Detecting entanglement from macroscopic measurements of the electric field and its fluctuations
Pedro Rosario Alan C. Santos Nicola Piovella Robin Kaiser André Cidrim Romain Bachelard

 



pedrorosario@estudante.ufscar.br



ac_santos@iff.csic.es





nicola.piovella@mi.infn.it
Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy & INFN Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy


robin.kaiser@univ-cotedazur.fr



andrecidrim@gmail.com



romain@ufscar.br

	
	

To address the outstanding task of detecting entanglement in large quantum systems, entanglement witnesses have emerged, addressing the separable nature of a state. Yet optimizing witnesses, or accessing them experimentally, often remains a challenge. We here introduce a family of entanglement witnesses for open quantum systems, based on the electric field — its quadratures and the total fluorescence. More general than spin-squeezing inequalities, it can detect new classes of entangled states, as changing the direction for far-field observation opens up a continuous family of witnesses, without the need for a state tomography. Their efficiency is demonstrated by detecting, from almost any direction, the entanglement of collective single-photon states, such as long-lived states generated by cooperative spontaneous emission. Able to detect entanglement in large quantum systems, these electric-field-based witnesses can be used on any set of emitters described by the Pauli group, such as atomic systems (cold atoms and trapped ions), giant atoms, color centers, and superconducting qubits.

	


Detecting entanglement from macroscopic measurements of the electric field and its fluctuations
    Romain Bachelard 0000-0002-6026-509X
    April 29, 2024
===============================================================================================






Introduction.—The detection of multipartite entanglement remains a challenging problem, due to the difficulty of realizing state tomography and the lack of tools to quantify it once the system state is known <cit.>. For low dimensional systems, for instance, entanglement witnesses such as the positive partial transpose of the density matrix <cit.>, the majorization criterion <cit.>, and many other proposals <cit.>, require state tomography. When the density matrix is not an available resource (and this is usually the case for quantum systems with a large Hilbert space), a different approach to the entanglement detection problem needs to be considered, typically relying on the direct measurement of quantum observables 𝒪̂ <cit.>. More specifically, if ρ̂ represents a quantum state, then 𝒪̂ is an entanglement witness if and only if a violation of the inequality Tr(𝒪̂ρ̂)≥ 0 implies an entangled nature of state ρ̂. This approach has put forward the investigation of multipartite entanglement in many-particle states <cit.>. In spin-like systems where the statistics (first and second moments) of a collective spin operator can be accessed experimentally, several sets of inequalities detecting spin squeezing have been proposed as entanglement witnesses <cit.>. These inequalities have, in turn, stimulated the definition of “metrologically-useful” squeezed states <cit.>, which enable reducing uncertainties in interferometry measurements beyond the so-called standard quantum limit <cit.>. To access larger classes of entangled states in systems where local measurements are particularly challenging, other entanglement witnesses taking into account the distance between particles have been proposed <cit.>. However, identifying experimentally accessible or more optimal witnesses remains a largely open challenge.

To address this problem, we introduce a continuous set of inequalities based on the measurement of the electric field — the quadratures, the total fluorescence, and their fluctuations, more precisely. While they reduce to spin-squeezing inequalities in particular geometries and observation angles, changing the direction of detection of the field allows one to probe an infinity of entanglement witnesses, as the optical path from the different atoms to the detection changes. This family of witnesses is thus more optimal than spin-squeezing ones and applies to quantum emitters described by the Pauli group — from natural two-level atoms to artificial ones with inhomogeneous broadening. 






Entanglement witnesses from the electric field.—Let us consider an ensemble of N two-level emitters, with |↑⟩_j and |↓⟩_j the excited and ground states for each atom j. Without local access to the emitters to realize state tomography, collective information on the system state can still be extracted from the radiated electric field. While the operator and the study of its momenta (intensity and field fluctuations) is a keystone of quantum optics <cit.>, we here show that it provides precious information on the atomic system state as well. In the far-field limit, the electric field operator for the two-level atoms reads

    Ê_𝐤^±=∑_j=1^Ne^∓ i𝐤.𝐫_jσ̂^∓_j,

with 𝐤 the direction of observation, σ̂^+_j=|↑⟩⟨↓|_j(σ̂^-_j=|↓⟩⟨↑|_j) the raising (lowering) Pauli operators of atom j, and 𝐫_j its position. Without loss of generality, the prefactor in front of the electric field operator has been set to unity. We then introduce the field quadratures X̂_𝐤 and Ŷ_𝐤, and inversion population operator Ẑ,



    X̂_𝐤=Ê_𝐤^++Ê_𝐤^-,
    
       Ŷ_𝐤=i(Ê_𝐤^+-Ê_𝐤^-), 
    
       Ẑ=∑_j=1^Nσ̂^z_j,


with σ̂^z_j=|↑⟩⟨↑|_j-|↓⟩⟨↓|_j the inversion population operator for atom j.

