title: On the edge of the set of no-signaling assemblages
authors: Michał Banacki Ricard Ravell Rodríguez Paweł Horodecki
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  International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
  Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland
  
  International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
  
  International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
  Faculty of Applied Physics and Mathematics, National Quantum Information Centre, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland 
  
  
  
  
  Following recent advancements, we consider a scenario of multipartite postquantum steering and general no-signaling assemblages. We introduce the notion of the edge of the set of no-signaling assemblages and we present its characterization. Next, we use this concept to construct witnesses for no-signaling assemblages without an LHS model. Finally, in the simplest nontrivial case of steering with two untrusted subsystems, we discuss the possibility of quantum realization of assemblages on the edge. In particular, for three-qubit states, we obtain a no-go type result, which states that it is impossible to produce assemblage on the edge using measurements described by POVMs as long as the rank of a given state is greater than or equal to 3.
  
  
  
  
  
  
  
  
  On the edge of the set of no-signaling assemblages
      Paweł Horodecki
      December 30, 2023
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  § INTRODUCTION
  
  
  Quantum theory provides us with phenomena going beyond any classical description or intuition. The most striking example of this statement is the possibility of obtaining correlations that cannot be explained by any local and realistic theory <cit.>. Another non-classical phenomenon of quantum mechanics is encapsulated in the idea of quantum steering, proposed by von Neumann <cit.> and reintroduced in a modern formulation in <cit.>. The mathematical description of multipartite steering is given by the notion of assemblage consisting of subnormalized states describing subsystem of a chosen party, indexed according to measurements performed by other parties, and fulfilling a set of no-signaling constraints. Forgetting about the quantum nature of this collection of subnormalized states, one can define a notion of an abstract no-signaling assemblage based only on no-signaling constraints <cit.>. As it is known that not all no-signaling assemblages admit quantum realization (they do not come from a steering scenario described by the rules of quantum mechanics), there is room for the idea of postquantum steering. While our knowledge of the set of no-signaling assemblages has been vastly expanded <cit.>, the structural relations between various convex subsets of this set (set of assembles which admit quantum realization or an LHS model) are not fully understood.
  
  This paper aims to discuss this structural relationship, following the general idea of the edge state <cit.>, defined concerning a given convex subset of a considered set of quantum states. We introduce a similar concept in the setting of postquantum steering. We characterize its structural properties and we provide some applications.
  
  In Section <ref> we recall the notion of a multipartite quantum steering and a generalized concept of no-signaling assemblage. Moreover, in Section <ref>, we introduce a definition of the edge of no-signaling assemblages and present its characterization. In Section <ref> we formulate a notion of witnesses for no-signaling assemblages which do not admit an LHS model. In particular, we provide construction of witnesses starting from particular edge assemblages. Section <ref> is dedicated to the problem of quantum realization of assemblages on the edge of no-signaling, in the simplest tripartite steering scenario with two uncharacteristic subsystems (each with binary settings and outcomes). Finally, Section <ref> consists of short discussion.
  
  
  
  
  § NO-SIGNALING ASSEMBLAGES
  
  
  Nowadays the idea of quantum steering <cit.> has become more relevant <cit.> -  for example as a tool for certification of entanglement in a setting with untrusted parties. Recently, a typical bipartite scenario of quantum steering has been generalized to accommodate many uncharacterized (untrusted) systems <cit.> and introduce assemblages even beyond quantum description <cit.>.
  
  Consider a (n+1)-partite steering scenario in which n distant untrusted subsystems A_i share a quantum state ρ with the distant trusted subsystem B (described by d dimensional Hilbert space) and subnormalized states of subsystem B conditioned upon uncharacterized measurements on subsystems A_i are given by 
  
      σ_a|x=Tr_A_1,…, A_n(M^(1)_a_1|x_1⊗…⊗ M^(n)_a_n|x_n⊗1ρ)
  where a|x=a_1… a_n|x_1… x_n and any M^(i)_a_i|x_i is an element of POVM (positive operator value measure) corresponding to the measurement outcome a_i∈{0,…, 𝒜_i-1} of the measurement setting x_i∈{0,…, 𝒳_i-1} related to subsystem A_i. We define quantum assemblage as any collection of subnormalized states on d dimensional space Σ={σ_a|x}_a,x, which can be obtained according to expression (<ref>). Note that Σ fulfills a set of no-signaling constraints, which may be used to formulate an abstract notion of no-signaling assemblage by forgetting about origin of Σ based on measurements performed on a quantum state.
  
   A no-signaling assemblage Σ={σ_a|x}_a,x is a collection of subnormalized states σ_a|x acting on d dimensional Hilbert space, for which
  
      ∀_x_1,…, x_n∑_a_1,…, a_nσ_a|x=ρ_B,
  where ρ_B is a state, and for any possible subset of indexes I={i_1,…, i_s} with 1≤ s<n
  
      ∀_x_1,…, x_n∑_a_j,j∉ Iσ_a|x=σ_a_i_1… a_i_s|x_i_1… x_i_s.
  
  It has been already shown <cit.>, that any such assemblage for n=1 admits a quantum realization like in (<ref>). Nevertheless, for n>1 this is no longer the case and the set of quantum assemblages is then a nontrivial subset of the set of all no-signaling assemblages abstractly defined as in Definition <ref>. 
  
  Inside this convex subset of quantum assemblages one can single out another nontrivial convex subset representing the steering scenarios with classically correlated systems <cit.> - the set of LHS (local hidden state) assemblages. We say that a no-signaling assemblage admits an LHS model if it can be represented by
  
      σ_a|x=∑_j q_j ∏_i^np^(A_i)_j(a_i|x_i)ρ_j
  where q_j≥ 0, ∑_j q_j=1, ρ_j are some states of characterized subsystem B and p^(A_i)_i(a_i|x_i) denotes conditional probability distributions for each of uncharacterized subsystems A_i respectively. Note that any LHS assemblage (<ref>) can be also described by σ_a|x=∑_j q_j p_j(a|x)ρ_j where each L_j={p_j(a|x)}_a,x denotes a deterministic conditional probability distribution (i.e. extremal point in the polytope of local correlations). Indeed, this follows from the fact that for any i all p^(A_i)_i(a_i|x_i) can be realized as convex combinations of deterministic distributions (on a single party).
  
  From now on we will omit subscript notation and we will write no-signaling assemblage simply as Σ={σ_a|x}. For further convenience, for any no-signaling box P={p(a|x)}, we define set of indexes I_P={a|x:p(a|x)≠ 0}. Note that if a|x=a_1… a_n|x_1… x_n where a_i∈{0,…, 𝒜_i-1} and x_i∈{0,…, 𝒳_i-1}, then the cardinality of I_L is equal to |I_L|=∏_i^n 𝒳_i, when L denotes some local deterministic box (I_L consists of indexes of positions in L occupied by probabilities equal to 1). Finally, let R_a|x stands for projection on the image of σ_a|x, i.e. Im(σ_a|x) .
  
  
  
  § EDGE OF THE SET OF NO-SIGNALING ASSEMBLAGES
  
  
  Let S_2 denotes the set of all no-signaling assemblages Σ={σ_a|x} related to the scenario with fixed number of untrusted parties and with fixed numbers of labels from settings and outcomes. Let S_1 denotes its subset consisting of all assemblages which are admitting an LHS model. Note that both sets are convex and compact. Following the general spirit considered in <cit.>, we introduce the following definition.
  
  
  We say that Σ∈ S_2 is on the edge of the set of no-signaling assemblages if for any convex decomposition Σ=ϵΣ_1+(1-ϵ)Σ_2 where Σ_1∈ S_1, Σ_2∈ S_2, we have ϵ=0.
  
  
  Observe that any extremal no-signaling assemblage is either LHS or it is on the edge, but not all edge assemblages are extremal (see example <ref>).
  Note that assemblage Σ is not on the edge if and only if there exists ϵ>0, some pure state |ψ⟩⟨ψ| and a deterministic box L={ p(a|x)} such that all operators σ̃_a|x=σ_a|x-ϵ p(a|x)|ψ⟩⟨ψ| are positive. If this is the case, we will say that it is possible to subtract an LHS part from a given assemblage Σ. It is so because collection {σ̃_a|x} by definition fulfills no-signaling conditions, the only obstruction for subtraction of an LHS part is the fact that operator related to the given position a|x will no longer be positive.
  
