A limit on nonlocality in any world in which communication complexity is not trivial
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Transcript of A limit on nonlocality in any world in which communication complexity is not trivial
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A limit on nonlocality in any
world in which communication
complexity is not trivial
IFT6195Alain Tapp
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In collaboration with… Gilles Brassard Harry Buhrman Naoh Linden André Allan Methot Falk Unger
Quant-ph/0508042
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Motivation What would be the consequences if the non local collerations in our world were stronger than the one given by quantum mechanics?
Theoretical computer science? Foundation of physics? Philosophy?
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Perfect Non Local Boxes
Alice Bob
NLB
yxba
a byx
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NLB and communication
One bit of communication is enough toimplement a NLB.
Alice sends a to Bob and output x=0 Bob outputs bay
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NLB and communication
NLBs does not allow for communication.
We can have a perfect box for which x and y are uniformly distributed and independent of (respectively) a and b.
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NLB, classical deterministic strategies
yes yes yes no
yes yes no yes
yes no yes yes
no yes yes yes
0 0 0 a b 0 ba0 01 00 11 1
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NLB classical implementation There is a probabilistic strategy with succes probability ¾ on all input.
There is no classical déterministic strategy with success proportion greater than ¾.
There is no probabilistic strategy with success probability greater than ¾. ¾
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Alice and Bob have the same strategy.If input=0 applies otherwiseMeasure and output the result.This strategy works on all inputs with probability:
)cos()sin(
)sin()cos()(
R
NLB quantum strategy
)16/( R
11002
1
)16/3( R
%85)8/(cos2
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NLB quantum strategy
%85)8/(cos2
Tsirelson proved in 1980 that this is optimal whatever the entanglement shared by the players.
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Bell theorem
The classical upper bound and the quantum lower bound do not match.
We can derive an inequality from this that provides a Bell theorem proof.
This is known as the CHSH inequality.
4/3 )8/(cos %85 2
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Classical Communication Complexity
Alice Bobk
Rz }1,0{
x y
),( yxf
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Quantum Communication Complexity
Alice Bobk
11
2
100
2
1
x y
),( yxf
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The classical/quantum probabilistic communication complexity of f, C(f)/Q(f) is the amount of classical communication required by the best protocol that succeeds on all input with probability at least when the players have unlimited prior classical/quantum correlation.
Communication Complexity
2/1
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Inner product (IP)
)2(mod
)()()(
),(
1
3211
2121
n
iii
nn
nn
yxyx
yxyxyxyx
yyyyxxxx
yxyxIP
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Inner product (IP)
)()(
)1()(
nIPQ
OnIPC
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Most functions are difficultFor most functions f
)1()(
)1()(
OnfQ
OnfC
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Equality
0),EQ(
1),EQ(
yxyx
yxyx
Alice and Bob each have a very large file and they want to know if it is exactly the same.
How much do they need to communicate?
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Equality
nRz 1,0
Alice Bobzymb yx
zxma
bm
ba mm Output
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Equality
2)EQ( C
2
1
1
yzxzPyx
yzxzPyx
By repeating the protocol twice we have success probability of at least ¾.
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Scheduling
0),(
)()()(),(
1
3211
2121
n
iii
nn
nn
yxyxS
yxyxyxyxS
yyyyxxxx
Alice and Bob want to find a time where they are both available for a meeting.
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Scheduling
)()(
)()(
nSQ
nSC
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Raz separationThere exists a problem such that:
))(log()(
)log()(
4/1
nOSQ
n
nSC
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IP using NLB
)()(
)()()(
)()()(
),(),(
2121
2211
3211
nn
nn
nn
iiii
iiii
BBBAAAyx
BABABAyx
yxyxyxyx
yxBA
yxNLBBA
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Perfect NLB implies trivial CC
ba xxx
ba xxx
0,1 ba xxx
Any function can be computed with a serie of AND gates and negations.
Distributed bit
Input bit
Negation:
AND Two NLBs
Outcome Bob sends to Aliceba yyy by
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AND
))(())((
)()()()(
)()(
),(),(
),(),(
2121
22
11
22
11
bbaa
bbabbaaa
baba
ba
ba
ba
ba
yxBBAAyxyx
yxyxyxyxyx
yyxxyx
xyBA
yxBA
xyNLBBA
yxNLBBA
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Main result
1)(, 6
1
2
1* fCfNLB
In any world where non local boxes can be implemented with accuracy larger than 0.91 communication complexity is trivial.
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CC with a bias We say that a function f can be computed with a bias if Alice and Bob can produce a distributed bit z such that
2
1]),([ zyxfP
ba zzz
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CC with a biasEvery function can be computed with a bias.
Alice’s input: xBob’s input: yAlice and Bob share z
Alice outputs a=f(x,z)Bob outputs b=0 if y=z and a random bit otherwise.
2
1
2
1
2
11
2
1 ]),([
nnbayxfP
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Idea We want a bounded bias.
Let’s amplify the bias.
Repetition and majority?
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IdeaMaj
Maj Maj Maj
Maj Maj Maj Maj Maj Maj Maj Maj Maj
)(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf )(~
)(~
)(~
xfxfxf
)(~xf
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Non local majority
),,(
2 iff 1),,(
332211
333
222
111
321321
babababa
ba
ba
ba
ba
xxxxxxMajyy
yyy
xxx
xxx
xxx
xxxxxxMajy
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NLM > 5/6 If NLM can be computed with probability stricly greather than 5/6 than every fonction can be computed with a bounded bias.
Below that treshold NLM makes things worst.
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NLM > 5/6
pphsp
s
pppqpppqph
q
p
)(2/1
2
1
312
3
2
1
))1()1(3)(1())1(3()(
0)( 6/5
)0( 2/1
3223
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Non local equality
),,(
iff 0),,(
332211
333
222
111
321321
babababa
ba
ba
ba
ba
xxxxxxNLEyy
yyy
xxx
xxx
xxx
xxxxxxNLEy
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NLE implies NLM
),,(
),,(
332211
321
321
332211
babababa
bbbbb
aaaaa
babababa
xxxxxxMajzz
xxxyz
xxxyz
xxxxxxNLEyy
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2 NLB implies NLE
213221
213221
33222211
3221
321
213222
322111
333222111
))()((
))()((
) () (
)()(
),,(
) ,1(
),(
bbbbbb
aaaaaa
babababa
bbaaba
bbaaba
bababa
zzxxxx
zzxxxx
xxxxxxx
xxxx
xxxNLE
xxxxNLBzz
xxxxNLBzz
xxxxxxxxx
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To conclude the proof
6
5
6
5
6
1
2
1 MajNLENLB
•Compute f several times with a bias•Use a tree of majority to improve the bias.•Bob sends his share of the outcome to Alice.
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Open question
Show some unacceptable consequences of correlations epsilon-stronger than the one predicted by quantum mechanics.