de Finetti reductions for correlations

When analysing quantum information processing protocols one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structure. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.


I. INTRODUCTION
The quantum de Finetti theorems [1][2][3][4] and post selection theorem [5] have a unique role in the analysis of quantum information processing tasks.These theorems give us a way to reduce permutation invariant quantum states to a more structured state, called the quantum de Finetti state.In general, we say that a state is of de Finetti-type if it is a convex combination of i.i.d.(independent identical distributed) states.de Finetti states are usually much easier to handle than general states due to their simple structure and therefore a reduction to such states can simplify calculations and proofs of various quantum information processing tasks.Indeed, one of the famous applications of reductions to de Finetti states is a proof which states that in order to prove security of quantum key distribution against general attacks it is sufficient to consider attacks on individual signals [5].Other applications include quantum tomography [6] or the quantum reverse Shannon coding [7].
It is thus interesting to see whether such a theorem is unique for quantum theory or can be proven for more general theories.More specifically, we are interested in a theorem which will allow us to do a similar de Finetti reduction when considering conditional probability distributions.Any physical system can be described by a conditional probability distribution P A|X where X is the input, or the measurement of the system, and A is the output.In particular, P A|X (a|x) is the probability for outcome a given that a measurement x was made.Note that the system may have as many inputs and outputs as required and therefore we do not restrict the structure of the system by describing it as a conditional probability distribution.
Conditional probability distributions can be used to describe the operational behaviour of physical systems, including systems that might not conform with quantum theory, such as non-signalling systems.Consider for example a system shared by two parties, P AB|XY , where X and A are, respectively, the input and output of the first party and Y and B of the second.We then say that the system is non-signalling if it cannot be used to commu-nicate, i.e., the output of one party is independent of the input of the other.The famous PR-box [8] is an example for a non-quantum bipartite system which can be written as a (non-signalling) conditional probability distribution.
One of the main motivations for a de Finetti reduction for conditional probability distributions in general, and non-signalling systems in particular, is device independent cryptography [9,10].In device independent cryptography we consider the devices as black boxes, about which we know nothing.The security of protocols based on such systems can therefore rely only on the observed statistics.When analysing these protocols the quantum de Finetti theorems cannot be applied for they depend on the dimension of the quantum systems, which is unknown in device independent protocols.Several different nonsignalling de Finetti theorems have been established recently [11][12][13], but they cannot be applied to these tasks either.
In this paper we prove a general de Finetti reduction theorem, from which we can derive several more specialized statements that are of interest for applications.The most simple and straightforward corollary is a de Finetti redcution which can be applied to permutation invariant conditional probability distributions.
The second corollary is a reduction which can be applied to a family of systems which is relevant for cryptographic protocols based on the CHSH inequality [14] or the chained Bell inequalities [15,16].In this corollary we connect any system out of this family of systems to a special non-signalling de Finetti system τ CHSH AB|XY .In contrast to the non-signalling de Finetti theorems of [11][12][13] we do not assume anything about the internal structure of P AB|XY , i.e., we do not need to assume non-signalling between any of the subsystems.
Up to date, almost all known device independent cryptographic protocols are based on the CHSH inequality or the more general chained Bell inequalities.For this reason we pay specific attention to systems which are relevant for such protocols.However, our theorem can be applied also to other families of systems which might be useful in future protocols.As an example of an application of our theorem we prove that for protocols which are based on the violation of the CHSH and chained Bell inequalities it is sufficient to consider the case where Alice and Bob share our de Finetti system τ CHSH AB|XY .We do this by bounding the distance between two channels which act on conditional probability distributions.
The remainder of this paper is structured as follows.In Section II we present and explain the general de Finetti reduction theorem and its corollaries.We then show in Section III how our de Finetti reductions can be used in applications and, in particular, how they can simplify security proofs of non-signalling cryptography.The proof for the de Finetti reduction theorem is given in the Appendix.We first explain an easier to follow proof that holds for systems with a CHSH symmetry (Appendix A) and then provide the general proof.

II. RESULTS
We start with some basic definitions.A and X denote random variables over a ∈ {0, 1, ..., l − 1} n and x ∈ {0, 1, ..., m − 1} n respectively.An n-partite system P A|X is a conditional probability distribution if for every x a P A|X (a|x) = 1 and for every a, x P A|X (a|x) ≥ 0. When we consider two different systems P A|X and Q A|X it is understood that both systems are over the same random variables X and A. The set {1, 2, . . ., n} is denoted by [n].
For stating the de Finetti reduction theorem we will need the following definitions.Definition 1.Given a system P A|X and a permutation π of its subsystems1 we denote by P A|X • π the system which is defined by ∀a, x P A|X • π (a|x) = P A|X (π(a)|π(x)) .
An n-partite system P A|X is permutation invariant if for any permutation π As mentioned in the previous section, we say that a system is a de Finetti system if it is a convex combination of i.i.d.systems.Formally, Definition 2. A de Finetti system τ A|X is a system of the form where dQ A1|X1 is some measure on the space of 1-party systems and Q ⊗n A1|X1 is a product of n identical 1-party systems Q A1|X1 .
