Simple and maximally robust processes with no classical common-cause or direct-cause explanation

Guided by the intuition of coherent superposition of causal relations, recent works presented quantum processes without classical common-cause and direct-cause explanation, that is, processes which cannot be written as probabilistic mixtures of quantum common-cause and quantum direct-cause relations (CCDC). In this work, we analyze the minimum requirements for a quantum process to fail to admit a CCDC explanation and present"simple"processes, which we prove to be the most robust ones against general noise. These simple processes can be realized by preparing a maximally entangled state and applying the identity quantum channel, thus not requiring an explicit coherent mixture of common-cause and direct-cause, exploiting the possibility of a process to have both relations simultaneously. We then prove that, although all bipartite direct-cause processes are bipartite separable operators, there exist bipartite separable processes which are not direct-cause. This shows that the problem of deciding weather a process is direct-cause process is not equivalent to entanglement certification and points out the limitations of entanglement methods to detect non-classical CCDC processes. We also present a semi-definite programming hierarchy that can detect and quantify the non-classical CCDC robustnesses of every non-classical CCDC process. Among other results, our numerical methods allow us to show that the simple processes presented here are likely to be also the maximally robust against white noise. Finally, we explore the equivalence between bipartite direct-cause processes and bipartite processes without quantum memory, to present a separable process which cannot be realized as a process without quantum memory.


Introduction
Common-cause and direct-cause relations are the building blocks of classical causal models, as a causal model of multiple variables consists of combinations of common-cause and direct-cause relations between them. Understanding the relationship between cause and effects is one of the fundamental goals of several physical theories and of statistical analysis [1]. Also, determining the causal relation behind a correlation of two objects is a fundamental problem in causal inference theory and plays a main role in topics such as hypothesis testing, social sciences, medicine, and machine learning [2,3]. In order to analyze causality in quantum phenomena, recent works proposed new frameworks of theory of Bayesian inference [4] and causal modeling [5], where un-derstanding common-cause and direct-cause relations in quantum mechanics plays a fundamental role.
Quantum causal processes consist of a sequence of quantum operations and may be analyzed from different equivalent perspectives, such as sequential quantum operations via non-Markovian processes [6][7][8][9], fragments of quantum circuits via quantum combs [10,11], quantum channels with memory [12], and quantum strategies for playing an n-turn game [13]. When referring to causal modeling, a process which can be written as a probabilistic mixture of common-cause (CC) and direct-cause (DC) processes is said to admit a classical common-cause or direct-cause (CCDC) explanation, since the causal relations in such process could be simulated classically, by sampling from a probability distribution and then implementing either a common-cause or directcause process.
In Ref. [14] the authors consider the different possibilities of combining CC and DC processes and, inspired by coherent mixture of quantum channels, the authors of Ref. [15] experimentally certify the existence of a quantum process with no classical CCDC explanation. Also, inspired by the Quantum Switch [16][17][18], Ref. [19] presents a coherent superposition of common-cause and direct-cause processes which cannot be explained by a classical CCDC.
In this work, we analyze quantum processes which cannot be decomposed as probabilistic mixtures of common-cause and direct-cause ones, hence admitting no classical CCDC explanation. We focus on the bipartite case, the simplest scenario where such processes can exist. In this simplest scenario, exploring the possibility of a process to have both CC and DC relations simultaneously, we present a process which can be simply realized by preparing a maximally entangled state and an identity channel, not needing an interpretation of it as a coherent mixture of causal relations. When attempting to work with the minimum non-trivial dimensions, we propose another process, which requires a controlled-NOT operation. We prove that these processes are the most robust ones against their worst possible noise, known as generalized noise. We also develop a semi-definite programming (SDP) numerical approach to quantify the non-classical CCDC property of a process, based on its robustness against white and general noise. Our numerical methods allow us to show that these simple processes presented here are also maximally robust against white noise when considering qubits, and to quantify the non-classical CCDC property in any bipartite ordered process.
We also investigate the differences and similarities between quantum entanglement and processes without a direct-cause explanation, showing that there exist separable processes which are not direct-cause. This example points out the limitations of purely entanglement-based methods to detect non direct-cause processes and answers a conjecture first raised in Ref. [20].
Finally, we connect our results with a different related field by proving that for the bipartite case, processes without quantum memory [20] are equivalent to processes having a direct-cause decomposition. With that, we present a bipartite separable process which cannot be realized as a process without quantum memory and contribute to the understanding of the relation between entanglement and of quantum memory [20][21][22][23][24].

The Choi-Jamiołkowski isomorphism
As is standard in several branches of quantum information theory, we will make use of the Choi-Jamiołkowski isomorphism to represent linear operators and linear maps [25][26][27]. We first present the "pure" version of the isomorphism, which allows us to represent linear operators as bipartite vectors. Here A I and A O are finite-dimensional complex vector spaces, which will late be identified with Alice's input and Alice's output, respectively.
where {|i } is the computational basis for A I and d A I is the dimension of A I In quantum information theory, linear transformations between operators are sometimes referred to as linear maps, or simply, as maps. In this paper, we identify linear maps with a single tilde. For instance, let L(A I ) be the set of linear operators mapping A I to itself, Λ : L(A I ) → L(A O ) is a map transforming operators from L(A I ) to L(A O ). We now present the standard version of the Choi-Jamiołkowski isomorphism, which allows us to represent linear maps as bipartite operators. (

2)
A common equivalent way to define the Choi operator of a map Λ is given by the equation where 1 is the identity operator and (·) T stands for the transposition in the computational basis. A linear map is a quantum channel iff it is completely positive (CP) and trace-preserving (TP). In the Choi-Jamiołkowski representation, a map Λ : L(A I ) → L(A O ) is CP iff Λ 0, i.e., Λ is positive semi-definite, and TP iff tr A O (Λ) = 1 A I .
For the sake of clarity, some equations in this manuscript will explicitly identify the input and output Hilbert spaces of a map with a superscript in the symbol representing the Choi operator of the map. In the above situation, for example, the Choi operator is equivalently represented by Λ A I /A O , indicating that the map Λ takes an operator from L(A I ) to one in L(A O ).

