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Games and Monogamy in the relativistically causal correlations

Micha l Kamo´ n

Gda´ nsk University of Technology National Quantum Information Center, University of Gda´ nsk

49 Symposium on Mathematical Physics, Toru´ n 17-18 June 2017

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Games and Monogamy in the relativistically causal correlations∗

∗Paper in preparation

In cooperation with:

Roberto Salazar1,2, Dardo Goyeneche4,5, Karol Horodecki3, Debashis Saha1,2 Ravishankar Ramanathan6, Pawel Horodecki2,4,

1) Institute of Theoretical Physics and Astrophysics, 2) National Quantum Information Centre, 3) Institute of Informatics Faculty of Mathematics, Physics and Informatics, University of Gda´ nsk, 80-308 Gda´ nsk, Poland 4) Faculty of Applied Physics and Mathematics, Gda´ nsk University of Technology, 80-233 Gda´ nsk, Poland 5) Institute of Physics, Jagiellonian University, 30-059 Krak´

• w

6) Laboratoire d’Information Quantique, Universit´ e Libre de Bruxelles, Belgium Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Games and Monogamy in the relativistically causal correlations

Presentation Plan:

1 Local deterministic, quantum and no-signaling measurements 2 Correlations of measurements outcomes 3 Polytope of RC correlations 4 Nonlocal games in RC 5 Security against RC eavesdropper 6 Conclusions Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Local deterministic, quantum and no-signaling measurements

General setting

1 [n] = {1, . . . , n} space-like separated parties 2 x = {x1, . . . , xn} inputs strings 3 a = {a1, . . . , an} outputs strings 4 P(a|x) joint probability of a given x Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Local deterministic, quantum and no-signaling measurements

General setting

1 [n] = {1, . . . , n} space-like separated parties 2 x = {x1, . . . , xn} inputs strings 3 a = {a1, . . . , an} outputs strings 4 P(a|x) joint probability of a given x

Local determinism

1 Each observable has predefined outcome 2 Space-like separation of subsystems

⇓ measurements independence

P(a|x)= λ π(λ)P(a1|x1,λ)P(a2|x2λ)...P(an|xn,λ)

Quantum measurements

1 Joint state represented by density matrix ρ 2 Measurement is given by Hermitian operator A P(a|x)=Tr(Ax ρa)

No-signaling

• None measurements xS = {xi}i∈S

performed locally by any subset of parties S ⊆ [n] can influence measurement statistics of other parties Sc, i.e.:

P(aSc |xSc )= a′S P(a′|x′)= a′′S P(a′′|x′′)

for all a′, a′′ with a′

Sc = a′′ Sc = aSc and for

all x′, x′′ with x′

Sc = x′′ Sc = xSc . Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Local deterministic, quantum and no-signaling measurements

General setting

1 [n] = {1, . . . , n} space-like separated parties 2 x = {x1, . . . , xn} inputs strings 3 a = {a1, . . . , an} outputs strings 4 P(a|x) joint probability of a given x

Local determinism

1 Each observable has predefined outcome 2 Space-like separation of subsystems

⇓ measurements independence

P(a|x)= λ π(λ)P(a1|x1,λ)P(a2|x2λ)...P(an|xn,λ)

Quantum measurements

1 Joint state represented by density matrix ρ 2 Measurement is given by Hermitian operator A P(a|x)=Tr(Ax ρa)

No-signaling

• None measurements xS = {xi}i∈S

performed locally by any subset of parties S ⊆ [n] can influence measurement statistics of other parties Sc, i.e.:

P(aSc |xSc )= a′S P(a′|x′)= a′′S P(a′′|x′′)

for all a′, a′′ with a′

Sc = a′′ Sc = aSc and for

all x′, x′′ with x′

Sc = x′′ Sc = xSc .