We are now ready to state the different de Finetti reductions.For simplicity we start by giving the first corollary of the more general theorem (Theorem 4).This corollary is the reduction for conditional probability distributions, which connects general permutation invariant systems to a specific de Finetti system.
Corollary 3 (de Finetti reduction for conditional probability distributions).There exists a de Finetti system τ A|X where x ∈ {0, 1, ..., m − 1} n and a ∈ {0, 1, ..., l − 1} n such that for every permutation invariant system P A|X ∀a, x P A|X (a|x) ≤ (n + 1) m(l−1) τ A|X (a|x) .The de Finetti system τ A|X is an explicit system which we construct in the proof of the general theorem in Appendix B. This theorem holds for every permutation invariant system P A|X , not necessarily quantum or nonsignalling.Note however that according to Definition 1 we consider permutations which permute the 1-party subsystems of P A|X 2 .Corollary 3 is relevant for the case in which one considers permutation invariant conditional probability distributions P A|X .However, if the systems one considers have additional symmetries S then we can prove a better de Finetti reduction -a reduction with a smaller factor and a special de Finetti system with the same symmetries S.
In the following we consider a specific family of symmetries -symmetries between different inputs and outputs of the subsystems of P A|X .Formally, the types of symmetries that we consider are described, among other things, by a number d ≤ m(l − 1) which we call the degrees of freedom of the symmetry (see Appendix B for details and formal definition of the symmetries).The general theorem then reads: Theorem 4 (de Finetti reduction for conditional probability distributions with symmetries S).There exists a de Finetti system τ S A|X where x ∈ {0, 1, ..., m − 1} n and a ∈ {0, 1, ..., l − 1} n such that for every permutation invariant system P A|X with symmetry S (with d degrees of freedon) For the case of no symmetry we have d = m(l − 1) from which Corollary 3 stated before follows.The proof of this theorem is given in Appendix B.
The symmetries S that we consider are of particular interest when considering cryptographic protocols which are based on non-signalling systems.For example, the systems which are being used in protocols which are 2 In contrast to systems P AB|XY which can also be permuted as P AB|XY • π (ab|xy) = P AB|XY (π(a)π(b)|π(x)π(y)), as is usually the case in cryptographic tasks.For dealing with such systems we will consider a different reduction, stated as Corollary 6. based on the violation of the CHSH inequality (such as [17,18]) have a great amount of symmetry.The additional symmetry allows us to prove a corollary of Theorem 4 which can be used to simplify such protocols.
Before we state the corollary for the CHSH case, let us define what do we mean when we say that a system has a CHSH-type of symmetry.In cryptographic protocols based on the CHSH inequality the basic systems that we consider are bipartite systems P AB|XY held by Alice and Bob where a, b, x, y ∈ {0, 1} n .Definition 5 (CHSH-type symmetry).A system P AB|XY where a, b, x, y ∈ {0, 1} n has a CHSH-type symmetry if for all i ∈ [n] there exists p i ∈ [0, 1  2 ] such that where a i = a 1 a 2 . . .a i−1 a i+1 . . .a n and b i , x i , y i are defined in a similar way.
A simple system P AB|XY which has this symmetry for example is a product system of 2-partite systems as in Figure 1 with different values of p.
Corollary 6 (de Finetti reduction for systems with the CHSH symmetry).There exists a non-signalling de Finetti system τ CHSH AB|XY where a, b, x, y ∈ {0, 1} n such that for every permutation invariant3 system P AB|XY with the CHSH symmetry as in Definition 5 ∀a, b, x, y P AB|XY (a, b|x, y) ≤ (n + 1) τ CHSH AB|XY (a, b|x, y) .
Note that we do not assume that the system P AB|XY satisfies any non-signalling conditions.Our theorem holds even when there is signalling between the subsystems.
Corollary 6 is derived from Theorem 4 by showing that d = 1 for the CHSH symmetry 4 .For pedagogical reasons, we also present a self-contained proof including an explicit construction of the system τ CHSH AB|XY in Appendix A.
Although the assumption about the symmetry of the systems in Corollary 6 appears to be rather restrictive, the statement turns out to be useful for applications.In the next section we show how the de Finetti reductions in general are useful and, in particular, how Corollary 6 can be used to simplify the analysis of protocols which are based on the violation of the CHSH inequality.

III. APPLICATIONS TO CRYPTOGRAPHY
To illustrate the use of the de Finetti reductions stated in the previous section, we start by considering the following simplified application.Let T be a test which interacts with a system P A|X and outputs "success" or "fail" with some probabilities.We denote by Pr fail (P A|X ) the probability that T outputs "fail" after interacting with P A|X .We consider permutation invariant tests, defined as follows.
Definition 7. A test T is permutation invariant if for all systems P A|X and all permutations π we have Pr fail (P A|X ) = Pr fail (P A|X • π) .
Using the de Finetti reduction in Corollary 3 we can prove upper bounds of the following type: Lemma 8. Let T be a permutation invariant test.Then for every system P A|X Pr fail (P A|X ) ≤ (n + 1) m(l−1) Pr fail (τ A|X ) .
