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P
LAYING

(
QUANTUM
)
GAMES

WITH

OPERATOR

SPACES


David Pérez
-
García

Universidad Complutense de Madrid

Bilbao 8
-
Oct
-
2011

O
UR

GROUP


2 postdoctoral
research

positions
opened

in
our

group

“Mathematics and Quantum
Information
”.

S
TRUCTURE

OF

THE

TALK


The object under study: 2P1R games


Examples. Why are they important?


Complexity theory I.
Inaproximability

results.


Complexity theory II. Parallel computation.


Position based cryptography.


Certifiable random number generation.


A bit (just one bit!) of history.


Where are the
maths
?


Our contribution. Operator Spaces.

2P1R
GAMES


2P1R
GAMES

x

y

a

b

1.
A set of possible
questions

for Alice and Bob (denoted by
x,y

resp.).

2.
A known
probability

distribution for the questions.

3.
A known
boolean

function V(
x,y,a,b
)
which decides, based on questions
and answers
a,b
,
whether they
witn

(=1) or loose (=0) the game.

4.
A
limitation in the communication
between Alice and Bob.


2P1R
GAMES

x

y

a

b


The
value of the game
is the largest probability of wining the game
while optimizing over the possible strategies of Alice and Bob.


It is assumed that Alice and Bob have free communication BEFORE the
game to coordinate an strategy .


Hence strategies can involve
shared randomness (classical value of
the game)

or
quantum entanglement

(quantum value of the game)
depending on the resources of Alice and Bob.


2P1R
GAMES

x

y

a

b

What is an strategy?

A probability distribution p(
ab|xy
)


Which are the possible strategies in the classical case?




)
|
(
)
|
(
)
(
)
|
(




y
b
p
x
a
p
q
xy
ab
p
B
A


And in the quantum one?



y
b
x
a
F
E
tr
xy
ab
p


)
|
(
.
0
]
,
[
,
,
0
,
)
(









y
b
x
a
H
b
y
b
H
a
x
a
y
b
x
a
F
E
y
Id
F
x
Id
E
F
E
H
S

EXAMPLES.
I
NAPROXIMABILITY


EXAMPLES
.

I
NAPROXIMABILITY

RESULTS

Theorem

(PCP theorem (
Arora
-
Safra
, 92)+ Parallel repetition
(
Raz
, 94)):


Unless P=NP, given e>0 and a polynomial
algorithm to determine the classical value of a game,
there exist games for which the value is 1 and the
algorithm outputs a value <e.

0

1

e

E
XAMPLES
.

I
NAPROXIMABILITY

RESULTS


It is the mother of most
inaproximability

results. For
instance:


Theorem

(
Hastad
, 1999):


Unless P=NP, given e>0 and a polynomial algorithm
to determine the MAX
-
CLIQUE of a graph, there
exist graphs of n vertices for which

e
n


1
algorithm

the
of
output
CLIQUE
-
MAX
Note that MAX
-
CLIQUE is always less or equal than n (!!)

E
XAMPLES
.

I
NAPROXIMABILITY

RESULTS

Connection with 2P1R games. Via LABEL
-
COVER.


Given a bipartite graph


And a set of colors


And a set of valid configurations for each edge



Find a coloring of the graph which maximizes the number of
edges with a valid configuration.

)
,
(
E
W
U
V



}
invalid

valid,
{
:

,
)
,
(







e
C
E
w
u
e
E
XAMPLES
.

I
NAPROXIMABILITY

RESULTS

Connection with 2P1R games. Via LABEL
-
COVER.


Colors

Number of valid edges = 4

Solution to LABEL
-
COVER = 5

E
XAMPLES
.

I
NAPROXIMABILITY

RESULTS

Connection with 2P1R games. Via LABEL
-
COVER.


Given and instance of LABEL
-
COVER, we define a 2P1R game
by:


Questions = vertices (from U to Alice and from W to Bob) with
uniform probability between the pairs which conform an edge
(and 0 in the rest).


Answers = colors.


They win the game if they give a valid coloring for the edge
which is asked.

E
XAMPLES
. P
ARALLEL

COMPUTATION


E
XAMPLES
.

