Flexible constrained de Finetti reductions
and parallel repetition of multi-player non-local games
joint work with Andreas Winter
Cécilia Lancien
Madrid - April 20th 2016
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
1 / 19
Outline
1
De Finetti type theorems
2
Multi-player non-local games
3
Using de Finetti reductions to study the parallel repetition of multi-player non-local games
4
Summary, open questions and perspectives
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
2 / 19
Outline
1
De Finetti type theorems
2
Multi-player non-local games
3
Using de Finetti reductions to study the parallel repetition of multi-player non-local games
4
Summary, open questions and perspectives
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
3 / 19
Classical and quantum finite de Finetti theorems
Motivation : Reduce the study of permutation-invariant scenarios to that of i.i.d. ones.
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
4 / 19
Classical and quantum finite de Finetti theorems
Motivation : Reduce the study of permutation-invariant scenarios to that of i.i.d. ones.
Classical finite de Finetti Theorem (Diaconis/Freedman)
Let P (n) be an exchangeable p.d. in n r.v.’s, i.e. for any π ∈ Sn , P (n) ◦ π = P (n) .
For any k 6 n, denote by P (k ) the marginal p.d. of P (n) in k r.v.’s.
2
(k ) Z ⊗k
6k .
P
−
Q
d
µ(
Q
)
Then, there exists a p.d. µ on the set of p.d.’s in 1 r.v. s.t. n
Q
1
→ The marginal p.d. (in a few variables) of an exchangeable p.d. is well-approximated by a
convex combination of product p.d.’s.
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
4 / 19
Classical and quantum finite de Finetti theorems
Motivation : Reduce the study of permutation-invariant scenarios to that of i.i.d. ones.
Classical finite de Finetti Theorem (Diaconis/Freedman)
Let P (n) be an exchangeable p.d. in n r.v.’s, i.e. for any π ∈ Sn , P (n) ◦ π = P (n) .
For any k 6 n, denote by P (k ) the marginal p.d. of P (n) in k r.v.’s.
2
(k ) Z ⊗k
6k .
P
−
Q
d
µ(
Q
)
Then, there exists a p.d. µ on the set of p.d.’s in 1 r.v. s.t. n
Q
1
→ The marginal p.d. (in a few variables) of an exchangeable p.d. is well-approximated by a
convex combination of product p.d.’s.
Quantum finite de Finetti Theorem (Christandl/König/Mitchison/Renner)
†
Let ρ(n) be a permutation-invariant state on (Cd )⊗n , i.e. for any π ∈ Sn , Uπ ρ(n) Uπ = ρ(n) .
For any k 6 n, denote by ρ(k ) = Tr(Cd )⊗n−k ρ(n) the reduced state of ρ(n) on (Cd )⊗k .
2
(k ) Z ⊗k
6 2kd .
Then, there exists a p.d. µ on the set of states on Cd s.t. ρ
−
σ
d
µ(σ)
n
σ
1
→ The reduced state (on a few subsystems) of a permutation-invariant state is well-approximated
by a convex combination of product states.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
4 / 19
De Finetti reductions (aka “Post-selection techniques”)
Motivation : In several applications, one only needs to upper-bound a permutation-invariant
object by product ones...
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
5 / 19
De Finetti reductions (aka “Post-selection techniques”)
Motivation : In several applications, one only needs to upper-bound a permutation-invariant
object by product ones...
“Universal” de Finetti reduction for quantum states (Christandl/König/Renner)
Let ρ(n) be a permutation-invariant state on (Cd )⊗n . Then,
Z
2
ρ(n) 6 (n + 1)d −1 σ⊗n d µ(σ), µ : uniform p.d. over the set of states on Cd .
σ
Canonical application : If f is an order-preserving linear form s.t. f 6 ε on 1-particle states, then
f ⊗n 6 poly(n)εn on permutation-invariant n-particle states.
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
5 / 19
De Finetti reductions (aka “Post-selection techniques”)
Motivation : In several applications, one only needs to upper-bound a permutation-invariant
object by product ones...