The proof of this lemma is given in Appendix C. The importance of the de Finetti reductions is already obvious from this simplified example -if one wishes to prove an upper bound on the failure probability of the test T , instead of proving it for systems P A|X it is sufficient to prove it for the de Finetti system τ A|X and "pay" for it with the additional polynomial factor of (n+ 1) m(l−1) .Since the de Finetti system has a nice i.i.d.structure this can highly simplify the calculations of the bound.Moreover, when considering security proofs one usually finds that the bound on Pr fail (τ A|X ) is exponentially small in n.If this is indeed the case then the polynomial factor of (n + 1) m(l−1) will not affect the bound.That is, an exponentially small bound on Pr fail (τ A|X ) implies an exponentially small bound on Pr fail (P A|X ).
While the notion of a test as discussed above allows for a simple treatment of cases where the failure can be directly defined as an event, it is unfortunately not directly applicable to security proofs for general cryptographic protocols, such as quantum key distribution.In order to prove security one usually needs to establish an upper bound on the distinguishing advantage between the applied protocol and an ideal protocol.Formally we describe the protocols by channels which act on the system and bound the distinguishing advantage between the two channels.
When considering quantum protocols this distinguishing advantage is given by the diamond norm [19].The distance between two channels E and F which act on quantum states ρ A is given by Informally the idea is that in order to distinguish two channels we are not only allowed to choose the input state to the channels ρ A , but also keep to ourselves a purifying system ρ C .
Although the definition of the diamond norm includes a maximisation over all states ρ AC , using the quantum post selection theorem, it was proven that when considering permutation invariant channels it is sufficient to calculate the distance for a specific quantum de Finetti state [5].Motivated by this we prove here a similar bound on a distance analogous to the diamond norm for channels which act on conditional probability distributions.
We consider here channels of the form E : {P A|X } → {P K } which interact with conditional probability distributions P A|X and output a classical bit string k ∈ {0, 1} t of some length t ≥ 0 with some probability P K (k).The probability distribution of the output depends on the channel E itself and is given by the following definition.Definition 9.The probability that a channel E will output a string k ∈ {0, 1} t when interacting with P A|X is where Pr E (x) is the probability that the channel E will input x to P A|X and E(a, x) is the function according to which the output of the channel is determined.Analogously, The connection between the channel and the system is illustrated in Figure 2.
Before considering the distance between two such channels we need the following definition.
The channel E ⊗ 1 acts on an extension P AC|XZ of P A|X and outputs a classical string k ∈ {0, 1} t according to the probability EK (k).
Definition 10.An extension5 of a system P A|X is a system P AC|XZ such that We say that an extension P AC|XZ is non-signalling if the second marginal, P C|Z is also properly defined, i.e., it does not depend on x.
For simplicity (and since it is the relevant scenario for cryptography) we consider here only non-signalling extensions P AC|XZ of P A|X .
Definition 11.The distance between two channels E, F : {P A|X } → {P K } according to the diamond norm is where the maximisation is over all systems P A|X and all possible extensions of them and F ⊗ 1(P AC|XZ ) is defined in a similar way.
A more explicit expression for the diamond norm is given in Equation (C2) in Appendix C.
As in Definition 7, we say that a channel is permutation invariant if for all permutations π, E(P A|X ) = E(P A|X • π).In a similar manner we can also consider channels which are S invariant.Definition 12.We say that a mapping µ of (a, x) to (a ′ , x ′ ) respects the symmetry S if for every system P A|X with this symmetry P A|X = P A|X • µ.
A channel E is S invariant if for every P A|X and every mapping µ which respects S we have E(P A|X ) = E(P A|X • µ).
Using these concepts and the de Finetti reduction given in Theorem 4 we can prove the following bound on the diamond norm.
Theorem 13.For any two permutation invariant and S invariant channels E, F : where d is the number of degrees of freedom of S and τ S AC|XZ is a non-signalling extension of the de Finetti system τ S A|X .
The proof is given in Appendix C.
In particular, for the case of CHSH symmetry we get the following corollary.
Corollary 14.For any two permutation invariant and CHSH invariant channels E, F : where τ CHSH ABC|XY Z is a non-signalling extension of the de Finetti system τ CHSH AB|XY .
Corollary 14 implies that when proving security of cryptographic protocols based on the CHSH inequality it is sufficient to consider the case where Alice and Bob share the de Finetti system τ CHSH AB|XY .However, one still needs to take into account all possible non-signalling extensions of this bipartite system to a tripartite system τ CHSH ABC|XY Z that includes the adversary, as can be seen from the maximisation over τ CHSH ABC|XY Z .These type of proofs can be done, as for example in [18].
We further emphasise that this does not imply that Alice and Bob's system is a convex combination of i.i.d systems when including the adversary's knowledge, but only from Alice and Bob's point of view.This is in contrast to the stronger result achieved by the quantum post selection theorem [5].However, due to the no-go theorems given in [20,21] we know that such a stronger result is not possible in the more general scenario that we consider here.
