Nonlocal correlations in the presence of signaling and their implications for topological zero modes

Bell's theorem renders quantum correlations distinct from those of any local-realistic model. Although being stronger than classical correlations, quantum correlations are limited by the Tsirelson bound. This bound, however, applies for Hermitian, commutative operators corresponding to non-signaling observables in Alice's and Bob's spacelike-separated labs. In this work we examine the extent of non-local correlations when relaxing these fundamental assumptions, which allows for theories with non-local signaling. We prove that, somewhat surprisingly, the Tsirelson bound in the Bell-CHSH scenario, and similarly other related bounds on non-local correlations, remain effective as long as we maintain the Hilbert space structure of the theory. Furthermore, in the case of Hermitian observables we find novel relations between non-locality, uncertainty, and signaling. We demonstrate that such non-local signaling theories are naturally simulated by systems of parafermionic zero modes. We numerically study the derived bounds in parafermionic systems, confirming the bounds' validity yet finding a drastic difference between correlations of signaling and non-signaling sets of observables. We also propose an experimental procedure for measuring the relevant correlations.


I. Introduction
In a Bell test [1,2], Alice and Bob measure pairs of particles (possibly having a common source in their past) and then communicate in order to calculate the correlations between these measurements. The strength of empirical correlations enables to characterize the underlying theory. In quantum mechanics, the above procedure corresponds to local measurements of Hermitian operators A 0 /A 1 on Alice's side and B 0 /B 1 on Bob's side. The correlators are defined using the quantum expectation value c ij = A i B j and it can be shown that the CHSH parameter obeys |B| ≡ |c 00 + c 10 + c 01 − c 11 which is known as the Tsirelson bound [3]. Stricter bounds on the correlators were proposed, e.g., by Uffink [4] and independently by Tsirelson, Landau and Masanes (TLM) [5][6][7]. Importantly, when calculating B in any local-realistic model it turns out that B ≤ 2, which is a famous variant of Bell's theorem known as the Clauser-Horne-Shimony-Holt (CHSH) inequality [8]. These bounds, however, are not enough for fully characterizing the Alice-Bob correlations. For the latter task, the Navascues-Pironio-Acin (NPA) hierarchical scheme of semidefinite programs was proposed [9].
All the above works plausibly assuming that Alice's and Bob's measurements are described by spatially local and Hermitian operators, implying that [A i , B j ] = 0 for all i, j. As such, they cannot lead to superluminal signaling between Alice and Bob.
Trying, on the one hand, to generalize some of the above results, and on the other hand to pin-point the core reason they work so well, we relax below these two assumptions and examine the consequences of complex-valued correlations emerging from non-Hermitian noncommuting Alice/Bob operators. We thus allow a restricted form of signaling between the parties, but we maintain the Hilbert space structure, as well as other core ingredients of quantum mechanics. Surprisingly, the Tsirelson bound and TLM inequality remain valid in this generalized setting. Apart from that, we find intriguing relations between nonlocality, uncertainty, and signaling in the case of Hermitian yet non-commuting observables.
Considering non-Hermitian non-commuting observables may seem far from any sensible model. To alleviate this impression, we study an explicit example of a parafermionic system that provides a setting for studying such observables. Parafermions (or rather parafermionic zero modes) are topological zero modes that generalize the better-known Majorana zero modes [10][11][12]. Parafermions can be realized in various quasi-one-dimensional systems [13][14][15][16][17][18][19], see [20] for a comprehensive review. Similarly to the case of Majoranas, observables in a system of parafermions are inherently non-local as they comprise at least two parafermionic operators hosted at different spatial locations. In the case of Majoranas, this nonlocality is known to have manifestations through the standard CHSH inequality [21]. We do not follow the investigation line of Ref. [21], but rather investigate a different aspect of nonlocality, However, as we explain in Sec. V, in order to measure their respective observables, Alice and Bob in our system must share a common region of space, which resolves the paradox. In this sense, Alice and Bob can be thought of as two experimenters acting on the same system. Therefore, the system of parafermions does not constitute a system in which the spatial and quantum mechanical notions of locality disagree. However, it simulates such a system (with spatial locality interpreted in a very naive way). Using these examples we investigate the theoretical bounds on correlations. We find that both systems obey the derived bounds.
