Towards "evidence based physics"

Towards “evidence based physics”

Richard Gill General Colloquium, 19 March 2015

Mathematical Institute Leiden

Rutherford: If you need statistics

you did the wrong experiment

John Stewart Bell (1928–1990)Bell’s theorem 1964–2014

http://www.math.leidenuniv.nl/~gill/einstein.m4a

http://www.math.leidenuniv.nl/~gill/einstein.m4a

> 50 years after Bell (1964) – still no loophole-free experimental proof

of quantum non-locality

– but the experimenters are getting damn close! (thanks to statistics)

Those damned loopholes…• Memory loophole

• Detection efficiency loophole

• Locality loophole

• Conspiracy loophole

• Coincidence loophole

• Finite statistics loophole

Weihs et al. (1998)Coming soon: Delft; Delft + Leiden

5.2. Verletzung der Bell-Ungleichungin einzelnenMessungen

Bob1 , + 1 , – 2 , + 2 , –

Zählungen 104122 100144 93348 908411 , + 77988 313 1728 1636 1791 , – 74935 1978 351 294 11432 , + 75892 418 1683 269 1100A

lice

2 , – 73456 1578 361 1386 156Koinzidenzen

Zeitverschiebung 3,8 ns Quantentheoretische VorhersageKoinzidenzfenster 4,0 nsKoinzidenzen gesamt 14573 ideal bei 94% -97% Kontrastρ(1,1) -0,70 ± 0,01 -0,71 0,68 ± 0,02ρ(2,1) -0,61 ± 0,01 -0,71 0,68 ± 0,02ρ(1,2) 0,71 ± 0,01 0,71 0,68 ± 0,02ρ(2,2) -0,71 ± 0,01 -0,71 0,68 ± 0,02S 2,73 ± 0,02 2,82 2,72 ± 0,04Verletzung 29,8 stddev

Ta belle 5. 3. : A uswertungvonlongdist35 , aufgenommen am 22.4.1998. Zus®atzlichsindnochdiequantentheo-retischenVorhersagenf®ur dieWertederK orrelationsfunktionenangegeben,einmalf®ur ein idealesE xperimenteinmalreduziertum denerzieltenK ontrast ( s. Tabelle5.2) . Weil dieserK ontrast nichtf®ur alle Winkelstellungengleichist,ergebensich gewisseFehler. A uflerdemmufl beimVergleichderDa tenmit denVorhersagenber®ucksichtigtwerden,dafl die im E xperimentrealisiertenAnalysatorwinkelnichtexakt denidealenWinkelnentsprechen.

Beobachter-Stationen die A nalysatorstellungen durcheine zus®atzliche G leichspannungum einenkon-stantenW inkelvon �� rotiert. Wenn dieseR otation bei Bob angewandtwird, dann entsprechen dieverschiedenenM odulatorstellungendenfolgendenW inkeln:

Ali cePolarisatorkanal / M odulatorstellung � � � � � � � �®Aquivalenter An alysatorwinkel � � � � � � � � �

BobPolarisatorkanal / M odulatorstellung � � � � � � � �®Aquivalenter An alysatorwinkel ��

Dies sind bekanntermaflenjeneW inkel, welchenachder Q uantenphysikgeradedie maximaleVer-letzungder CHSH Ungleichungergebensollten. U nsereD atens®atze liefern f®ur denB ell-Parameter ,dessenB etragnach lokal realistischen M odellen kleinerals � seinm®uflte, Wertevon � � � �� (long-dist35) bzw. � � �� (�rstlong3). Die beidenM essungendauertenjeweils s. Die Verletzungder C HSH-Unglei chungbetr®agt im ersten Fall �� im zweiten �� , gemessenin S tandardabwei-chungendes Bell-Parameters . D ie angegebenenFehlerwurdenrein aus denberechneten Fehlern derZ ®ahlraten ermittelt, wie in A bschnitt 5.1 beschrieben,unterder Annahme,dafl es sich um poissonscheZ ®ahlprozessehandelt.