Two additional remarks are in order.First, to use this corollary we must consider protocols which are invariant under the CHSH symmetry (and therefore the channel describing them will also be invariant under the relevant mappings).Fortunately, this invariance can be ensured by performing an additional step in the beginning of the protocol, called depolarisation [22].The depolarisation procedure will not affect the correctness of the protocol and will make it invariant under the appropriate mappings µ.Such depolarisation procedures can also be constructed for other types of protocols such as protocols which are based on the chained Bell inequalities.
Second, note that since the system τ CHSH AB|XY is not quantum but non-signalling, this result cannot be applied in a trivial manner to proofs where it is assumed that Alice and Bob's statistics is restricted by quantum theory.

CONCLUDING REMARKS AND OPEN QUESTIONS
In this paper we introduced a general de Finetti-type theorem from which various more specialised variants can be derived.Interestingly, such theorems can be formulated even without relying on assumptions regarding the non-signalling conditions between the subsystems or the underlying dimension.In the general theorem, Theorem 4, we can also see how additional symmetries of the systems can affect the factor in the de Finetti reduction.This suggests that the same relationship might exist in the quantum variant of the theorem as well [5].
As an example for an application we showed how our theorems can be used to simplify the analysis of cryptographic protocols.Specifically, we explained how our theorem can be used in device independent protocols in which the parties are not assumed to be restricted by quantum theory.We hope that this approach will also be useful for quantum device independent information processing protocols in the future.
The techniques used to prove our theorems are different from the techniques used in previous papers to establish general de Finetti theorems.We therefore hope that our techniques will shed new light on de Finetti reductions in general as well as applications in device independent scenarios.
In this section we give a direct proof of Corollary 6.The proof for the general theorem, Theorem 4, is given in the next section.Corollary 6 can also be proven easily by using the general theorem, however, we choose to give also this direct proof to help the interested reader understand the main ideas and insights of the proof which are already present in this simpler version.
In order to prove Corollary 6 we construct a specific de Finetti system τ CHSH AB|XY .As our de Finetti system we choose τ CHSH AB|XY = Q ⊗n A1B1|X1Y1 dQ A1B1|X1Y1 to be a convex combination of systems Q ⊗n A1B1|X1Y1 where Q A1B1|X1Y1 is the basic system given in Figure 1.As a density measure we choose dQ to be uniform over all systems Q A1B1|X1Y1 of this form, i.e., we integrate uniformly over different values of p ∈ [0, 1  2 ].We can write τ CHSH AB|XY explicitly by using the following notation.Definition 15. τ CHSH AB|XY is the non-signalling system defined by The de Finetti system is non-signalling since the systems Q A1B1|X1Y1 are non-signalling for every value of p (see Figure 1).Proof.The integral above can be solved explicitly: .
where B is the Beta function.Recall that N CHSH is a functions of a, b, x and y although we do not write it explicitly.
The following lemma gives us an upper bound on any entry P AB|XY (ab|xy) of every permutation invariant system P AB|XY with the CHSH symmetry.
Lemma 17.For every permutation invariant system P AB|XY with the CHSH symmetry The idea behind the proof of this lemma is to bound the value of a specific entry P AB|XY (ab|xy) by counting how many entries P AB|XY (ã b|xy) must have the same value as P AB|XY (ab|xy) due to the symmetry of P AB|XY .Since the sum of all entries with particular inputs x, y is 1 this will give us a bound on P AB|XY (ab|xy).
Proof.Given a, b, x, y imagine that we are placing a colored ball above each foursome (a i , b i , x i , y i ) as in Figure 3.If the foursome fulfils the CHSH condition we label it with a blue ball, otherwise with a red ball.With this picture in mind, the CHSH symmetry as in Definition 5 actually says that by changing two balls of the same color we do not change the value according to the probability distribution P AB|XY .
Given a specific entry P AB|XY (ab|xy) we would like to know how many entries with the same inputs x, y have to have the same value as the given entry.Formally, we would like to have a lower bound on How small can N (a, b, x, y) be?Or in other words, in how many ways can we change a and b while getting an entry P AB|XY (ã b|xy) with the same value?We now prove One way of changing (a, b, x, y) to (ã, b, x, y) without changing the value of the entry is to change (a, b, x, y) to (ã, b, x, y) such that both will have the same sequence of colored balls.For example, in Figure 3 we can change (a 1 , b 1 , x 1 , y 1 ) = (0, 0, 0, 0) to (ã 1 , b1 , x 1 , y 1 ) = (1, 1, 0, 0) since they have the same inputs (x 1 , y 1 ) = (0, 0) and both will be denoted by a blue ball (therefore according to the symmetry this change will not affect the overall value of the entry).In how many such different ways can we change a and b?For every index i ∈ [n] and every input bits x i , y i there are exactly two a i , b i for which the CHSH conditions holds (i.e.blue ball) and two for which it does not (red ball).Therefore there are exactly 2 n different pairs of strings (ã, b) such that (a, b, x, y) and (ã, b, x, y) have the same sequence of colored balls and therefore P AB|XY (ã b|xy) = P AB|XY (ab|xy).