However, the maximal achievable correlations in the truly local system (first example) are significantly weaker than those of the non-local one.
The outline of the paper is as follows: In Sec. II we derive new bounds on the CHSH parameter in the case of non-Hermitian non-commutative operators. Further characterization of the non-local correlations in such cases is presented in Sec. III. Then, in Sec. IV we discuss the connection between nonlocality, uncertainty and signaling in the case of Hermitian non-commutative operators. After presenting these theoretical results, we turn to a concise introduction of parafermions in Sec. V, followed by a numerical analysis of the correlations in these systems and comparison with the theory in Sec. VI. We conclude in Sec. VII. An experimental scheme for measuring the complex correlations in a parafermionic system is discussed in Sec. V B together with Appendices A and B.

II. CHSH inequalities for complex-valued correlations
In what follows we derive the quantum bound on a complex-valued Bell-CHSH parameter (for similar bounds on the real-valued Bell-CHSH parameter see [22]).
Let C be an n × n Hermitian matrix whose ij-th entry is where ∆ X = XX † − | X | 2 is the uncertainty in X (which is assumed to be non-zero).
Proof. Denote |ψ , the underlying quantum state. For any n-dimensional vector, v T = where D is a (positive semidefinite) diagonal matrix whose entries are D ii = ∆ X i , and Therefore, DCD T 0 and so is C 0.
as the complex-valued Bell-CHSH parameter of the non-commuting pairs, A i and B j . The following holds where η is either C(A 0 , A 1 ) or C(B 0 , B 1 ).
Proof. Construct the matrix C for the operators A 0 , A 1 , and B j , where η def = C(A 0 , A 1 ). By the Schur complement condition for positive semidefiniteness this is equivalent to Let v T j = [(−1) j , 1]. The above inequality implies This together with the triangle inequality yield which completes the proof. Note that by swapping the roles of A and B, a similar inequality is obtained where η = C(B 0 , B 1 ).

III. Tighter, complex-valued Q 1 correlations
The set of possible quantum correlations Q can be well approximated by another (larger) set dubbed as Q 1 [9]. This set can be derived from a certain positive-semidefiniteness condition, but for reaching tighter approximations of the quantum set Q, some more conditions have to be fulfilled [9]. These increasingly harder conditions then form a hierarchy of positive-semidefinite programs, but we shall stop at Q 1 for the purposes of this paper.
In the case of vanishing one-point correlators A i = B j = 0, the characterization of Q 1 can be given in the form of the so-called TLM inequality [5][6][7], which is a nonlinear bound on C(A i , B j ). This bound is known to be necessary and sufficient for the correlators to be realizable in quantum mechanics [5][6][7]9].
Below we prove a complex-valued analog of the TLM inequality. It is emphasized that this inequality is stronger than the CHSH inequality and was only approximated in experiments [23], so it might be interesting to test it numerically for the case of parafermions as we shall do in Sec. VI.

Theorem 2 The following holds for any operators
which coincides with the characterization of Q 1 for real-valued C(B j , A i ).
Proof. The inequality (6) implies which follows from the non-negativity of the determinant of the matrix obtained by subtracting the right hand side from the left hand side in (6). Therefore, This and the triangle inequality give rise to the theorem, IV. The relation between nonlocality, uncertainty, and signaling For Hermitian observables, the deviation of Im(C(B j , A i )) from 0 implies non-commutation of A i , B j and is thus a signature of signaling. In the case of non-Hermitian observables, Im(C(B j , A i )) = 0 does not require non-commutativity and thus is not a signature of signaling. The theorem below is valid in both cases. However, only in the Hermitian case it can be interpreted as the subject matter of this section.
