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Experimenter‘s Freedom in Bell‘s Theorem and Quantum Cryptography

Johannes Kofler, Tomasz Paterek, and Časlav Brukner

Non-local Seminar Vienna–BratislavaVienna, February 3rd 2006

archiv:quant-ph/0510167, accepted for Phys. Rev. A

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Contents

1. Bell’s Theorem

2. Local Realism and the Freedom of Choice

3. The CHSH and the Mermin Inequality

4. Quantum Cryptography

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Bell‘s Theorem

Locality: The result of a measurement performed on system A is unaffected by operations on a space-like separated system B

Realism: Measurement results are determined by “hidden variables” which exist prior to and independent of observation

Bell’ Theorem [Bell (1964)]:

Local realism is in conflict with quantum mechanics

Famous experiments: Freedman and Clauser (1972)

Clauser and Horne (1974)

Aspect (1981, 1982)

Tittel et al. (1997)

Weihs et al. (1998)

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Loopholes in Bell Tests

1. Fair Sampling: the detected pairs are statistically significant (fair)representatives of all the emitted pairs

experimental problem: detection efficiency

2. Locality: no causal mechanism whatsoever can bring information from one side to the other

experimental problem: space-like separation

3. Freedom: the experimenter is free to choose the measurement settings

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Agenda

- assume local realism

- for consistency with experiments: give up freedom

- characterize (quantify) the “insane” consequences of such a program

- show consequences in quantum cryptography

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Freedom of Choice

- 2 separated partners, A and B, space-like separated experiments

- A: settings: k = 1,2 B: settings: l = 1,2

outcomes: X = +1,–1 outcomes: Y = +1,–1

- measured probability for correlation: P(X=Y|kl)

- local realism assumes a quadruple q = {X1,X2,Y1,Y2} produced by the source for each run (pair), existing independently of whether any or which measurements are performed

X {X1,X2}, Y {Y1,Y2}

- mathematical probability for correlation: P(Xk=Yl)

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Freedom of Choice

- measured probability for correlation: P(X=Y|kl)

- mathematical probability for correlation: P(Xk=Yl)

Freedom: [Gill et al. (2003)]

The setting choice (k,l) is statistically independent of

the quadruple q = {X1,X2,Y1,Y2}

in many thought repetitions of the experiment the probability of every possible value of q remains the same for any choice of the settings

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Lack of Freedom

Imagine: freedom is an illusion

both choice of settings and results are consequences of a common local realistic mechanism

experimenter’s choice is determined in advance

or, e.g., the parity of the number of cars passing the laboratory within the next n seconds, where n is given by the cube of the fourth decimal of the actual temperature in degrees Fahrenheit, is correlated with the local realistic source which emits the particles

Measure of the lack of freedom:

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The CHSH Inequality

The mathematical probabilities satisfy a set-theoretical constraint:

CHSH inequality [Clauser et al. (1969)]:

In terms of measured probabilities, with

whereadapted bound

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CHSH Inequality with No Freedom

Example: Local realistic mechanism with no freedom:

- source “knows” (in advance) the measurement settings of A and B

- whenever A and B both (will) measure their second setting (k = 2, l = 2), the source sends perfectly anti-correlated pairs:

P(X=Y|22) = 0

- in the other three cases it sends perfectly correlated pairs

P(X=Y|11) = P(X=Y|12) = P(X=Y|21) = 1

- then the logical bound of 3 can be reached:

still satisfied as

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CHSH Inequality with total Freedom

Example: Local realistic mechanism with total freedom

- complete freedom

- the two inequalities become identical

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Abandonment of Freedom

- consider experiment with maximally entangled state

- for perfect setting angles the measured CHSH expression can become

- therefore, to keep a local realistic view which is in agreement with the experimental results, we have to have

- hence the freedom has to be restricted (abandoned) at least by the amount

- if we assume

then

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Generalization to N parties

- N space-like separated parties

- settings: kj {1,2}

outcomes: X(j) {1,–1}

- Local realism assumes existence of {X1(1),X2

(1),…,X1(N),X

2(N)}

- mathematical probability for correlation

- measured probability for correlation

- lack of freedom

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Mermin Inequality

The mathematical probabilities satisfy a set-theoretical constraint, equivalent to the Mermin inequality [Mermin (1990), Żukowski et al. (2002)]:

In terms of the measured probabilities:

with

adapted bound

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Abandonment of Freedom

- take maximally entangled N-party GHZ state

- gap between non-adapted bound and experimental result increases exponentially with the number of partners

- to explain experimental result local realistically, the freedom of each party has to be abandoned by the (exponentially fast saturating) amount

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Intermediate Summary

Local realistic picture with restricted experimenter’s freedom

- local realistic bound increases

- why is the (experimental) bound in the CHSH inequality 2.414 and not, e.g., the logical bound 3?

- we had to introduce purely theoretical and experimentally not accessible entities, i.e., the mathematical probabilities P(Xk=Yl)

- Occam’s razor?

Popper’s falsifiability principle?

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Quantum Cryptography

Motivation

- the violation of Bell’s inequality is necessary and sufficient for efficient extraction of a secret key [Gisin et al. (2002)]

- if an eavesdropper Eve has (partial) knowledge about the settings of Alice and Bob (e.g., bad random-number generator), she can simulate a violation of Bell’s inequality and successfully eavesdrop [Hwang (2005)]

- Connection:

lack of freedom in a local realistic picture

is equivalent to the fact that an

eavesdropper has partial setting knowledge in a quantum experiment

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The BBM–CHSH Protocol

(1,1), (2,2) key establishing measurement

(1,3), (2,3), (14), (2,4) CHSH measurement

orthogonal combinations ignore

CHSH: ?

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Eve’s Setting Knowledge

- source sends a (maximally entangled)singlet pair in each run

- Eve has partial setting knowledge

- model:

before each run Eve gets the probabilities for all 8 settings to be chosen

qij is the probability for (i,j) with i = 1,2; j = 1,2,3,4

- for simplicity:

one setting always has high probability: Q ≥ 1/8

the others have equal low probability: (1–Q)/7

- Q = 1: E: total knowledge, A and B: no freedom

Q = 1/8: E: no knowledge, A and B: total freedom

- Lack of freedom:

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Eve’s Attack Algorithm

- if qij = Q, Eve attacks with the product state

maximizes: P(X=–Y|13) 1

P(X=–Y|23) 1

P(X=–Y|24) 1

know the key for 11 and 22

- exception: q14 = Q

minimizes P(X=–Y|14) 0

- Eve always sends product states: local realism with restricted freedom

or

or

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CHSH Violation

- CHSH can be violated for Q > 0.44

- but what about the secret key?

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Bit Error Rate and Mutual Informations

- bit error rate

- mutual informations

- best mutual information for a givenD under the condition of no settingknowledge

- secret key agreement iff (never fulfilled)

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- Secret key extraction is never possible:

- Critical error rate

D0 = 14.6 % at Q = Q0 = 0.63

- Q > Q0

protocol insecure: IAB ≤ IBE and

D < D0 and

CHSH violated

- Q ≤ Q0

protocol secure: IAB < IBE but

D ≥ D0

Alice and Bob will not use their key because they find D ≥ D0

D > D0 D < D0

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Consequences

- the one-to-one correspondence between secret key extraction and violation of a Bell’s inequality is lost

- depending on the amount of setting knowledge (Q) which is leaking out of the laboratories of A and B, they have to calculate (via an optimization procedure over all possible attacks) a new bound for CHSH

- a violation of this adapted bound corresponds again to the possibility of secret key extraction (as this is equivalent to a violation of the “original” CHSH expression)

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Conclusions

- Bell’s theorem

a local realistic description of the world can only be maintained if the experimenter’s freedom is partly abandoned

- we quantified the degree of the lack of freedom for the CHSH and the Mermin inequality

- quantum cryptography:

the lack of freedom in a local realistic world is equivalent to a situation in which the eavesdropper has setting knowledge

- if a certain knowledge threshold is beaten, the eavesdropper can find out the key without being revealed (neither by the error rate nor by the CHSH inequality)

- the one-to-one correspondence between secret key extraction and violation of the CHSH inequality can only be restored by adapting the bound of the inequality