Changing a and b in different ways than the way given above will necessarily change the colors sequence.However, we can still prove using the permutation invariance of P AB|XY that for some specific changes the value of the entry will still stay the same.The specific changes that we consider are determined by permutations of the colored balls.
In order to understand how every different permutation of the balls π is realised as a permutation on x, y, a, b consider the example drawn in Figure 4. On the left side we see a permutation of the balls from Figure 3.We start by filling up the columns for which there is no change in the color of the ball with the original columns.Then we pair the permuted balls such that each blue ball is replaced with a red ball, and permute the columns according to this paring.The permutation in the figure, for example, is just the permutation of indices (1, 2) and (4, 7).In general, every permutation of the balls can be described by such pairing and between every two different permutation we will have at least one index which will be permuted in one of them and not in the other.
For every permutation π as described above we have since the system permutation invariant.We will now show that due to the CHSH symmetry we also have where ãπ = a and bπ is derived from b by negation of all the bits which are being permuted in π.
To see that Equation (A2) is correct recall that the permutation π permuted two columns i, j only if for one of them the CHSH condition holds and for the other not.Therefore, if for example we had a i ⊕ b i = x i • y i (and the index i was permuted by π) then (π(a)) i ⊕ (π(b)) i = (π(x)) i • (π(y)) i .By definition (ã π ) i = a i and ( bπ ) i = b i and therefore we also know that a i ⊕ ( bπ ) i = x i • y i .Combining this with the CHSH symmetry and proceeding in the same way for all the indices that π permutes, we get Equation (A2).
Any different permutation π will result in a different bπ and therefore for any different permutation π we have a different entry P AB|XY (ã, b, x, y) with the same value as the original entry P AB|XY (a, b, x, y).Since there are n NCHSH different permutations of the balls we have n NCHSH different ways of changing (a, b, x, y) to (ã, b, x, y).
We can now answer our original question and bound N (a, b, x, y).We can combine both of the ways given above to change a and b without changing the value of the entry according to P AB|XY (with or without changing the colors sequence).This implies that in total there are at least 2 n × n NCHSH different ways of changing a and b and we can conclude that Since for all x, y a,b P AB|XY (ab|xy) = 1, we get from Equation (B18) the following bound on the value of P AB|XY (ab|xy): We can now prove Corollary 6 directly.
A direct proof of Corollary 6.By combining Lemma 16 and Lemma 17 we get Corollary 6.
The above proof for systems which have the CHSH symmetry can be also applied to systems which have the symmetry induced by the more general chained Bell inequalities in a similar way.Since the number of measurements of the basic systems Q A1B1|X1Y1 does not play a role in the structure of our de Finetti system (see Lemma 16) the same bounds exactly will hold for systems with the chained Bell inequality symmetry.

Appendix B: Proof of the general de Finetti reduction
In this section we prove our most general de Finetti reduction, given in Theorem 4. The proof proceeds along the same lines as the direct proof of Corollary 6 in the previous section.We start by explaining the types of symmetries S that we deal with and how to construct the appropriate de Finetti system τ S A|X .We then give a lower bound on the entries of the de Finetti system, analogously to Lemma 16, and an upper bound on the entries of a permutation invariant system P A|X with the symmetry S, analogously to Lemma 17.Using these two bounds we get Theorem 4.

Symmetries and de Finetti systems
A symmetry S is a set of conditions.We say that a system P A|X has a symmetry S if it fulfils all of these conditions.For any symmetry S that we consider we define a different de Finetti system τ S A|X of the form Q ⊗n A1|X1 dQ A1|X1 .When defining such a de Finetti system for a specific type of symmetry S we are free to choose the measure dQ A1|X1 as we like.The key idea is to choose the structure of the systems Q A1|X1 on which we integrate in such a way that it "encodes" the symmetry S that we consider.
For example, assume we consider a family of systems P A|X , with a, x ∈ {0, 1} n , which has the following type of symmetry S: ∀i ∈ [n] ∀a i , x P A|X (a i 0, |x) = P A|X (a i 1|x) (that is, given a i = a 1 , . . ., a i−1 , a i+1 , . . ., a n and x the probability that the i'th bit a i will be 0 or 1 is the same).We then want Q A1|X1 to have the following property: and we say that Q A1|X1 encodes the symmetry S.
For the more general treatment it will be easier to start by defining the allowed structure of the system Q A1|X1 and from it deduce the different types of symmetries and systems that we consider.
Consider a 1-party system Q A1|X1 where x 1 ∈ {0, 1, . . ., m − 1} and a 1 ∈ {0, 1, . . .l − 1}.We can think about Q A1|X1 as m vectors of size l.We call each of the l long vectors an input vector, since it describes the probability distribution of the outputs, given a specific input (see Figure 5).Defining a system Q A1|X1 then reduces to defining its input vectors.