P
ARALLEL

COMPUTATION

Given a
boolean

function f(
x,y
), minimize c in:


x

y

c bits of
communication

f(
x,y
)

f(
x,y
)

E
XAMPLES
. P
OSITION

BASED

CRYPTOGRAPHY
.

(
CHANDRAN

ET

AL
, 2009)


E
XAMPLES
.

P
OSITION

BASED

CRYPTOGRAPHY

1D for simplicity

x

y

a

b

AIM
: That only someone in position P could answer with
probability 1 to the challenge. It would allow unconditional
secure communication !!!

Coordinated

Position
P

E
XAMPLES
.

P
OSITION

BASED

CRYPTOGRAPHY

Relation with 2P1R games. Since the verifiers act coordinated,
we can assume there is just one of them.

Based on answering times, we have:

x

y

a

b

Communication “independent
-
one
-
way”

E
XAMPLES
.

P
OSITION

BASED

CRYPTOGRAPHY

Hence, the aim is to find a challenge which can be won always
with arbitrary communication (all classical challenges are like
that) but not with “independent
-
one
-
way” communication.

x

y

a

b

The honest case is the
one of arbitrary
communication, since
there is only a single
prover
.

E
XAMPLES
.

P
OSITION

BASED

CRYPTOGRAPHY

This is impossible classically. Both models of communication
are the same. To see it, just copy and send the received
question.

In the quantum case (with no entanglement) it is indeed
possible (Buhrman et al., 2010). The key idea lies on the fact
that it is NOT possible to copy quantum states by the NO
-
CLONING theorem.

Question: Is it also possible when a polynomial amount of
entanglement is allowed?


Partial answers (
Beigi

et al.,
Burhman

et al, 2011):

LINEAR = YES, EXPONENTIAL = NO.

E
XAMPLES
.
R
ANDOM

NUMBER

GENERATION

(
PIRONIO

ET

AL
., 2010)


E
XAMPLE
.

R
ANDOM

NUMBER

GENERATION

PROBLEM: How to construct an apparatus which
generates perfect random numbers (and hence secret) in
a certifiable way?

001110101001010101
….
.

could have a copy of
001110101001010101…..

But in quantum mechanics copying is not allowed !!!

E
XAMPLE
.

R
ANDOM

NUMBER

GENERATION

Theorem

(
Pironio

et al.,
Colbeck

et al., 2010):

If (after many rounds in the game) one gets a value
strictly larger than the classical one, there is a
classical “deterministic” post
-
processing of the outputs
a, b which produces numbers which are perfectly
random and secret.

x

y

a

b

DONE EXPERIMENTALLY ALREADY !!!

E
XAMPLE
.

R
ANDOM

NUMBER

GENERATION

The key is, hence, the existence of quantum
strategies which are NOT classical. This
guarantees an intrinsic randomness
.

)
|
(
xy
ab
p
Classical

Quantum

Non
-
signaling

HISTORICAL NOTE

The existence of this intrinsic randomness is
precisely what Einstein did not believe in his
criticism of quantum mechanics in the 30’s.


The first experiment showing that, indeed, this
randomness does
ocurr

was done by A. Aspect in
the 80’s.


The experiment was based precisely on the
analysis of the value of a particular game, known
as CHSH (
Clauser
, Horne,
Shimony
, Holt).

O
UR

CONTRIBUTION

T
HE

PROBLEM

WE

WANT

TO

ATTACK

Estimate D.

Parameters:

Number of questions = N

Number of answers = M

Size (dimension) of the quantum system = d

D

Quantum
strategies

Classical

strategies

)
|
(
xy
ab
p
How large can D be?

value
classical
value
quantum

D
V
IOLATION

OF

A

BELL

INEQUALITY
. F
ORMAL

DEFINITION

OF

D.

Bell
inquality

T



abxy
xy
ab
c
xy
ab
p
T
T
V
)
|
(
sup
)
(
classical

p






abxy
xy
ab
d
q
xy
ab
p
T
T
V
)
|
(
sup
)
(

d
dim
quantum
q




(T)
V
(T)
V
sup
1
c
d
q
T

D
BELL INEQUALITY= 2P1R GAME (
generalized
)

)
,
,
,
(
)
,
(
T
xy
ab
y
x
b
a
V
y
x


O
PERATIONAL

INTERPRETATION

OF

D

Bell
Inequality

T

p
-
1
p
1
D


Where p is the maximum (classical)
noise which a quantum strategy can
withstand before getting classical.

