{'fraction_non_alphanumeric': 0.03968184599248317, 'fraction_numerical': 0.010401188707280832, 'mean_word_length': 4.412488174077578, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Bell violations from arbitrary joint measurability structures

Measurement incompatibility is necessary but not sufficient for violation of a Bell inequality. The structure of (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph with its vertices representing measurements and its hyperedges representing all (and only) compatible sets of measurements. We show that for any joint measurability structure that admits at least a pair of incompatible vertices (i.e., a non-trivial joint measurability structure), there exists a quantum realization with a set of measurements that enables a Bell violation. In other words, we identify, for any joint measurability structure, an incompatible set of measurements that is sufficient for a Bell violation.


I. INTRODUCTION
Bell inequality violations perhaps constitute the strongest evidence for the nonclassicality of quantum theory. These violations hinge on the existence of entangled states and incompatible measurements in quantum theory. In recent years, the relationship between measurement incompatibility and the possibility of Bell inequality violations has been an active area of research [1][2][3][4][5].
The most general quantum measurements are represented by positive operator-valued measures (POVMs). A set of POVMs is said to be jointly measurable (or compatible) if there exists a single POVM that can be marginalized in different ways to realize each of the POVMs in the set. Otherwise, it's said to be not jointly measurable (or incompatible). A question that has been of interest is the following: given any set of incompatible POVMs that Alice can implement on her quantum system, does there always exist a bipartite entangled state that she can share with Bob and a set of POVMs that Bob can implement on his part of the state such that their joint statistics violates a Bell inequality? This question was recently settled in the negative [4,5]. Thus, measurement incompatibility does not generically imply Bell nonlocality, similar to how there exist entangled states that do not yield a Bell violation (for any choice of local measurements) [6,7].
Here we ask a related but distinct question, focussing on a qualitative feature of incompatibility. In general, a set of POVMs will exhibit various incompatibility relations among its constituents. Any particular pattern of such incompatibility relations can be represented by * sy215@duke.edu † nikola.andrejic@pmf.edu.rs ‡ rkunjwal@ulb.ac.be a hypergraph that we term a joint measurability structure: the vertices of this hypergraph represent POVMs in the set of POVMs and the hyperedges represent all (and only) the jointly measurable subsets of POVMs in this set. The question, then, is the following: given any joint measurability structure, can Alice always identify a set of incompatible POVMs satisfying it such that there exists a bipartite entangled state that she can share with Bob and a set of POVMs that Bob can implement on his part of the state so that their joint statistics violates a Bell inequality? In other words, does every joint measurability structure enable a Bell violation? Note that we want a Bell violation that actually uses the incompatibility of all the measurements in a joint measurability structure and not merely restricted subsets of them, i.e., Alice must implement all of the measurements in the joint measurability structure on her wing of the Bell experiment and the statistics of all these measurements should play a non-trivial role in the Bell inequality being violated. 1 We answer our question in the affirmative by providing an explicit recipe for constructing a quantum realization of any joint measurability structure that also leads to a Bell violation. To do this, we combine the results of Refs. [1,4] with the methods of Ref. [8], leading to a straightforward recipe for Bell-violating quantum realizations of arbitrary joint measurability structures. 1 Note that any joint measurability structure exhibiting some incompatibility necessarily admits a pair of incompatible POVMs and, as such, can be used to demonstrate a violation of the CHSH inequality if Alice chooses to implement only those two measurements and ignores the rest in the joint measurability structure [1]. The question we are interested in is whether there exists a Bell inequality involving the statistics of all of Alice's measurements (and not just a pair of them) that admits a violation. In this section we define the basic notions that we will need such as the joint measurability of POVMs, joint measurability structures and their quantum realizations, and N -Specker scenario as a special case of a joint measurability structure. Our definitions are equivalent to those in Refs. [8,9]. Given a set of POVMs, its different subsets may or may not be compatible.
All of these different (in)compatibility relations between the POVMs in a given set are contained in what we call the joint measurability structure on that set, a notion we now formally define. A joint measurability structure is said to be quantumrealizable if its vertices represent quantum measurements that satisfy all the (in)compatibility relations dictated by it. The question of quantum realizability of a particular joint measurability structure boils down to the question of the existence of such quantum measurements. In Ref. [8] it was shown that all joint measurability structures admit quantum realizations, a result we will exploit in this note. Crucial to the construction in [8] is a particular class of joint measurability structures called N -Specker scenarios.
Definition 3 (N -Specker scenario). An N -Specker scenario is a joint measurability structure on a set of N ≥ 2 incompatible measurements where every (N − 1)-element subset of the set is compatible.
Hence, a 2-Specker scenario corresponds to a pair of incompatible measurements.
We will here qualitatively describe our approach to the problem and in the next section we will fill in the quantitative details. The logic of our approach is as follows: 1. We know that arbitary joint measurability structures can be realized with POVMs using the method of Ref. [8].
2. The technique of Ref. [8] has two parts: Firstly, to decompose a given joint measurability structure into substructures corresponding to different N -Specker scenarios (referred to as "minimal incompatible sets" of vertices in [8]).
Secondly, using quantum realizations for N -Specker scenarios (for any N ≥ 2), we can use a 'direct sum trick' to construct a quantum realization for the given joint measurability structure.
3. We use the first part of the technique of Ref. [8] and combine it with the quantum realizations of N -Specker scenarios (for N ≥ 3) recently presented in Ref. [4], where these realizations enable Bell violations. For the 2-Specker scenario, we can use any pair of incompatible POVMs to achieve a realization violating the CHSH inequality, following the results of Wolf et al. [1].
4. Using the same 'direct sum trick' as in Ref. [8], we show that, in fact, one can leverage the construction of Ref. [4] to show quantum Bell violations from arbitrary joint measurability structures. The Bell inequality that's violated by our construction for any joint measurability structure with v vertices is the I vv22 Bell inequality [10].

