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A physical approach to Tsirelson’s problem

M. Navascués, T. Cooney, D. Pérez-Garćıa and N. Villanueva
Facultad de Matemáticas, Universidad Complutense de Madrid

Abstract

Tsirelson’s problem deals with how to model separate measurements in quantum

mechanics. In addition to its theoretical importance, the resolution of Tsirelson’s prob-

lem could have great consequences for device independent quantum key distribution

and certified randomness. Unfortunately, understanding present literature on the sub-

ject requires a heavy mathematical background. In this paper, we introduce quansality,

a new theoretical concept that allows to reinterpret Tsirelson’s problem from a foun-

dational point of view. Using quansality as a guide, we recover all known results on

Tsirelson’s problem in a clear and intuitive way.

1 Introduction

Standard (i.e., non-relativistic) quantum mechanics describes bipartite systems by associat-
ing them to a tensor product of Hilbert spaces H = HA⊗HB. Measurements and operations
corresponding to party A (B) are then described by linear operators on H that act trivially
over HB (HA). Things are different, though, in Algebraic Quantum Field Theory (AQFT),
where the Haag-Kastler axioms only require that operators corresponding to measurements
performed in space-like separated bounded regions of Minkowski space-time commute [1].

Tsirelson’s problem consists in finding out if the bipartite correlations generated one way
or the other are essentially the same, i.e., if any set of AQFT-like correlations admits a non-
relativistic approximation. The problem of bounding the set of tensor bipartite correlations
arises naturally in device-independent quantum key distribution [2] and certified random-
ness generation [3]. Since current theoretical tools can only bound the set of AQFT-like
correlations [4], we may be unnecessarily limiting the range of all such protocols.

Tsirelson himself proved that commutative and tensor sets of correlations coincide if the
underlying Hilbert space H where measurement operators act is finite dimensional (see [5]).
Sholtz and Werner then showed that this result still holds under the weaker assumption
that Alice’s or Bob’s operator algebra is nuclear [5]. Such is the case if, for instance, Alice
is restricted to perform two different dichotomic measurements. More recently, Tsirelson’s
problem was related to Connes’ embedding conjecture in C∗-algebras [6]. The above results
rely on very heavy mathematical machinery; their proofs (and sometimes even statements)
thus appear incomprehensible to the average physicist.

1

http://arxiv.org/abs/1105.3373v1


Tsirelson’s problem has recently gained popularity in the Quantum Information commu-
nity, not only due to the practical consequences of its resolution, but also for the aforemen-
tioned connections with sophisticated areas of mathematics. Looking at it from the outside,
though, that Tsirelson’s problem is related to a conjecture which hundreds of mathemati-
cians have attempted to solve over the course of 35 years is not good news at all. It implies
that a physicist aiming at solving it should study advanced mathematics for several years
in order to find himself as lost as the very brilliant mathematical minds who are currently
trying to solve Connes’ embedding conjecture.

In this paper we will introduce a new foundational concept called quansality, related
to the possibility of finding a local quantum description for experiments in a bounded re-
gion of space-time. We will show that, despite its intuitive nature, quansality can be proven
mathematically equivalent to a tensor product description of bipartite quantum correlations.
Therefore, if there existed an AQFT-type scenario giving rise to bipartite correlations impos-
sible to reproduce in non-relativistic quantum systems, then any quantum model attempting
to explain the measurement statistics of one the parts could be experimentally falsified.

This intuition will allow us to recover all known results on Tsirelson’s theorem, with
physical insight and basic linear algebra as our only tools. With this, we thus hope to make
Tsirelson’s problem accessible to any physicist with a reasonable mathematical knowledge.

The paper is organized as follows: in Section 2 we will introduce the notation to be used
along the article. Then we will define and motivate the concept of quansality. In Section 4 we
will present Tsirelson’s problem and derive Tsirelson’s finite dimensional result by explicitly
constructing a local quantum model for one of the party’s observations. We will show how
to extend some results to the infinite dimensional case in Section 6. Finally, we will present
our conclusions.

2 Mathematical notation

Along this article, given a separable Hilbert space H, we will regard quantum states as
positive elements of S1 = {A ∈ B(H) : ‖A‖1 < ∞}, that we will suppose normalized unless
otherwise specified. Note that S1 ⊂ S2 = {A ∈ B(H) : tr(AA†) < ∞}. Both S1 and S2 will
play an important role in the discussion of Tsirelson’s theorem.