Keeping in mind that we will need to integrate over Q A1|X1 to get a de Finetti system, we fill in the entries Q A1|X1 (a|x) of the input vectors with different parameters {p 1 , p 2 , . . .p d }, while making sure that the sum of the input vector is 1 for every value of the parameters p i .The number of parameters d that we use to define Q A1|X1 quantifies the number of degrees of freedom that Q A1|X1 has, and it is bounded by (l − 1)m. Figure 6 shows two different ways of filling in an input vector of length 3 with parameters.
We can now define a specific set of allowed systems Q A1|X1 .
Definition 18.A system Q A1|X1 is said to be allowed if given any two of its input vectors, they are either a permutation of one another or they have a completely different set of parameters.
In Figure 7 we give 2 examples for allowed Q A1|X1 systems with 2 inputs and 3 outputs.
The symmetry S behind When considering a specific system Q A1|X1 it is easy to say which set of conditions it fulfils, i.e., which symmetry S it encodes.For example, the system to the right in Figure 7 encodes the following symmetry of a system P A|X with a ∈ {0, 1, 2} n and x ∈ {0, 1} n , ∀i ∈ [n] ∀x i , a i P A|X (a i 0|x i 0) = P A|X (a i 1|x i 0) P A|X (a i 0|x i 1) = P A|X (a i 2|x i 1) .
More generally, the symmetry S can be constructed from Q A1|X1 as follows.
Definition 19.Given a system Q A1|X1 as above the symmetry S is defined by the following symmetry conditions: For all i ∈ [n], for all x, a and x ′ , a ′ where a ′ = a 1 . . .a i−1 a ′ i a i+1 . . .a n and x ′ is defined in a similar way, if That is, if we change the pair (a i , x i ) to some (a ′ i , x ′ i ) of the "same type" according to Q A1|X1 , then the probability according to P A|X does not change.
In Definition 19 we started from the system Q A1|X1 and derived the symmetry S.However, given a set of conditions S one can also try to construct a system Q which fulfils them.For every symmetry S for which a system Q A1|X1 can be constructed such that the condition in Definition 19 holds our proof can be applied.In other words, the only thing needed for our theorem to hold is a pair S, Q A1|X1 with the desired relationship.

The de Finetti system -integration over
Given a specific structure of Q A1|X1 as previously described, we can now perform the integration over a tensor product of n such systems and get a de Finetti system.As mentioned before, we are free to choose the measure dQ A1|X1 with which we perform the integration.
For simplicity, we only consider Q A1|X1 systems in which all the input vectors are permutations of one another (recall Definition 18).It will later become clear, that if we have more independent input vectors, then we can just multiply the different integrals by one another.In general, due to our proof technique, our entire proof can be applied independently for each set of permuted input vectors and then combined in the end to one proof by multiplying the results.In the rest of the proof, we use the following notations.Given the system Q A1|X1 we denote by t i , for 0 ≤ i ≤ d, the number of times the parameter p i appears in each input vector of Q A1|X1 .In addition, we define t d+1 to be the number of times the "unfree" entry appears in the input vector.Using this notation, we can set the range of the parameter p i to be 0, c i ≡ 1 ti 1 − j<i t j p j .As an example, consider the input vector in Figure 8. p 1 appears two times and therefore t 1 = 2 and c 1 = 1 2 .Indeed, in order for this input vector to be a valid probability distribution we must have p 1 ∈ 0, 1 2 .For p 2 we have t 2 = 3 and c 2 = 1 3 (1 − 2p 1 ), and t 3 = 1.Next we define the following "coloring" function: For every pair of strings (a, x), where a ∈ {0, 1, . . ., l − 1} n and x ∈ {0, 1, . . ., m − 1} n , we denote by N i the number of indices j ∈ [n] for which C(a j , x j ) = i.Using this definition we have . Using all the notation above, we can now define our measure to be7 and use it to define our de Finetti-type system.
Definition 20.For any symmetry S and the matching system Q A1|X1 as above, the de Finetti system τ S A|X is defined by .
Lower bounding the de Finetti system The following lemma is the analogous of Lemma 16 in the previous section.
Lemma 21.The following lower bound on τ S A|X (a|x) holds Before we continue to the proof of Lemma 21, note that although we have chosen a specific ordering of the parameters in the integration in Definition 20, this ordering does not affect the bound in Lemma 21.Moreover, this bound is optimal in the sense that there is always at least one pair of strings (a, x) for which the equality is reached, and this pair is independent of the chosen order of the integration.
Proof.In the proof we use the following formula: where B is the Beta function.We also need the following identities: We start by proving the following by induction: This follows from Equation (B2) while noting that for the first index we have c 1 = 1 t1 by definition.Induction hypothesis for d-1: where we used Equation (B2) to get from (B8) to (B9), Equation (B3) to get from (B9) to (B10), Equation (B4) to get from (B10) to (B11), the induction hypothesis (B7) to get from (B11) to (B12) and Equation (B5) in the last line.
Upper bounding a permutation invariant system P A|X with symmetry S The following lemma gives us an upper bound on any permutation invariant system P A|X (a|x) with the symmetry S.This lemma is the analogous of Lemma 17 in the previous section.