Any arbitrary separable state of N particles can be written as ρ̂=∑_l=1^Lp_lρ̂^(l)_1⊗ρ̂^(l)_2⊗⊗ρ̂^(l)_N with ∑_l=1^Lp_l=1 <cit.>, where the superscript (l) denotes a local state of the statistical mixture. If the state cannot be written in this form, there is thus entanglement between at least two particles. Following this statement, we now introduce our electric-field-based witness for entanglement <cit.>.

Theorem:  If there exists a wavevector 𝐤 for which a quantum state ρ̂ satisfies the inequality 

    W_𝐤=min{w_1,𝐤, w_2,𝐤, w^α,β,γ_3,𝐤, w^α,β,γ_4,𝐤} < 0,

then ρ̂ is an entangled state. 

The witness W_𝐤 encompasses the following series of entanglement witnesses



    w_1,𝐤    =  N(2+N)  - ⟨X̂^2_𝐤⟩ - ⟨Ŷ^2_𝐤⟩ - ⟨Ẑ^2⟩,
     w_2,𝐤   = (ΔX̂_𝐤)^2+(ΔŶ_𝐤)^2+(ΔẐ)^2  -2N,
     w^Â,B̂,Ĉ_3,𝐤   = 2N +(N-1)(ΔÂ)^2-⟨B̂^2⟩ -⟨Ĉ^2⟩ ,
      w^Â,B̂,Ĉ_4,𝐤   =(N-1)[(ΔÂ)^2+(ΔB̂)^2]-⟨Ĉ^2⟩
       -N (N-2),


where (Δ∙ )^2=⟨∙ ^2⟩ - ⟨∙⟩ ^2 corresponds to the variance, and superscript {Â,B̂,Ĉ} to the cyclic permutations over the set {X̂_𝐤,Ŷ_𝐤,Ẑ}.

The derivation of the witness relies on showing that all separable states fulfill the inequalities w_n,𝐤≥ 0 for n=1...4 (using a concavity argument <cit.>). Hence, a state satisfying W_𝐤<0 violates at least one of these inequalities and is thus entangled. 

While the argument is similar to the one used to derive spin-squeezing inequalities, we point out that our electric-field witnesses represent a much broader family of witnesses. More specifically, the original inequality proposed by Sørensen et al. <cit.> was generalized to a finite set of inequalities by Tóth et al. <cit.>, to account for the different components of the collective spin operators. The relation between (<ref>) and the spin squeezing-inequalities of Refs. <cit.> is obtained by setting 𝐤=0, so they are hereafter denoted by W_0. Differently, the present family of inequalities (<ref>) is infinite, and one can span the witnesses by changing the direction of detection of the light. Changing the light wavenumber k also provides a broader class of witness <cit.>. We here focus on close-to-resonance witnesses. Indeed, far from resonance, one enters the fully dispersive regime, often used for quantum non-demolition measurements <cit.>. Nevertheless, this limit is beyond the scope of this work.











Three-atom case.—Let us now illustrate the advantage of electric-field-based inequalities by considering the light scattered by a linear arrangement of three atoms along the x̂-axis, prepared in the following state

    |ψ⟩=1/√(3)(|↑↑↓⟩+e^iΛ|↓↑↑⟩+e^i2Λ|↓↓↑⟩),

equally spaced by d=0.3/k and a relative phase Λ=π/3. Figure <ref> exhibits the behavior of the entanglement witness W_𝐤 for this system along different directions of observation 𝐤, based on the measurement of the field operators (<ref>) and their fluctuations. In this particular one-dimensional configuration, the spin-squeezing inequality W_0 can be directly probed with the electric field in directions orthogonal to the chain (dashed circumference on the yz-plane in Fig. <ref>), since the optical path until the detector is the same for all atoms [𝐤·(𝐫_j-𝐫_m)≡ 0]. Note that W_0> 0, meaning that spin-squeezing inequalities do not detect the entanglement of this state. Differently, the statistics of the scattered light can capture it in some directions of observation. The colored area corresponds to the directions where the field measurement allows for the detection of entanglement, and the gray-colored one to directions where W_𝐤≥0, so the entanglement is not detected by witness (<ref>). This demonstrates how the degree of freedom of the phase terms in the electric field allows for a more optimal entanglement detection.