Being the trace norm the natural norm of the set S1, we will say that a sequence of
quantum states (ρN ) converges to ρ in S1 if limN→∞ ‖ρN −ρ‖1 = 0. Analogously, we will say
that (ρN) converges to ρ in S2 if limN→∞ tr{(ρN −ρ)(ρN −ρ)†} = 0. Notice that convergence
in S1 implies convergence in S2; the contrary is generally false.

3 Quansality

Assume that we are in a scenario where two separated parties, Alice and Bob, are each
able to tune interactions x and y in their respective labs (see Figure 1). Such interactions
(measurements) will in turn produce two pieces of experimental data a and b (the outcomes).

2



Figure 1: Sketch of a bipartite experiment of non-locality.

We will make the realistic assumption that the sets A and B of possible interactions which
Alice and Bob can produce, as well as the set of possible experimental results they can
observe, are finite. That way, if Alice and Bob compare their measurement results after
many repetitions of the experiment, they will be able to estimate the set of probability
distributions {P (a, b|x, y) : x ∈ A, y ∈ B}. Following Tsirelson’s terminology, we will call
such a set behavior, and we will denote it by P (a, b|x, y). This notation will not lead to
confusion, since along the text it will always be clear whether by P (a, b|x, y) we refer to a
probability or a behavior.

Now, suppose that Alice performs a measurement x with outcome a, and denote by
P x
a (b|y) Bob’s resulting single behavior. Then the fact that Bob’s statistics can be described

locally is equivalent to the condition

∑

a

P (a|x)P x
a (b|y) = P (b|y), (1)

where P (b|y) is a probability distribution independent of x. This condition ensures that
Bob’s statistics are not affected by Alice’s measurement choice x, and so Bob can study his
system without introducing Alice as a variable. Taking into account that P (a|x)pxa(b|y) =
P (a, b|x, y), we have that the last condition is equal to

∑

a

P (a, b|x, y) = P (b|y), (2)

the no-signaling condition [7].

Suppose now that we seek a description of Bob’s part that is not only independent of
Alice, but also compatible with Quantum Mechanics (QM). How should we modify condition
(1) to account for this extra condition? The answer requires a detailed discussion.

To build a quantum model for his system, Bob must find a set of projector operators
{F y

b } ⊂ B(HB), with
∑

b F
y
b = IHB

, and a quantum state σ ∈ S1(HB) such that P (b|y) =
tr(σF y

b ). Suppose he does.

3



Now, imagine that Alice switches interaction x and reveals Bob both x and her mea-
surement results in all realizations of the experiment. If Bob’s model is truly independent
of Alice, her distant actions should not alter the previous picture. Consequently, if his ini-
tial model was correct, then Bob should be able to find states {σ̄x

a} ⊂ S1(HB) such that
P (b|y, a, x) = tr(σ̄x

aF
y
b ).

If for some choice of {σ̄x
a} we have that

∑

a P (a|x)σ̄x
a = σ, then Bob’s model prior to

Alice’s communications can be considered correct; Alice is just completing such a model with
supplementary information (x, a) about Bob’s physical states. If, however,

∑

a P (a|x)σ̄x
a = σ

cannot be enforced for any choice of {σ̄x
a} compatible with the statistics P (b|y, a, x), then

Bob’s model was flawed from the very beginning, and Alice has helped him to falsify it.
This way, we arrive at the notion of quansality.

Definition 1. Quansality
Let P (a, b|x, y) be a bipartite behavior. We say that P (a, b|x, y) is quansal (QUANtum
cauSAL) if there exist a Hilbert space HB, projector operators {F y

b :
∑

b F
y
b = IHB

} and
a set of subnormalized quantum states {σx

a} ⊂ S1(HB) such that P (a, b|x, y) = tr(F y
b σ

x
a) and

∑

a

σx
a = σ 6= σ(x). (3)

From the above discussion, it follows that quansality is a necessary and sufficient condition
for Bob to find an independent quantum description of his physical system.

Note that quansality implies the no-signaling condition (from Alice to Bob). Condition
(3), though, is not necessary to prevent signaling from Alice, since it could well be that,
although

∑

a σ
x
a 6=

∑

a σ
x′

a for x 6= x′, both states exhibit the same statistics when measured
with {F y

b }. In the next section, we will explain how quansality relates to Tsirelson’s problem.
We will also prove that quansality is a symmetric property, i.e., if Bob’s subsystem admits
a quantum model, so does Alice’s.