Die wahrenFehlerder Meflgr®ofle sind in unseremFalle schwierig zu ermitteln, denn die wahr-scheinlich gr®oflten Fehlerquellensind die Drif t der Modulatorenund unkorrelierte W inkelfehlerdurch

113

Prob(≠): 0.85, 0.15, 0.81, 0.85Prob(=): 0.15, 0.85, 0.19, 0.15

Weihs et al. (1998)5.2. Verletzung der Bell-Ungleichungin einzelnenMessungen

Bob1 , + 1 , – 2 , + 2 , –

Zählungen 104122 100144 93348 908411 , + 77988 313 1728 1636 1791 , – 74935 1978 351 294 11432 , + 75892 418 1683 269 1100A

lice

2 , – 73456 1578 361 1386 156Koinzidenzen








113

Estimated Prob(≠): 0.85, 0.15, 0.81, 0.85Estimated Prob(=): 0.15, 0.85, 0.19, 0.15

Ursin et al. (2006)

Distance no worries for spooky particles

Stephen Pincock

ABC Science Online

Friday,!8!December!2006

A message sent using entangled, or spooky, particles of light has been beamed across

the ocean (Image: iStockphoto)

Scientists have used quantum physics to zap an encrypted message more than 140 kilometres between two Spanish islands. Professor Anton

Zeilinger from the University of Vienna and an international team of scientists used 'spooky' pulses of light to send the message. They say this is

an important step towards making international communications more secure. Zeilinger described the study this week at the Australian Institute of

Physics meeting in Brisbane. The photons they sent were linked together through a process known as quantum entanglement. This means that

their properties remained tightly entwined or entangled, even when separated by large distances, a property Einstein called spooky. The group's

achievement is important for the emerging field of quantum cryptography, which aims to use properties such as entanglement to send encrypted

messages. Research groups around the world are working in this field. But until now they have only been able to send messages relatively short

distances, limiting their usefulness. Zeilinger's team wants to be able to beam the messages to satellites in space, so they could theoretically be

relayed anywhere on the planet.

To test their system, the team went to Tenerife in the Canary Islands, where the European Space Agency operates a telescope specifically designed

to communicate with satellites. Instead of pointing the telescope at the stars, Zeilinger says, the scientists turned it to the horizontal and aimed it

towards a photon sending station 144 kilometres away on the neighbouring island of La Palma. "Very broadly speaking, we were able to establish

a quantum communication connection," he says. "We worried a lot about whether atmospheric turbulence would destroy the quantum states. But it

turned out to work much better than we feared." The results suggest it should be possible to send encrypted photons to a satellite orbiting 300 or

400 kilometres above the Earth, he says. "This is our hope. We believe that such a system is feasible." The next step is to try the system out with

an actual satellite, a project which is likely to involve the European Space Agency and others. "This is about developing quantum communications

on a grand scale," Zeilinger says. His team expects to publish its results soon.

J.S. Bell, “Bertlmann’s socks and the nature of reality”

Timed settings

Outputs (outcomes)

Journal de Physique, Colloque C2, suppl. au no 3, Tome 42 (1981), C2 41–61

Timed settings α, β; values 1, 2

Outputs: outcomes A, B, values ± 1

Alice’s (binary) measurement outcome registered before Bob’s setting could become available, and vice-versa

Binary settings chosen independently, completely at random

α β

A B

Local realism• Realism: we can define (in the model) what Alice and

Bob’s outcomes would have been, for all possible values of their settings

• Locality: Alice’s outcomes can’t depend on which setting is chosen by Bob, and vice-versa

• Freedom: Alice and Bob have free choice of settings

⇒ existence of counterfactual outcomes A1, A2, B1, B2; observed outcomes A, B determined by independent random

setting choices (A = A1 or A2; B = B1 or B2)

A1

A2

B1

B2A1

A2

B1

B2

Let A1, B1, A2, B2 be members of {"red", "green"}.