“Universal” de Finetti reduction for quantum states (Christandl/König/Renner)
Let ρ(n) be a permutation-invariant state on (Cd )⊗n . Then,
Z
2
ρ(n) 6 (n + 1)d −1 σ⊗n d µ(σ), µ : uniform p.d. over the set of states on Cd .
σ
Canonical application : If f is an order-preserving linear form s.t. f 6 ε on 1-particle states, then
f ⊗n 6 poly(n)εn on permutation-invariant n-particle states.
Drawback : All permutation-invariant states are upper-bounded by the same mixture of tensor
power states → Any other information is lost...
“Flexible” de Finetti reduction for quantum states
Let ρ(n) be a permutation-invariant state on (Cd )⊗n . Then,
Z
2
2
ρ(n) 6 (n + 1)3d −1 F ρ(n) , σ⊗n σ⊗n d µ(σ), µ : uniform p.d. over the set of states on Cd .
σ
√ √
→ Follows from pinching trick. [Notation : F (ρ, σ) = k ρ σk1 , fidelity between states ρ, σ.]
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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What is the “flexible” de Finetti reduction good for ?
ρ(n) 6 poly(n)
Z
F ρ(n) , σ⊗n
2
σ⊗n d µ(σ)
σ
State-dependent upper-bound : Amongst states of the form σ⊗n , only those which have a high
fidelity with the state of interest ρ(n) are given an important weight.
→ Useful when one knows that ρ(n) satisfies some additional property : only states σ⊗n
approximately satisfying this same property should have a non-negligible fidelity weight...
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
6 / 19
What is the “flexible” de Finetti reduction good for ?
ρ(n) 6 poly(n)
Z
F ρ(n) , σ⊗n
2
σ⊗n d µ(σ)
σ
State-dependent upper-bound : Amongst states of the form σ⊗n , only those which have a high
fidelity with the state of interest ρ(n) are given an important weight.
→ Useful when one knows that ρ(n) satisfies some additional property : only states σ⊗n
approximately satisfying this same property should have a non-negligible fidelity weight...
Canonical examples of applications : Symmetric states satisfying some linear constraint
n
• If N ⊗n (ρ(n) ) = τ⊗
0 , for some CPTP map N and state τ0 , then
ρ(n) 6 poly(n)
Z
2n
F (τ0 , N (σ)) σ⊗n d µ(σ).
σ
→ Exponentially small weight on states σ⊗n s.t. N (σ) 6= τ0 .
• If N ⊗n (ρ(n) ) = ρ(n) , for some CPTP map N , then there exists a p.d. e
µ over the range of
ρ(n) 6 poly(n)
Z
σ
N s.t.
2
F ρ(n) , σ⊗n σ⊗n d e
µ(σ).
→ No weight on states σ⊗n s.t. σ ∈
/ Range(N ).
Particular case : N : σ 7→ ∑x hx |σ|x i|x ihx | = ∑x Pσ (x )|x ihx | quantum-classical channel.
If X is finite and P (n) is a permutation-invariant
p.d. on X n , then there exists a p.d. e
µ over the set
Z
2
(n )
(n)
⊗n
⊗n
of p.d.’s on X s.t. P
6 poly(n) F P , Q
Q de
µ(Q ).
Q
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
6 / 19
Outline
1
De Finetti type theorems
2
Multi-player non-local games
3
Using de Finetti reductions to study the parallel repetition of multi-player non-local games
4
Summary, open questions and perspectives
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
7 / 19
`-player non-local games
` cooperating but separated players. Each player i receives an input xi ∈ Xi (“query”) and
produces an output ai ∈ Ai (“answer”). They win if some binary predicate V (a1 , . . . , a` , x1 , . . . , x` )
on the answers & queries is satisfied. To achieve this, they can agree on a joint strategy before
the game starts, but then cannot communicate anymore.
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
8 / 19
`-player non-local games
` cooperating but separated players. Each player i receives an input xi ∈ Xi (“query”) and
produces an output ai ∈ Ai (“answer”). They win if some binary predicate V (a1 , . . . , a` , x1 , . . . , x` )
on the answers & queries is satisfied. To achieve this, they can agree on a joint strategy before
the game starts, but then cannot communicate anymore.