4 Tsirelson’s problem

Definition 2. Let P (a, b|x, y) be a bipartite behavior. P (a, b|x, y) is non-relativistic (or
admits a tensor representation) if there exist a pair of Hilbert spaces HA,HB, a state ρAB ∈
S1(HA ⊗HB) and two sets of projectors {Ea

x ∈ B(HA)}, {F
b
y ∈ B(HB)} such that

1.
∑

aE
x
a = IA,

∑

b F
y
b = IB, for all x, y.

2. tr(ρABE
x
a ⊗ F y

b ) = P (a, b|x, y).

The set of all behaviors of this form will be denoted by Q.

Definition 3. Let P (a, b|x, y) be a bipartite behavior. We will say that P (a, b|x, y) is rel-
ativistic (or admits a field representation) if there exists a separable Hilbert space H, a
quantum state ρ ∈ S1(H) and a set of projectors {Ex

a , F
y
b : x ∈ A, y ∈ B}, such that

4



1.
∑

aE
x
a =

∑

b F
y
b = I, for all x, y.

2. [Ex
a , F

y
b ] = 0, for all a, b, x, y.

3. tr(ρEx
aF

y
b ) = P (a, b|x, y).

We will call Q′ the set of all such behaviors. In [4] it was proven that Q′ is a closed set.

Remark 1. It can be proven (see [6]) that in the definitions of Q and Q′ we can relax the
condition that measurement operators are projectors and demand instead that such opera-
tors Ex

a , F
y
b are positive semidefinite, i.e., elements of a Positive Operator Valued Measure

(POVM).

Since [Ex
a ⊗ IB, IA ⊗ F y

b ] = 0, we have that Q̄ ⊂ Q′, i.e., any behavior approximable
by non-relativistic behaviors admits a field representation. Tsirelson’s problem consists of
deciding whether such an inclusion is strict, that is, whether Q̄ 6= Q′ [5]. In addition to
its relevance in Algebraic Quantum Field Theory (AQFT), Tsirelson’s problem acquires a
foundational nature due to the following lemma (inspired by the constructions presented in
[8, 9]).

Lemma 4. P (a, b|x, y) is non-relativistic iff it is quansal.

Proof. The right implication is immediate: if P (a, b|x, y) satisfies the conditions of Definition
2, take σx

a ≡ trA(ρABE
x
a ⊗ IB) and note that

∑

a σ
x
a = ρB and that tr(σx

aF
y
b ) = tr(ρABE

x
a ⊗

F y
b ) = P (a, b|x, y).
For the left implication, define

Ex
a ≡ [σ−1/2σx

aσ
−1/2]T , (4)

where T denotes the transpose with respect to an orthonormal basis {|j〉}j for H. Clearly,
Ex

a ≥ 0, and

∑

a

Ex
a =

∑

a

[σ−1/2σx
aσ

−1/2]T = I, (5)

and so Alice’s operators are complete POVM elements. Define now the quantum state
ρAB ≡ |φσ〉〈φσ| with

|φσ〉 =

dim(H)
∑

j=1

|j〉 ⊗ σ1/2|j〉. (6)

Here, we allow for the possibility that dim(H) = ∞ (we are working under the assumption
that H is separable).

Obviously, ρAB ≥ 0. Also, tr(ρAB) = trB(σ
1/2

Iσ1/2) = tr(σ) = 1. The operator ρAB thus
corresponds to a normalized quantum state. Finally, we have that

5



tr(ρABE
x
a ⊗ F y

b ) = trB(σ
1/2(Ex

a )
Tσ1/2F y

b ) =

= tr(σx
aF

y
b ) = P (a, b|x, y). (7)

Therefore, if Q̄ 6= Q′, then there exist multipartite scenarios in Algebraic Quantum Field
Theory where each part cannot describe its own local quantum system in a way consistent
with future communications from other parties. Such a situation would be certainly unde-
sirable: if that were the case, then any quantum theory attempting to explain the results of
a particular experiment would have to include the rest of the universe in its formulation!

In the following sections we will show how to recover all existing results on Tsirelson’s
theorem by building upon this intuition. That is, given a relativistic bipartite distribution,
we will find a local quantum model consistent with Bob’s measurement statistics, thereby
proving that such a distribution also admits a tensor representation.