Notice that (A1 = B2) & (B2 = A2) & (A2 = B1) ⇒ (B1 = A1)

Therefore (A1 ≠ B2) or (B2 ≠ A2) or (A2 ≠ B1) ⇐ (B1 ≠ A1 )

Therefore Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1) ≥ Prob(B1 ≠ A1)

Just feasible: three LHS probabilities each equal to 25%, RHS equal to 75%

QM allows three LHS probabilities equal to 15%, RHS equal to 85%.

15% = 0.1464…= ½ – ¼ √2; 85% = 0.8536…= ½ + ¼ √2

A1

A2

B1

B2

Bell’s inequality:

Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1) ≥ Prob(B1 ≠ A1)

More Bell inequalities by flipping outcomes, e.g. flipping B1 and B2

Prob(A1 = B2) + Prob(B2 = A2) + Prob(A2 = B1) ≥ Prob(B1 = A1)

More inequalities by picking different side of square …

Theorem (Fine, 1982). Set of 8 inequalities is complete and minimal

A1

A2

B1

B2

STsirelson’s bound

deterministic local-realistic model

S’

4

2

-2

-4

-4 -2 2 4

completelyrandom

22

22-

22- 22

a generalizedBell inequality

another

another

2 x 2 x 2 p x q x r

General case: p parties, each with q measurement settings, each measurement with r outcomes

Generalised Bell inequalities (a cartoon)

How QM can do it

• Random variables σx(1), σy(1), σx(2), σy(2)

• Values ±1

• (σx(1)σy(2)) (σy(1)Sx(2)) (σy(1)σy(2)) = + (σx(1)σx(2))

How QM can do it

• ∃ self-adjoint operators σx(1), σy(1), σx(2), σy(2)

• eigenvalues ± 1

• “(1) operators” commute with “(2) operators”

• “x operator” and “y operator” anti-commute

• (σx(1)σy(2)) (σy(1)σx(2)) (σy(1)σy(2)) = – (σx(1)σx(2))

How QM can do it• (σx(1)σy(2)) (σy(1)σx(2)) (σy(1)σy(2)) = – (σx(1)σx(2))

• (σx(1)σy(2)), (σy(1)σx(2)), (σy(1)σy(2)), (σx(1)σx(2)) do not commute; have eigenvalues ±1

• Find vector as close as possible to an eigenvector with eigenvalue –1 of (σx(1)σy(2)), and of (σy(1)σx(2)), and of (σy(1)σy(2)), and as close as possible to an eigenvector of (σx(1)σx(2)) with eigenvalue +1

• The closest you can get leads to 0.85…

Tsirelson, 1980

Why not better? Information causality!

• Principle of information causality: if Bob sends m bits of information to Alice, then Alice gains at most m bits of information of Bob’s that she didn’t know before

• No action at a distance = no-signalling = information causality with m = 0

• Pawlowski et al. (2009; Nature). (1) QM satisfies the principle of information causality. (2) No physical theory which satisfies the principle of information causality can do better than 0.85…

Why more experiments after Weihs 1998? Or even: after Aspect 1982?

• Memory loophole






5.2. Verletzung der Bell-Ungleichungin einzelnenMessungen

Bob1 , + 1 , – 2 , + 2 , –

Zählungen 104122 100144 93348 908411 , + 77988 313 1728 1636 1791 , – 74935 1978 351 294 11432 , + 75892 418 1683 269 1100A

lice

2 , – 73456 1578 361 1386 156Koinzidenzen








113

Prob(≠): 0.85, 0.15, 0.81, 0.85Prob(=): 0.15, 0.85, 0.19, 0.15

Towards “Evidence based physics”