The link product operation
First defined in Ref. [11], the link product is a useful mathematical tool to deal with composition of elements in a quantum circuit presented in the Choi-Jamiołkowski representation. Definition 1.3 (Link product). Let Λ 1 ∈ L(A ⊗ A ) and Λ 2 ∈ L(A ⊗A ) be linear operators. The link product between Λ 1 and Λ 2 is defined as where Λ T A 1 is the partial transpose of Λ 1 on the space A .
As stated previously, the link product is useful for composing linear maps and quantum objects. If Λ 1 : L(A) → L(A ) and Λ 2 : L(A ) → L(A ) are linear maps with Choi operators Λ 1 ∈ L(A ⊗ A ) and Λ 2 ∈ L(A ⊗ A ), respectively, it can be shown that the Choi operator of the composition In particular, when ρ ∈ L(A), is a linear operator with no components on A , and Λ ∈ L(A⊗A ), we have Hence, we can use the link product to represent quantum operations being performed on quantum states. Also, if ρ 1 ∈ L(A 1 ) and ρ 2 ∈ L(A 2 ), acting on different spaces, we have ρ 1 * ρ 2 := ρ 1 ⊗ ρ 2 . Therefore, the link product of independent systems is simply the tensor product.
Additionally, when ρ, M ∈ L(A) act in the same linear space, the link product is given by ρ * M := tr(M T · ρ), which is simply the trace of their product with an extra transposition. This form can also be used to write Born's rule.

The CCDC scenario
The main objects analyzed in this work are the bipartite ordered processes, which may be understood as a physical dynamics which flows from Alice to Bob. For all definitions in this section, we consider a scenario where Alice is a quantum physicist in a laboratory who can perform any quantum operation that transforms states from system A I (Alice input) to A O (Alice output), and Bob is a quantum physicist in a laboratory who can perform any quantum measurement on states defined on system B I (Bob input).

Markovian processes
We start by presenting the definition of bipartite Markovian processes, which admits a direct-cause interpretation [6,28,29] and are also known in the literature as quantum processes with no memory [20], and cause-effect causal maps [14]. In a Markovian scenario, Alice receives a known quantum state ρ ∈ L(A I ) which acts on her input system space. Alice can then perform an arbitrary quantum operation 1 Λ : to obtain a quantum state on system A O , which will then be subjected to a known deterministic dynamics described by a channel D : L(A O ) → L(B I ). Finally, the state D Λ(ρ) ∈ L(B I ) arrives to Bob, who can perform an arbitrary measurement on it. See Fig. 1 for a circuit based pictorial illustration.
A bipartite Markovian process can then be described by the known quantum state ρ ∈ L(A I ) and the known dynamics represented by the channel D : L(A O ) → L(B I ). In the Choi-Jamiołkowski representation, this can be conveniently described by Under this definition, for any quantum operation Λ : L(A I ) → L(A O ) performed by Alice, the quantum state arriving in Bob's input space can be obtained via the link product as where ρ A I is a quantum state i.e., ρ 0, tr(ρ) = 1 and D A O /B I is the Choi operator of a quantum channel from the output of Alice to Bob's input i.e., D 0, tr B I (D) = 1 A O . We denote the set of Markovian processes by L Markov .
1 Which may be a quantum channel (deterministic quantum operation) or a quantum instrument (probabilistic quantum operation). Alice receives a fixed state ρ ∈ L(H A I ) on which she can perform an arbitrary quantum operation. After Alice's operation, the system is subjected to a fixed dynamics described by a quantum channel D : L(A O ) → L(B I ), then arriving at Bob's input space laboratory. No auxiliary system is used in this circuit, which implies that correlations between the parties are only due to the direct communication from Alice to Bob.

Direct-cause processes
In a Markovian process, all correlations between Alice and Bob admit a direct-cause interpretation, since no auxiliary system or environment is required at any point. In a scenario where Alice and Bob may share prior classical correlations, it is natural to define direct-cause processes as any process which can be written as a probabilistic mixture of Markovian processes. Hence, the correlations arising from these processes can always be explained by classical mixture of direct-cause ones.
is a classical mixture of Markovian processes, that is, W DC can be written as We denote the set of direct-cause processes by L DC .
In the bipartite scenario, an ordered process is direct-cause if and only if it is a process without quantum memory, as defined and shown in Ref. [20]. See Appendix A for details. Although Ref. [19] analyzes tripartite ordered processes, when bipartite processes are considered, their definition is equivalent to the one presented in Refs. [14,15], which is also equivalent to the one used here. We discuss this in more details in Section 6.2.
2.2.1 Quantum separability as an outer approximation for the set of DC processes Which coincides with the definition of a DC process (Def. 2.2) without the trace normalization constraints. It follows directly from Eq. (9b) that every direct-cause process is separable in the bipartition A I |A O B I . Conversely, if a process is entangled in the bipartition A I |A O B I , such process cannot have a DC explanation. This allows us to employ techniques from entanglement theory to certify whether a process is not direct-cause. A simple outer approximation for the set of separable states is the set of states with positive partial transpose (PPT) [30,31]. Also, it is known that the set of states with a PPT k-symmetric extension [32,33] forms a hierarchy of sets which converges to the set of separable states when k → ∞. For every fixed k, checking if a state has a PPT symmetric k-extension can be done via SDP. The precise definition of a PPT k-symmetric extension is provided in Appendix E. One could be tempted to assume that all process which are separable in A I |A O B I bipartition have a direct-cause explanation. However, in Section 7, we prove that this is not the case by constructing an explicit example. This ensures that the problem of certifying if a given process is not direct-cause is not equivalent to certifying entanglement.

Common-cause processes
Common-cause processes are those where the correlations are not due to communication from Alice to Bob, but only due to a bipartite quantum state ρ ∈ L(A I ⊗ B I ) initially shared by them. In this way, a common-cause process does not involve any particular quantum dynamic and can be fully characterized by a fixed bipartite state. In order to establish a notation which dialogues to general scenarios which may have non-trivial dynamics between Alice and Bob, we describe common-cause processes by where ρ A I B I is a quantum state shared between Alice and Bob, i.e., ρ A I B I 0, tr(ρ A I B I ) = 1, and 1 A O is the identity operator acting on A O . We denote the set of common-cause processes by L CC .

Classical CCDC processes
Classical CCDC processes are processes which can be decomposed as a convex combination of a common-cause and a direct-cause ones. Such processes can always be understood as simple classical statistical mixtures between quantum common-cause and quantum direct-cause processes [14,19].
where W CC ∈ L CC , W DC ∈ L DC , and p ∈ [0, 1]. We denote the set of classical CCDC processes by L CCDC , which is the convex hull of L CC ∪ L DC .