Figure: Schematic representation of space of local deterministic L,

quantum Q and no-signaling N S sets. J. Barrett et al. Phys. Rev. A 71, 022101 (2005). Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Correlations of measurements

• utcomes

“Boxes” - families of joint probability distributions {P(a|x)}a,x Two parties, two inputs, two outputs (2,2,2)-scenario

CHSH:=A0B0+A0B1+A1B0−A1B1

Figure: Schematic presentation of values of CHSH-like game

within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016).

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Correlations of measurements

• utcomes

“Boxes” - families of joint probability distributions {P(a|x)}a,x Two parties, two inputs, two outputs (2,2,2)-scenario

CHSH:=A0B0+A0B1+A1B0−A1B1

Figure: Schematic presentation of values of CHSH-like game

within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/

Relativistic causality1

Main assumption: No causal loops!

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016).

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Correlations of measurements

• utcomes

“Boxes” - families of joint probability distributions {P(a|x)}a,x Two parties, two inputs, two outputs (2,2,2)-scenario

CHSH:=A0B0+A0B1+A1B0−A1B1

Figure: Schematic presentation of values of CHSH-like game

within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/

Relativistic causality1

Main assumption: No causal loops! Figure: Violation of causality by “point to point” signaling in

two-party scenario1.

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016).

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Correlations of measurements

• utcomes

“Boxes” - families of joint probability distributions {P(a|x)}a,x Two parties, two inputs, two outputs (2,2,2)-scenario

CHSH:=A0B0+A0B1+A1B0−A1B1

Figure: Schematic presentation of values of CHSH-like game

within local deterministic, quantum and no-signaling theory. R. Gill, Epidemiology Meets Quantum: Statistics, Causality, and Bell’s Theorem, www.slideshare.net/

Relativistic causality1

Main assumption: No causal loops! Figure: Violation of causality by “point to point” signaling in

two-party scenario1.

In (2,2,2)-scenario no-signaling conditions are necessary and sufficient conditions for relativistic causality NS(2, 2, 2) ≡ RC(2, 2, 2)

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016).

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

General necessary condition for random variable XM to signaling to a correlation

• f random variables XP and XQ

J +[XP] ∩ J +[XQ] ⊆ J +[XM]

Three party, two input, two output (3,2,2)-scenario

Figure: A particular spacetime configuration of measurement events in the

three-party case, where “point to region” signaling is allowed by RC1.

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016)

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

General necessary condition for random variable XM to signaling to a correlation

• f random variables XP and XQ

J +[XP] ∩ J +[XQ] ⊆ J +[XM]

Three party, two input, two output (3,2,2)-scenario

Figure: A particular spacetime configuration of measurement events in the

three-party case, where “point to region” signaling is allowed by RC1.

Necessary and sufficient conditions for relativistic causality1

P(b,c|y,z) =

• a P(a,b,c|x,y,z)

=

• a P(a,b,c|x′,y,z)

∀x,x′,y,z,b,c P(a,b|x,y) =

• c P(a,b,c|x,y,z)

=

• c P(a,b,c|x,y,z′)

∀z,z′,x,y,a,b P(a,c|x,z) =

• b P(a,b,c|x,y,z)

=

• b P(a,b,c|x,y′,z)

∀y,y′,x,z,a,c P(a|x) =

b,c P(a,b,c|x,y,z)

=

• b,c P(a,b,c|x,y′,z′)

∀y,y′,z,z′,x,a P(c|z) =

a,b P(a,b,c|x,y,z)

=

• a,b P(a,b,c|x′,y′,z)

∀x,x′,y,y′,z,c

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781 (2016)

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Extremal boxes of (3,2,2) polytope

RC polytope Boxes Classes Total 153 600 196 “Pure RC” 151 392 190 N S 2144 5 L 64 1

Table: Characterization of extremal

points of (3,2,2) RC polytope. Calculations were carried out at the Academic Computer Center in Gda´ nsk

Figure: Schematic representation of space of

local deterministic L, quantum Q, no-signaling N S and relativistically causal RC sets. N S polytope Boxes Classes Total 53 856 46 N S 53 792 45 L 64 1

Table: Characterization of extremal

points of (3,2,2) N S. J. Barrett et al.