• Randomisation, “blinding” deals with memory, conspiracy, finite statistics, …

• “Intention to treat principle” deals with detection inefficiency, coincidence loophole, …

• “Experimental unit” is time-slot, not: pair of entangled particles

1(B1 ≠ A1) ≤ 1(A1 ≠ B2) + 1(B2 ≠ A2) + 1(A2 ≠ B1)

Define Alice and Bob’s settings, α, β ∈ {1, 2 }, completely random

Counterfactual outcomes A1, A2, B1, B2 fixed ∈ {–1, +1}

Define Iij = 1(α = i & β = j)

E( I12 1(A1 ≠ B2) + I22 1(A2 ≠ B2) + I21 1(A2 ≠ B1) – I11 1(A1 ≠ B1) ) ≥ 0

Deterministic physics for measurement outcomes; random measurement settings!

Towards “evidence based physics”(1) Randomisation

A1

A2

B1

B2

Theorem (Gill 2003)

• N consecutive trials, new random settings in each trial, arbitrary “local realistic” physics for outcomes (arbitrary memory effects, …)

• Define Z = N12(≠) + N22(≠) + N21(≠) – N11(≠)

• We expect Z to be positive

• Using martingale Hoeffding inequality: for c > 0, Prob( Z / N ≤ – c /√N) ≤ exp(– ½ c2)

QM best (in expectation): Z ≈ (0.15 + 0.15 + 0.15 – 0.85) N / 4

A new generation of experiments – and a priority dispute

• Giustina et al (2013, Nature)

• Christensen et al. (2013, PRL)

Towards “evidence based physics”(2) “Intention to treat principle” Experimental unit = time-slot, not particle pair

Giustina et al. 2013

3/6/2015 Physics - Closing Quantum Loopholes

http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.111.130406 1/1

American Physical Society

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B. G. Christensen et al., Phys. Rev. Lett. (2013)

DetectionLoopholeFree Test of Quantum Nonlocality, and ApplicationsB. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E.Lita, A. Miller, L. K. Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W. Lim, N.Gisin, and P. G. KwiatPhys. Rev. Lett. 111, 130406 (2013)Published September 26, 2013

Synopsis: Closing Quantum Loopholes

The oddness of quantum mechanics came to the fore in an epic faceoff between Einstein and Bohr in the 1930s. On one side,Einstein said that quantum mechanics discards accepted notions of localism (a measurement couldn’t be influenced by a remoteexperiment) and realism (outcomes of measurements only depend on the thing being measured, not on the measurer). On theother side, Bohr said that’s right, and you’d better get used to it. Writing in Physical Review Letters, Bradley Christensen at theUniversity of Illinois at UrbanaChampaign and colleagues report an experiment that solves one of the major experimentallimitations that have denied a definitive win for quantum mechanics.

In the 1960s, John Bell put the debate on a quantitative footing by publishing mathematical inequalities that, if violated, wouldconclusively rule out any classical theory, local and realistic, as an alternative to quantum mechanics and thus explainobservations like entanglement. Many experiments have shown that quantum systems violate Bell’s inequalities, but they are allhampered by at least one of two loopholes. One, the locality loophole, might allow some kind of information to travel from oneparticle to another. The second is the detection loophole: if the measurement uses a lowefficiency detector, then you might notbe sampling all the entangled photons properly.

A recent paper by another group claiming to have closed the detection loophole is disputed by Christensen et al., who havecarried out new measurements with a highefficiency source of entangled photons and highefficiency detectors. This eliminatesassumptions about properly sampling the entangled photons and, the authors say, firmly closes the door on the detectionloophole. And it’s not just of academic interest: Bell tests are currently used to verify the security of proposed quantumcommunications systems, so ensuring their reliability is important. – David Voss