Bipartite ordered processes
Previously, we have only considered processes which do not require any explicit use of auxiliary systems or environments. In a more general process, the initial state ρ received by Alice may be defined in a larger space, and Alice's lab can only operate on part of it. In order to tackle this situation, we now consider that the initial state is defined not only on Alice's input system, but also in some auxiliary system with unconstrained finite dimension, that is, ρ ∈ L(A I ⊗ aux). Now, Alice can only operate on the part of ρ that enters her laboratory, that is, she can perform any operation Λ : Definition 2.6 (Bipartite ordered process). A linear operator W ∈ L(A I ⊗ A O ⊗ B I ) 2 Here 1 aux stands for the identity map on the auxiliary space, that is, for any operator A ∈ L(aux) we have is a bipartite ordered process if there exists a quantum state ρ A I aux ∈ L(A I ⊗ aux),i.e., ρ A I aux 0, tr(ρ A I aux ) = 1 and a quantum channel D : We denote the set of bipartite ordered processes by L A→B .
Bipartite ordered process may be seen as quantum objects transforming a quantum operation into a quantum state 3 and are also known in the literature as quantum non-Markovian processes [7], causal maps [14], quantum co-strategies [13], and ordered process matrices [35]. Also, ordered processes may also be seen as a particular instance of quantum channels with memory [12] and quantum combs [11].
In this general definition of bipartite ordered processes, Markovian processes form the particular case where there is no auxiliary space aux. Common-cause processes can also be represented as bipartite ordered processes. Indeed, if W CC = ρ A I B I ⊗ 1 A O , we can set the auxiliary space H aux as isomorphic to H B I and the initial state ρ A I aux as isomorphic to ρ A I B I . Then, one can set the channel D as an operation that sends the auxiliary system to B I and discard the system in A O . This can be done formally via where U SWAP := ij |ij ji| is the swap operator. A circuit illustration of common-cause process as bipartite ordered ones is presented in Fig. 3. With all the important processes defined, Fig. 4 shows the relationship between all the sets defined in this section.
In addition to its definition, bipartite ordered processes admit a characterization in terms of linear and positive semi-definite constraints, which Figure 3: A general bipartite common-cause process can be represented by a circuit of this form. The shared quantum state between Alice and Bob comes from an initial correlated state between Alice and an auxiliary system. The state of the auxiliary system is exchanged with the system coming from the output of Alice and sent to Bob, and the auxiliary system is then discarded, which makes the output state of Alice to have no impact on the input of Bob. This makes only the input states of Alice and Bob to be relevant for the process, which is why the correlations between them stem from the common-cause correlations on ρ.
will be useful later in this work. If follows from direct inspection that all bipartite ordered process where σ A I is a quantum state. Interestingly, this condition is also sufficient [11]. For any linear operator W respecting the conditions in Eq. (15), one can find a quantum state ρ ∈ L(A I ⊗aux) and a quantum channel D : One possible construction is done by setting the auxiliary space aux to be isomorphic to A I and ρ ∈ L(A I ⊗ aux) be be isomorphic to a purification of σ A I , for instance, Now, one can define a quantum channel via where √ σ is the unique positive semi-definite square root of σ and σ −1 is the Moore-Penrose inverse of σ, that is, the inverse of σ on its range. In this way, direct inspection shows that D is the Choi operator of a quantum channel and that W = ρ * D.
Inspired by Ref. [36], we now present another characterization for bipartite ordered processes which will be useful for proving our results. A linear operator W ∈ L(A I ⊗ A O ⊗ B I ) is a bipartite ordered process if and only if it respects where with X (·) = tr X (·)⊗ 1 X d X and L A→B (W ) is the map projecting an operator W into the linear space spanned by bipartite ordered processes. Eqs. (15) and (18) are equivalent conditions for an operator W ∈ L(A I ⊗ A O ⊗ B I ) to be a valid bipartite ordered process. In particular, conditions given by Eqs. (18) are rather useful for implementing our numerical methods, due to the use of the projector L A→B (W ).

Detecting and quantifying nonclassical CCDC
Inspired by the robustness of entanglement [37,38], one can quantify the violation of a classical CCDC decomposition in a process W ∈ L(A I ⊗ A O ⊗ B I ) in terms of its generalized robustness of non-classical CCDC 4 [19] by where L A→B is the set of bipartite ordered processes defined in Eqs. (18). The generalized robustness R G (W ) corresponds to how resistant is the non-classical CCDC property of the process W against its worst possible noise.
In Appendix B, we show that the generalized robustness of all bipartite processes is upper- , which is attainable with the examples of non-classical CCDC processes we present in Section 5.
Another quantification of non-classical CCDC is in terms of its white-noise robustness, given by The value R W N (W ) corresponds to how resistant is the non-classical CCDC property of the process W against white noise. Figure 5 gives a representation of both types of robustness.
In Appendix C, we show that the white noise robustness of all bipartite processes is upper- Such bound is not tight, but is useful to demonstrate that the convex optimization problem of Eq.(21) respects strong duality (see Appendix D for more details). 4 Note that the definition of generalized robustness RG here has a one-to-one relation to the "non-classicality of causality" C presented in Ref. [19] via RG(W ) = C(W ) 1+C(W ) . Figure 5: Representation of the problem of determining if a given process W is non-classical CCDC. The above figure illustrates an ordered process operator Ω ∈ L A→B such that the minimum convex combination of it with the process W results in a classical CCDC process.

Witnessing non-classical CCDC processes
A non-classical CCDC witness is a Hermitian op- [19]. Non-classical CCDC witnesses that are not trivial are the ones having tr(SW ) < 0 for some non-classical CCDC process W , as the main purpose of a witness is to certify that a given process is non-classical CCDC. Since the set of classical CCDC processes is closed and convex, similarly to entanglement [39], every non-classical CCDC process may be certified by a non-classical CCDC witness. Also, as proven in Appendices D and E, the convex optimization problems used to define generalized and white noise robustness of non-classical CCDC respect a strong duality condition and the dual formulation of our outer approximation can be used to explicitly construct non-classical CCDC witness. Similarly to entanglement witnesses, verifying if an arbitrary operator S ∈ L( is a non-classical CCDC witness is likely to be computationally hard [40], as it would require, in principle, to verify that tr(SW CCDC ) ≥ 0 for all classical CCDC processes W CCDC . Despite the hardness of the general problem, we now provide simple sufficient (but not necessary) analytical conditions to ensure that S is a non-classical CCDC witness. Any 0 is a non-classical CCDC witness. This claim follows from the fact that every CC process respects W CC = A O W CC 0 and every DC process respects W 4 Beyond the entanglement approximation: SDP hierarchies for tight nonclassical CCDC certification As stated in Sec. 2.2.1, the set of separable processes in the bipartition A I |A O B I provides an outer approximation for the set of DC processes.
As pointed in Ref. [19,20], this provides a SDP approach that gives lower bounds for the non-DC and non-classical CCDC robustness of a bipartite ordered process. Indeed, since the relation L DC ⊆ L PPT k DC holds for every natural number k, we can define the convex hull L out,PPT k CCDC := conv(L CC , L out,PPT k DC ), and the quantity which gives a lower-bound for the actual generalized robustness, since the relation R low,PPT k G (W ) ≤ R G (W ) holds for every W . Analogously, we can also obtain a lower-bound for the white noise robustness by defining R As we show in Sec. 7, approximating the set of DC processes by the set of separable process is rather limited. In particular, there exist processes which are separable in the bipartition A I |A O B I but are non-classical CCDC. Hence, such processes would never be detected by a method solely based in quantum entanglement techniques. In this section, we overcome this problem by presenting two novel SDP hierarchies of sets which converge to the set of CCDC processes. Also, one hierarchy is based in an inner approximation, whereas the other constitutes an outer ap-proximation. Hence, we can obtain a sequence of converging upper and lower bounds on the nonclassical CCDC robustnesses. All the sets discussed in this section are illustrated in Fig. 6.