Single-excitation states.—We now consider an N-atom single-excitation Dicke state 

    |𝒟_N⟩=1/√(N)∑_n=1^N e^iϕ_n|↑_n⟩,

where |↑_n⟩ represents a state where atom n is excited and all the others are in the ground state. These states are detected by electric field witnesses in almost all directions. Indeed, one can show that, for these states <cit.>

    w_2,𝐤 =-w_3,𝐤^Ẑ,Ŷ_𝐤,X̂_𝐤=S_𝐤,

with S_𝐤≡ -4/N∑_j=1^N∑_s≠ j^Ncos[ϕ_s-ϕ_j+𝐤·(𝐫_j-𝐫_s)]. A negative S_𝐤 violates inequality (<ref>), while a positive S_𝐤 violates inequality (<ref>). Hence, only the directions of observation 𝐤 which satisfy S_𝐤=0 are inadequate to detect the entanglement from the field fluctuations: It is a set of measure zero. 

Let us illustrate this feature by considering a regular chain of N=100 atoms, again along the x̂-axis, with spacing d=π/2k. Taking into account the rotational symmetry around the x̂-axis, the light is monitored in the xy-plane, using the polar angle θ, with 𝐤=k(cosθ,sinθ,0). At an angle θ=π/2, the double sum S_𝐤 simplifies into S_0=∑_j=1^N∑_s≠ j^Ncos(ϕ_s-ϕ_j). For S_0=0, spin-squeezing inequalities do not capture entanglement. Focusing on states with phases ϕ_n=n δ, this condition is reached when δ is a solution of T_N(cosδ)-Ncosδ+(N-1)=0, with T_N the Chebyshev polynomial of the first kind <cit.>. The angular dependence of the witness W_𝐤 for such a case is shown in Fig. Fig:Decay(a): The witness is negative for any direction of observation, thus detecting entanglement, except for θ=π/2 where spin-squeezing witness W_0 is measured. We point out that the detection of these single-excitation states by the electric field witness is valid for arbitrary particle numbers.









Dynamical generation of entanglement.—Beyond the issue of entanglement detection, let us now discuss the generation of entanglement via collective spontaneous emission. We consider two-level atoms driven resonantly, whose interaction through the vacuum modes results in an effective dipole-dipole interaction between them. In three dimensions, this interaction can be described by the following master equation <cit.>

    dρ̂/dt   =∑^N_j,m≠ jiΔ_jm[σ̂^+_jσ̂^-_m,ρ̂]+ ℒ(ρ̂),

where we have set ħ≡ 1. The first part represents the coherent component of the dipole-dipole interaction, while the second term corresponds to the dissipative part, ruled by the Lindbladian ℒ(ρ̂)=∑^N_j,mΓ_jm(σ̂^-_jρ̂σ̂^+_m-1/2{σ̂^+_mσ̂^-_j,ρ̂}). The excitation-exchange term Δ_jm and crossed decay rates Γ_jm are given by the Green's tensor, Δ_jm≡ -ϵ̂_j^*·{𝐆(r⃗_jm)}·ϵ̂_m and Γ_jm≡ϵ̂_j^*·{𝐆(r⃗_jm)}·ϵ̂_m, with ϵ̂_l is the polarization of the l-th dipole, and r⃗_jm=r⃗_j-r⃗_m the relative position of atoms j and m.  In free space, the tensor for dipoles with transition frequency ω=kc=2π c/λ and linewidth Γ
is given by 𝐆(r⃗_jm)=3Γ/4e^ik r_jm/(kr_jm)^3[(k^2r^2_jm+ikr_jm-1)_3-(k^2r^2_jm+i3kr_jm-3)r⃗_jmr⃗^T_jm/r^2_jm] for j≠ m, and 𝐆(r⃗_jj)=iΓ/2_3.