• Phys. Rev. A 71, 022101 (2005).
• 2M. L. Almeida, et al. Phys. Rev. Lett. 104.230404.

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Extremal boxes of (3,2,2) polytope

RC polytope Boxes Classes Total 153 600 196 “Pure RC” 151 392 190 N S 2144 5 L 64 1

Table: Characterization of extremal

points of (3,2,2) RC polytope. Calculations were carried out at the Academic Computer Center in Gda´ nsk

Figure: Schematic representation of space of

local deterministic L, quantum Q, no-signaling N S and relativistically causal RC sets. N S polytope Boxes Classes Total 53 856 46 N S 53 792 45 L 64 1

Table: Characterization of extremal

points of (3,2,2) N S. J. Barrett et al.

• Phys. Rev. A 71, 022101 (2005).

Nonlocal games in RC

Guess Your Neighbor Input (GYNI)2

Input promise: x ⊕ y ⊕ z = 0 Winning condition: ω = 1 4 [P(000|000) + P(110|011) + P(011|101) + P(101|110)]

• Win. Prob. (ω)

L 1 4 Quantum 1 4 N S 1 3 RC 1 2

Harmonic progression

ωc= 1

4 <ωns= 1 3 <ωrc= 1 2

P(abc|xyz)=

1

2 ,

(1⊕b⊕c⊕y)(1⊕a⊕b⊕x)=1 0,

• therwise
• 2M. L. Almeida, et al. Phys. Rev. Lett. 104.230404.

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Examples of other games

Game 100%

Game 100% winning condition: xy ⊕ yz = a ⊕ c

Potential communication task:

1 costly communication channel to one party 2 no communication necessary

GHZ game

GHZ game winning condition: A0B1C1+A1B0C1+A1B1C0−A0B0C0 ≤ 2 ≤ 4

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Examples of other games

Game 100%

Game 100% winning condition: xy ⊕ yz = a ⊕ c

Potential communication task:

1 costly communication channel to one party 2 no communication necessary

GHZ game

GHZ game winning condition: A0B1C1+A1B0C1+A1B1C0−A0B0C0 ≤ 2 ≤ 4

• Win. Prob.

100% GHZ L

3 4 3 4

Quantum

3 4

1 NS

3 4

1 RC 1 1

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Examples of other games

Game 100%

Game 100% winning condition: xy ⊕ yz = a ⊕ c

Potential communication task:

1 costly communication channel to one party 2 no communication necessary

GHZ game

GHZ game winning condition: A0B1C1+A1B0C1+A1B1C0−A0B0C0 ≤ 2 ≤ 4

• Win. Prob.

100% GHZ L

3 4 3 4

Quantum

3 4

1 NS

3 4

1 RC 1 1

General Hierarchy

For any probabilistic game classical, quantum, NS and RC winning probabilities can be organized as follows: ωc ≤ ωq ≤ ωns ≤ ωrc ≤ 1

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness No randomness extraction based on n-party GHZ-Mermin inequalities1

1 RC polytope is larger than NS 2 More attack strategies 3 Adversary can always prepare RC box s.t.: P(a|x)=1

for any (x, a).

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness No randomness extraction based on n-party GHZ-Mermin inequalities1

1 RC polytope is larger than NS 2 More attack strategies 3 Adversary can always prepare RC box s.t.: P(a|x)=1

for any (x, a). Cryptography based on monogamy Example:

CHSH(AB):=A0B0+A0B1+A1B0−A1B1 L ≤2 N S ≤ 4

CHSH monogamy:

CHSH(AB)+CHSH(BC) N S ≤ 4

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness No randomness extraction based on n-party GHZ-Mermin inequalities1

1 RC polytope is larger than NS 2 More attack strategies 3 Adversary can always prepare RC box s.t.: P(a|x)=1

for any (x, a). Cryptography based on monogamy Example:

CHSH(AB):=A0B0+A0B1+A1B0−A1B1 L ≤2 N S ≤ 4

CHSH monogamy:

CHSH(AB)+CHSH(BC) N S ≤ 4 RC ≤ 8

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness No randomness extraction based on n-party GHZ-Mermin inequalities1

1 RC polytope is larger than NS 2 More attack strategies 3 Adversary can always prepare RC box s.t.: P(a|x)=1

for any (x, a). Cryptography based on monogamy Example:

CHSH(AB):=A0B0+A0B1+A1B0−A1B1 L ≤2 N S ≤ 4

CHSH monogamy:

CHSH(AB)+CHSH(BC) N S ≤ 4 RC ≤ 8

Unique games monogamy:

a=πxy (b),

where πxy is some permutation depending on the inputs (x, y)

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Security against RC eavesdropper1

Device independent randomness amplification

1 Parties receive random inputs from

imperfect randomness source

2 Test for violation of Bell inequality is

performed

3 If there exist input-output pair (x, a) for

any box {P} and 0 < η < 1 s.t.:

η≤{P(a|x)}≤1−η,

then there exist secure protocol of amplification of randomness No randomness extraction based on n-party GHZ-Mermin inequalities1

1 RC polytope is larger than NS 2 More attack strategies 3 Adversary can always prepare RC box s.t.: P(a|x)=1

for any (x, a). Cryptography based on monogamy Example:

CHSH(AB):=A0B0+A0B1+A1B0−A1B1 L ≤2 N S ≤ 4

CHSH monogamy:

CHSH(AB)+CHSH(BC) N S ≤ 4 RC ≤ 8

Unique games monogamy:

a=πxy (b),

where πxy is some permutation depending on the inputs (x, y) No monogamy in RC for any unique game For uniformly distributed input sets X, Y , Z and

• utputs set {0, . . . , d − 1}, the RC box:

P(abc|xyz)=

1

d , if a=πxy (b), c=πzy (b) 0,

• therwise

can achieve no-signaling maximum success probability for both pairs of players simultaneously,

• 1P. Horodecki, R. Ramanathan, arXiv: 1611.06781

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Hierarchy of Monogamies

RC monogamy NS Monogamy ≺ RC Monogamy Any RC Monogamy is a monogamy in NS but not any NS Monogamy in monogamy in RC.

• 3D. Saha, R. Ramanathan, Phys. Rev. A 95, 030104 (2017)

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Hierarchy of Monogamies

RC monogamy NS Monogamy ≺ RC Monogamy Any RC Monogamy is a monogamy in NS but not any NS Monogamy in monogamy in RC. Activation of RC monogamy 3 CHAIN3 monogamy:

CHAIN(3)BA+CHAIN(3)BC N S ≤ 8 RC ≤ 12

Activation:

CHAIN(3)BA+CHAIN(3)BC +CYC[6]B RC ≤ 12,

where

CHAIN(3)BA = B1A1+B2A1+B2A2+

B3A2+B3A3−B1A3

CYC(6)BA = B1B′ 1

• +

B′ 1B2

• +

B2B′ 2

• +
• B′

2B3

• +

B3B′ 3

• −

B′ 3B1

• Figure: Chordal decomposition of correlations graph for activation
• f CHAIN(3) monogamy. Solid, dashed and dotted lines represent

correlated, anticorrelated and compatible observables respectively

• 3D. Saha, R. Ramanathan, Phys. Rev. A 95, 030104 (2017)

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Conclusions We study relativistically causal correlations in the (3,2,2)-scenario in view of:

1 it’s geometric properties

giving extremal points of RC polytope

2 computational properties

characterizing hierarchy of nonlocal games in RC

3 security against RC adversary

restoring monogamy for potential cryptographic protocols

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations

Thank you for your attention!

Micha l Kamo´ n Games and Monogamy in the relativistically causal correlations