Theorem 3 Let B be the complex-valued Bell-CHSH parameter. Then, Namely, the local uncertainty relations represented by Re(η)/2 [24], the extent of nonlocality naturally represented in terms of the CHSH parameter Re(B)/(2 √ 2), and an expression linked to quantum signaling, Im(B)/(2 √ 2), are confined to the unit ball. See Figure 1.
Proof. We have seen that Therefore, Because, from which the theorem follows.
The maximal value that B achieves in the standard CHSH scenario corresponds to "isotropic" correlators of the form c ij = (−1) ij / √ 2 (i.e., all having the same absolute value). This behavior seems to be true even in more general cases [25]. If we assume that the correlators are isotropic, we can prove a stronger version of the above theorem: Theorem 4 Let B be the complex-valued Bell-CHSH parameter. In the isotropic case, where Proof. In case the isotropy holds, (10) reads and thus Averaging both sides in this inequality over j = 0, 1, and rearranging give Finally, substituting ̺ = B/4 into (20) yields the theorem.

V. Parafermions: physics, algebra, and observables
In Sec. V A, we summarize the physics of parafemrions. In Sec. V B, we discuss how to measure parafermionic observables. In Sec. V C, we introduce two examples of Alice's and Bob's observables that we later investigate numerically in Sec. VI.

A. Parafermion physics and algebra
Parafermionic zero modes can be created in a variety of settings [13][14][15][16][17][18][19]. In different settings, the systems have subtly different properties. We focus on parafermions implemented with the help of fractional quantum Hall (FQH) edges proximitized by a superconductor [13,16,17]. The setup employs two FQH puddles of the same filling factor ν (grey regions in Fig. 2a) separated by vacuum. This gives rise to two counter-propagating chiral FQH edges. The edges can be gapped either by electron tunneling between them (T domains) or by proximity-induced superconducting pairing of electrons at the edges (SC domains).
Domain walls between the domains of two types host parafermionic zero modes α s,j with s = R/L = ±1 denoting whether a parafermion belongs to the right-or left-propagating edge respectively, and j denoting the domain wall number.
The physics of parafermions is associated with degenerate ground states of the system.
One can show that α s,j α † s,k d = −e 2iπ/ν , which implies that it has d distinct eigenvalues, all having the form −e iπν(r+1/2) with r ∈ Z.
Unitary operators α s,j α † s,k are thus natural "observables" in the system despite being non-Hermitian. The permutation relations of such operators immediately follow from Eqs. (21)(22)(23). Despite being spatially disconnected, such operators composed of different pairs of parafermions may not commute, e.g., It is interesting to note that in the case of Majoranas (ν = 1), none of these two unique properties would hold: the operators iα s,j α † s,k would be Hermitian, while two such operators having no common Majoranas would commute.

B. Measuring parafermionic observables
A system combining parafermions with charging energy was introduced in Ref. [26]. In such a system there is a charging energy associated with the total system charge Q tot = j Q j + Q 0 , where Q 0 = 2en C is the charge of the proximitizing superconductor, and n C is the number of Cooper pairs in it. However, no energy cost is associated with different distributions of a given total charge over different SC domains. Therefore, the ground state of such a system has degeneracy d N SC −1 , where the reduction by a factor of d corresponds to fixing the system's total charge. The properties of operators α s,j α † s,k acting in this reduced subspace are identical to those in the original system of parafermions with unrestricted total charge.