III. CONSTRUCTION OF BELL-VIOLATING QUANTUM REALIZATIONS OF ARBITRARY JOINT MEASURABILITY STRUCTURES
Consider any joint measurability structure J with v vertices. We resolve J into N -Specker scenarios embedded in the hypergraph for N ∈ {2, 3, . . . , N max }, where N max ≤ v, following the technique of Ref. [8]. We label the set of all N -Specker scenarios in this resolution of the joint measurability structure by {Sp s } s . An example of a joint measurability structure with its resolution into N -Speckers is shown in Fig. 1 with v = 4.
A. Bell violation from any N -Specker scenario, N ≥ 3 We have the following situation: Alice is given some arbitrary joint measurability structure J and she has to come up with POVMs realizing this structure that also enable a Bell violation. Given an N -Specker scenario in the resolution of J, labeled Sp s , we assign POVMs {M where {|j } N j=1 is an orthonormal basis of C N , η = 1/(N − 1), and A xj are defined as entries of the N × N where q 2 1 + q 2 0 = 1 and q 2 k+1 = 1 − 1 (N −k) 2 q 2 k for k ≥ 1. This means that each row of A, denoted A x , is a unit vector, i.e., N j=1 |A xj | 2 = 1, as expected [4,11]. Hence, Alice has access to the POVMs M where B yj are entries of the N × N matrix where Again, each row of B, de-noted B y , is a unit vector, i.e., N j=1 |B yj | 2 = 1, as expected [4,11].
The quantum state |ψ ∈ C N ⊗ C N shared between Alice and Bob is given by The Bell scenario therefore consists of two parties, each with N dichotomic measurements. The I (s) N N 22 Bell inequality [10] for this scenario is given by The probabilities in I p (s) The quantum violation is maximized , so that the probabilities p (s) (00|x, y) with y > x ≥ 1 cancel out. Note that 0 ≤ ≤ 1 for 0 ≤ q 0 ≤ 1. Then and η = 1   We consider any Sp s in the resolution of J that is just a pair of incompatible vertices. Under the no-signalling conditions, the expression for I (s) N N 22 Bell inequality reduces to the CHSH inequality for N = 2. That is, the inequality (also, referred to as the CH inequality, [12]) is equivalent to where A 1 is the same measurement as M y=2 . We refer the reader to Ref. [12] for a proof of this equivalence (see also [13] for a more modern treatment). Now, to obtain a violation of the CHSH inequality from a 2-Specker scenario, all we need is any pair of incompatible dichotomic POVMs. Wolf et al. [1] showed that Alice can use any such pair POVMs to violate the CHSH inequality for some choice of entangled state shared with Bob and some choice of POVMs for Bob. For example, one could take the shared state ρ s to be a two-qubit maximally entangled state (hence H s ∼ = C 2 ) and the measurements {M (s) to be those that achieve Tsirelson's bound [14]. The assignment of trivial {0, I s } POVMs to the rest of the v − 2 vertices in J can proceed as we did in the N ≥ 3 case. y } v y=1 associated with Alice and Bob, respectively. For each s, these states and measurements act on the Hilbert space H s ⊗ H s (s runs through all the N -Specker scenarios in the decomposition of J, cf. Fig. 1). We now 'stack' these constructions together so that the Hilbert space on each side becomes H = s H s and we have states and measurements defined on the tensor product space H ⊗ H ≡ ( s H s ) ⊗ ( s H s ) such that we obtain a Bell violation for the joint measurability structure J on Alice's side for an appropriate choice of Alice's measurements, Bob's measurements, and the shared quantum state between them. In doing this, we will follow, again, the prescription of Ref. [8].
We is also Bell violating. Note that x, y ∈ {1, . . . , v} and we will show that I vv22 > 0 for this choice of shared state and local measurements. The statistics relevant for I vv22 is determined by the following probabilities: Hence, given any joint measurability structure required of Alice's measurement settings, we can construct POVMs respecting it such that there exist measurement settings (POVMs) for Bob and a quantum state shared between Alice and Bob yielding a Bell inequality violation.