Description of an `-player non-local game G
• Input alphabet : X = X1 × · · · × X` . Output alphabet : A = A1 × · · · × A` . (both finite)
• Game distribution = P.d. on X : {T (x ) ∈ [0, 1], x ∈ X }.
• Game predicate = Binary predicate on A × X : {V (a, x ) ∈ {0, 1}, (a, x ) ∈ A × X }.
• Players’ strategy = Conditional p.d. on A given X : {P (a|x ) ∈ [0, 1], (a, x ) ∈ A × X }.
→ Belongs to a set of “allowed strategies”, depending on the kind of correlation resources that
the players have (e.g. shared randomness, quantum entanglement, no-signalling boxes etc.)
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
8 / 19
`-player non-local games
` cooperating but separated players. Each player i receives an input xi ∈ Xi (“query”) and
produces an output ai ∈ Ai (“answer”). They win if some binary predicate V (a1 , . . . , a` , x1 , . . . , x` )
on the answers & queries is satisfied. To achieve this, they can agree on a joint strategy before
the game starts, but then cannot communicate anymore.
Description of an `-player non-local game G
• Input alphabet : X = X1 × · · · × X` . Output alphabet : A = A1 × · · · × A` . (both finite)
• Game distribution = P.d. on X : {T (x ) ∈ [0, 1], x ∈ X }.
• Game predicate = Binary predicate on A × X : {V (a, x ) ∈ {0, 1}, (a, x ) ∈ A × X }.
• Players’ strategy = Conditional p.d. on A given X : {P (a|x ) ∈ [0, 1], (a, x ) ∈ A × X }.
→ Belongs to a set of “allowed strategies”, depending on the kind of correlation resources that
the players have (e.g. shared randomness, quantum entanglement, no-signalling boxes etc.)
Value of a game G over a set of allowed strategies AS (A |X )
Maximum winning probability for players playing G with strategies P ∈ AS (A |X ) :
)
(
ωAS (G) = max
∑
T (x )V (a, x )P (a|x ) : P ∈ AS (A |X )
a∈A ,x ∈X
→ Bell functional of particular form : all coefficients in [0, 1]
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Flexible constrained de Finetti reductions
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Some usual sets of allowed strategies
Classical correlations : P ∈ C (A |X ) if
∀ x ∈ X , ∀ a ∈ A , P (a|x ) =
∑
Q (m)P1 (a1 |x1 m) · · · P` (a` |x` m),
m∈M
for some p.d. Q on M and some p.d.’s Pi (·|xi m) on Ai .
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Flexible constrained de Finetti reductions
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Some usual sets of allowed strategies
Classical correlations : P ∈ C (A |X ) if
∀ x ∈ X , ∀ a ∈ A , P (a|x ) =
∑
Q (m)P1 (a1 |x1 m) · · · P` (a` |x` m),
m∈M
for some p.d. Q on M and some p.d.’s Pi (·|xi m) on Ai .
Quantum correlations : P ∈ Q (A |X ) if
∀ x ∈ X , ∀ a ∈ A , P (a|x ) = hψ|M (x1 )a1 ⊗ · · · ⊗ M (x` )a` |ψi,
for some state |ψi on H1 ⊗ · · · ⊗ H` and some POVMs M (xi ) on Hi .
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Flexible constrained de Finetti reductions
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Some usual sets of allowed strategies
Classical correlations : P ∈ C (A |X ) if
∀ x ∈ X , ∀ a ∈ A , P (a|x ) =
∑
Q (m)P1 (a1 |x1 m) · · · P` (a` |x` m),
m∈M
for some p.d. Q on M and some p.d.’s Pi (·|xi m) on Ai .
Quantum correlations : P ∈ Q (A |X ) if
∀ x ∈ X , ∀ a ∈ A , P (a|x ) = hψ|M (x1 )a1 ⊗ · · · ⊗ M (x` )a` |ψi,
for some state |ψi on H1 ⊗ · · · ⊗ H` and some POVMs M (xi ) on Hi .
No-signalling correlations : P ∈ NS (A |X ) if
∀ I ( [`], ∀ x ∈ X , ∀ aI ∈ AI , P (aI |x ) = Q (aI |xI ),
for some p.d.’s Q (·|xI ) on AI .