Introducing charging energy allows for designing a relatively simple protocol for measuring α s,j α † s,k (both parafermions have the same s!) [26]. A sketch of the measurement setup is shown in Fig. 2b. Two additional FQH edges (belonging to one of the puddles) are required in this setup. Tunneling of FQH quasiparticles is allowed directly between the two edges with tunnelling amplitude η ref or between each edge and the corresponding parafermion α s,j/k with amplitude η j/k . As changing the charge of the parafermionic system is energetically costly, the leading non-trivial process resulting from coupling of the edges to the parafermions is co-tunneling of quasiparticles: a quasiparticle is transferred between the edges, while the parafermion state is changed via α s,j α † s,k and the effective tunneling amplitude is η cot ≃ −η k η * j /E C , where E C is the charging energy. The two processes, direct and parafermionmediated tunneling of a quasiparticle between the edges, interfere quantum-mechanically.
When a voltage bias V is applied between the edges, the tunneling current between the edges is sensitive to this interference: where κ is the interference suppression factor due to finite temperature and other effects, Re [A] = A + A † /2, and |V | is assumed to be much larger than the temperature T of the probing edges. As a result, by measuring I T , one can measure the operator Re e iϕ α s,j α † s,k with phase ϕ depending on the phases of η ref and η cot . Thus, one can measure the system in the eigenstates of α s,j α † s,k employing the fact that the eigenvalues of the α s,j α † s,k are discrete: for a generic ϕ, distinct eigenvalues of α s,j α † s,k correspond to distinct eigenvalues of Re e iϕ α s,j α † s,k . Alternatively, through tuning the phase ϕ, one can measure independently Re α s,j α † s,k and Im α s,j α † s,k = Re e −iπ/2 α s,j α † s,k , and combine the measurement results for calculating the expectation value α s,j α † s,k .
C. Alice's and Bob's observables For a parfermionic system with three SC domains (as in Fig. 2)  The fact that natural parafermionic observables involve parafermions at different spatial locations gives an opportunity to study a unique manifestation of quantum non-local correlations. To this end, we introduce several sets of observables, cf. Fig. 3a. We introduce observables accessible to Alice, and two different sets of observables accessible to Bob: and Naively, Alice's observables are local with respect to either set of Bob's observables.
Indeed, A and either the B or B ′ set use different parafermions, which can be made arbitrarily distant from each other. However, the locality issue in this system is subtler as in order to probe an observable of the form α s,j α † s,k , one needs to enable FQH quasiparticle tunneling to both parafermions simultaneously. At the same time, quasiparticles can tunnel to a parafermion only from the FQH puddle corresponding to the parafermion index s, not through vacuum and not from the other puddle. Therefore, as can be seen from Fig. 3b, the A and B sets are indeed mutually local, while A and B ′ are not. The ability of Alice to measure observables in A and of Bob to measure observables in B ′ , requires them to have access to a common region of the upper FQH puddle.
These locality properties are reflected in the permutation relations of the observables, which are as follows: The sets B and B ′ are thus identical as long as only Bob's side is concerned. However, the non-commutativity of B ′ with A may and does affect the cross-correlations observed by Alice and Bob.
The standard tool for studying quantum correlations is given by Bell inequalities. However, since the observables considered here have more than two eigenvalues, we require CHSH-like inequalities suitable for multi-outcome measurements. In Sec. VI, we study the inequalities introduced in Secs. II-IV, as well as an inequality from Ref. [27]. These inequalities involve correlators of the form A j B † k and A j (B ′ k ) † . Since [A j , B k ] = 0, A j B † k can be experimentally obtained by performing strong measurements of A j and B k separately according to the protocol of Sec. V B and then calculating the correlations. The non-commutativity of A j and B ′ k does not allow for such an approach in the case of A j (B ′ k ) † . However, this correlator can be measured with the help of weak measurements [28,29] as described in Appendix A.