  Note that assemblages on the edge are precisely that which admit the maximal value of so-called steerable (steering) weight <cit.>. Therefore, the membership question regarding the edge of the set of no-signaling assemblages can be stated in the form of the SDP problem. Here we present an analytical procedure that determines an answer for the evoked question (without the SDP approach). To do this, we invoke the following lemmas.
  
  
  Consider a positive operator σ∈ M_d(ℂ)_+ and a pure state |ψ⟩⟨ψ|∈ M_d(ℂ)_+, then σ - ϵ |ψ⟩⟨ψ| ≥ 0 for some ϵ>0 if and only if |ψ⟩∈Im(σ).
  
  
  By Lemma <ref> and previous discussion it is obvious that one can subtract an LHS part from a given assemblage Σ if and only if there exists a local deterministic box L and a normalized vector |ψ⟩ such that |ψ⟩∈⋂_a|x∈ I_LIm(σ_a|x).
  
   Note that because of that, in the particular
   case of d=2, one can subtract an LHS part from assemblage Σ if and only if there exists L such that {σ_a|x: a|x∈ I_L} does not contain 0 or two elements proportional to different rank one states.
  
  
  The above observation may be reformulated as the next lemma.
  
  
  An LHS part can be subtracted from assemblage Σ if and only there exists a local deterministic box L such that det(∏_a|x∈ I_LR_a|x - 1)= 0.
  
   Fix given local deterministic box L and enumerate elements of the set {σ_a|x: a|x∈ I_L} as {σ_i:i=1,… ,|I_L|}. Accordingly to that enumeration let us put R_i instead R_a|x.
  
  Assume that an LHS part (related to L) can be subtracted from a given assemblage. Then there exists normalized vector |ψ⟩∈⋂_i^|I_L|Im(σ_i). From this we have R_i|ψ⟩=|ψ⟩ for any i and as a consequence R_1R_2… R_|I_L| |ψ⟩ = |ψ⟩, so det(∏_i^|I_L|R_i - 1)= 0 since |ψ⟩≠ 0.
  
  For the opposite implication assume that det(∏_i^|I_L|R_i - 1)=0. If so then there exists a normalized vector |ψ⟩ such that R_1R_2...R_|I_L||ψ⟩ = |ψ⟩. Define |ϕ⟩ =R_2...R_|I_L||ψ⟩.
  Due to that we have |||ϕ⟩|| ≤ |||ψ⟩|| and R_1 |ϕ⟩ = |ψ⟩. This implies the following decomposition |ϕ⟩= |ψ⟩ + |ψ⟩_⊥ with |ψ⟩⊥ |ψ⟩_⊥ and opposite inequality |||ϕ⟩|| ≥ |||ψ⟩||. Finally, |ϕ⟩ = |ψ⟩ and R_1|ψ⟩ = |ψ⟩. Iteration of this procedure provides that |ψ⟩ is an eigenvector (corresponding to eigenvalue 1) of R_i for each i=1, …, |I_L|.
  
  
  
  This leads to the final theorem.
  
  
  An assemblage Σ is on the edge of the set of no-signaling assemblages if and only if det(∏_L∈ℒ(∏_a|x∈ I_LR_a|x - 1))≠ 0, where ℒ denotes a set of all local deterministic boxes.
  
  
  There is yet another simple way to check that a given assemblage is not on the edge.
  
   
  Let Σ be a no-signaling assemblage of operators σ_a|x acting on d-dimensional space. Then it is not on the edge if there exists a local deterministic box L such that ∑_a|x∈ I_Lrank(σ_a|x)>(|I_L|-1)d.
  
  
  Enumerate all elements by {σ_i:i=1,…,|I_L|}. Because ∑_i^|I_L|rank(σ_i)>(|I_L|-1)d we have rank(σ_1)+rank(σ_2)>d and therefore V_1=Im(σ_1)∩Im(σ_2)≠{0} with dim V_1≥rank(σ_1)+rank( σ_2)-d. For similar reason we also have dim V_1+rank(σ_3)>d (as rank(σ_1)+rank(σ_2)+rank(σ_3)>2d) and V_2=V_1∩Im(σ_3)≠{0}. Iterating this argument, we derive at V_|I_L|-1=⋂_i^|I_L|Im(σ_i)≠{0}, so there exists a non-zero vector which belongs to the images of all operators σ_i.
  
  
  As a simple consequence of the previous lemma, we obtain the following corollary.
  
  Let Σ be a no-signaling assemblage of operators σ_a|x acting on d-dimensional space. If Σ is on the edge, then ∑_a|xrank(σ_a|x)≤(∏_i^n 𝒳_i-1)(∏_i^n 𝒜_i)d.
  
  Consider the set Λ of all positions a|x in the assemblage Σ. Note that |Λ|=∏_i^n 𝒳_i∏_i^n 𝒜_i. Now observe that one can define ∏_i^n 𝒜_i disjoint subsets Λ_i of Λ such that Λ=∪_i Λ_i and Λ_i=I_L_i for some local deterministic box L_i. By the Lemma <ref> we see that if Σ is on the edge, sum of ranks of elements σ_a|x related to a given Λ_i cannot exceed (|I_L_i|-1)d=(∏_i^n 𝒳_i-1)d.
  
  
  Let us consider a particular steering scenario with n=2 and a,b,x,y∈{0,1}, it is convenient to see any such no-signaling assemblage Σ={σ_ab|xy} as the following box 
  
      Σ=[ [ σ_00|00 σ_01|00 σ_00|01 σ_01|01; σ_10|00 σ_11|00 σ_10|01 σ_11|01; σ_00|10 σ_01|10 σ_00|11 σ_01|11; σ_10|10 σ_11|10 σ_10|11 σ_11|11 ] ].
  No-signaling conditions (<ref>) have now simple interpretation. Namely, in each row sum of two operators on the right-hand side must be equal to the sum of two operators on the left-hand side (i.e. ∑_bσ_ab|xy=∑_bσ_ab|xy'), while similarly in each column the sum of two operators in the upper part must be equal to the sum of two operators in the lower part (i.e. ∑_aσ_ab|xy=∑_aσ_ab|x'y). Normalization condition (<ref>) is encoded into the fact that Tr(∑_a,bσ_ab|xy)=1 for all pairs x,y.
  Note that in considering setting (i.e. a,b,x,y∈{0,1}) there are 16 deterministic boxes L_αβγδ={p_αβγδ(ab|xy)}, defined by conditional probabilities (⊕ stands here for addition modulo 2 and α,β, γ, δ∈{0,1}).
  
      p_αβγδ(ab|xy)=
      
      1   for a=α x⊕β, b=γ y⊕δ
      
      0    otherwise.
  Using graphical presentation like in (<ref>), any L_αβγδ can be seen as a box with four positions occupied by 1 forming a rectangle (and other positions occupied by 0). Therefore, one can subtract an LHS part from Σ if and only if there exist a rectangle (one of 16 possible) with vertexes given by positions occupied by operators with common vector in their images, as it was stated in Lemma <ref> (see for example a graphical presentation of possible rectangle given by color red in (<ref>)).
  
      Σ=[ [ σ_00|00       … σ_00|01       …;       …       …       …       …; σ_00|10       … σ_00|11       …;       …       …       …       … ] ].
  In particular if d=2 according to Remark <ref> this can be done if and only if among operators from rectangle there is no 0 nor two operators proportional to different rank one states - we will use graphical interpretation (<ref>) extensively.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § CONSTRUCTION OF WITNESSES
  
  
  Consider a no-signaling assemblage Σ={σ_a|x} with σ_a|x∈ M_d(ℂ)_+⊂ M_d(ℂ)_sa where M_d(ℂ)_sa stands for a space of hermitian operators. Then Σ can be seen as an element in a real Hilbert space ⊕_a|xM_d(ℂ)_sa. Using theorem of Riesz and Hahn-Banach type of reasoning <cit.> for sets S_1⊂ S_2, one can introduce a notion of a witness for no-signaling assemblages beyond LHS description. We say that W∈⊕_a|xM_d(ℂ)_sa is a witness if Tr(WΣ_1)≥ 0 for all Σ_1∈ S_1 and there exists Σ_2∈ S_2∖ S_1 such that Tr(WΣ_2)< 0. Note that Σ∈ S_2∖ S_1 if and only if there exist a witness W such that Tr(WΣ)< 0. In fact any assemblage belongs to the nontrivial subspace V⊂⊕_a|xM_d(ℂ)_sa defined by no-signaling conditions, so one may consider only witnesses from V - however we will not restrict our attention to V, as the following construction of witnesses out of edge assemblages may lead to witnesses beyond V (discussion on optimality <cit.> of witnesses will be considered elsewhere).
  