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Christensen et al. 2013

Christensen et al. 2013

(H þ V=H " V) basis and a canonical Clauser-Horne-Shimony-Holt Bell violation [30] of 2:827# 0:017—within error of the maximum violation allowed by quantummechanics (2

ffiffiffi2

p$ 2:828). These values are on par with the

highest reported violations of Bell’s inequality everreported [31], but unlike all previously reported resultsinclude no accidental subtraction or postprocessing of anykind. As a result, this source provides not only the bestexperimental evidence to date that local realistic theoriesare not viable, but also provides the best test so far of theupper limits for quantum correlations; some superquantumtheories [32] actually predict that the upper limit for theBellinequality can be greater than 2

ffiffiffi2

p, a prediction constrained

by the results reported here [19].The high entanglement quality, along with the detection-

loophole-free capability, offers interesting possibilities forapplications, notably for device-independent quantuminformation processing. Here the goal is to implement acertain protocol, and to guarantee its security, without rely-ing on assumptions about the internal functioning of thedevices used in the protocol. Being device independent, thisapproach is more robust to device imperfections comparedto standard protocols, and is in principle immune to side-channel attacks (which were shown to jeopardize the secur-ity of some experimental quantum cryptography systems).

One prominent example is device-independent random-ness expansion (DIRE) [33–36]. By performing local

measurements on entangled particles, and observing non-local correlations between the outcomes of these measure-ments, it is possible to certify the presence of genuinerandomness in the data in a device-independent way.DIRE was recently demonstrated in a proof-of-principleexperiment using entangled atoms located in two trapsseparated by 1 m [33]; however, the resulting 42 randombits required a month of data collection. Here we show thatour setup can be used to implement DIRE much moreefficiently. The intrinsic randomness of the quantum sta-tistics can be quantified as follows. The probability for anyobserver (hence, for any potential adversary) to guess themeasurement outcome (of a given measurement setting) isbounded by the amount of violation of the CH inequality:

pguess % ½1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2" ð1þ 2BÞ2

p)=2 [33], neglecting finite

size effects. In turn, this leads to a bound on the min-entropy per bit of the output data, Hmin ¼ "log2ðpguessÞ.Finally, secure private random bits can be extracted fromthe data (which may, in general, not be uniformly random)using a randomness extractor [37]. At the end of the pro-tocol, a final random bit string of length L $ NHmin " S isproduced, where N is the length of the raw output data andS includes the inefficiency and security overhead of theextractor.Over the 4450 measurement blocks (each block

features 25 000 events), we acquire 111 259 682 datapoints for 3 h of data acquisition. The average CH viola-tion of B ¼ 5:4+ 10"5 gives a min-entropy of Hmin ¼7:2+ 10"5. Thus, we expect ,8700 bits of private ran-domness, of which one could securely extract at least 4350bits [19]. The resultant rate (0:4 bits=s) improves by morethan 4 orders of magnitude over the bit rate achieved in[33] (1:5+ 10"5 bits=s). This shows that efficient andpractical DIRE can be implemented with photonicsystems.We have presented a new entangled photon pair creation,

collection, and detection apparatus, where the high systemefficiency allowed us to truly violate a CH Bell inequalitywith no fair-sampling assumption (but still critically rely-ing on the no-signaling assumption that leaves the causalityloophole open). Because photonic entanglement is particu-larly amenable to the types of fast, random measurementand distributed detection needed to close the locality loop-hole, this experiment (together with efforts by other groups[8,12,16]) represents the penultimate step towards a com-pletely loophole-free test of Bell’s inequality and advanceddevice-independent quantum communication applications;here, we demonstrated production of provably securerandomness at unprecedented rates. Finally, our highsource quality enables the best test to date of the quantummechanical prediction itself.This research was supported by the DARPA InPho

program, U.S. Army Research Office AwardNo. W911NF-10-1-0395, NSF PHY 12-05870, the NISTQuantum Information Science Initiative, the Swiss NSF

FIG. 2 (color online). A plot of the Bell parameter B0 [Eq. (3)]as a function of the produced entangled state. B0 > 1 (red dashedline) is not possible for any local realistic theory. Data points in

black are the measured B0 as r is varied in the state ðrjHHiþjVViÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p; r ¼ 1 (0) corresponds to a maximally entangled

(separable) state. For this plot, every data point was measured for30 sec at each measurement setting; the particular settings wereoptimally chosen based on the model of our source for eachvalue of !. The blue line represents the Bell parameter we expectfrom the model of our source. Here, to improve the statistics, wedid not pulse our source with the Pockels cell. We see violationsfor 0:20< r < 0:33.