Inner approximation for direct-cause processes
The following inner approximation leads to a hierarchy that converges to the set of direct-cause processes and is inspired by the SDP approach for entanglement detection introduced in [41].
corresponds to the inner approximation of the set L DC and is composed by every process W that can be written as and D are the Choi operators of quantum channels, i.e., Note that in Def. 4.1, we imposed the partial trace quantum channel normalization condition which does not appear in an entanglement-based approach where we only impose positivity and the full trace constraint. As we will see in the further sections, this corresponds to a fundamental difference, since the set of separable quantum states is not equivalent to the set of direct-cause processes.
By construction, for any set of quantum states , and the quantity which provides an upper-bound for the generalized robustness R G (W ), since the relation R G (W ) ≤ R up,E N G (W ) holds for every bipartite ordered process W and E N .
As discussed in details in Appendix E, the quantity R up,E N G (W ) can be evaluated by an SDP. Moreover, this SDP converges to the exact value of R G (W ) when the set of states E N contains all pure states in L(A I ). Analogously, we can also obtain an upper-bound for the white noise robustness by defining R up,E N W N (W ).

Outer approximation for direct-cause processes
For the outer approximation for the set of directcause processes, we will make use of an outer approximation for the set of quantum states. Since the set of quantum states is convex, we can always construct an outer approximation given by a polytope, similarly to the strategy used to construct a polytopes providing outer approximations for the set of quantum measurements [42][43][44][45][46]. Let S d be the set of quantum states with dimension d, there exists a set of linear operatorsρ i ∈ L(C d ) that satisfy tr(ρ i ) = 1 and such that the convex hull of {ρ i } i contains S d . Note that the operatorsρ i are not required to be positive semi-definite, hence they do not correspond to quantum states. For the moment, let us assume that we know a set of trace-one operators We can then use this finite set to construct an outer approximation for the set of DC processes, similarly to the inner approximation in Def. 4.1.
be a fixed set of linear operators with tr(ρ i ) = 1 ∀i, such that the convex hull ofÊ N contains the set of all quantum states acting on A I . A bipartite ordered process W is in the set L out,Ê N DC if it can be written as and D By construction, we have that L DC ⊆ L out,Ê N DC and the set L out,Ê N CCDC := conv(L CC ∪ L out,Ê N DC ) is an outer approximation for L CCDC . The generalized and white noise robustnesses are defined in a way analogous to what is done in Sec. 4.1, thus being denoted by R low,Ê N G (W ) and R low,Ê N W N (W ), and these are lower bounds for the robustness R G (W ) and R W N (W ) respectively.
We now address the problem of finding sets of trace-one operators which contain the set of quantum states. Let us first start with the simple qubit case (d = 2) where the set of quantum states can be faithfully represented in terms of the Bloch sphere [47]. For this case, we can construct a set of trace-one operators {ρ i } i such that conv({ρ i } i ) ⊇ S 2 simply by finding a polyhedron that includes a sphere of unit radius. One method to find such polyhedron goes as following: in R 3 . These will be the vertices of a polyhedron inscribed by the Bloch sphere; 2. Find the radius r in < 1 of the largest sphere inscribed by the polyhedron; 3. Construct a polyhedron with vertices given by v i r in . This polyhedron includes the Bloch sphere.
The last step is ensured by the symmetry of the problem, a sphere of radius r in is contained in the polytope with norm-one vertices if and only if a sphere of radius one contains the polytope with vertices v i r in . Also, the radius of the largest sphere inscribed by a polytope (required in step 2) can be made by first finding the facet representation of this polytope with ver- , step which can be done via Fourier-Motzkin elimination and can be tackled with the aid of numerical packages such as lrs [48] or the Matlab code vert2lcon [49]. With the facet representation, the radius of the largest inscribed sphere can be found by evaluating the distance of the origin to the hyperplane represented by each facet. A concrete example for the qubit case can be found in the Appendix B: "A family of polyhedra", of Ref. [50]. In such work, the authors provide a family of polyhedra parameterized by n ∈ N where the radius of the larger inscribed sphere is greater than or equals to r n := cos 2 π 2n and if n is an odd number, the number of vertices is given by N = 2n 2 . This family of polyhedra can be used for a systematic approach to the problem.
For the general d > 3, we start by pointing out that every convex set can be approximated by a polytope. Now, in order to obtain a concrete set of operators {ρ i } i , we refer to the methods described in Appendix A: "Calculating shrinking factors", of Ref. [43] (see also Red. [51]). In a nutshell, the method starts by sampling random pure (which are extremal) states |ψ i ψ i | in the set S d and by obtaining the facet representation for the polytope associated to this set. Then, for any fixed "shrinking factor" η, we can verify whether a noisy quantum state ηρ + (1 − η)1/d can be written as a convex combination of {|ψ i ψ i |} i by means of an SDP. If all η-noisy states can be written as convex combinations of {|ψ i ψ i |} i , the convex hull of operators is given byρ i := 1 η |ψ i ψ i | + 1 − 1 η 1/d. We observe that we can tighten the outer approximation presented in Def. 4.2 by imposing that the process W should be separable in the bipartition A I |A O B I . Also, if we do this by means of the P P T k-symmetric extension (as in Def. 2.3), the problem can still be tackled by means of an SDP.
We present a summary of all the sets defined as approximations to the set L DC in Table 1. For obtaining the corresponding inner or outer approximations for the set of classical CCDC processes L CCDC , it is necessarily to take the convex hull of L CC with the chosen set for approximating L DC . Before finishing this section, we state that, throughout this work, whenever we calculate the non-classical CCDC robustnesses of the inner and outer approximations for L CCDC , the first lower bounds to be calculated are always provided by the outer approximation L out,PPT k CCDC . If the lower bounds obtained with such approximation does not match the upper bounds, then we take the outer approximation provided by L out,Ê N CCDC .