5 Finite dimensions and the Heat Vision effect

In the following, we will prove that definitions 2 and 3 are equivalent when we demand the
Hilbert spaces H,HA,HB to be finite dimensional.

Theorem 5. Tsirelson’s Theorem
Let P (a, b|x, y) admit a finite dimensional field representation. Then, P (a, b|x, y) is non-
relativistic.

Proof. From lemma 4, it is enough prove that P (a, b|x, y) is quansal, i.e., that there exist
operators {F y

b } and quantum states {σx
a} giving rise to P (a, b|x, y) and satisfying (3).

To grasp some intuition about the problem, picture the following scenario: Alice performs
a measurement and then gives Bob control over the whole space H. Consequently, HB → H,
σx
a → ρxa ≡ Ex

aρE
x
a and Bob’s measurement operators are {F y

b }. Let us see how this rough
approach performs. Indeed, we have that P (a, b|x, y) = tr(ρxaE

y
b ). Nevertheless, the states

∑

a ρ
x
a ≡ ρx will be generally x-dependent. The problem of this latter strategy is thus that

Alice’s initial measurement leaves a ‘mark’ x on the state, and this mark is to be erased if
we want to recover Tsirelson’s theorem.

Let us then complicate slightly our previous scheme, see Figure 2: as before, Alice is
going to measure, store the pair of values (x, a) and give Bob total control of H. However,
this time she’s going to introduce noise to her system before leaving the house. Note that
the Kraus operators of any Completely Positive (CP) map Alice applies to the states ρxa will
commute with Bob’s measurement operators. Thus the statistics of any perturbed state ρ̃xa
will be the same, i.e., tr(ρ̃xaF

y
b ) = tr(ρxaF

y
b ) = P (a, b|x, y). On the other hand, it could be

that ρ̃x is closer to ρ̃x
′

than ρx to ρx
′

.
What could Alice do? Well, we just know that she is capable of implementing a num-

ber of measurements. Suppose, then, that after performing a measurement x and storing

6



Figure 2: How to build a quansal representation for P (a, b|x, y) ∈ Q′. 1) Alice interacts
with her system in some way x and stores the outcome a. 2) She adds noise to her part of
the lab in order to erase the information of her measurement choice. Since she is restricted
to act with her algebra of observables, her operations do not affect Bob’s statistics. 3) She
gives Bob full control of the bipartite system.

the outcome a, Alice measures randomly her system N times in the hope of erasing the
information about x.

Define the completely positive maps Ωx : B(H) → B(H)

Ωx(M) ≡
∑

a

Ex
aMEx

a . (8)

The map

Ω(M) ≡
1

|A|

∑

x∈A

Ωx(M) (9)

therefore represents the channel that results when Alice applies any interaction x with prob-
ability 1/|A|. We will next see that the limit Ω∞(ρ) ≡ limN→∞ΩN(ρ) always exists, and,
indeed, it erases the information on x. That is, Ω∞(Ωx(ρ)) = Ω∞(ρ), for all x ∈ A.

Through the invertible transformation |i〉〈j| → |i〉|j〉, we can view ρ as a vector in
S2

∼= H ⊗ H and Ω as a super operator Ω̄ acting over H ⊗ H. The expression of Ω thus
changes to

Ω̄ =
1

|A|

∑

x

Ω̄x, with Ω̄x =
∑

a

Ex
a ⊗ (Ex

a )
T . (10)

Note that Ω̄ is a hermitian positive semidefinite operator, and so it can be diagonalized,
i.e., Ω̄ =

∑

i λi|Ψi〉〈Ψi|. Moreover, since Ω̄ is a convex combination of projectors Ω̄x, we have
that λi ∈ [0, 1]. It follows that Ω̄N =

∑

i λ
N
i |Ψi〉〈Ψi|, and thus

Ω̄∞ = lim
N→∞

Ω̄N =
∑

i|λi=1

|Ψi〉〈Ψi|. (11)

The existence of the map Ω∞ in S2 has been proven. Since in finite dimensions S1 = S2

(i.e., the norms ‖ · ‖1, ‖ · ‖2 are equivalent), limN→∞ΩN (ρ) exists for all ρ ∈ S1. From
tr{ΩN(ρ)} = 1 and (I⊗ ΩN)(ρ) ≥ 0, we thus have that Ω∞ is completely positive and trace
preserving.