Sub-no-signalling correlations : P ∈ SNOS (A |X ) if
∀ I ( [`], ∀ x ∈ X , ∀ aI ∈ AI , P (aI |x ) 6 Q (aI |xI ),
for some p.d.’s Q (·|xI ) on AI .
Remark : To check that a conditional p.d. P is NS, it is enough to check that it satisfies the NS
conditions on subsets of the form I = [`] \ {i }, i.e. that for each 1 6 i 6 `, the marginal of P on
A \ Ai |X does not depend on Xi . But this is probably false for SNOS.
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Flexible constrained de Finetti reductions
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Some remarks on no-signalling and sub-no-signalling correlations
Players sharing (sub-)no-signalling correlations : no limitation is assumed on their physical power,
apart from the fact that they cannot signal information instantaneously from one another. In the
no-signalling case, players are forced to always produce an output, whatever input they received,
while in the sub-no-signalling case they are even allowed to abstain from doing so.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
10 / 19
Some remarks on no-signalling and sub-no-signalling correlations
Players sharing (sub-)no-signalling correlations : no limitation is assumed on their physical power,
apart from the fact that they cannot signal information instantaneously from one another. In the
no-signalling case, players are forced to always produce an output, whatever input they received,
while in the sub-no-signalling case they are even allowed to abstain from doing so.
Relating the NS and the SNOS values of games
Clearly, for any game G, ωNS (G) 6 ωSNOS (G). And there are examples of games G s.t.
ωSNOS (G) = 1 while ωNS (G) < 1 (e.g. anti-correlation game).
If G is a 2-player game, then ωNS (G) = ωSNOS (G) (reason : for any 2-party SNOS
correlation, there exists a 2-party NS correlation dominating it pointwise).
If G is an `-player game whose distribution T has full support, then
ωNS (G) < 1 ⇒ ωSNOS (G) < 1 (more quantitatively : there exists Γ > 1, which depends on
T , such that ωSNOS (G) > 1 − δ ⇒ ωNS (G) > 1 − Γδ).
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Flexible constrained de Finetti reductions
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Parallel repetition of multi-player games
The ` players play n instances of G in parallel : Each player i receives its n inputs
(1)
(n)
(1)
(n )
∈ Xi together and produces its n outputs ai , . . . , ai ∈ Ai together.
Product game distribution on X n : T ⊗n x n = T x (1) · · · T x (n) .
xi
, . . . , xi
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Parallel repetition of multi-player games
The ` players play n instances of G in parallel : Each player i receives its n inputs
(1)
(n)
(1)
(n )
∈ Xi together and produces its n outputs ai , . . . , ai ∈ Ai together.
Product game distribution on X n : T ⊗n x n = T x (1) · · · T x (n) .
xi
, . . . , xi
Game Gn : The players win if they win all n instancesof G.
→ Product game predicate on A n × X n : V ⊗n an , x n = V a(1) , x (1) · · · V a(n) , x (n) .
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
11 / 19
Parallel repetition of multi-player games
The ` players play n instances of G in parallel : Each player i receives its n inputs
(1)
(n)
(1)
(n )
∈ Xi together and produces its n outputs ai , . . . , ai ∈ Ai together.
Product game distribution on X n : T ⊗n x n = T x (1) · · · T x (n) .
xi
, . . . , xi
Game Gn : The players win if they win all n instancesof G.
→ Product game predicate on A n × X n : V ⊗n an , x n = V a(1) , x (1) · · · V a(n) , x (n) .
t/n
Game G
: The players win if they win any t (or more) instances
of G amongst the
n.
→ Game predicate on A n × X n defined as : V t /n an , x n = 1 if ∑ni=1 V a(i ) , x (i ) > t and
V t /n an , x n = 0 otherwise. In particular : Gn/n = Gn .
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Parallel repetition of multi-player games
The ` players play n instances of G in parallel : Each player i receives its n inputs
(1)
(n)
(1)
(n )
∈ Xi together and produces its n outputs ai , . . . , ai ∈ Ai together.
Product game distribution on X n : T ⊗n x n = T x (1) · · · T x (n) .
xi
, . . . , xi
Game Gn : The players win if they win all n instancesof G.