The superradiant cascade from interacting dipoles prepared in an initially fully-excited state was first explored by Dicke <cit.> using collective states with a given photon number, which are therefore entangled. Yet, later studies showed that a semi-classical approach described properly the accelerated radiance <cit.>, and entanglement is not generated in that process <cit.>. Differently, the long-lived (subradiant) modes that eventually emerge <cit.> become very close to mixtures of ground and collective single-excitation states, and thus entangled <cit.>. While the fully-excited state decays mostly toward the symmetric (superradiant) states, subradiant states are most efficiently generated by sending the system to a statistical mixture <cit.>. This scenario is confirmed in our simulations by analyzing the electric field and its fluctuations. As illustrated in Fig. Fig:Decay(b) for the regular chain, entanglement is detected at a later time for an initially fully excited system (Γ t≈ 3), as compared to an initially maximally mixed state ρ̂=/2^N (Γ t≈ 1): This observation is consistent with entanglement being generated only at late times, and being generated more efficiently from a statistical mixture than from a fully-excited state. 

As an alternate probe for the presence of entanglement, let us now monitor the global (pairwise) concurrence, defined as 𝒞_glob=∑_j,s≠ j𝒞(ρ̂_js)/N(N-1), where 𝒞(ρ̂_js) corresponds to the pairwise concurrence for the pair (j,s), derived from its reduced density matrix ρ̂_js <cit.>. As observed in Fig. Fig:Decay(b), the concurrence becomes finite at a later time than the entanglement detected through the field fluctuations. The electric-field-based detection may thus be a particularly promising tool to study critical systems, where concurrence has been used to detect phase transitions <cit.>.




Three-dimensional clouds.—While the one-dimensional chain offers access to a direct measurement of spin squeezing W_0 through the electric field in directions orthogonal to the chain, three-dimensional clouds do not possess a symmetry that allows for this feature. Let us now demonstrate how the family of witnesses based on the electric field extends the detection of entanglement for these systems as well. We now consider a disordered cloud of N=8 two-level atoms in three dimensions, prepared in the antisymmetric product state |A⟩=|+-+-+-+-⟩, where we have introduced the single-atom state |±⟩=(|↑⟩±|↓⟩)/√(2). Note that while the atoms, and thus the antisymmetric state, are ordered by positions in the linear chain, for the three-dimensional cloud the ordering in |A⟩ is arbitrary. The evolution of the witness W_𝐤 during the decay dynamics is presented in Fig. Fig:Decay(c): Similar features for the 3D (gray curve) and 1D (purple curve) configurations are observed, with entanglement being detected at an early time (Γ t≈ 10^-1) in both cases. This suggests that the electric field witness is equally efficient to probe entanglement in arbitrary geometries.

The emergence of entanglement on this short (superradiant) time scale can be understood as follows: The antisymmetric separable state |A⟩ has a strong projection on both superradiant and subradiant entangled states. Due to the short distance between the atomic dipoles, the strong Hamiltonian interactions are responsible for shifting their relative phases on a short timescale, thus sending the system toward an entangled state. This is akin to the case of emitters with different energies <cit.>, but here with a shift induced by dipole-dipole interactions. 


Resorting to collective observables is all the more critical when the size of the quantum system increases, since tomography, even if local measurements are available <cit.>, is no longer possible due to the size of the Hilbert space. The ability of such observables to capture the entanglement may nonetheless be affected by the system size since the number of remote, weakly interacting pairs of particles will be relatively larger. To investigate how the entanglement detection scales with the system size, we resort to the second-order cumulant approximation, where three-point and higher correlations are factorized in terms of one- and two-point ones <cit.>. The regular atomic chain is initially in state |A⟩, from which it evolves in the presence of dipole-dipole interactions. The evolution of W_𝐤 reveals that entanglement is detected at earlier times as the lattice spacing is reduced, and as the system size N is increased, see Fig. Fig:Decay(d). While the former effect is quite intuitive, since the interactions become stronger at shorter inter-particle distances, the latter is rather a signature of long-range interactions. But this observation also demonstrates the ability of the field-based witness to detect entanglement efficiently in large quantum systems. 