5 Simplest non-classical CCDC processes We now present a "simple" process which has the highest generalized robustness in the scenario where d A I (see Fig. 7). This process is mathematically represented by We argue that this process has a conceptually simple realization, since it only requires the preparation of a maximally entangled state and the identity channel. The process W ddd 2 has the interesting property of transforming a quantum channel Λ : L(A I ) → L(A O ) into a state which is proportional to its Choi operator Λ = ij |i j| ⊗ Λ(|i j|), that is forming an outer approximation for the set of quantum states, there exist quantum channels D   Despite the simplicity, we now show that W ddd 2 is non-classical CCDC and, moreover, it is the one with highest generalized robustness in the sce- Theorem 5.1. The bipartite ordered process attains the maximum generalized robustness of all processes with the same dimensions. That is, Proof. We start the proof by defining the operator and showing that S is a valid non-classical CCDC witness, i.e., every CCDC process W CCDC satisfies tr(SW CCDC ) ≥ 0. To ensure that S is a witness, we start by pointing that where the last inequality holds because the smallest eigenvalue of |φ where the last inequality holds because the eigenvalues of U SWAP are +1 or −1. Conditions (31) and (32) together ensure that S is a witness, as shown in Eqs. (22). Direct calculation shows that tr(SW ddd 2 ) = d− d 2 = d(1 − d) and that, for any bipartite ordered process Ω, we have thus r ≥ 1 − 1 d . We finish the proof by invoking Theorem B.3, which states that r ≤ 1 − 1 d .
We now analyze the robustness of W ddd 2 against the white noise process. For that, we use the SDP formulation to numerically tackle the case where d = 2 and d = 3. For the inner approximation, we have used the set L in,E N CCDC with E N being a set with N = 10 4 uniformly random pure states for d = 2 and N = 200 uniformly random pure states for d = 3. For the outer approximation, it was enough to use the loose approximation L out,PPT CCDC . The results obtained were where equations hold up to numerical precision.
Since the upper and lower-bound coincides, these values are the actual values of robustnesses.
For d = 2, we believe that W 222 2 is the process with maximum white noise robustness on its scenario. Our conjecture is based on a heuristic seesaw technique inspired by [52,53], which suggests that the highest value of white noise robustness in the scenario where d = 2 is 0.8421.
In a nutshell, our see-saw technique goes as follows: for a fixed scenario d A I = d A O = d and d B I = d 2 , we sample a random bipartite ordered process W . Then, we obtain its optimal non-classical CCDC witness S from the dual formulation of the PPT white noise robustness (see Appendix E). We then find the bipartite ordered process which maximally violates the witness S, then finding its optimal witness after that. This process is re-iterated until it converges to a robustness value, which we expect to be considerably greater than the robustness of the initially sampled process W . The see-saw method is described in details in Appendix F.
Moreover, in the scenario where d = 3, our see-saw method could find a bipartite ordered process W max which has R up,E 200 This result shows the power of the see-saw method in finding processes which are robust against white noise. It also shows that, even though W ddd 2 has maximum generalized robustness for every d, which we proved analytically, in the case where d = 3, W 333 2 is not the most robust process against white noise. This leads us to conjecture that W ddd 2 may be not the most robust process against white noise for d > 3.
Before finishing this section we mention that, following the same steps from the demonstration of Theorem 5.1, we can also obtain an analytical lower-bound for white noise robustness of W ddd 2 , which is Differently from the lower-bound presented for generalized robustness, the above inequality is not tight.
For instance, when d = 2, this lower-bound provides R W N (W ddd 2 ) ≥ 8 15 ≈ 0.5333, which is considerably lower than R low,PPT W N (W ddd 2 ) = 0.8421. However, it is interesting to point out that the above expression shows that R W N (W ddd 2 ) → 1 when d → ∞.
A maximally entangled state is initially shared between Alice and the auxiliary system. After Alice's operation, a control-NOT is applied, then aux is discarded.

The scenario with minimum dimensions:
We now consider the scenario where , that is, the minimum non-trivial dimensions. For this scenario, we propose the process composed by a two-qubit maximally entangled state between Alice and the auxiliary system, and the decoder channel from Alice output to Bob's input being a control-NOT channel, where the auxiliary system is later discarded (see Fig. 8). Mathematically, such process is described by where |U CNOT is the Choi vector of the control-NOT unitary gate U CNOT , given by Direct calculation shows that the process W 222 can also be written in terms of a un-normalized GHZ state |GHZ := |000 + |111 ∈ L(A I ⊗ A O ⊗ B I ) : where σ A O X is the Pauli matrix σ X applied on subsystem A O . The decomposition of Eq. (38) expresses W 222 as a probabilistic mixture of pure "GHZ-like" non-normalized states 5 and it will be useful to prove that W 222 attains the maximum generalized robustness value for its scenario.
Another decomposition of the process W 222 is in terms of Pauli matrices: with implicit tensor product between the operators.
is outside LA→B. This ensures that, as illustrated in Fig. 9, W222 is on the boundary of LA→B.

is,
Proof. The proof of this theorem follows similar steps to the proof of theorem 5.1. We start by defining the operator and showing that S is a non-classical CCDC witness. For this, we need to show that tr A O (S) 0 and S T A I 0, as shown in Eqs. (22).
Now, for the second condition, we have where the last inequality holds true because |GHZ GHZ| T A I and and the eigenvalues of ij |jii ijj| and x are +1 and −1. Direct calculation shows that tr(SW 222 ) = −2 and, for every process Ω in this scenario, we have tr(SΩ) ≤ 2, by an argument analogous to Eq. (33b).
Since S is a non-classical CCDC witness, if (1− r)W 222 + rΩ is a CCDC process, it holds that tr (S [(1 − r)W 222 + rΩ]) ≥ 0. It is also true that thus r ≥ 1 2 . We finish the proof by invoking Theorem B.3, which states that r ≤ 1 2 in this scenario.
We also evaluated the robustness of W 222 against white noise, obtaining evidences that W 222 attains the maximal white noise robustness on its scenario.

