
2.8 3.0 3.2 3.4 3.6 3.8 4.0 β 0.25 0.50 0.75 1.00 P´ al et al. [PVN14] current work
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Mermin3 self-testing: proof in 4 steps
2.8 3.0 3.2 3.4 3.6 3.8 4.0 β 0.25 0.50 0.75 1.00 P´ al et al. [PVN14] current work
tight :)
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Outline
What is self-testing? Previous results and new findings Self-testing from operator inequalities Two examples: the CHSH and Mermin3 inequalities Summary and future work
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Summary and future work
Summary self-testing from operator inequalities improvements for the CHSH and Mermin3 inequalities first provably tight self-testing statement
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Summary and future work
Summary self-testing from operator inequalities improvements for the CHSH and Mermin3 inequalities first provably tight self-testing statement Future work Merminn
?
= ⇒ GHZn state (preliminary numerics) tilted CHSH
?
= ⇒ non-maximally entangled 2-qubit states [project in progress with Tim Coopmans and Christian Schaffner] beyond Jordan’s lemma? apply this approach to steering?
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