VI. Numerical analysis of the correlation bounds for a parafermionic system
Below we investigate the effects of locality on quantum-mechanical correlations using the parafermionic observables introduced in Sec. V C. We first investigate a CHSH-like inequality for three-outcome measurements that was derived in Ref. [27]. The inequality states that for a local-realistic system where . The observables are assumed to have eigenvalues e 2πir/3 , r ∈ Z, which is the case for the observables defined in Eqs. (31)(32)(33)(34). In quantum mechanics (where the expectation values are defined in the standard way A j B k = ψ|A j B k |ψ ), for observables such that [A j , B k ] = 0, the maximum attainable value is known to be ≈ 2.91 [30]. Searching through different states |ψ , we have maximized numerically the value of I 3 . In both the local (using Bob's B k observables defined in Eq. (32)) and non-local cases (Bob's observables are B ′ k defined in Eqs. (33,34)), we have found the same max(I 3 ) ≈ 2.60. Therefore, this correlation parameter does not distinguish between the mutually local and non-local Alice's and Bob's observables. In other words, quantum nonlocality does not seem to have any effect on the maximal attainable correlations provided that the local properties of operators are preserved.
This is not what we find when studying the inequalities introduced in Secs. II-IV, cf. Table I. Before we write our results, there is an important technical remark to make. Correlation functions C(X i , X j ) defined in Eq. (2) are not well-defined in all of the Hilbert space as the denominator can turn out to be zero. However, the points where it does, constitute a set of measure zero among all the states. Moreover, in the vicinity of these special points, C(X i , X j ) does not diverge but stays bounded; however, the limiting value as one approaches the special point depends on the direction of approach. Therefore, with careful treatment, these special points do not constitute a problem for investigation.
The maximization of the l.h.s. of (4) yields |B| max ≈ 2.44 for the local case and |B| max ≈ 2.82 for the non-local case. Both are below the bound of 2 √ 2 ≈ 2.83, however, the non-local case is clearly much closer to saturating the bound. To investigate the reason, we turned to the middle part of the inequality, which provides a tighter observable-specific bound.
The value of Reη A = Re [C(A 0 , A 1 )], as well as the corresponding expressions for B and B ′ depend on the system state |ψ , as a function of which they change between −1 and 1.
Therefore, the maximum of the middle part over all states yields the same bound of 2 √ 2.
Therefore, the correlations do distinguish between the local and non-local case. The fact that the bound in (4)  Summarizing, we find that the two different CHSH-like inequalities studied are not equivalent in terms of distinguishing quantum correlations of mutually commuting observables from those of non-commuting observables. All the inequalities derived in Secs. II-IV are clearly saturated much better in the non-local case. This raises a natural question: can tighter versions of these bounds be derived using the assumption of [A j , B k ] = 0?

VII. Discussion
In the first part of the paper (Secs. II-IV) we have proven a number of new bounds on quantum correlations that can be applied to non-Hermitian, multiple-outcome observables.
These bounds are also applicable to the case when Alice's and Bob's observables do not commute, thus providing relations between signaling, uncertainty, and nonlocality.
In the second part of the paper we numerically studied these bounds with the help of observables in a system of parafermions. Such systems were shown to accommodate non-Hermitian, non-commutative multiple outcome observables hosted at (naively) spatially disjoint locations, thus simulating the above scenario of non-local signaling theories. We compared two cases: when Alice's and Bob's observables commute and when they do not.
Apart from this difference, the observables had identical properties. We have found that the non-commutative observables saturate the bounds derived in (Secs. II-IV) much better.