PRL 111, 130406 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

27 SEPTEMBER 2013

130406-4

(H þ V=H " V) basis and a canonical Clauser-Horne-Shimony-Holt Bell violation [30] of 2:827# 0:017—within error of the maximum violation allowed by quantummechanics (2

ffiffiffi2

p$ 2:828). These values are on par with the

highest reported violations of Bell’s inequality everreported [31], but unlike all previously reported resultsinclude no accidental subtraction or postprocessing of anykind. As a result, this source provides not only the bestexperimental evidence to date that local realistic theoriesare not viable, but also provides the best test so far of theupper limits for quantum correlations; some superquantumtheories [32] actually predict that the upper limit for theBellinequality can be greater than 2

ffiffiffi2

p, a prediction constrained

by the results reported here [19].The high entanglement quality, along with the detection-

loophole-free capability, offers interesting possibilities forapplications, notably for device-independent quantuminformation processing. Here the goal is to implement acertain protocol, and to guarantee its security, without rely-ing on assumptions about the internal functioning of thedevices used in the protocol. Being device independent, thisapproach is more robust to device imperfections comparedto standard protocols, and is in principle immune to side-channel attacks (which were shown to jeopardize the secur-ity of some experimental quantum cryptography systems).

One prominent example is device-independent random-ness expansion (DIRE) [33–36]. By performing local

measurements on entangled particles, and observing non-local correlations between the outcomes of these measure-ments, it is possible to certify the presence of genuinerandomness in the data in a device-independent way.DIRE was recently demonstrated in a proof-of-principleexperiment using entangled atoms located in two trapsseparated by 1 m [33]; however, the resulting 42 randombits required a month of data collection. Here we show thatour setup can be used to implement DIRE much moreefficiently. The intrinsic randomness of the quantum sta-tistics can be quantified as follows. The probability for anyobserver (hence, for any potential adversary) to guess themeasurement outcome (of a given measurement setting) isbounded by the amount of violation of the CH inequality:

pguess % ½1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2" ð1þ 2BÞ2

p)=2 [33], neglecting finite

size effects. In turn, this leads to a bound on the min-entropy per bit of the output data, Hmin ¼ "log2ðpguessÞ.Finally, secure private random bits can be extracted fromthe data (which may, in general, not be uniformly random)using a randomness extractor [37]. At the end of the pro-tocol, a final random bit string of length L $ NHmin " S isproduced, where N is the length of the raw output data andS includes the inefficiency and security overhead of theextractor.Over the 4450 measurement blocks (each block

features 25 000 events), we acquire 111 259 682 datapoints for 3 h of data acquisition. The average CH viola-tion of B ¼ 5:4+ 10"5 gives a min-entropy of Hmin ¼7:2+ 10"5. Thus, we expect ,8700 bits of private ran-domness, of which one could securely extract at least 4350bits [19]. The resultant rate (0:4 bits=s) improves by morethan 4 orders of magnitude over the bit rate achieved in[33] (1:5+ 10"5 bits=s). This shows that efficient andpractical DIRE can be implemented with photonicsystems.We have presented a new entangled photon pair creation,