Theorem 5.3. The white noise robustness of the bipartite ordered processes W 222 is
Proof. We provide a lower-bound for R W N (W 222 ) by using similar steps to the proof of theorem 5.2, starting with the non-classical CCDC witness We have tr(SW 222 ) = −2 and tr(S) = 2 3 − 4 = 4. As S is a non-classical CCDC wit- thus r ≥ 2 3 . We now show, using techniques which are similar to the proof of Theorem B.3, that the above lower-bound can be attained. First notice that the qubit depolarizing channel D η (ρ) = (1−η)ρ+ ηtr(ρ) 1 2 is entanglement breaking when η = 2 3 [54,55]. Also, notice that, since tr A I (W 222 ) = , if we apply the depolarizing channel on the subspace A I of the process W 222 , we obtain It is worth to mention that every witness introduced in the proofs of Theorems 5.1 and 5.2 are the optimal witnesses for these particular processes.
Similarly to the scenario with d A I = d A O = d and d B I = d 2 , we implemented the see-saw algorithm that indicates that, in the scenario with , the highest white noise robustness is exactly 2 3 . This suggests that W 222 has also maximum white noise robustness in its scenario.
6 Relation with previous research 6. 1 Comparison with the non-classical CCDC process of Ref. [15] We now compare our proposed process with the non-classical CCDC process presented and experimentally implemented in Ref. [15]. The process consists of the preparation of a maximally entangled state |φ + shared between Alice and the auxiliary system, a partial swap channel from L(A O ⊗ aux) to L(B I ⊗ aux ), which is simply a unitary composed by a coherent mixture of an identity and a SWAP gate, and a partial trace on the output of the auxiliary system. This process can be explicitly written as where |U PS A O aux/B I aux is the Choi vector of the unitary partial SWAP 6 gate U PS , given by with U SWAP being the SWAP gate for qubits, given by The label MRSR makes reference to the names MacLean, Ried, Spekkens and Resch, authors of Ref. [15]. Fig. 10 ilustrates the process W MRSR . Using our numerical methods, we can evaluate the values of robustnesses for W MRSR , which are The authors from Ref. [15] use an equivalent way to represent the unitary partial SWAP gate, which is (50) In the second term, the identity channel exchanges the outputs BI and aux , in comparison to the identity channel in the first term. This part of the channel does exactly the same as U SWAP does, differing only by the fact that the output labels are not explicitly exchanged. Figure 10: Circuit representation of the process W MRSR presented in Ref. [15]. A maximally entangled state is initially shared between Alice and the auxiliary system. After Alice's operation, the partial-SWAP gate U PS is applied, then aux is discarded.
When comparing the robustness values of W MRSR with W 222 (see Section 5.2 and Eqs. (53)), which is a process defined in the same scenario, we verify that W 222 is strictly more robust against both generalized and white noise than W MRSR .
For the case of the process W 222 2 , we argue that the construction of W 222 2 is simpler than W MRSR . Both processes require the preparation of a maximally entangled qubit state, but W MRSR requires the implementation of the control swap operation, which is a coherent superposition between the identity channel and the swap channel, while W 222 2 only requires the identity channel.
6.2 Comparison with the non-classical CCDC process of Ref. [19] In Ref. [19], the authors proposed to study nonclassical CCDC by considering quantum superpositions of both relations. Their example is a the tripartite process ordered as can be written as where D A and D B are Choi operators of quantum channels.
In a tripartite scenario, common-cause and direct-cause relations can be more complex than in the bipartite scenario. In the case of commoncause relations, more general situations are discussed in Refs. [56,57]. However, following the definitions presented in Ref. [19], we restrict the attention to the same common-cause and directcause relations of the bipartite case, recovering them when taking d B O =d C I =1. The following definitions are taken considering the ones presented in Ref. [19].
A tripartite ordered process W CC is commoncause if where ρ A I B I is a quantum state in L(A I ⊗ B I ).
A tripartite ordered process W DC is direct-cause if where ρ A I i are quantum states in L(A I ) and D A tripartite ordered process W CCDC is classical CCDC if it can be decomposed in a convex combination of a common-cause process and a direct-cause process. Note that when bipartite processes are considered, i.e., the dimensions of B O and C I are equal to one, their definition are equivalent to the ones presented in section 2.
The example of non-classical CCDC process presented by the authors is the operator 7 The label FB makes reference to the names Feix and Brukner, authors of Ref. [19].
This process can be seen as a superposition of a common-cause and a direct-cause processes, as the first term corresponds to a common-cause process, whereas the second term corresponds to a direct-cause process. We can also represent W FB in terms of ordered quantum circuits 8 as illustrated in Fig. 11. 7 The state |φ + appearing in the expression of W FB is originally written, in Ref. [19], as generic bipartite state |ψ . However, we verified that the numerical robustness results obtained in Ref. [19] are reproduced when |ψ is a maximally entangled state, which lead us to take |ψ = |φ + , as this state is used in every other process mentioned before in this work. 8 Indeed, every ordered quantum process can be represented in terms of ordered quantum circuits by concatenating quantum states and quantum operations [11]. Figure 11: Circuit representation of the process W FB presented in Ref. [19]. A fixed qubit |+ = 1 I controls the swap operation from the auxiliary system in C 2 I and A O to B I and from C 2 I and B O to C 1 I . As the control qubit is fixed, the swapping operation occurs in the same way in both channels. This combination of operations generate the pure non-classical CCDC process W FB In Ref. [19], the authors numerically obtained a lower-bound to the generalized robustness of W FB , using the PPT outer approximation of DC processes, obtaining 9 Using our inner approximation method with N = 200, we could obtain, up to numerical precision, the upper-bound showing that R G (W FB ) = 0.1855, up to numerical precision. For completeness, we have also computed the white noise robustness, obtaining

The role of coherent mixture of causal relations
As discussed in this section, previous research has shown that one way to obtain processes with the non-classical CCDC property is by coherently superposing causal relations. For instance, the tripartite processes presented in Ref. [19] and discussed in Section 6.2 is constructed in a way to 9 Strictly speaking, the authors have evaluated quantity before mentioned here, the non-classicality of causality C, which has a one-to-one relation with the generalized robustness via R low,PPT be a coherent superposition of a purely commoncause and a purely direct-cause processes. Also, the bipartite process presented in Ref. [15] is inspired by a coherent mixture of causal relations which is mathematically formalized by the application of the partial swap operation (see Eq. (50)).
Differently from previous works, we have shown in this work that the connection between nonclassical CCDC and coherent superpositions of causal relations may be more subtle than it seems at first glance. In particular, although the nonclassical CCDC process W ddd 2 presented in section 5 may be viewed as a superposition of a process which is initialized in |00 with a process which is initialized in state |11 , it also admits a natural interpretation as a process with both common-cause and direct-cause relations simultaneously, without explicitly considering any superposition of causal relations. We then argue that the process W ddd 2 does not need to be interpreted as a coherent superposition of causal relations.