  Let us introduce a set 𝒵 which consists of all Z∈⊕_a|xM_d(ℂ)_sa such that Tr(ZΣ)≥ 0 for any no-signaling assemblage Σ. Any witness can be expressed in a canonical form related to particular assemblage on the edge (compare with Theorem 4 in <cit.>).
  
  Let W be a witness, then W=Z-ϵ1 where Z∈𝒵 such that Tr(ZΣ)=0 for some edge assemblage Σ and ϵ>0.
  
  
  Note that Σ is on the edge if and only if for any local deterministic box L, there is no pure state |ψ⟩∈ℂ^d such that |ψ⟩∈⋂_a|x∈ I_LIm(R_a|x). Let Σ be on the edge. Define Z∈⊕_a|xM_d(ℂ)_sa by Z_a|x=1_d-R_a|x. Obviously Tr(ZΣ)=0 and there exists ϵ such that 0<ϵ≤inf_LHSTr(ZΣ_LHS). Indeed, for any LHS assemblage of the form L⊗ |ψ⟩⟨ψ|, there exists (a|x)∈ I_L such that Tr(|ψ⟩⟨ψ|Z_a|x)≠ 0, because if not, then |ψ⟩∈⋂_a|x∈ I_LIm(R_a|x) which is in contradiction with the fact that Σ in on the edge.
  If so, we may define a witness detecting Σ as W=Z-ϵ/Tr(Σ)1_⊕ where 1_⊕=⊕_a|x1_d and Tr(Σ) does not depend on particular Σ. Note that by putting maximal ϵ we obtain W such that there exists a Σ_LHS for which Tr(WΣ_LHS)=0.
  
  
  Consider the following rank two state ρ=1/2|ϕ_1⟩⟨ϕ_1| + 1/2 |ϕ_2⟩⟨ϕ_2| where |ϕ_1 ⟩ =|0⟩⊗1/√(2) (|00⟩ + |11⟩), |ϕ_2⟩ =|1⟩⊗1/√(2) (|00⟩ - |11⟩) and assemblage Σ={σ_ab|xy} given by σ_ab|xy=Tr_AB(P_a|x⊗ Q_b|y⊗1ρ) with projective measurements P_0|0=Q_0|0=|0⟩⟨ 0| and  P_0|1=Q_0|1=|+⟩⟨ +|
  
      Σ=1/8[ [  2|0⟩⟨ 0|  2|1⟩⟨ 1|  2|+⟩⟨ +|  2|-⟩⟨ -|;  2|0⟩⟨ 0|  2|1⟩⟨ 1|  2|-⟩⟨ -|  2|+⟩⟨ +|;  2|0⟩⟨ 0|  2|1⟩⟨ 1|         1         1; 2|0 ⟩⟨ 0|  2|1⟩⟨ 1|         1         1 ] ].
  It can be easily seen that this assemblage is on the edge (compare with Remark <ref>). Moreover, as it has two rows which are equal, Σ=1/2(Σ_1+Σ_2) where Σ_i for i=1 (i=2) is constructed out of Σ by exchanging fourth row (third row) with zeros and multiplying third row (fourth row) by a factor of two. As Σ_1≠Σ_2 given assemblage is not extremal. Note that for 0≤ p ≤ 1 each assemblage Σ_p=pΣ_1+(1-p)Σ_2 is still on the edge - this provides an example of a flat region inside the edge.
  Define Z as
  
      Z =[ [ |1⟩⟨ 1| |0⟩⟨ 0| |-⟩⟨ -| |+⟩⟨ +|; |1⟩⟨ 1| |0⟩⟨ 0| |+⟩⟨ +| |-⟩⟨ -|; |1⟩⟨ 1| |0⟩⟨ 0|       0       0; |1⟩⟨ 1| |0⟩⟨ 0|       0       0 ] ].
  Looking at symmetries of Z we can see that computation of ϵ is equivalent to the following minimization problem
  min_ρTr((2|0⟩⟨ 0|+|+⟩⟨ +|) ρ)= 3-√(5)/2. By fixing this ϵ we define a witness W= Z - 3-√(5)/81_⊕ detecting Σ_p.
  
  
  
  
  § EDGE ASSEMBLAGES WITH QUANTUM REALIZATION
  
  
  In this section we will restrict our attention to the case of assemblages Σ={σ_ab|xy} with a,b,x,y ∈{0,1} and σ_ab|xy acting on d dimensional space. In what follows we will use A,B,C for description of subsystems (where subsystem C will be characterized).
  
  As not all no-signaling assembles in this setting admits quantum realization it is natural to ask whether there exist edge assemblages that are not postquantum - in this section we provide a positive answer to this question. Note the difference with the related case of the edge of the polytope of bipartite no-signaling boxes with binary outcomes/settings (where S_1 is given by boxes of local type, and S_2 stands for all boxes). In that case, it can be easily shown that any box on the edge must be non-local and extremal in the polytope of all no-signaling boxes (i.e. it must be a PR-box up to relabeling) - any such extreme point does not admit quantum realization <cit.>. From this, any assemblage obtained by measurements on tripartite state separable in AB|C cut is never on the edge.
  
  
  For any pure tripartite state |ψ_ABC⟩∈ℂ^2⊗ℂ^2⊗ℂ^d entangled in a cut AB|C, there exists a pair of projective measurements {P_a|0}_a=0^1,{P_a|1}_a=0^1 on the subsystem A and  {Q_a|0}_a=0^1,{Q_a|1}_a=0^1 on the subsystem B respectively, such that a no-signaling assemblage Σ={σ_ab|xy} obtained by σ_ab|xy=Tr_AB(P_a|x⊗ Q_b|y⊗1|ψ_ABC⟩⟨ψ_ABC|) is on the edge of the set of no-signaling assemblages. 
  
   Assume that |ψ_ABC⟩ is genuinely entangled (i.e. entangled in any bipartite cut).  By result given in <cit.>, one can adjust projective measurements in such a way that resulting assemblage Σ={σ_ab|xy} is both extremal and not LHS, thus it is on the edge of no-signaling. To conclude the proof we may without loss of generality assume that |ψ_ABC⟩=|φ_A⟩|φ_BC⟩ with |φ_BC⟩∈ℂ^2⊗ℂ^d being entangled. Put P_a|0=P_a|1 with P_0|0=|φ_A⟩⟨φ_A| and define measurements on subsystem B {Q_b|0}_b=0^1,{Q_b|1}_b=0^1  in such a way that any two operators of the form Tr_B(Q_b|y⊗1|φ_BC⟩⟨φ_BC|) are not proportional (this is always possible - see for example <cit.>). It is then obvious that one cannot subtract an LHS part from this assemblage (there is no appropriate rectangle), hence Σ is on the edge.
  
  
  In the case of quantum no-signaling assemblages arises a natural question concerning the maximal rank of initial state ρ from which, by appropriate choice of local measurements, one can obtain assemblage which is on the edge - we address this issue in a three-qubit setting.
  
  
  
  Let ρ_ABC∈ M_2(ℂ)⊗ M_2(ℂ)⊗ M_2(ℂ) be a tripartite state. Consider two pairs of POVMs {M_a|0}_a=0^1,{M_a|1}_a=0^1 on the subsystem A and two pairs of POVMs {N_b|0}_b=0^1,{N_b|1}_b=0^1 on the subsystem B. Define a no-signaling assemblage via σ_ab|xy=Tr_AB(M_a|x⊗ N_b|y⊗1ρ_ABC). If rank(ρ_ABC)≥ 3 then Σ={σ_ab|xy} is not on the edge of the set of no-signaling assemblages. 
  
