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(
abxy
)
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
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