Conclusion.—In this work, we have discussed how measuring the electric field quadratures, the total fluorescence, and their fluctuations can serve to detect entanglement between the quantum emitters. To this end, we used a witness related to spin squeezing which incorporates the phases present in the electric field. This witness captures, in particular, single-excitation collective states, in almost any direction of observation. Derived for systems of qubits described by the Pauli group, the witness is thus also valid for ensembles with inhomogeneous broadening, a common case for artificial qubits such as nitrogen-vacancy centers <cit.>. An interesting prospect is to generalize these electric-field-based inequalities to quantum emitters with arbitrary spin <cit.>, offering tools to probe multilevel entanglement in systems with qudits <cit.>. 

While finding an optimal witness remains an outstanding challenge <cit.>, the proposed family of witnesses presents the strong advantage of not relying on state tomography, but rather on measuring collective observables which will be particularly practical in some experiments. Note that the inversion population operator Ẑ, and its fluctuations, can be measured from the evolution of the fluorescence in an integrating sphere, for example — the radiated power corresponds to a decrease of the excited population, or by probing the atomic populations using an auxiliary ancilla state.

The electric field operators in different directions actually correspond to specific relative phases between the atomic operators, set by the term 𝐤 in Eq. (<ref>). While this corresponds to an already infinite family of witnesses since 𝐤 spans the 4π solid angles, a much broader family of collective operators and associated witnesses can be introduced, by setting arbitrary relative phases between the atomic operators. This ensemble of witnesses, which would span a parameter space [0; 2π]^N-1 (one of the phases can be set arbitrarily) is much broader than the one based on the electric field operators, yet accessing it may bring back the requirement of challenging local measurements.


Acknowledgments.—P.R., A.C.S., A.C., and R.B. acknowledge the financial support of the São Paulo Research Foundation (FAPESP) (Grants No. 2018/15554-5, 2019/13143-0, 2019/12842-2, 2022/06449-9, 2023/07463-8, 2022/12382-4, and 2023/03300-7), from the Brazilian CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Grant No. 313632/2023-5. R.B. and R.K. received support from the project STIC-AmSud (Ph879-17/CAPES 88887.521971/2020-00), CAPES-COFECUB (CAPES 88887.711967/2022-00). R.K. acknowledges the support by QuantERA ERA-NET Cofund in Quantum Technologies (GA No. 731473), project PACE-IN, and the European project ANDLICA, ERC Advanced grant No. 832219. A.C.S acknowledges the support by the Proyecto Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM) from the Comunidad de Madrid.