collection, and detection apparatus, where the high systemefficiency allowed us to truly violate a CH Bell inequalitywith no fair-sampling assumption (but still critically rely-ing on the no-signaling assumption that leaves the causalityloophole open). Because photonic entanglement is particu-larly amenable to the types of fast, random measurementand distributed detection needed to close the locality loop-hole, this experiment (together with efforts by other groups[8,12,16]) represents the penultimate step towards a com-pletely loophole-free test of Bell’s inequality and advanceddevice-independent quantum communication applications;here, we demonstrated production of provably securerandomness at unprecedented rates. Finally, our highsource quality enables the best test to date of the quantummechanical prediction itself.This research was supported by the DARPA InPho

program, U.S. Army Research Office AwardNo. W911NF-10-1-0395, NSF PHY 12-05870, the NISTQuantum Information Science Initiative, the Swiss NSF

FIG. 2 (color online). A plot of the Bell parameter B0 [Eq. (3)]as a function of the produced entangled state. B0 > 1 (red dashedline) is not possible for any local realistic theory. Data points in

black are the measured B0 as r is varied in the state ðrjHHiþjVViÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p; r ¼ 1 (0) corresponds to a maximally entangled

(separable) state. For this plot, every data point was measured for30 sec at each measurement setting; the particular settings wereoptimally chosen based on the model of our source for eachvalue of !. The blue line represents the Bell parameter we expectfrom the model of our source. Here, to improve the statistics, wedid not pulse our source with the Pockels cell. We see violationsfor 0:20< r < 0:33.

PRL 111, 130406 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

27 SEPTEMBER 2013

130406-4

Application: DIRE Device Independent Randomness Extraction

Loopholes

Giustina et al forgot about coincidence loophole. On re-analysing data appropriately, experiment fortunately still successful! (Larsson et al, 2014, PRA)

• Memory loophole






What is the experimental unit?

• Physicist’s answer: the entangled pair of particles

• Statistician’s answer: the time-slot!

• Local coarse-graining forces binary outcome per time-slot

• e.g.: if multiple events, take first; merge “–1” and “no event”

The coincidence loopholeEvents paired because detection times close enough

Events not paired because detection times in different time-slots

The problem with Giustina et al: initially analysed as above.

Fortunately, after reanalysis as below, still a significant violation of Bell inequality

Alice

Bob

Alice

Bob

The detection loopholeBy imposing grid of fixed time-slots (preferably: new random settings per time slot) we have an experiment with (at least) ternary outcome: {“red”, “green”, “no detection”}

Classical solution: discard time-slots without “coincident events”

Correct solution: coarse-grain locally – also for “resolving” multiple events in one time-slot

Alice

Bob

Christensen vs Giustina11 12 21 22

dd 1054015 1140690 1178589 124696

dn 467388 377147 3509092 4557443

nd 639434 3336595 508696 4344146

nn 297803163 295109568 294767623 290937715

tot 299964000 299964000 299964000 299964000

11 12 21 22dd 0.003513804989932130.003802756330759690.003929101492179060.00041570321771946

dn 0.001558146977637320.00125730754357190.011698377138590.0151932998626502

nd 0.002131702470963180.01112331813150910.001695856836153670.0144822245336107

nn 0.9927963455614670.9838166179941590.9826766645330770.96990877238602

11 12 21 22dd 29173 34146 34473 1869

dn 16897 13931 116364 148844

nd 17029 112097 12975 142502

nn 27089919 28192171 27663499 27633773

tot 27153018 28352345 27827311 27926988

11 12 21 22dd 0.00107439254082180.001204344825798360.001238818943016090.0000669245104412979

dn 0.0006222881007186750.0004913526553094640.004181647303255420.00532975485934967

nd 0.00062714943878430.003953711765287840.0004662685517835340.00510266270032415

nn 0.9976761699196750.9943505907536040.9941132652019450.989500657929885

> (sum(chris[2:3, 4]) - sum(chris[2:3, 1:3])) / sqrt(sum(chris[2:3, ]))[1] 2.694237> (sum(giust[2:3, 4]) - sum(giust[2:3, 1:3])) / sqrt(sum(giust[2:3, ]))[1] 15.01396