→ Product game predicate on A n × X n : V ⊗n an , x n = V a(1) , x (1) · · · V a(n) , x (n) .
t/n
Game G
: The players win if they win any t (or more) instances
of G amongst the
n.
→ Game predicate on A n × X n defined as : V t /n an , x n = 1 if ∑ni=1 V a(i ) , x (i ) > t and
V t /n an , x n = 0 otherwise. In particular : Gn/n = Gn .
The value ωAS (Gn ), resp. ωAS (Gt /n ), is the maximum winning probability for players playing Gn ,
resp. Gt /n , with strategies P ∈ AS (A n |X n ).
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
11 / 19
Parallel repetition of multi-player games
The ` players play n instances of G in parallel : Each player i receives its n inputs
(1)
(n)
(1)
(n )
∈ Xi together and produces its n outputs ai , . . . , ai ∈ Ai together.
Product game distribution on X n : T ⊗n x n = T x (1) · · · T x (n) .
xi
, . . . , xi
Game Gn : The players win if they win all n instancesof G.
→ Product game predicate on A n × X n : V ⊗n an , x n = V a(1) , x (1) · · · V a(n) , x (n) .
t/n
Game G
: The players win if they win any t (or more) instances
of G amongst the
n.
→ Game predicate on A n × X n defined as : V t /n an , x n = 1 if ∑ni=1 V a(i ) , x (i ) > t and
V t /n an , x n = 0 otherwise. In particular : Gn/n = Gn .
The value ωAS (Gn ), resp. ωAS (Gt /n ), is the maximum winning probability for players playing Gn ,
resp. Gt /n , with strategies P ∈ AS (A n |X n ).
Question : For AS being either C, Q, NS or SNOS, we clearly have
ωAS (G)n 6 ωAS (Gn ) 6 ωAS (G).
But in the case where ωAS (G) < 1, what is the true behavior of ωAS (Gn ) ? Does it decay to 0
exponentially (in n), and if so at which rate ? More generally, does ωAS (Gt /n ) as well decay to 0
exponentially as soon as t /n > ωAS (G) ?
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Flexible constrained de Finetti reductions
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Intuitively, why should de Finetti reductions be useful to understand the parallel
repetition of multi-player games ?
⊗n
Observation : Obviously, the game distribution TX⊗n and the game predicate VAX
of Gn are both
permutation-invariant.
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Flexible constrained de Finetti reductions
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Intuitively, why should de Finetti reductions be useful to understand the parallel
repetition of multi-player games ?
⊗n
Observation : Obviously, the game distribution TX⊗n and the game predicate VAX
of Gn are both
permutation-invariant.
Consequence : One can assume w.l.o.g. that the optimal winning strategy PA n |X n , in the set of
allowed strategies AS (A n |X n ), for Gn is permutation-invariant as well. And hence,
Z
2
⊗n
⊗n
TX⊗n PA n |X n 6 poly(n)
F TX⊗n PA n |X n , QAX
QAX
dQAX .
QAX
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Flexible constrained de Finetti reductions
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Intuitively, why should de Finetti reductions be useful to understand the parallel
repetition of multi-player games ?
⊗n
Observation : Obviously, the game distribution TX⊗n and the game predicate VAX
of Gn are both
permutation-invariant.
Consequence : One can assume w.l.o.g. that the optimal winning strategy PA n |X n , in the set of
allowed strategies AS (A n |X n ), for Gn is permutation-invariant as well. And hence,
Z
2
⊗n
⊗n
TX⊗n PA n |X n 6 poly(n)
F TX⊗n PA n |X n , QAX
QAX
dQAX .
QAX
⊗n
Goal : Show that the only p.d.’s QAX for which the fidelity weight is not exponentially small are
those s.t. QAX is close to being of the form TX RA |X with RA |X ∈ AS (A |X ). Because what
⊗n
is trivially understood.
happens when playing Gn with such strategy RA
|X
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
12 / 19
Outline
1
De Finetti type theorems
2
Multi-player non-local games
3
Using de Finetti reductions to study the parallel repetition of multi-player non-local games
4
Summary, open questions and perspectives
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
13 / 19
Parallel repetition of (sub-)no-signalling multi-player games : some results
Parallel repetition of sub-no-signalling `-player games
Let G be an `-player game s.t. ωSNOS (G) 6 1 − δ for some 0 < δ < 1. Then, for any n ∈ N and
t > (1 − δ + α)n, ωSNOS (Gn ) 6 1 − δ2 /5C`2
n
and ωSNOS (Gt /n ) 6 exp −nα2 /5C`2 , where
C` = 2`+1 − 3.