  
  
  We may restrict to the case rank(ρ_ABC)=3 (see Appedix). Assume that all POVMs are given by PVMs. 
  Note that certain structures (defined by certain positions occupied by operators with assumed ranks) of assemblage may be excluded as rank(ρ_ABC)=3 provides constraints on ranks for operators obtained after measurements on one of (or both) subsystems. Careful analysis of all remaining and relevant structures shows (see Appedix) that in each case one can either subtract an LHS part or considered structure is impossible when rank(ρ_ABC)=3. Finally, one can see (Appendix) that any assemblage Σ with POVMs realization can be written as a nontrivial convex combination Σ=pΣ_1+(1-p)Σ_2 where Σ_1 admits a quantum realization with PVMs.
  
  
  Theorem <ref> may be compared with a case of the edge of quantum states where the set of all three-qubit states plays a role of S_2 and a subset of all fully separable three-qubit states in natural way stands for S_1. Note that there is no product vector |ϕ_A⟩|ϕ_B⟩|ϕ_C⟩ in the image of a state τ_ABC=1/4(1-Π) where Π stands for projection on space spanned by four vectors forming an unextendible product basis (UPB) <cit.>. Therefore, τ_ABC is on the edge but on the other hand rank(τ_ABC)=4.
  
  Note that due to Example <ref>, rank three is the smallest rank for which theorem like the one above may be formulated, since there is a choice of states with rank two from which one can obtain assemblages on the edge. The following theorem provides sufficient conditions for such situations.
  
  
  Let ρ_ABC∈ M_2(ℂ)⊗ M_2(ℂ)⊗ M_d(ℂ) with rank(ρ_ABC)=2. Assume that there exists a POVM {M_a|x}_a=0^1 on the subsystem A such that both ρ_a|x=Tr_A(M_a|x⊗1⊗1ρ_ABC) are rank one and entangled. Then one can choose additional projective measurements such that Σ given by σ_ab|xy=Tr_AB(M_a|x⊗ Q_b|y⊗1ρ_ABC) is on the edge. 
  
  
  Since ρ_a|0 are pure and entangled for a=0,1, one can define a pair of nontrivial projective measurements {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 on subsystem B in such a way, that for a given a any two operators of the form σ_ab|0y=Tr_B(Q_b|y⊗1ρ_a|0) are not proportional (this is always possible - see for example <cit.>) - this concludes the proof.
  
  
  
  
  § DISCUSSION
  
  
  Motivated by the relation between sets of separable states and all states we have introduced the notion of the edge of no-signaling assemblages constructed concerning the subset of assemblages that admit an LHS model. We have provided a full characterization of edge assemblages and we have stated some necessary criteria for being on the edge. Moreover, we have discussed the notion of witnesses for no-signaling assemblages beyond LHS realization and we have related this discussion to the previously evoked concept of the edge. In the case of tripartite states (with two qubits), we have characterized pure states steerable to edge assemblages and we have provided sufficient conditions for this when states have rank 2. Finally, we have proved a no-go type result for three-qubit states with a rank greater than or equal to 3.
  
  While presented results better describe the set of no-signaling assemblages, many interesting questions are remaining. In particular, it would be natural to ask for a full characterization of witnesses, at least in a case of a specific setting. Besides, further generalizations of Theorem <ref> may be performed by taking arbitrary dimension d or changing the number of uncharacterized parties or possible settings and outcomes.
  
  The concept of the edge of the no-signaling assemblages represents - in analogy to the standard entanglement picture - the strongest form of correlations within the set of all no-signaling assemblages since edge assemblages do not admit any admixture of local hidden states. We hope that because of the latter unique property they will be an important resource for the typical security tasks in semi-device independent quantum information. The explicit analysis presented here should also provide tools for the construction of practical witnesses of the edge assemblages in the future.
  
  
  Acknowledgments- M.B., R.R.R., and P.H. acknowledge support by the Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). M.B. is grateful to Marcin Marciniak for the discussion.
  
  
  
  
  § PROOF OF THEOREM <REF>
  
  
  To adjust convection, consider two pairs of POVMs with binary outputs {M_a|0}_a=0^1,{M_a|1}_a=0^1. We say that they are the same up to relabeling if M_0|0=M_0|1 or M_0|0=M_1|1. In other case we say that POVMs are different up to relabeling. In particular the same convention will be used for pairs of projective measurements (PVMs).
  
  In order to prove the no-go results of Theorem <ref>, we will need to introduce a series of simple lemmas. In what follows we will use observations stated in discussion after Corollary <ref> (in the main text) - possibility of subtraction of an LHS part will be stated in terms of the existence of certain rectangles within considered assemblages.
  
  
  Let ρ_AB∈ M_2(ℂ)⊗ M_d(ℂ) be a bipartite state and ρ_a=Tr_A(P_a⊗1ρ_AB) for some nontrivial projective measurement {P_a}_a=0^1 on the subsystem A. If rank(ρ_1)=rank(ρ_2)=1, then rank(ρ_AB)≠ 3. Moreover, if d≥3, rank(ρ_AB)=3 and rank(ρ_1)=0 imply rank(ρ_2)=3 and if d=2 none of ρ_a have rank zero. 
  
  
   Without loss of generality we may put P_0=|0⟩⟨ 0| and P_1=|1⟩⟨ 1|. Assume that on the contrary rank(ρ_AB)=3. Observe that ρ_AB can be expressed by spectral decomposition
  
      ρ_AB=∑_i=1^3λ_i|ψ_i⟩⟨ψ_i|
  with 
  
      |ψ_i⟩=α_i|0⟩|h_i⟩+β_i|1⟩|g_i⟩
  where λ_i>0 and |ψ_i⟩, |h_i⟩, |g_i⟩ are normalized for any i. If rank(ρ_1)=rank(ρ_2)=1, then for any i=1,2,3 we have |h_i⟩=|h⟩ and |g_i⟩=|g⟩ (since possible phase factors may be always incorporated in α_i or β_i respectively and if α_i or β_i is equal to zero form of |h_i⟩ or |g_i⟩ may be arbitrary). Therefore, all pure states (<ref>) belong to the same two-dimensional subspace. Since {|ψ_i⟩}_i is an orthonormal system we obtain a contradiction - this concludes the first part of the proof. 
  
  Now assume once more that rank(ρ_AB)=3 and d≥ 3. If rank(ρ_1)=0 then by formula (<ref>) and (<ref>) we get that each |ψ_i⟩ is proportional to |1⟩|g_i⟩. Orthogonality of {|ψ_i⟩}_i imply then orthogonality of {|g_i⟩}_i (it is possible if and only if d≥ 3). Since 
  
      ρ_2=∑_i=1^3λ_i|g_i⟩⟨ g_i|
  we have rank(ρ_2)=3.
  
  
  
   Let ρ_AB∈ M_2(ℂ)⊗ M_d(ℂ) be a bipartite state such that rank(ρ_AB)=2. Consider two pairs of nontrivial projective measurements {P_a|0}_a=0^1,{P_a|1}_a=0^1 on the subsystem A and define ρ_a|x=Tr_A(P_a|x⊗1ρ_AB). Assume that {P_a|0}_a=0^1 and {P_a|1}_a=0^1 are different (up to relabeling). If one of ρ_a|x has rank zero then the others have rank two and they are proportional. If x_1≠ x_2 and ρ_0|x_1,ρ_1|x_1 have rank one then either ρ_0|x_2 and ρ_1|x_2 have rank two or all ρ_a|x have rank one and they are proportional.
  
  
  Without loss of generality we may put P_0|0=|0⟩⟨ 0| and P_1|0=|1⟩⟨ 1|. Assume that rank(ρ_AB)=2. Observe that ρ_AB can be expressed by spectral decomposition
  
      ρ_AB=∑_i=1^2λ_i|ψ_i⟩⟨ψ_i|
  with 
  
      |ψ_i⟩=α_i|0⟩|h_i⟩+β_i|1⟩|g_i⟩
  where λ_i>0 and |ψ_i⟩, |h_i⟩, |g_i⟩ are normalized for any i.
  