Lemma 22.For every permutation invariant system P A|X (a|x) with symmetry S we have The idea behind the proof is identical to the idea behind the proof of Lemma 17.We bound the value of a specific entry P A|X (a|x) by counting how many entries P A|X (ã|x) in the same input vector must have the same value as P A|X (a|x) due to the symmetry of P A|X .Since the sum of any input vector is 1 this will give us a bound on P A|X (a|x).
Proof.For our counting arguments we use here the same notation of the coloring function C given in Equation (B1) and the definition of N i thereafter.That is, for any a ∈ {0, 1, . . ., l − 1} n and x ∈ {0, 1, . . ., m − 1} n , we denote by N i the number of indices j ∈ [n] for which C(a j , x j ) = i.We can imagine this as placing a colored ball for each pair (a j , x j ) as in Figure 9.With this picture in mind, the symmetry S actually says that by changing two balls of the same color we do not change the value according to the probability distribution P A|X .
Let N (a, x) = {ã ∈ {0, 1, ..., l − 1} n |P A|X (ã|x) = P A|X (a|x)} .In how many ways can we change a while not changing the value of the entry according to P A|X ?We now prove As in the proof of Lemma 17 in the previous section, we have two different ways of changing a to ã: with and without changing the color sequence of the balls.Indeed, the first possible way to change a without changing the value of the entry is to a change a pair (a j , x j ) to a pair ( ãj , x j ) of the same color (note that we do not change x j since we want to stay in the same input vector of P A|X , i.e., not to change the input x).In the example of Figure 9 we can change the first pair (a 1 , x 1 ) = (3, 0) to ( ã1 , x 1 ) = (0, 0) for example.How many different strings ã can we create this way?In each input vector of Q A|X we have t j entries of the j'th color8 and we can choose a entry with this color for each one of the N j indices with this color.Therefore, there are exactly d+1 j=1 t Nj j different strings ã with the same color sequence as a and hence, according to the symmetry S, with the same value P A|X (a, x) = P A|X (ã, x).
Changing a in different ways than the way given above will necessarily change the colors sequence.However, we can still prove, using the permutation invariance of P A|X , that for some specific changes the value of the entry will stay the same.The specific changes that we consider are derived by permutations of the colored balls.
In order to understand how every different permutations of the balls π is realised as a permutation on x and a consider the example drawn in Figure 10.On the left side we see a permutation of the balls from Figure 9.We start by filling up the columns for which there is no change in the color of the ball with the original columns.Then we fill in the blank columns in such a way that each of the original columns appears once.The permutation in the figure for example, is just the permutation of indices (3,4) and (6,7,8).In general, there might be several ways to choose the permutation on x and a, but they are all equivalent for our purpose and therefore we can just choose one.
The important thing to note is that between every two different permutations of the balls we always have at least one index in which we have a different colored ball in the end.That is, we can write that for every π, π ′ = π, there exists j ∈ [n] such that C(π(a) j , π(x) j ) = C(π ′ (a) j , π ′ (x) j ).We use this to construct from every permutation π a different string ãπ for which P A|X (a|x) = P A|X (ã π |x), as follows.For any index j ∈ [n] that π permutes we change a j to (some) ãj π such that C( ãj π , x j ) = C(π(a) j , π(x) j ) .
(B17) This is always possible since the input vectors of Q A|X are permutations of one another, i.e., if C(π(a) j , π(x) j ) = k then there must be some a ′ j for which C(a ′ j , x j ) = k9 .We are now left to show that P A|X (a, x) = P A|X (ã π , x).Since P A|X is permutation invariant, we have Combining these two equations together we get P A|X (a, x) = P A|X (ã π , x) as desired.
Since for every two different permutations of the balls we always have at least one index in which we have a different colored ball in the end, we get different ãπ 's from different permutations π.There are exactly n N1,•••,N d ,N d+1 different permutations of the balls, and therefore the same number of different ãπ when proceeding this way.
We can now answer our original question and bound N (a, x).We can combine both of the ways given above to change a without changing the value of the entry according to P A|X .This implies that in total, there are at least ways of changing a and we can conclude that Since for all x a P A|X (a|x) = 1, we get from Equation (B18) the following bound on the entry value P A|X (a|x): By combining Lemma 21 and Lemma 22 we get Theorem 4.

Deriveing the corollaries from the general theorem
As mentioned before, for every symmetry S for which a system Q A1|X1 can be construct such that the condition in Definition 19 holds our proof can be applied.In order to derive the corollaries we just need to describe the type of symmetry that we consider and the relevant Q A1|X1 that we use to construct the de Finetti system.In Corollary 3 for example the systems P A|X that we consider have no special symmetry.Therefore we can derive this corollary from Theorem 4 by choosing Q A1|X1 without any internal symmetry (see for example Figure 11).In this case we have d = m(l − 1) degrees of freedom, hence we get Corollary 3.