It is hence desirable to have a large D. How does D
scale with the parameters N,M, d?

MAIN THEOREM

Theorem

(Junge, Palazuelos, Pérez
-
García, Villanueva,
Wolf, CMP + PRL, 2010).

D can be arbitrarily large, This requires:

N= D^2

M= EXP(D)

d= D^2

Theorem

(Junge, Palazuelos, CMP, 2011).

D can be arbitrarily large, This requires:

N= D^2, M= D^2, d= D^2

Recent

improvements

Theorem

(Buhrman et al, 2011).

D can be arbitrarily large, This requires:

N= D, M= EXP(D), d= D.

A
N

INTERESTING

APPLICATION

Theorem

(Junge, Palazuelos, Pérez
-
García, Villanueva,
Wolf, CMP + PRL, 2010).

There exist quantum
estrategies

with EXP(N) questions, N
answers and dimension of the quantum system N which
need the communication of log(N) bits to be simulated
classically
.

Quantum entanglement can save communication!!!

T
HE

TOOLS
: O
PERATOR

SPACES

O
PERATOR

S
PACES

An
operator space

is a complex vector space E with
a sequence of norms defined on such that:


Given a C*
-
algebra, there exists a unique norm
which makes a C*
-
algebra. With these
norms, A is an operator space.

)
(
E
M
n
mn
nm
M
m
M
n
b
x
a
axb



}
,
max{
m
n
m
n
b
a
b
a



)
(
A
M
n
O
PERATOR

SPACES

In particular:

Given

)
,
(
max
k
k
C



i
i
n
A
x
max

O
PERATOR

SPACES

The
morphisms

in this category are the completely
bounded maps:

n
n
cb
u
u
F
E
u
sup
,
:


)
(
)
(
:
1
F
M
E
M
u
u
n
n
n
n



CB(E,F) is an operator space via

))
(
,
(
))
,
(
(
F
M
E
CB
F
E
CB
M
n
n

In particular E* is an operator space

C
ONNECTION

WITH

THE

2P1R
GAMES

Theorem

(Junge, Palazuelos, Pérez
-
García, Villanueva,
Wolf, CMP + PRL, 2010).

Given a 2P1R game (or more generally a Bell inequality)


The classical value is given (with the order
a,x,b,y
) by the
norm:




The quantum value, by the norm
:




xy
ab
T
)
)
,
(
),
,
(
(
*
N
M
N
M
B
B
B








)
)
,
(
),
,
(
(
*
N
M
N
M
CB
CB
CB








Operator Spaces are the natural mathematical framework
to analyze 2P1R games.

Some references in this direction:

1.
D. Pérez
-
García, M.M. Wolf, C. Palazuelos, I. Villanueva, M. Junge,
Unbounded violations of Bell inequalities, Comm. Math. Phys.
279, 455
(2008)

2.
M. Junge, C. Palazuelos, D. Pérez
-
García, I. Villanueva, M.M. Wolf,
Operator Space theory: a natural framework for Bell inequalities,
Phys. Rev.
Lett
. 104, 170405 (2010).

3.
M. Junge, C. Palazuelos, D. Pérez
-
García, I. Villanueva, M.M. Wolf,
Unbounded
violations of bipartite Bell Inequalities via Operator
Space theory, Comm. Math. Phys.
300, 715

739 (2010).

4.
M. Junge, M.
Navascués
, C. Palazuelos, D. Pérez
-
García, V.B.
Scholtz
, R.F.
Werner,
Connes
' embedding problem and
Tsirelson's

problem, J. Math. Phys.
52, 012102 (2011)

5.
A.Salles
, D.
Cavalcanti
, A.
Acín
, D. Pérez
-
García., M.M. Wolf, Bell inequalities
from
multilinear

contractions, Quant. Inf. Comp. 10, 0703
-
0719 (2010) .

6.
M. Junge, C. Palazuelos, Large violation of Bell inequalities with low
entanglement, Comm. Math. Phys. 306 (3), 695
-
746 (2011) .

W
HAT

WE

LEARNT