  Firstly, without loss of generality assume that ρ_0|0=0, then α_1=α_2=0 and |g_1⟩⊥ |g_2⟩. If so then the other ρ_a|x are proportional to ∑_i=1^2λ_i|g_i⟩⟨ g_i| which has rank two.
  
  
   If without loss of generality ρ_0|0, ρ_1|0 have rank one, then we can put |h_i⟩ =|h⟩ and |g_i⟩ =|g⟩. If |h⟩ and |g⟩ are equal (up to irrelevant phase), then ρ_0|1, ρ_1|1 are proportional to |g⟩⟨ g|. If this is not the case then ρ_0|1, ρ_1|1 have rank two.
  
  
  
  
  Consider a collection of orthonormal vectors |c_i⟩∈ℂ^d and a two-dimensional subspace H⊂ℂ^d. Projection of each |c_i⟩ on H is proportional to the same vector if and only if there exists a vector |c⟩∈ H othonormal to each |c_i⟩. 
  
  
   If |c⟩∈ H, then projection of each |c_i⟩ is proportional to normalized |c_⊥⟩∈ H, where |c_⊥⟩⊥ |c⟩.
  
  On the other and, let Q be a projection on H and let Q|c_i⟩=α_i|c_⊥⟩ with α_i≥ 0 for some normalized vector |c_⊥⟩∈ H and all i. Take normalized vector |c⟩∈ H such that |c_⊥⟩⊥ |c⟩. Then Q=|c⟩⟨ c| + |c_⊥⟩⟨ c_⊥| and ⟨ c|c_i⟩=0 for all i.
  
  
  
   Let ρ_ABC∈ M_2(ℂ)⊗ M_2(ℂ)⊗ M_2(ℂ) be a tripartite state. Consider two pairs of nontrivial projective measurements {P_a|0}_a=0^1,{P_a|1}_a=0^1 on the subsystem A and two pairs (different up to relabeling) of nontrivial projective measurements {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 on the subsystem B. Define a no-signaling assemblage via  σ_ab|xy=Tr_AB(P_a|x⊗ Q_b|y⊗1ρ_ABC). If rank(ρ_ABC)= 3 then it is impossible that Σ={σ_ab|xy} takes one of the following forms
  
      [ [ ϕ_00 ϕ_01    X    X; ϕ_10 ϕ_11    X    X;    X    X  φ_1  φ_2;    X    X  φ_3  φ_4 ] ]
  ,
  
      [ [ ϕ_00    X  φ_1    X; ϕ_10    X  φ_3    X;    X ϕ_01    X  φ_2;    X ϕ_11    X  φ_4 ] ],
  where X stands for any rank two operator and ϕ_i stands for rank one operator proportional to |ϕ_i⟩⟨ϕ_i|.
  
  
  
  Without loss of generality we may put P_0|0=Q_0|0=|0⟩⟨ 0|, P_1|0=Q_1|0=|1⟩⟨ 1| and P_0|1=|0'⟩⟨ 0'|, P_1|1=|1'⟩⟨ 1'|, Q_0|1=|0”⟩⟨ 0”|, Q_1|1=|1”⟩⟨ 1”| where 
  
      |0'⟩=α |0⟩ +β |1⟩, |1'⟩=β |0⟩ -α |1⟩,
  
  
      |0”⟩=γ |0⟩ +δ |1⟩, |1”⟩=δ |0⟩ -γ |1⟩.
  Note that γ,δ≠ 0 due to general assumption (different pair of PVMs on the subsystem B). Observe that since rank(ρ_ABC)=3 we have a spectral decomposition
  
      ρ_ABC=∑_i=1^3λ_i|ψ_i⟩⟨ψ_i|.
  Finally, define vectors |x⟩, |y⟩∈ℂ^4 via equations
  
      |ϕ_kl⟩=x_kl|0⟩+y_kl|1⟩.
  Observe that for any pair k,l coefficients x_kl,y_kl cannot be both equal to zero (due to normalization).
  
  In the first part of the proof, assume on the contrary that Σ is of the form (<ref>). If so, then α,β,γ,δ≠ 0 and |ψ_i⟩ from (<ref>) have the following form
  
      |ψ_i⟩=c^i_00|00⟩|ϕ_00⟩+c^i_01|01⟩|ϕ_01⟩+c^i_10|10⟩|ϕ_10⟩+c^i_11|11⟩|ϕ_11⟩,
  
  where |c_i⟩=(c^i_00, c^i_01, c^i_10, c^i_11)∈ℂ^4 are normalized and |c_i⟩⊥ |c_j⟩ for i≠ j.  Obviously dim(span{|x⟩, |y⟩})=2 (the last statement is true because by the no-signaling condition for the first row of (<ref>), |ϕ_00⟩ and |ϕ_01⟩ are not proportional, so (x_00,x_01), (y_00,y_01) are linearly independent). Finally, let us introduce operators L_j for j=1,2,3,4
  
      L_1=diag(αγ, αδ, βγ,βδ), L_2=diag(αδ, -αγ,βδ, -βγ),
  
  
      L_3=diag(βγ, βδ, -αγ,-αδ), L_4=diag(βδ, -βγ, -αδ,αγ).
  Note that operators L_j represents action of measurements P_a|1, Q_b|1 (related to positions in (<ref>) occupied by φ_j) on vectors |ψ_i⟩. Indeed, for example we have
  
      Tr_AB(P_0|1⊗ Q_0|1⊗1|ψ_i⟩⟨ψ_i|)=|φ̃_1,i⟩⟨φ̃_1,i|
  where
  
  
      |φ̃_1,i⟩=c^i_00αγ|ϕ_00⟩+c^i_01αδ|ϕ_01⟩+c^i_10βγ|ϕ_10⟩+c^i_11βδ|ϕ_11⟩=⟨c_i|L_1|x⟩ |0⟩+⟨c_i|L_1|y⟩ |1⟩
  and similarly
  
      |φ̃_2,i⟩=c^i_00αδ|ϕ_00⟩-c^i_01αγ|ϕ_01⟩+c^i_10βδ|ϕ_10⟩-c^i_11βγ|ϕ_11⟩=⟨c_i|L_2|x⟩ |0⟩+⟨c_i|L_2|y⟩ |1⟩,
  
  
      |φ̃_3,i⟩=c^i_00βγ|ϕ_00⟩+c^i_01βδ|ϕ_01⟩-c^i_10αγ|ϕ_10⟩-c^i_11αδ|ϕ_11⟩=⟨c_i|L_3|x⟩ |0⟩+⟨c_i|L_3|y⟩ |1⟩,
  
  
      |φ̃_4,i⟩=c^i_00βδ|ϕ_00⟩-c^i_01βγ|ϕ_01⟩-c^i_10αδ|ϕ_10⟩+c^i_11αγ|ϕ_11⟩=⟨c_i|L_4|x⟩ |0⟩+⟨c_i|L_4|y⟩ |1⟩,
  
  with |c_i⟩=(c^i_00, c^i_01, c^i_10, c^i_11). By assumption we see that operators in the intersection of the third and the fourth rows with the third and the fourth columns in (<ref>) are rank one, so for any fixed j=1,2,3,4, all |φ̃_j,i⟩ are proportional to some |φ_j⟩ (some but not all of them may be equal to zero). Observe that by (<ref>-<ref>) this is true if for any fixed j=1,2,3,4, projection of each |c_i⟩ on span{L_j|x⟩,L_j|y⟩} is the same up to normalization or equal to zero. From Lemma <ref> we see that this is true only if there exists a common vector |c_4 ⟩∈ℂ^4 orthonormal to others |c_i⟩ and such that |c_4 ⟩∈span{L_j|x⟩, L_j|y⟩} for all j (note that this is true since |c_4 ⟩ is unique up to phase due to orthogonality condition in ℂ^4).
  
  To conclude the proof it is enough to show that it is impossible to find a vector |c_4 ⟩ with such property that L_j^-1|c_4 ⟩∈span{|x⟩, |y⟩} for all j. Put |c_4 ⟩=(c_1,c_2,c_3,c_4). Observe that if |c_4 ⟩ has three coefficients equal to zero, then all |c_i ⟩ for i=1,2,3 have a common zero coefficient, i.e. there is a pair k,l such that c_kl^i=0 for all i=1,2,3. Therefore, |c_4 ⟩ has at most two coefficients equal to zero. 
  