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Parallel repetition of (sub-)no-signalling multi-player games : some results
Parallel repetition of sub-no-signalling `-player games
Let G be an `-player game s.t. ωSNOS (G) 6 1 − δ for some 0 < δ < 1. Then, for any n ∈ N and
t > (1 − δ + α)n, ωSNOS (Gn ) 6 1 − δ2 /5C`2
n
and ωSNOS (Gt /n ) 6 exp −nα2 /5C`2 , where
C` = 2`+1 − 3.
Parallel repetition of no-signalling 2-player games
Let G be an 2-player game s.t. ωNS (G) 6 1 − δ for some 0 < δ < 1. Then, for any n ∈ N and
t > (1 − δ + α)n, ωNS (Gn ) 6 1 − δ2 /27
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n
and ωNS (Gt /n ) 6 exp −nα2 /33 .
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Parallel repetition of (sub-)no-signalling multi-player games : some results
Parallel repetition of sub-no-signalling `-player games
Let G be an `-player game s.t. ωSNOS (G) 6 1 − δ for some 0 < δ < 1. Then, for any n ∈ N and
t > (1 − δ + α)n, ωSNOS (Gn ) 6 1 − δ2 /5C`2
n
and ωSNOS (Gt /n ) 6 exp −nα2 /5C`2 , where
C` = 2`+1 − 3.
Parallel repetition of no-signalling 2-player games
Let G be an 2-player game s.t. ωNS (G) 6 1 − δ for some 0 < δ < 1. Then, for any n ∈ N and
t > (1 − δ + α)n, ωNS (Gn ) 6 1 − δ2 /27
n
and ωNS (Gt /n ) 6 exp −nα2 /33 .
Parallel repetition of no-signalling `-player games with full support
Let G be an `-player game whose input distribution T has full support, and s.t. ωNS (G) 6 1 − δ
n
for some 0 < δ < 1. Then, for any n ∈ N and t > (1 − δ + α)n, ωNS (Gn ) 6 1 − δ2 /5C`2 Γ2 and
ωNS (Gt /n ) 6 exp −nα2 /5C`2 Γ2 , where C` = 2`+1 − 3 and Γ is a constant which only depends
on T .
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Flexible constrained de Finetti reductions
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Parallel repetition of (sub-)no-signalling multi-player games : proof ingredients
Starting point : The optimal winning strategy PA n |X n ∈ SNOS (A n |X n ) for Gn satisfies
Z
e (QAX )2n Q ⊗n dQAX ,
TX⊗n PA n |X n 6 poly(n)
F
AX
QAX
e QAX = min max F TX RA |X , QA X .
where F
I
I
I
06/ =I [`] RAI |XI
→ Follows from monotonicity of F under taking marginals + specific form of marginals of P +
universal de Finetti reduction for conditional p.d.’s (Arnon-Friedman/Renner).
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Parallel repetition of (sub-)no-signalling multi-player games : proof ingredients
Starting point : The optimal winning strategy PA n |X n ∈ SNOS (A n |X n ) for Gn satisfies
Z
e (QAX )2n Q ⊗n dQAX ,
TX⊗n PA n |X n 6 poly(n)
F
AX
QAX
e QAX = min max F TX RA |X , QA X .
where F
I
I
I
06/ =I [`] RAI |XI
→ Follows from monotonicity of F under taking marginals + specific form of marginals of P +
universal de Finetti reduction for conditional p.d.’s (Arnon-Friedman/Renner).
Separating the “very-signalling” and the “not-too-signalling” parts in the integral :
1
Fix 0 < ε < 1 and define Pε = QAX : max min kTX RAI |XI − QAI X k1 6 ε .