It is known that the standard CHSH parameter has distinct bounds for classical local (|B| ≤ 2) and non-local (|B| ≤ 4) hidden variable theories. Our variation of the CHSH (4) measurements [28,29]. For the observables defined in Sec. V C, the following permutation relation holds: The rest of this appendix is dedicated to designing weak measurements of the required type and adapting the protocol of Appendix B to measuring parafermionic observables. We start with the measurement protocol discussed in Sec. V B. Suppose one of the additional FQH edges involved in the protocol has voltage V applied to it, while the other edge is grounded. The current injected to the first edge is I in = νe 2 V /h, while the tunneling current between the edges is I T , cf. Eq. (25). Suppose one measures the current for time t, so that the number of quasiparticles injected into the system is N = I in t/(νe). The number of quasiparticles q tunneling within the time window will be fluctuating around the average q = pN = I T t/(νe) with p = I T /I in . The expression for I T in Eq. (25) is valid as long as |I T | ≪ |I in |. In this regime, tunneling of different quasiparticles can be considered independent, and thus the probability of observing tunneling of q quasiparticles should be approximated well by the binomial distribution If one measures for a sufficiently long time, i.e., N ≫ 1, the binomial distribution is wellapproximated by the Gaussian distribution Depending on the eigenvalue −e iπν(r+1/2) of the measured observable α s,j α † s,k , the tunneling probability p = p 0 + δp r , with where ϕ = arg(η * ref η cot ), cf. Eq. (25). Denoting the initial state of parafermions as r ψ r |r and using some approximations, one can derive the state of the system after switching on the tunnel couplings for time t, where λ represents additional quantum numbers of the edges. It follows from Eq. (A2) Having not performed the calculation, we make a plausible assumption that also Further assuming ηcot η ref p 0 N ≪ 1, we can neglect η cot e iπν(r+1/2) in Eq. (A5) and obtain that for our purposes one can replace |Φ with which brings us to weak measurements of the type considered in Appendix B.
Consider now two weak measurements accessing A j and (B ′ k ) † performed one after the other, with the number of quasiparticles tunneled in each of the measurements being q 1 and q 2 . Repeating the calculation of Appendix B, we obtain Using Eq. (B8), one sees that by choosing different phases ϕ, ϕ ′ , one can measure {A j , (B ′ k ) † } = 2 A j (B ′ k ) † e iπν cos πν.
uses essentially the same measurement procedure as in Refs. [31][32][33], and is similar in spirit (yet has important differences) to Refs. [34,35]. We first discuss how to measure correlations of Hermitian non-commuting observables, and then generalize the scheme to non-Hermitian observables.
Suppose one wants to measure the average {A, B} = ψ|{A, B}|ψ , where A and B are Hermitian non-commuting operators, and |ψ is some quantum state. Introduce the eigenbases of A and B: A|a = a|a , B|b = b|b . Any system state |ψ can then be written as |ψ = a ψ a |a = a,b ψ a |b b|a with some coefficients ψ a . We assumed that there is no degeneracy in the spectra of A and B; generalization of the below consideration for the case with degeneracy is straightforward.
Consider two detectors, D 1 and D 2 each having coordinate Q j and momentum P j operators, [P j , Q k ] = −iδ jk , with j and k having values 1 and 2. Prepare the system and detectors in initial state where |q j is an eigenstate of Q j with eigenvalue q j , and N = (πσ 2 ) −1/4 .
The Hamiltonian describing the system and the detectors is where the coupling constants λ j (t) = 0 except for λ 1 (t) = g/T for t ∈ (0; T ) and λ 2 (t) = g/T for t ∈ (T ; 2T ). Then after the system has interacted with the detectors, their state is |Φ = e −igH 2 e −igH 1 |Φ in = N 2 a,b dq 1 dq 2 ψ a b|a exp − (q 1 − ga) 2 2σ 2 − (q 2 − gb) 2 2σ 2 |b |q 1 |q 2 . (B5) Measuring Q 1 and Q 2 of the detectors and calculating their correlations then yields the desired quantity. Indeed, Provided that g |a − a ′ | ≪ 2σ for all a, a ′ (which is the condition for weakness of the measurement), one obtains Φ|Q 1 Q 2 |Φ = g 2 2 a,a ′ ,b ψ * a a| (a|b b b| + |b b b|a ′ ) ψ a ′ |a ′ = Each of the averages in the r.h.s. can be measured using the protocol for Hermitian observables outlined above. Then combining them according to Eq. (B8) yields the desired correlation of non-Hermitian non-commuting observables.