7



Suppose now that |Ψ〉 is such that Ω̄|Ψ〉 = |Ψ〉. Then,

〈Ψ|Ω̄|Ψ〉 =
1

|A|

|A|
∑

x=1

〈Ψ|Ω̄x|Ψ〉 = 1. (12)

This can only be true if 〈Ψ|Ω̄x|Ψ〉 = 1 for all x, and so Ω̄x|Ψ〉 = |Ψ〉 for all x. This implies
that Ω̄xΩ̄

∞|φ〉 = Ω̄∞|φ〉 for any vector |φ〉. We thus have that Ω̄∞Ω̄x = Ω̄∞, i.e., for any x
and any M ∈ S2,

Ω∞(
∑

a

Ex
aMEx

a ) = Ω∞(M). (13)

In summary, we have that the states

σx
a ≡ Ω∞(Ex

aρE
x
a ) (14)

satisfy

∑

a

σx
a =

∑

a

Ω∞(Ex
aρE

x
a ) = Ω∞(

∑

a

Ex
aρE

x
a ) = Ω∞(ρ) ≡ σ, (15)

and so generate the quansal behavior

tr(σx
aF

y
b ) = lim

N→∞
(ΩN (Ex

aρE
x
a )F

y
b ) = tr(Ex

aρE
x
aF

y
b ) = P (a, b|x, y). (16)

Remark 2. The previous proof can be extended to the case where Ex
a , F

b
y are general POVM

elements rather than projectors. The maps Ωx should then be defined as

Ωx(M) ≡
∑

a

√

Ex
aM

√

Ex
a (17)

Seen as operators in S2, these maps are also positive semidefinite, and its norm is smaller
or equal than 1, since they are unital [10]: the eigenvalues of Ω are thus in the interval [0, 1].
From this point on, the rest of the proof is identical.

Note that in order to prove Tsirelson’s theorem we just invoked finite dimensionality to
argue that S2 = S1, and thus that the limiting state limN→∞Ω(ρ) exists in S1 for any initial
ρ. In infinite dimensions, S2 6= S1 in general, so it can happen that repeated independent
random measurements do not thermalize, i.e., that the map limN→∞ΩN is well defined in
S2

∼= H⊗H but nevertheless the sequence of states limN→∞ΩN (ρ) does not converge to any
quantum state (any element of S1). Such a non-trivial phenomenon is called Heat Vision
[11], and, under some fair assumptions on the structure of the hamiltonian operator of the
system1, is always accompanied by an unbounded energy increase. As shown in [11], the

1More concretely, that the spectrum of the hamiltonian H does not contain accumulation points. This

is always the case if H describes a finite number of trapped non-relativistic particles subject to a potential

bounded from below.

8



Heat Vision effect can be observed even in infinite dimensional systems with just two von
Neumann measurements of two outcomes each.

Tsirelson’s theorem can therefore be extended in the following way:

Theorem 6. Tsirelson’s Theorem (bis)
Let P (a, b|x, y) be a bipartite behavior, attainable in a separable Hilbert space by means
of a commutative setup. If Alice’s or Bob’s measurement operators are such that random
independent measurements do not produce Heat Vision, then P (a, b|x, y) is non-relativistic.

6 Infinite dimensions-general results

The previous theorem makes one wonder whether arguments based on Heat Vision are really
necessary in order to guarantee that Tsirelson’s theorem holds in infinite dimensions. The
next result shows that such is not the case.

Lemma 7. Suppose that |A| = 2, and that both measurement settings x1, x2 have associated
two outcomes each. Then, any behavior P (a, b|x, y) admitting a field representation can be
approximated arbitrarily well by non-relativistic behaviors.

Proof. First, note that

∑

a=1,2

Ex
aρE

x
a =

1

2
(UxρUx + ρ), (18)

where Ux = Ex
1 − Ex

2 . It is immediate that Ux = U †
x, and U2

x = I. The operators U1, U2 are
thus unitary and self-adjoint, and, of course, commute with Bob’s measurement operators.
We will denote by {Λx : x = 1, 2} the CP maps Λx(ρ) = UxρUx.

Although we will follow the lines of the proof of Theorem 5, there will be two important
changes: 1) rather than using a single channel Ω∞ to erase the information on x, we will

consider two families of channels Γ
(N)
1 ,Γ

(N)
2 , one for each measurement setting x. 2) The

limits limN→∞ Γ
(N)
1 (ρ1),Γ

(N)
2 (ρ2) will not exist in general. Instead, we will demand the much

weaker condition that the operator Γ
(N)
1 (ρ1)− Γ

(N)
2 (ρ2) tends to 0 in trace norm.