06/ =I [`] RAI |XI 2
2
• QAX ∈
/ Pε ⇒ Fe QAX 6 1 − ε2 .
• QAX ∈ Pε ⇒ ∃ RA |X ∈ SNOS (A |X ) : 12 kTX RA |X − QAX k1 6 C` ε.
→ Technical lemma behind : If a conditional p.d. approximately satisfies each of the NS
constraints, up to an error ε, then it is C ε-close to an exact SNOS p.d.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Parallel repetition of (sub-)no-signalling multi-player games : proof ingredients
Starting point : The optimal winning strategy PA n |X n ∈ SNOS (A n |X n ) for Gn satisfies
Z
e (QAX )2n Q ⊗n dQAX ,
TX⊗n PA n |X n 6 poly(n)
F
AX
QAX
e QAX = min max F TX RA |X , QA X .
where F
I
I
I
06/ =I [`] RAI |XI
→ Follows from monotonicity of F under taking marginals + specific form of marginals of P +
universal de Finetti reduction for conditional p.d.’s (Arnon-Friedman/Renner).
Separating the “very-signalling” and the “not-too-signalling” parts in the integral :
1
Fix 0 < ε < 1 and define Pε = QAX : max min kTX RAI |XI − QAI X k1 6 ε .
06/ =I [`] RAI |XI 2
2
• QAX ∈
/ Pε ⇒ Fe QAX 6 1 − ε2 .
• QAX ∈ Pε ⇒ ∃ RA |X ∈ SNOS (A |X ) : 12 kTX RA |X − QAX k1 6 C` ε.
→ Technical lemma behind : If a conditional p.d. approximately satisfies each of the NS
constraints, up to an error ε, then it is C ε-close to an exact SNOS p.d.
Putting everything together : The winning probability when playing Gn with strategy PA n |X n is
upper-bounded by poly(n) (1 − ε2 )n + (1 − δ + 2C` ε)n .
It then just remains to choose ε = C`
in order to conclude.
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1 + δ/C`2
1/2
− 1 and get rid of the polynomial pre-factor
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
15 / 19
Outline
1
De Finetti type theorems
2
Multi-player non-local games
3
Using de Finetti reductions to study the parallel repetition of multi-player non-local games
4
Summary, open questions and perspectives
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Summary
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Summary
If ` players sharing sub-no-signalling correlations have a probability at most 1 − δ of winning
a game G, then their probability of winning a fraction at least 1 − δ + α of n instances of G
played in parallel is at most exp(−nc` α2 ), where c` > 0 is a constant which depends only
on `.
→ Optimal dependence in α, even in the special case α = δ.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Summary
If ` players sharing sub-no-signalling correlations have a probability at most 1 − δ of winning
a game G, then their probability of winning a fraction at least 1 − δ + α of n instances of G
played in parallel is at most exp(−nc` α2 ), where c` > 0 is a constant which depends only
on `.
→ Optimal dependence in α, even in the special case α = δ.
In the case ` = 2, this is equivalent to the analogous concentration result for the
no-signalling value of G (cf. Holenstein).
Cécilia Lancien
Flexible constrained de Finetti reductions
Madrid - April 20th 2016
17 / 19
Summary
If ` players sharing sub-no-signalling correlations have a probability at most 1 − δ of winning
a game G, then their probability of winning a fraction at least 1 − δ + α of n instances of G
played in parallel is at most exp(−nc` α2 ), where c` > 0 is a constant which depends only
on `.
→ Optimal dependence in α, even in the special case α = δ.
In the case ` = 2, this is equivalent to the analogous concentration result for the
no-signalling value of G (cf. Holenstein).
In the case where the distribution of G has full support, this implies a similar concentration
result for the no-signalling value of G, but with a highly game-dependent constant in the
exponent (cf. Buhrman/Fehr/Schaffner and Arnon-Friedman/Renner/Vidick).
→ What about games where some of the potential queries are never asked to the players ?
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Broader perspectives
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Broader perspectives
Classical case : Exponential decay of ω(Gn ) and ω(Gt /n ) under parallel repetition for any
2-player game G (Raz, Holenstein, Rao).