  Assume first that |c_4 ⟩ has no zero coefficient. Then L_j^-1|c_4 ⟩ for j=1,2,3 form a linearly independent set. Indeed, we have 
  
  
      det[ [  α^-1γ^-1c_1  α^-1δ^-1c_1  β^-1γ^-1c_1;  α^-1δ^-1c_2 -α^-1γ^-1c_2 β^-1δ^-1 c_2;  β^-1γ^-1c_3  β^-1δ^-1c_3 -α^-1γ^-1c_3 ] ]=c_1c_2c_3α^-1γ^-1(|α|^-2+|β|^-2)(|γ|^-2+|δ|^-2)≠ 0.
  
  
  Therefore, L_j^-1|c_4 ⟩∈span{|x⟩, |y⟩} cannot be true for all j=1,2,3, as dim(span{|x⟩, |y⟩})=2. This leads to a contradiction. 
  
  Assume then that |c_4 ⟩ has one or two coefficients equal to zero - for example let c_1,c_2≠ 0 and c_3=0. then we have
  
  
      det[ [  α^-1γ^-1c_1  α^-1δ^-1c_1;  α^-1δ^-1c_2 -α^-1γ^-1c_2 ] ]=-c_1c_2α^-2(|γ|^-2+|δ|^-2)≠ 0,
  
  so in particular L_1^-1|c_4 ⟩ and L_2^-1|c_4 ⟩ are linearly independent. Note that if L_j^-1|c_4 ⟩∈span{|x⟩, |y⟩} for all j=1,2, then span{L_1^-1|c_4 ⟩, L_2^-1|c_4 ⟩}=span{|x⟩, |y⟩} and by c_3=0 we have x_10=y_10=0 which is a contradiction with normalization of |ϕ_10⟩. By similar calculations based on properties of L_j^-1, the same conclusion can be shown for any |c_4 ⟩ with one or two coefficients equal to zero. In the end, by negation of condition from lemma <ref>, it is impossible to obtain assemblage of the form (<ref>) starting from nontrivial projective measurement on rank three state ρ_ABC.
  
  To conclude the proof, consider a second possibility, namely assume that Σ is of the form (<ref>). If so, then α,β,γ,δ≠ 0. Define a unitary U_AB acting on subsystem AB 
  
      U_AB=1_A⊗|0⟩⟨ 0|+ U_A⊗|1⟩⟨ 1|
  with U_A being a unitary such that U^†_A|0⟩=|0'⟩, U^†_A|1⟩=|1'⟩. Obviously, ρ̃_ABC=U_ABρ_ABCU^†_AB is still a state with rank equal to three. By the properties of U_A and assumed form of Σ, one can see that Σ̃={σ̃_ab|xy} defined by σ̃_ab|xy=Tr_AB(P_a|x⊗ Q_b|y⊗1ρ̃_ABC) have the following form
  
      [ [ ϕ_00 ϕ_01    …    …; ϕ_10 ϕ_11    …    …;    X    …    …    …;    X    …    …    … ] ].
  
  If so, then similarly to the first part of the proof, one can write a spectral decomposition ρ̃_ABC=∑_i=1^3λ_i|ψ̃_i⟩⟨ψ̃_i| with |ψ̃_i⟩=U_AB|ψ_i⟩ and 
  
      |ψ̃_i⟩=c^i_00|00⟩|ϕ_00⟩+c^i_01|01⟩|ϕ_01⟩+c^i_10|10⟩|ϕ_10⟩+c^i_11|11⟩|ϕ_11⟩,
  where |c_i⟩ for i=1,2,3 are like in the first part of the proof. Using once more properties of U_A, we see that orthonormal vectors from spectral decomposition of initial state ρ_ABC (see (<ref>)) can be given as 
  
  
      |ψ_i⟩=c^i_00|00⟩|ϕ_00⟩+c^i_01α|01⟩|ϕ_01⟩+c^i_11β|01⟩|ϕ_11⟩ +c^i_10|10⟩|ϕ_10⟩+c^i_01β|11⟩|ϕ_01⟩-c^i_11α|11⟩|ϕ_11⟩
  
  
  We can represents action of appropriate measurements P_a|x, Q_b|y (related to positions in (<ref>) occupied by φ_j) on vectors |ψ_i⟩ by introduction of vectors |φ̃_j,i⟩ given as
  
  
      |φ̃_1,i⟩=c^i_00γ|ϕ_00⟩+c^i_01δα|ϕ_01⟩+c^i_11δβ|ϕ_11⟩=⟨c_i|S_1|x⟩ |0⟩+⟨c_i|S_1|y⟩ |1⟩.
  
  
      |φ̃_2,i⟩=c^i_00δα|ϕ_00⟩-c^i_01γ|ϕ_01⟩+c^i_10δβ|ϕ_10⟩=⟨c_i|S_2|x⟩ |0⟩+⟨c_i|S_2|y⟩ |1⟩.
  
  
      |φ̃_3,i⟩=c^i_01δβ|ϕ_01⟩+c^i_10γ|ϕ_10⟩-c^i_11δα|ϕ_11⟩=⟨c_i|S_3|x⟩ |0⟩+⟨c_i|S_3|y⟩ |1⟩.
  
  
      |φ̃_4,i⟩=c^i_00δβ|ϕ_00⟩-c^i_10δα|ϕ_10⟩-c^i_11γ|ϕ_11⟩=⟨c_i|S_4|x⟩ |0⟩+⟨c_i|S_4|y⟩ |1⟩.
  
  where
  
      S_1=diag(γ, δα,0,δβ), S_2=diag(δα,-γ,δβ,0),
  
  
      S_3=diag(0,δβ,γ, -δα), S_4=diag(δβ,0, -δα,-γ).
  
  
  Obviously (<ref>) implies that for any fixed j=1,2,3,4, all |φ̃_j,i⟩ are proportional to some |φ_j⟩ (some but not all of them may be equal to zero). Due to no-signaling conditions for the first and the second column in (<ref>), |ϕ_00⟩ and |ϕ_10⟩ are linearly independent and the same is true for |ϕ_01⟩ and |ϕ_11⟩. If so, then pairs (x_00,x_10), (y_00,y_10) and (x_01,x_11), (y_01,y_11) are linearly independent as well. Note that from this, for each j=1,2,3,4, dim(span{S_j|x⟩, S_j|y⟩})=2. Therefore, we see that the structure of (<ref>) implies (according to Lemma <ref>) that (unique up to the phase in ℂ^4) vector |c_4 ⟩ orthonormal to each |c_i ⟩ for i=1,2,3 must satisfy |c_4 ⟩∈span{S_j|x⟩, S_j|y⟩} for any j=1,2,3,4. However, by the form of diagonal matrices S_j, |c_4 ⟩=(0,0,0,0) which is a contradiction and the second part of the proof is concluded.
   
   
  
  
  Now we are ready to present a proof of Theorem <ref>.
  
   It is enough to show the statement in the case of rank(ρ_ABC)=3. Indeed, let rank(ρ_ABC)> 3, then ρ_ABC=αρ_ABC^(1)+βρ_ABC^(2) where α,β>0 and ρ_ABC^(1),ρ_ABC^(2) are states such that rank(ρ_ABC^(1))=3. Observe that 
  σ_ab|xy=ασ^(1)_ab|xy+βσ^(2)_ab|xy with σ^(i)_ab|xy=Tr_AB(M_a|x⊗ N_b|y⊗1ρ_ABC^(i)) for i=1,2. Therefore, if any assemblage obtained from rank three state in not on the edge, the same must be true for assemblage obtained from state with higher rank. Therefore we set rank(ρ_ABC)=3.
  