Separable process without a directcause explanation
As mentioned in previous sections, the definition of direct-cause processes (Def.2.2) reminds one of the definition of separable quantum states. More precisely, if we do not impose that the operators D have to respect the quantum channel condition tr B I D is precisely the definition of a separable state on the bipartition A I |A O B I . We now show that, despite being related, these two definitions are not equivalent.
Consider the process which is a separable operator by construction. First, notice that which shows that W is a valid bipartite ordered process.
The process W SEP can be physically realized by preparing a maximally entangled qubit state between A I and aux, and using a "decoder" described by In this way we have Note that the channel D A O aux/B I can be implemented as follows: first, perform a computational basis measurement on A O . If the outcome is 0, perform a computational basis measurement on aux and send the output qubit to B I . If the outcome is 1, perform a measurement on aux in the X-basis instead.

Theorem 7.1. The bipartite ordered process W SEP is separable in the bipartition
Proof. Equation (61) represents W SEP as a convex combination of product states, ensuring that W SEP is separable in the bipartition A I |A O B I . In order to show that W SEP is not a direct-cause process, let us assume that W SEP can be written as a convex combination are normalized quantum states and every D i has non-trivial overlap with, at least, 3 out of the 4 states |0 0|, |1 1|, |+ +|,|− −|. Indeed, let ρ = 1 2 (1 + i α i σ i ) be an arbitrary state where {α i } are real numbers that satisfy i α 2 i ≤ 1 and σ i are Pauli matrices. Suppose that ρ has zero overlap with some pure state |ψ ψ| = 1 2 (1 + i β i σ i ) with i β 2 i = 1, then we have that ( α, β) = −1, where (·, ·) is the Euclidean inner product. By the Cauchy-Schwarz inequality, we have with equality if and only if α is a multiple of β. This shows that α = − β and therefore ρ cannot be orthogonal to any other pure quantum state. Let us choose some fixed index j in the sum and suppose, without lack of generality, that ρ j has non-zero overlap with |1 , |+ , |− . Then, from the above definition of W SEP , we calculate By positivity of D A O /B I and ρ A I , each term in the sum has to be zero. Since, by assumption, tr(|1 1|ρ j ) = 0, it must be the case that D j obeys Similarly, by calculating other projectors we get By construction, W SEP is separable in the bipartition A I |A O B I , implying that W SEP has a PPT k-symmetric extension for every k ∈ N and R low, PPT k G This means that the non-classical CCDC property of W SEP cannot be certified by any purely entanglement-based criterion, such as the ones explored in Refs. [15,19,20].
Using the inner and outer approximations presented in Secs. 4.1 and 4.2 with the family of qubits presented in the Appendix B of Ref. [50] (n = 171, which corresponds to N = 2 * 171 2 states), we obtain upper and lower bounds for the generalized and white noise robustnesses of W SEP with equality holding up to numerical precision. In Ref. [20], it was conjectured that the set of separable processes and the set of processes without quantum memory are not the same, the latter being a strict subset of the first. Since, the definitions of bipartite processes without quantum memory and bipartite direct-cause processes are equivalent (see Appendix A), we have then proven the conjecture presented in Ref. [20] by explicitly constructing the bipartite separable process W SEP , which cannot be realized by processes without quantum memory.
It is interesting to point that our numerical methods allowed us to obtain a relatively high robustness of R W N (W SEP ) = 0.2930, but techniques exclusively based on entanglement would lead to the trivial lower bound R sep W N (W SEP ) ≥ 0. Since R W N (W SEP ) = 0.2930 is considerably greater than zero, we see that the approximating the set of direct-cause processes by separable processes may lead to very unsatisfactory results.

Certifying non-classical CCDC on PPT processes
In this section, we present an example of a non-classical CCDC process which has R low,PPT G (W ) = R low,PPT W N (W ) = 0, i.e., its nonclassical CCDC property cannot be certified by the PPT approximation used in Refs. [15,19]. Such a process can be obtained by exploiting a class of entangled states with positive partial transpose, presented in Ref. [58]. The class of states of our interest is being entangled for a ∈ (0, 1) and separable for a = 0 or a = 1. We now set a = 1 2 and use ρ 2×4 1 2 to define:  'We will now ensure that W PPT is a nonclassical CCDC process by using a better approximation L out,PPT k DC . In particular, we set k = 2, then we obtain

Summary of generalized and white noise robustnesses
In previous sections, we presented several examples of non-classical CCDC processes with different values of generalized and white noise robustness. Table 2 summarizes the non-classical CCDC robustnesses of several processes presented in this work and compare the actual value of robustness with the values obtained with methods based on entanglement criteria.