Quantum case : Exponential decay of ω(Gn ) under parallel repetition for any 2-player
game G with full support (Chailloux/Scarpa), only polynomial decay is known in general (Yuen).
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Broader perspectives
Classical case : Exponential decay of ω(Gn ) and ω(Gt /n ) under parallel repetition for any
2-player game G (Raz, Holenstein, Rao).
Quantum case : Exponential decay of ω(Gn ) under parallel repetition for any 2-player
game G with full support (Chailloux/Scarpa), only polynomial decay is known in general (Yuen).
“Standard” proof techniques : Show that conditioned on the event “the players have
already won k instances of the game”, the probability is high that :
i) they lose in at least 1 of the n − k remaining instances → exponential decay of ω(Gn ),
ii) they lose in most of the n − k remaining instances → exponential decay of ω(Gt /n ).
But not so straightforward and not easily generalizable to more than 2 players...
→ What about tackling the problem via de Finetti reductions ? Obstacle : classical and
quantum conditions cannot be read off on the marginals...
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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Broader perspectives
Classical case : Exponential decay of ω(Gn ) and ω(Gt /n ) under parallel repetition for any
2-player game G (Raz, Holenstein, Rao).
Quantum case : Exponential decay of ω(Gn ) under parallel repetition for any 2-player
game G with full support (Chailloux/Scarpa), only polynomial decay is known in general (Yuen).
“Standard” proof techniques : Show that conditioned on the event “the players have
already won k instances of the game”, the probability is high that :
i) they lose in at least 1 of the n − k remaining instances → exponential decay of ω(Gn ),
ii) they lose in most of the n − k remaining instances → exponential decay of ω(Gt /n ).
But not so straightforward and not easily generalizable to more than 2 players...
→ What about tackling the problem via de Finetti reductions ? Obstacle : classical and
quantum conditions cannot be read off on the marginals...
Using flexible de Finetti reductions to prove the multiplicative or additive behavior of certain
quantities appearing in QIT (e.g. output norms or entropies of quantum channels).
Example : Given a sequence of convex sets of states K (n) on H⊗n , n ∈ N, equivalence
between the multiplicative behavior under tensoring of the maximum fidelity function
F (·, K (n) ) and the support function hK (n) (·).
→ In the case where K (n) is the set of separable states on (A ⊗ B)⊗n (across the bipartite
min
cut A⊗n :B⊗n ), link with soundness gap amplification of QMA(2) and weak additivity of S∞
.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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References
R. Arnon-Friedman, R. Renner, “de Finetti reductions for correlations”, arXiv[quant-ph] :1308.0312.
R. Arnon-Friedman, R. Renner, T. Vidick, “Non-signalling parallel repetition using de Finetti reductions”,
arXiv[quant-ph] :1411.1582.
A. Chailloux, G. Scarpa, “Parallel repetition of free entangled games : simplification and improvements”,
arXiv[quant-ph] :1410.4397.
M. Christandl, R. König, G. Mitchison, R. Renner, “One-and-a-half quantum de Finetti theorems”,
arXiv :quant-ph/0602130.
M. Christandl, R. König, R. Renner, “Post-selection technique for quantum channels with applications to quantum
cryptography”, arXiv[quant-ph] :0809.3019.
P. Diaconis, D. Freedman, “Finite exchangeable sequences”.
T. Holenstein, “Parallel repetition : simplifications and the no-signaling case”, arXiv :cs/0607139.
H. Buhrman, S. Fehr, C. Schaffner, “On the Parallel Repetition of Multi-Player Games : The No-Signaling Case”,
arXiv[quant-ph] :1312.7455.
C. Lancien, A. Winter, “Flexible constrained de Finetti reductions and applications”, in preparation
C. Lancien, A. Winter, “Parallel repetition and concentration for (sub-)no-signalling games via a flexible constrained
de Finetti reduction”, arXiv[quant-ph] :1506.07002.
A. Rao, “Parallel repetition in projection games and a concentration bound”.
R. Raz, “A parallel repetition theorem”.
H. Yuen, “A parallel repetition theorem for all entangled games”, arXiv[quant-ph] :1604.04340.
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Flexible constrained de Finetti reductions
Madrid - April 20th 2016
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