Giustina et al: 5 times as many standard deviations (10 times as much data)

Christensen et al: 20 runs with total of 6000 random setting changes Giustina et al: four runs with fixed settings

Giustina et al: Prob(A1 ≠ B1), Prob(A1 ≠ B2), Prob(A2 ≠ B1), Prob(A2 ≠ B2)

37, 124, 134, 297 per thousand

37 + 124 + 134 = 295 < 297

Christensen et al: Prob(A1 ≠ B1), Prob(A1 ≠ B2), Prob(A2 ≠ B1), Prob(A2 ≠ B2)

12, 44, 46, 104 per 10 thousand

12 + 44 + 46 = 102 < 104

Why not 15%, 15%, 15%, 85% ?

Why not 15%, 15%, 15%, 85%?

• “Detector efficiency” & “Less is more”

• At QM’s best, we would need somewhat better detectors than when we’re rather far from QM’s best! We don’t have them yet.

• A much less entangled state turns out to be much less sensitive to imperfections in measurement!

• Giustina, Christensen aimed at best quantum state given their available detector efficiency

An infinite family of Bell inequalities

Remember Bell inequality Prob(B1 ≠ A1) ≤ Prob(A1 ≠ B2) + Prob(B2 ≠ A2) + Prob(A2 ≠ B1)

Rewrite as:

Prob(A ≠ B | 11) ≤ Prob(A ≠ B | 12) + Prob(A ≠ B | 21) + Prob(A ≠ B | 22)

We also have the “no-signalling” equalities

Prob(A = “red” | 11) = Prob(A = “red” | 12) Prob(A = “red” | 21) = Prob(A = “red” | 22) Prob(B = “red” | 11) = Prob(B = “red” | 21) Prob(B = “red” | 12) = Prob(B = “red” | 22)

Take linear combinations to get new inequalities Some of the resulting family have names: Clauser-Horne, Eberhard

A statistician easily computes the optimal test (= Bell inequality minus linear combination of no-signalling equalities, with smallest variance)

It seems that Giustina et al. and Christensen et al. each used a close to optimal test!

Conclusion

• Maybe in a year or two, we will have a definitive (positive outcome) experiment

• Maybe it will be done in Delft

• The physicists have learnt that they do need sharp statistical thinking for this experiment

• Accardi contra Bell (cum mundi): The Impossible Coupling. Växjö proceedings; quant-ph/0110137.

• Comment on "Exclusion of time in the theorem of Bell" by K. Hess and W. Philipp. With Gregor Weihs, Anton Zeilinger and Marek Zukowski; Europhysics Letters; quant-ph/0204169.

• No time loophole in Bell's theorem; the Hess-Philipp model is non-local. With Gregor Weihs, Anton Zeilinger and Marek Zukowski; PNAS; quant-ph/0208187.

• Time, Finite Statistics, and Bell's Fifth Position. Växjö proceedings; quant-ph/0301059.

• The statistical strength of nonlocality proofs. With Wim van Dam and Peter Grunwald; IEEE-IT; quant-ph/0307125.

• Bell's inequality and the coincidence-time loophole. With Jan-Ake Larsson; Europhysics Letters; quant-ph/0312035.

• Optimal Bell tests do not require maximally entangled states. With Antonio Acin and Nicolas Gisin; PRL; quant-ph/0506225.

• Maximal violation of the CGLMP inequality for infinite dimensional states. With Stefan Zohren, Paul Reska, and Willem Westra; PRL; quant-ph/0612020.

• A tight Tsirelson inequality for infinitely many outcomes. With Stefan Zohren; Europhysics Letters; arXiv: 1003.0616.

Survey paper by yours truly: Statistics, Causality and Bell’s Theorem

Statistical Science (2014)

Towards "evidence based physics"

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