  
  From now on unless stated otherwise, let us, instead of general POVMs, consider only nontrivial projective measurements with elements P_a|x,Q_b|y (different from 0 or 1) acting on subsystem A and B respectively. Therefore we are interested in structural analysis of no-signaling assemblage Σ={σ_ab|xy}  given by σ_ab|xy=Tr_AB(P_a|x⊗ Q_b|y⊗1ρ_ABC). For two given pairs of nontrivial projective measurement {P_a|0}_a=0^1,{P_a|1}_a=0^1 on the subsystem A we define ρ_a|x=Tr_A(P_a|x⊗1⊗1ρ_ABC) and we put r_1=rank(ρ_0|0), r_2=rank(ρ_1|0), r_3=rank(ρ_0|1), r_4=rank(ρ_1|1). In similar way we introduce π_b|y=Tr_B(1⊗ Q_b|y⊗1ρ_ABC) for measurements on the subsystem B and q_i, i=1,2,3,4 for related ranks of this operators. In principal r_1,r_2,r_3,r_4∈{0,1,2,3} and the same is true for q_i. Without loss of generality we assume that r_1≥ r_2 and r_3≥ r_4 (it is possible due to relabeling). According to Lemma <ref> we may restrict our attention to (r_1,r_2), (r_3,r_4) of the form (3,0), (3,1),(3,2),(3,3), (2,1),(2,2). 
  
  In the remaining part of the proof, we will show that all forms of Σ compatible with all possible values of r_i,q_j (i.e. not excluded by rank(ρ_ABC)=3) are either not on the edge or are simply impossible to obtain (note that some of the considered structures, from which one can subtract an LHS part, could still be impossible to obtain due to some reasons which are not discussed in this proof - this is, however, irrelevant for the considered question, if a given structure is not an obstruction for the subtraction of an LHS part, then its existence is not important).
  
  Observe that if one can show that a given assemblage, defined with measurements {Q_b|0}_b=0^1,{Q_b|1}_b=0^1  is not on the edge, then assemblage, defined with the same {Q_b|0}_b=0^1 and  {Q_b|1}_b=0^1 equal to {Q_b|0}_b=0^1 (up to relabeling), must be outside the edge as well (the same observation is true for measurements on the subsystem A), since the initial rectangle from which one can subtract an LHS part will be changed into rectangle with positions occupied by two of operators presented in the previous rectangle - as it can be seen in the following example
  
      [ [ σ_00|00       … σ_00|01       …;       …       …       …       …;       …       …       …       …; σ_10|10       … σ_10|11       … ] ]→[ [ σ_00|00       … σ_00|00       …;       …       …       …       …;       …       …       …       …; σ_10|10       … σ_10|10       … ] ].
  Therefore, if we show that for considered values of r_i certain form of assemblage given by {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 different up to relabeling is not on the edge, then the same is true when we omit assumption of difference of measurements (for assemblage of this certain form).
  
  
  
  We will show that if r_1=3 and r_3=3 or r_3=2, then considered assemblages coming from measurements on rank three states are not on the edge. Observe that by performing measurements {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 on ρ_a|x we obtain some particular row from assemblage Σ
  
      [  σ_a0|x0  σ_a1|x0  σ_a0|x1 σ_a1|x1. ]
   Since ρ_a|x∈ M_2(ℂ)⊗ M_2(ℂ) and rank(ρ_a|x)=3 or rank(ρ_a|x)=2, we may use Lemma <ref> and Lemma <ref> in order to state what are possible ranks of operators in a considered row (<ref>). From now on ψ_i∈ M_2(ℂ) denotes some operator with rank one proportional to state |ψ_i⟩⟨ψ_i|, 
  ψ'_i∈ M_2(ℂ) denotes either ψ_i or 0 and X∈ M_2(ℂ) denotes any operator with full rank (i.e. rank two). Note that only possible structures of rows which are relevant for certification that a given assemblage is not on the edge, are those which have the largest number of operators with rank zero and rank one (i.e. if one can subtract an LHS part from assemblage with row of a certain structure, one can also subtract an LHS part from assemblages with row which arise from the previous structure by changing operator on some position to operator with rank two). 
  
  Firstly, assume that {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 are different (up to relabeling). According to Lemma <ref>, when rank(ρ_a|x)=3, only relevant (up to relabeling y→ y⊕ 1 and b_1→ b_1⊕ 1,b_2→ b_2⊕ 1) rows are given as
  
      [ ψ_1   X ψ_2  X. ]
  
  In the analogous way, according to Lemma <ref>, when rank(ρ_a|x)=2, only relevant (up to relabeling y→ y⊕ 1 and b_1→ b_1⊕ 1,b_2→ b_2⊕ 1) rows are given as
  
  
      [  0  X  X X, ]
  
  
      [ ψ_1 ψ_2   X  X. ]
  
  
      [  ψ  ψ  ψ ψ. ]
  
  
      [ ψ_1   X ψ_2  X. ]
  
  
  As r_1=3 and r_3=2,3, we should analyze all assemblages with the first row and the third row described by (<ref>) or (<ref>-<ref>) respectively (up to relabeling). For example by taking (<ref>) with relabeling and (<ref>) we get
  
      [ [   X ψ_1 ψ_2   X;   …   …   …   …;   ψ   ψ   ψ   ψ;   …   …   …   … ] ]
  and one can see that there exists a choice of rectangle such that it does not contain any rank zero operator or any pair of rank one operators which are not proportional, i.e. such assemblage is not on the edge. Analysis of all possibilities like the one above (with relabeling) shows that there is only one problematic case given by
  
      [ [ ψ_1   X ψ_2   X;   …   …   …   …;   X ψ_3   X ψ_4;   …   …   …   … ] ].
  When (<ref>) is obtained by the same (up to relabeling) pair of measurements on the subsystem B, then ψ_1=ψ_2 and there is an LHS part that one can subtract from (<ref>), therefore we may still assume that {Q_b|0}_b=0^1,{Q_b|1}_b=0^1 are different (up to relabeling). Note that with (<ref>), neither r_2 nor r_4 can be equal to 0 as in this case there would be a column in (<ref>) with operators with rank two, rank one and rank zero, which would be in contradiction with a possible form of this column given by (<ref>-<ref>) (as in that case {P_a|0}_a=0^1,{P_a|1}_a=0^1 should be different up to relabeling and column must be relater to some q_i=2,3). In similar fashion, if assemblage is of the form (<ref>), and r_2 or r_4 is equal to 1, then considered assemblage is not on the edge. Indeed, it is enough to consider r_2=1. In that case once more {P_a|0}_a=0^1,{P_a|1}_a=0^1 are different (up to relabeling) an since q_i=2,3 for each i=1,2,3,4, (<ref>-<ref>) imply the following structure 
  
      [ [ ψ_1   X ψ_2   X; ϕ_1 ϕ_2 ϕ_3 ϕ_4;   X ψ_3   X ψ_4;   X   X   X   X ] ]
  from which one can certainly subtract an LHS part.
  
  As a results of this considerations we see that analysis of (<ref>) boils down to r_2,r_4=2,3. The only situation problematic to subtraction of an LHS part from an assemblage is related to the case when the second and the fourth rows are given by (<ref>) and assemblage (<ref>) is of the form given by (<ref>) - this structure is however impossible if rank(ρ_ABC)=3 according to Lemma <ref> (with different up to relabeling measurements on the subsystem B). This concludes the first part of the proof.
  
  Due to the whole above analysis, it is enough to discuss only the case when (r_1,r_2), (r_3,r_4) are of the form (2,1),(2,2). Without the loss of generality we may put r_1, r_3=2. Analysis of (<ref>-<ref>) shows all possible obstructions for subtraction of an LHS part. The first potential obstruction is related to the case where both the first and the third row are like (<ref>) and assemblage is given by (<ref>) - this obstruction is excluded due to the previous discussion. Another potential obstruction is related to the case where both the first and the third row have zero operators but in different columns. If both measurements on the subsystem A are the same up to relabeling one can subtract an LHS part from assemblage with mentioned property, so the only relevant case is without loss of generality given by the following structure (we use here form (<ref>) for the first row and the third row) 
  
      [ [ 0 X X X; X X … …; X 0 X X; X X … … ] ]
  where at least two unspecified positions are occupied by a nonzero operator and an LHS part can be subtracted as well.
  