Conclusions
In this work, we have introduced a class of bipartite ordered processes, and a process with dimension of three qubits, which are maximally robust against general noise and very likely to be the most robust against white noise for the qubit case. This class of processes can be implemented by preparing a pair of maximally entangled states and an identity channel, admitting a natural interpretation of a process with both common-cause and direct-cause relations simultaneously. Hence, in contrast to previously known non-classical CCDC processes [14,19], the class presented here does not require either the construction or the interpretation directly based on coherent superposition of causal relations. Several analytical results proved in this work employed general convex analysis arguments based on witness hyper-planes, combined with entanglement theory concepts, such as entanglement breaking channels. We believe that the techniques developed here may find applications in related problems.
We have also presented a systematic semidefinite approach to characterize the set of nonclassical CCDC processes. In particular, we provided a hierarchy of inner and outer approximations that converge to the set of classical CCDC processes, and an entanglement-based hierarchy which, despite not converging to the set of CCDC processes, provides us several tight and non-trivial bounds.
In order to tackle situations where we could not prove the value of the highest robustness of a given scenario analytically, we constructed a heuristic see-saw method, which provided numerical evidence of the highest robustnesses attainable on such scenario, also providing a valid lower-bound for the value of the highest robustness.
Finally, we have shown that, although all bipartite processes that are entangled in the bipartition A I |A O B I do not have a direct-cause decomposition, the converse does not hold. Our proof consists in explicitly constructing a process which is separable on the bipartition A I |A O B I , but does not have a direct-cause decomposition. Since bipartite processes without quantum memory are equivalent to bipartite direct-cause ones, our results prove a conjecture first raised in Ref. [20] and contributes towards a better understanding of the particularities of quantum memory, spacial entanglement and temporal entanglement [20][21][22][23][24].
All SDP optimization problems presented in this manuscript were implemented using MAT-LAB, the convex optimization package Yalmip [59] and CVX [60], the solvers MOSEK, SeDuMi and SDPT3 [61][62][63], and the toolbox for quantum information QETLAB [64]. All our codes are available in the public repository [65] and can be Process (Eq.)  . Lower-bounds that match the actual robustnesses are highlighted in green, while lower bounds that are rather different from the actual robustnesses are highlighted in red. We can observe that entanglement based criteria could never detect W SEP as a non-classical CCDC process, while the PPT k-symmetric extension bound for W PPT provides the loose lower bounds R low, PPT k A Ordered processes without quantum memory are equivalent to direct-cause processes on the bipartite case We now present the definition of ordered (non-Markovian) processes without quantum memory, first introduced in Ref. [20].
Definition A.1 (Process without quantum memory [20]). A linear operator W ∈ L(A I ⊗ A O ⊗ B I ) is a bipartite ordered process without quantum memory if it can be written as where ρ A I aux SEP is a separable state and D A O aux/B I is the Choi operator of a quantum channel.
We now show that all bipartite processes without quantum memory are bipartite direct-cause, viceversa. This proof was first presented in Appendix A.3.2 of Ref. [20], but we reproduce it here for the sake of completeness.
where D Now, we need to show that every direct-cause process is a process without quantum memory in the bipartite case. Let us assume that W is direct-cause. Then W can be written as

B A tight upper bound for the generalized robustness
In this section, we prove that the generalized robustness of any process is upper-bounded by . This bound is saturated by the processes W ddd 2 (Eq. Proof. By definition, any bipartite ordered process can be written as W = ρ A I aux * D auxA O /B I . Since Λ is entanglement breaking, it holds that Λ A I ⊗ 1 aux (ρ A I aux ) is a separable state. Therefore, we can use the same argument presented in the proof of theorem A.1 to ensure that Λ A I ⊗ 1 A O B I (W ) is a direct-cause process.
is an entanglement breaking channel.
Proof. A necessary and sufficient condition [55] for Λ to be entanglement-breaking is that its Choi operator Λ is separable between input and output spaces. We now show that Λ is separable by is CCDC, with Λ A I \ ⊗ 1 A O B I (W ) being a valid bipartite ordered process.
By analyzing the definition of generalized robustness presented in Eq. (20), we see that setting holds for any bipartite ordered process W .

C An upper bound for the white noise robustness
In this section, we present an upper bound for the white noise robustness. Differently from the generalized robustness case, this bound is not tight, but is useful for proving the strong duality relation for the problem of evaluating the white noise robustness for non-classical CCDC processes in Section D.

E SDPs for CCDC separability
In this section, we present the explicit forms of the SDPs mentioned in Section 3 to obtain the generalized and white noise robustnesses of a bipartite ordered process W .
Consider the problem of obtaining the PPT k generalized robustness of a process W , i.e., finding the robustnesses of a process against its worst noise Ω, so the resulting combination process lies in L out,PPT k A→B . This is represented by the following optimization program: (92) This follows from the fact that, computationally, imposing the Bose k-symmetric extension condition instead of the k-symmetric extension condition does not add any complexity to the problem, but the Bose k-symmetric extension condition detects more entangled states than the standard k-symmetric one [67].
Since Eqs. (88) contains products of optimization variables, such as qW CC , the optimization program above is not linear, hence not an SDP. This issue can be circumvented by rewriting the problem in the following equivalent form: From the Lagrangian of the SDPs, we can obtain their dual problems, which generate the optimal non-classical CCDC witnesses S for the given process W [66]. For instance, considering k = 1, the dual form 10 of the PPT generalized robustness is given by which is the one we use in our heuristic see-saw presented in Section F. The variable S ⊥ DC in Eq. (96c) is associated with an orthogonal projection onto L A→B . Also, the same methods presented in Section D to prove that the robustness optimization problem satisfies strong duality can be used to show that this upper bound problem also respects strong duality. We, then, have R low, PPT G = −tr(SW ), when S is the optimal witness for W . 10 Strictly speaking, the dual objective function is to "maximize −tr(SW )", which results in R(W ) = tr(SW ). Here, we adopted a convention of changing the sign of S and replacing this objective function by "minimize tr(SW )". With this convention, the operator S is a non-classical CCDC witnesses as defined in Section 3.1, that is, it satisfies tr(SW CCDC ) ≥ 0 for all classical CCDC processes W CCDC .
We can see that Eq. (96c) together with Eqs. (96e) and (96f) ensure that S is a non-classical CCDC witness. Also, Eq. (96b) corresponds to the normalization condition, which determines that it is a witness for the generalized robustness measure.
Analogously, the dual form of the PPT white noise robustness is given by min tr (SW ) (97a) where Eq. (97b) is the normalization condition for the witness S, which corresponds to the white noise robustness measure. A similar method can be used to characterize our inner approximation L in,E N CCDC . In this case we just need to absorb the probabilities p i of Eq. (93d) into the quantum channel D i to obtain D i := p i D i . This allows us to replace the constraint of Eq. (93d) by We can also obtain the dual form of the inner approximation problems of robustnesses from the Lagrangian, which results in a similar SDP to Eqs. (96) and Eqs. (97), but replacing Eqs. (96c) and (96e) by where {S i } is a set of variables in L(A I ⊗ A O ⊗ B I ).
F Heuristic algorithm that seeks for the maximum robustness on a given scenario  [52,53], we now present a heuristic iterative method to seek for the highest robustness values for a given scenario. This method works either for the generalized or white noise robustness, which is why in the following we do not make explicit which robustness quantifier to work with. Our heuristic algorithm works as follows. First, we sample a bipartite ordered process W 1 using one of the methods we describe in Appendix G. Then, we perform the dual robustness problem for W 1 and obtain its optimal non-classical CCDC witness S 1 , which gives R(W 1 ) = −tr(S 1 W 1 ). Next, we find a process W 2 which maximally violates this first witness S 1 , a problem that can be solved by the 3. Output the operator: which is a valid bipartite ordered process.
which is a valid bipartite ordered process.