Define the families of maps

Γ
(N)
1 ≡

1

N + 1

N
∑

k=0

(Λ1 ◦ Λ2)
k; Γ

(N)
2 ≡ Γ

(N)
1 ◦ Λ1. (19)

Then it can be checked that

Γ
(N)
1 (Ω1(ρ)) +

1

N + 1
(Λ1 ◦ Λ2)

N+1(ρ) = Γ
(N)
2 (Ω2(ρ)) +

1

N + 1
ρ. (20)

Given an arbitrary 1-partite behavior q(a|x) for Alice we thus have that the states

9



σ1
a(N) ≡

N + 1

N + 2
Γ
(N)
1 (ρ1a) + q(a|1)

1

N + 1
(Λ1 ◦ Λ2)

N+1(ρ),

σ2
a(N) ≡

N + 1

N + 2
Γ
(N)
2 (ρ2a) + q(a|2)

1

N + 1
ρ (21)

satisfy condition (3) and so give rise to the non-relativistic behavior

PN(a, b|x, y) =
N + 1

N + 2
P (a, b|x, y) +

1

N + 2
q(a|x)p(b|y). (22)

As you can see, limN→∞ PN(a, b|x, y) = P (a, b|x, y), i.e., P (a, b|x, y) can be approximated
arbitrarily well by a non-relativistic behavior.

Unfortunately, the former trick does not work in scenarios other than the two inputs-
two outputs case on Alice’s side. Imagine, for instance, that Alice can apply two different
interactions 1 and 2, the former having two possible outputs and the latter, three outputs.
Let P (a, b|x, y) ∈ Q′ be Alice and Bob’s relativistic distribution. Then, using a similar
construction one arrives at

PN(a, b|x, y) ≡ (1− pN)P (a, b|x, y) + pNq(a|x)p(b|y) ∈ Q, (23)

with pN ≡
1+ 3

N+1

7+ 3

N+1

. Note that this time limN→∞ pN = 1/7 6= 0, i.e., we cannot prove that

P (a, b|x, y) ∈ Q̄.
However, this last result has a clear physical interpretation: suppose that Alice and Bob

are performing an AQFT experiment of non-locality, with Alice having two interactions with
two and three outcomes to play with; and Bob, an arbitrary number of interactions with
an arbitrary number of outcomes. Then, if Alice only measures her system with probability
6/7− ǫ and with probability 1/7+ ǫ outputs a random result, the resulting behavior admits
a tensor representation.

7 Conclusion

In this article we have introduced the axiom of quansality, a generalization of the no-signalling
principle where local statistics are constrained to be quantum. In the bipartite case, we
showed that quansality is mathematically equivalent to the existence of a non-relativistic
model to account for Alice and Bob’s observations. We then showed how to exploit such an
equivalence in order to recover Tsirelson’s theorem and its extension to infinite dimensional
systems where Alice has a two-setting/two-outcome scenario. The particular strategy that
we described to ‘quansalize’ Bob’s system does not work if Alice’s number of measurement
settings or outcomes is bigger, though. Any further generalization of Tsirelson’s theorem will
then have to modify Bob’s measurement operators as well as Alice’s in order to move from Q′

10



to Q. It is also worth mentioning that the approach described here cannot be extended to the
tripartite case. Indeed, reference [9] gives examples of tripartite distributions P (a, b, c|x, y, z)
which admit local quantum descriptions but such that P (a, b, c|x, y, z) 6∈ Q,Q′.

On a more positive side, the fact that bipartite results can be derived from such simple
notions suggests that the full solution of the problem may be achieved with physical intuition
alone, without resorting to sophisticated mathematics. So now that they have no excuse for
not coping with the current state-of-the-art on Tsirelson’s problem, we would thus like to
encourage physicists around the world to actually work on it.

References

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11

http://arxiv.org/abs/1009.1833
http://arxiv.org/abs/0812.4305
http://arxiv.org/abs/1008.1168
http://arxiv.org/abs/1010.4983

	1 Introduction
	2 Mathematical notation
	3 Quansality
	4 Tsirelson's problem
	5 Finite dimensions and the Heat Vision effect
	6 Infinite dimensions-general results
	7 Conclusion