  
  The remaining situation is related to the case where the first and the third rows are like (<ref>-<ref>). Take a convention in which 
  
      [  σ_1/σ̃_1  σ_2/σ̃_2  σ_3/σ̃_3 σ_4/σ̃_4, ]
  stands for one of the following rows
  
      [ σ_1 σ_2 σ_3 σ_4 ], [  σ̃_1  σ̃_2  σ̃_3 σ̃_4. ]
  Firstly put r_4=1. There are two relevant cases to consider 
  
      [ [  ψ_1/ψ_I  ψ_2/ψ_I    X/ψ_I    X/ψ_I;        …        …        …        …; ψ_3/ψ_II ψ_4/ψ_II   X/ψ_II   X/ψ_II;     ϕ'_1     ϕ'_2     ϕ'_3     ϕ'_4 ] ]
   and
  
      [ [ ψ_1/ψ_I ψ_2/ψ_I   X/ψ_I   X/ψ_I;       …       …       …       …;       X       X     ψ_3     ψ_4;    ϕ'_1    ϕ'_2    ϕ'_3    ϕ'_4 ] ]
  where ϕ'_i stands for operator proportional to state |ϕ_i⟩⟨ϕ_i| or operator equal to 0. If assemblages (<ref>-<ref>) are obtained with {P_a|0}_a=0^1,{P_a|1}_a=0^1 which are the same (up to relabeling), one can subtract an LHS part. In other case form of the first two columns in (<ref>) implies that q_1=q_2=1 which is in contradiction with Lemma <ref> (as rank(ρ_ABC)=3). Similar contradiction is obtained for (<ref>) if the first row consists of rank one operators. Due to (<ref>-<ref>) for columns only relevant case for (<ref>) is given by 
  
      [ [  ψ_1  ψ_2    X    X;    X    X    X    X;    X    X  ψ_3  ψ_4; ϕ'_1 ϕ'_2 ϕ'_3 ϕ'_4 ] ]
  which cannot be on the edge.
  
  To consider the remaining case for r_1=r_3=2, put r_4=2. Without loss of generality, we may consider three particular examples for such situation
  
      [ [   ψ_1/ψ_I   ψ_2/ψ_I     X/ψ_I     X/ψ_I;         …         …         …         …;  ψ_3/ψ_II  ψ_4/ψ_II    X/ψ_II    X/ψ_II; ψ_5/ψ_III ψ_6/ψ_III   X/ψ_III   X/ψ_III ] ],
  
  
      [ [   ψ_1/ψ_I   ψ_2/ψ_I     X/ψ_I     X/ψ_I;         …         …         …         …;         X         X       ψ_3       ψ_4; ψ_5/ψ_III ψ_6/ψ_III   X/ψ_III   X/ψ_III ] ]
  and 
  
      [ [ ψ_1 ψ_2   X   X;   …   …   …   …;   X   X ψ_3 ψ_4;   X   X ψ_5 ψ_6 ] ].
  Once more, if assemblages (<ref>-<ref>) are obtained with {P_a|0}_a=0^1,{P_a|1}_a=0^1 which are the same (up to relabeling), one can subtract an LHS part - assume that it is not the case. Similar to the previous discussion the first two columns in (<ref>) imply that q_1=q_2=1 which is in contradiction with Lemma <ref>. The same is true for (<ref>) if the first row and the fourth row consists of rank one operators. If the first row in (<ref>) consists only of rank one operators and the fourth row have rank two operators, then the second row consists of rank two operators and one can subtract an LHS part from this structure (the same is true is the role of the first and the fourth rows are reversed) - this is true since in that case q_3,q_4=2,3 and related columns must be given by (<ref>-<ref>).
  
  In the end, if both the first and the fourth row have rank two operators, only obstruction for not being on the edge would be given by 
  
      [ [ ψ_1 ψ_2   X   X;   X   X ϕ_1 ϕ_2;   X   X ψ_3 ψ_4; ψ_5 ψ_6   X   X ] ].
  However, note that (<ref>) is the same as (<ref>) up to exchange of measurements on subsystems A and B (and relabeling). Therefore, by Lemma <ref>, this case can be neglected.
  
  Finally, by analogous arguments, one can see that the only structure of type (<ref>) which may provide an obstacle for the subtraction of an LHS part is given by formula (<ref>), which once more according to Lemma <ref> is irrelevant for ρ_ABC with rank three. The proof is now completed in the case of nontrivial projective measurement.
  
  
  Now let us consider the case of trivial projective measurements given by 0, 1. Firstly, assume that P_0|0=Q_0|0=0. Then Σ has the first column and the first row which consists of zeros and due to no-signaling and normalization conditions one can see that there always exists an LHS part that can be subtracted. Therefore, without loss of generality if there exist at least one trivial measurement on both subsystems, obtained assemblage is not on the edge. If so, assume that measurement on the subsystem B are given by nontrivial projective measurements (we may put them to be different up to relabeling). Now put P_0|0=P_0|1=0. Clearly assemblage obtained by this measurements have the first row and the third row which consists of zeros, while the remaining rows are the same and they consist of operators obtained after measurements on the subsystem B performed on reduced state ρ_BC=Tr_A(ρ_ABC). Note that rank(ρ_BC)≥ 2 since ρ_BC=ρ_0|0+ρ_1|0 and max{r_1,r_2}≥ 2 by previous discussion (here both ρ_a|x are assumed to be obtained by nontrivial projective measurement). Without loss of generality we may then assume that rank(ρ_BC)=2. In this case the relevant row is given by one of (<ref>-<ref>) and it is obvious that one can subtract an LHS part from the considered assemblage. To conclude this part of the proof, consider the case in which P_0|0=1 and {P_a|1}_a=0^1 is nontrivial. We may discuss only relevant case for which r_3=2 and the third row is given by one of (<ref>-<ref>). By no-signaling constraints for columns we see that assemblage obtained with such measurements is not on the edge, for example 
  
      [ [ ψ_1+σ_10|10   X+σ_11|10 ψ_2+σ_10|11   X+σ_11|11;           0           0           0           0;         ψ_1           X         ψ_2           X;     σ_10|10     σ_11|10     σ_10|11     σ_11|11 ] ].
  Therefore, any no-signaling assemblage obtained from projective measurements performed on rank three state is not on the edge.
  
  
  Finally, we can extend the proof in order to cover the case of all POVMs. Consider once more an assemblage σ_ab|xy=Tr_AB(M_a|x⊗ N_b|y⊗1ρ_ABC) where M_a|x, N_b|y are elements of nontrivial POVMs acting on the subsystem A and B respectively. In this case 
  
      M_0|0   = m_0|0P_0|0 + m_1|0P_1|0
       
         M_1|0   = (1-m_0|0)P_0|0 + (1-m_1|0)P_1|0
      
         M_0|1   = m_0|1P_0|1+m_1|1P_1|1
      
         M_1|1   = (1-m_0|1)P_0|1 + (1-m_1|1)P_1|1
  
  where 0<m_0|0, m_1|0, m_0|1, m_1|1 <1 and P_a|x are orthogonal projections coming from spectral decomposition of appropriate M_a|x. Define α= min{m_0|0,1-m_1|0,m_0|1,1-m_1|1}>0. Using previous equalities we can write 
  M_a|x=α P_a|x + M̃_a|x for all a,x=0,1 where each M̃_a|x is a positive operator. By analogous reasoning we may write N_b|y = β Q_b|y + Ñ_b|y for all b,y =0,1 and some β > 0. Inserting the above expressions for the POVMs into the definition of σ_ab|xy, we get σ_ab|xy=αβσ̃_ab|xy + σ_ab|xy^junk where σ̃_ab|xy = Tr_AB(P_a|x⊗ Q_b|y⊗1ρ_ABC) is an assemblage obtained by nontrivial projective measurements realized on a tripartite state ρ_ABC with rank (ρ_ABC) ≥ 3 and σ_ab|xy^junk is some irrelevant no-signaling assemblage (up to normalization equal to 1- αβ). We know that σ̃_ab|xy is not on the edge (we can subtract an LHS part from it) and therefore the initial assemblage σ_ab|xy is neither on the edge. Using similar reasoning in the case with some trivial POVMs, we obtain the same outcome based on previous results. Therefore, the proof of Theorem <ref> is concluded.
  
  
  
  
  
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