For deriving Corollary 6 we use the system Q A1B1|X1Y1 given in Figure 1.For this system we have d = 1 and therefore Corollary 6 follows.This lemma states that there exists a de Finetti system τ A|X and a non-signalling extension of it τ AC|XZ such that any permutation invariant system P A|X can be post selected from it with probability ≥ 1 (n+1) m(l−1) .When we say that P A|X can be post selected we mean that there exists an input z to τ AC|XZ and an output of this measurement c z such that with probability τ C|Z (c z |z) ≥ 1 (n+1) m(l−1) the post-measurement system is P A|X (see Figure 12).Note that we consider a specific extension τ AC|XZ of the system τ A|X , and by choosing different inputs z we can post select different systems P A|X .
It is easy to see how to derive Lemma 23 from Corollary 3 by using the formalism introduced in [18,23] of partitions of a conditional probability distribution.We repeat here the relevant statements.where p c ≥ 0, c p c = 1 and the systems P c A|X are such that Lemma 25 (Lemma 9 in [18]).Given a system P A|X , there exists a partition with element p c , P c A|X if and only if ∀a, x p c • P c A|X (a|x) ≤ P A|X .
Lemma 26 (Lemma 3.2 in [23]).Given a system P A|X let Z be the set of all partitions p cz , P cz A|X cz of P A|X .
There exists an extension system P AC|XZ of P A|X and an input z to it such that ∀a, x P AC|XZ (a, c z |x, z) = p cz • P cz A|X (a|x) .
Moreover, the system P AC|XZ does not allow signalling between the A/X and the C/Z interfaces10 .
Using the lemmas above and Corollary 3 we can now prove Lemma 23.
Proof.The above lemmas together with Corollary 3 imply that in our case for any permutation invariant system P A|X , 1 (n+1) m(l−1) , P A|X is an element of a partition of τ A|X .Moreover, there exists a system τ AC|XZ and an input z such that with probability 1 (n+1) m(l−1) the post-measurement system is P A|X : ∀a, x τ AC|XZ (a, c z |x, z) = 1 (n + 1) m(l−1) P A|X , i.e., Lemma 23 holds.
A proof of Lemma 8 We repeat here Lemma 8 for convenience.
Lemma.Let T be a permutation invariant test as in Definition 7 and Pr fail (P A|X ) be the probability that T outputs "fail" on P A|X .Then for any system P A|X Pr fail (P A|X ) ≤ (n + 1) m(l−1) Pr fail (τ A|X ) .
Proof.We follow here a similar proof given in [24] while using the quantum post selection theorem [5].
First, since the test T is permutation invariant it is sufficient to consider only permutation invariant systems.To see this recall that for any system P A|X and permutation π we have Pr fail (P A|X ) = Pr fail (P A|X • π) according to Definition 7. Therefore we also have by linearity (C1) The system 1 n! π P A|X • π is permutation invariant and therefore without loss of generality we can consider only permutation invariant systems.
Next we define the following probabilities.Let Pr fail∧cz (τ AC|XZ ) be the probability that the second part of the system, τ C|Z is measured with z and the output is c z and that the first part of the system, τ A|X fails the test T at the same time.That is, Pr fail∧cz (τ AC|XZ ) = Pr fail (τ A|X ) • τ C|Z (c z |z) .
In a similar way we define Pr fail|cz (τ AC|XZ ) to be the probability the τ A|X fails the test T given that c z is the outcome measurement of τ C|Z .According to probability theory we have and that Pr fail|cz (τ AC|XZ ) = Pr fail (P A|X ) (given that the outcome measurement was c z , the post measurement system is P A|X ).All together we get Pr fail (P A|X ) ≤ (n + 1) m(l−1) Pr fail (τ A|X ) as required.

Diamond norm for conditional probability distributions
We start by writing the diamond norm defined in Definition 11 in an explicit way: where the third equality is due to the explicit form of the trace distance previously given in [17,18].Using this explicit form of the diamond norm we can now prove the following lemma.where τ AC|XZ is a non-signalling extension of τ S A|X .
For every a, b, x, y ∈ {0, 1} n denote by N CHSH the number of indices m ∈ [n] for which the foursome (a m , b m , x m , y m ) fulfils the CHSH condition, i.e. a m ⊕ b m = x m • y m .n − N CHSH is then the number of indices for which the foursome (a m , b m , x m , y m ) does not fulfil the CHSH condition.We can now formally define τ CHSH AB|XY :

Lemma 16 . 8 FIG. 3 .
FIG. 3. Partition to CHSH quartets.If the foursome (ai, bi, xi, yi) fulfills the CHSH condition it is denoted by a blue ball, otherwise by a red ball.

FIG. 12 .
FIG. 12. Post selecting system P A|X from an extension of τ A|X .If the outcome of the measurement z is cz then the post measurement system is P A|X .

Definition 24 .
A partition of a system P A|X is a family of pairs p c , P c A|X c Pr fail (P A|X ) = 1 n! π Pr fail (P A|X • π) = Pr fail 1 n! π P A|X • π .

EPx
K|C • P C|Z − F K|C • P C|Z 1 C|Z (c|z) E K|C (k|c) − F K|C (k|c) Pr E (x) a|E(a,x)=k P A|XC (a|xc) − x Pr F (x) a|F (a,x)=k P A|XC (a|xc) (C2)