M. Markham, D.J. Twitchen

C. Abellán, W. Amaya, V. Pruneri, M.W. Mitchell

B. Hensen, H. Bernien, A.E. Dréau, A. Reiserer, N. Kalb, M.S. Blok, J. Ruitenberg, R.F.L. Vermeulen, R.N. Schouten, D. Elkouss, S. Wehner, T.H. Taminiau, R. Hanson

Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km arXiv:1508.05949

• Introduction • Setup at TU Delft • Data evaluation • Discussion

• Introduction • Setup at TU Delft • Data evaluation • Discussion

Bell’s theorem

1. 2. 3.

Communication is limited to speed of light Settings a,b can be freely chosen Local hidden variable model (LHVM) governs the experiment

3’. Quantum theory predicts

Bell’s game Game score 1 if −1 𝑎𝑎𝑎𝑎 𝑥𝑥𝑥𝑥 = 1 C= 0 else 1. 2. 3.

Communication is limited to speed of light Settings a,b can be freely chosen uniformly at random Local hidden variable model (LHVM) governs the experiment

P(𝐶𝐶 = 1) ≤ 3/4

3’. Quantum theory predicts

P(𝐶𝐶 = 1) ≤(√2+2)/4≈0.85

Unless…

Locality loophole

Time t

x ∈ {−1,1}

A

a ∈ {0,1}

y ∈ {−1,1} Distance L

B

b ∈ {0,1}

- If L is so short that Alice can signal to Bob before y is produced then y=f(a,b) - Strategy with winning probability one: x=1, 𝑦𝑦 = −1

𝑎𝑎𝑏𝑏

Detection loophole

A

a ∈ {0,1}

y ∈ {−1,1} Distance L

B

b ∈ {0,1}

- Outcomes are not always conclusive  data discarded - Strategy with winning probability one: if a=0 output x=1, if b=0 output y=1 else no output

efficiency η

Time t

x ∈ {−1,1}

Bell experiments Locality loophole addressed (photons)

Aspect (1982), Weihs (1998), Tittel (1999)

Detection loophole closed (ions, atoms, superconducting qubits, photons)

Rowe (2001), Matsukevich (2008), Ansmann (2009), Giustina (2013), Christensen (2013)

Detection loophole closed AND locality loophole addressed = “loophole-free” Bell test

• Introduction • Setup at TU Delft • Data evaluation • Discussion

Time

Setup for closing the loopholes

Space

Bell, J. Phys. (Paris) Colloq. C2, 41 (1981).

GO / NOT GO

𝑥𝑥 ∈ {−1,1}

𝑥𝑥 ∈ {−1,1}

Matter qubits: “easy” high fidelity readout

𝑎𝑎 ∈ {0,1}

L=? limited by 3 us readout  L>1 km

𝑏𝑏 ∈ {0,1}

Phys. Rev. Lett. 91, 110405 (2003).

Loophole-free Bell test campus ✗

Bob lab Alice lab ✗

Beamsplitter lab

Experimental scheme Alice

time

space

Beam splitter station

Bob

Experimental scheme Initialization Initialization of the qubits into a well defined state Nature 477, 574 (2011)

time

space

Alice

Beam splitter station

Bob

Experimental scheme Spin-photon entanglement Create spin-photon entanglement on both sides. Send photons to beam splitter.

Alice

Beam splitter station

Bob

Experimental scheme Alice Entanglement generation

* click!

Photon detection projects spins into entangled state Barrett and Kok, PRA 71, 060310 (2005) Nature 497, 86 (2013)

time

space

Beam splitter station

Bob

Experimental scheme Random basis choice

1! Fresh Quantum Random Number Generator Optics Express 22, 1645 (2014) arXiv:1506.02712 (2015)

time

space

Alice

Beam splitter station

Bob

Experimental scheme Readout Distance such that signaling is not possible:

Alice

Beam splitter station

Bob

end

Detection loophole closed Nature 477, 574 (2011)

start

• Introduction • Setup at TU Delft • Data evaluation • Discussion

Statistical analysis LHVM imply limitations in probability or expectation

Finite number of events

Hypothesis test - Null hypothesis: the experiment was governed by a LHVM - p-value: Probability of data at least as extreme as that observed given a LHVM maximized over all LHVMs. - p-value is not the probability that our experiment is ruled by a LHVM

Rigorous bound • No unfounded assumptions: i.i.d. • No approximations: Gaussian

Gill, Found. of Prob. and Phy., 179-206 (2003) Zhang, Glancy, Knill, PRA 84 062118 (2011) Bierhorst, Found. Phys. 44, 736-761 (2014)

Challenges for modelling the experiment t=1(YES) or 0(NO)

T

go?

1 if t −1 C= 0 else

𝑎𝑎𝑎𝑎 𝑥𝑥𝑥𝑥

=1

Local hidden variable model (LHVM) governs the experiment: 1. Setting choices are independent conditioned on the history 2. Inputs have a bias from uniform 𝝉𝝉 (𝟏𝟏𝟏𝟏−𝟓𝟓 ) 3. Settings and outcomes at both sites are conditionally independent on the history 4. Arbitrary statistics at heralding station

P(𝐶𝐶 = 1) ≤ 3/4+3(𝜏𝜏 + 𝜏𝜏 2 )

Upper bound on the p-value Experimental Data - n = number of GO rounds - k = number of wins - 𝜏𝜏 = bias of the RNGs

p-value ≔ max 𝑃𝑃 𝑎𝑎𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑘𝑘 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖 𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿) 𝐿𝐿𝐻𝐻𝐻𝐻𝑀𝑀

Remarks - Upper bound is the tail of an i.i.d. distribution -

Elkouss, Wehner in preparation (2015)

Evaluation of the experiment Gathered data - 1 event per hour - 245 events over more than 220 hours - 196 “wins” over the 245 events

How do we interpret it?

Hypothesis test - Null hypothesis: the experiment was governed by a LHVM - p-value: probability of data at least as extreme as that observed given a LHVM maximized over all LHVMs.

p-value ≔ max 𝑃𝑃 𝑎𝑎𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 196 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖 245 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿) ≤ 0.039 𝐿𝐿𝐻𝐻𝐻𝐻𝑀𝑀

• Introduction • Setup at TU Delft • Data evaluation • Discussion

Conclusions

First loophole-free Bell inequality violation arXiv:1508.05949 (2015)

Next: • Independent confirmation • Proof-of-principle of device-independent security … • Quantum networks

Thank you!

どうもありがとうございます！

## march meeting

Experimental loophole-free violation of a Bell inequality ... Communication is limited to speed of light. 2. Settings a,b can ... addressed = âloophole-freeâ Bell test.

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