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@bib@innerbibempty

[Bennett et al.(1996)Bennett, DiVincenzo, Smolin, and Wootters]Bennett_1996
  author author C. H. Bennett, author D. P. DiVincenzo, author J. A. Smolin, and author W. K. Wootters, title title Mixed-state entanglement and quantum error correction, https://doi.org/10.1103/PhysRevA.54.3824 journal journal Phys. Rev. A volume 54, pages 3824 (year 1996)NoStop
[Hill and Wootters(1997)]Hill_1997
  author author S. Hill and author W. K. Wootters, title title Entanglement of a pair of quantum bits, https://doi.org/10.1103/physrevlett.78.5022 journal journal Phys. Rev. Lett. volume 78, pages 5022 (year 1997)NoStop
[Wootters(1998)]Wootters_98
  author author W. K. Wootters, title title Entanglement of formation of an arbitrary state of two qubits, https://doi.org/10.1103/PhysRevLett.80.2245 journal journal Phys. Rev. Lett. volume 80, pages 2245 (year 1998)NoStop
[Vidal and Werner(2002)]Vidal_2002
  author author G. Vidal and author R. F. Werner, title title Computable measure of entanglement, https://doi.org/10.1103/PhysRevA.65.032314 journal journal Phys. Rev. A volume 65, pages 032314 (year 2002)NoStop
[Gühne and Tóth(2009)]G_hne_2009
  author author O. Gühne and author G. Tóth, title title Entanglement detection, https://doi.org/10.1016/j.physrep.2009.02.004 journal journal Physics Reports volume 474, pages 1 (year 2009)NoStop
[Peres(1996)]Peres_1996
  author author A. Peres, title title Separability criterion for density matrices, https://doi.org/10.1103/PhysRevLett.77.1413 journal journal Phys. Rev. Lett. volume 77, pages 1413 (year 1996)NoStop
[Nielsen and Kempe(2001)]Nielsen_2001
  author author M. A. Nielsen and author J. Kempe, title title Separable states are more disordered globally than locally, https://doi.org/10.1103/PhysRevLett.86.5184 journal journal Phys. Rev. Lett. volume 86, pages 5184 (year 2001)NoStop
[Horodecki(1997)]Horodecki_1997
  author author P. Horodecki, title title Separability criterion and inseparable mixed states with positive partial transposition, https://doi.org/10.1016/s0375-9601(97)00416-7 journal journal Physics Letters A volume 232, pages 333 (year 1997)NoStop
[Hofmann and Takeuchi(2003)]Hofmann_2003
  author author H. F. Hofmann and author S. Takeuchi, title title Violation of local uncertainty relations as a signature of entanglement, https://doi.org/10.1103/PhysRevA.68.032103 journal journal Phys. Rev. A volume 68, pages 032103 (year 2003)NoStop
[Shchukin and Vogel(2005)]Shchukin_2005
  author author E. Shchukin and author W. Vogel, title title Inseparability criteria for continuous bipartite quantum states, https://doi.org/10.1103/PhysRevLett.95.230502 journal journal Phys. Rev. Lett. volume 95, pages 230502 (year 2005)NoStop
[Horodecki et al.(2006)Horodecki, Horodecki, and Horodecki]Horodecki_2006
  author author M. Horodecki, author P. Horodecki, and author R. Horodecki, title title Separability of mixed quantum states: Linear contractions and permutation criteria, https://doi.org/10.1007/s11080-006-7271-8 journal journal Open Systems and Information Dynamics volume 13, pages 103 (year 2006)NoStop
[Terhal(2000)]Terhal_2000
  author author B. M. Terhal, title title Bell inequalities and the separability criterion, https://doi.org/10.1016/s0375-9601(00)00401-1 journal journal Physics Letters A volume 271, pages 319 (year 2000)NoStop
[Bruß et al.(2002)Bruß, Cirac, Horodecki, Hulpke, Kraus, Lewenstein, and Sanpera]Bru_2002
  author author D. Bruß, author J. I. Cirac, author P. Horodecki, author F. Hulpke, author B. Kraus, author M. Lewenstein, and author A. Sanpera, title title Reflections upon separability and distillability, https://doi.org/10.1080/09500340110105975 journal journal Journal of Modern Optics volume 49, pages 1399 (year 2002)NoStop
[Horodecki et al.(1996)Horodecki, Horodecki, and Horodecki]Horodecki_1996_1
  author author M. Horodecki, author P. Horodecki, and author R. Horodecki, title title Separability of mixed states: necessary and sufficient conditions, https://doi.org/10.1016/s0375-9601(96)00706-2 journal journal Physics Letters A volume 223, pages 1 (year 1996)NoStop
[Sperling and Vogel(2013)]2013_Sperling_Vogel_prl
  author author J. Sperling and author W. Vogel, title title Multipartite entanglement witnesses, https://doi.org/10.1103/PhysRevLett.111.110503 journal journal Phys. Rev. Lett. volume 111, pages 110503 (year 2013)NoStop
[Friis et al.(2018a)Friis, Vitagliano, Malik, and Huber]2018_Friis_Huber_natrevphys
  author author N. Friis, author G. Vitagliano, author M. Malik, and author M. Huber, title title Entanglement certification from theory to experiment, https://doi.org/10.1038/s42254-018-0003-5 journal journal Nature Reviews Physics volume 1, pages 72–87 (year 2018a)NoStop
[Friis et al.(2018b)Friis, Marty, Maier, Hempel, Holzäpfel, Jurcevic, Plenio, Huber, Roos, Blatt, and Lanyon]Friis2018
  author author N. Friis, author O. Marty, author C. Maier, author C. Hempel, author M. Holzäpfel, author P. Jurcevic, author M. B. Plenio, author M. Huber, author C. Roos, author R. Blatt, and author B. Lanyon, title title Observation of entangled states of a fully controlled 20-qubit system, https://doi.org/10.1103/PhysRevX.8.021012 journal journal Phys. Rev. X volume 8, pages 021012 (year
  2018b)NoStop
[Sun et al.(2024)Sun, Zhou, Tavakoli, Xu, and Yu]2024_Sun_Yu_prl
  author author L.-L. Sun, author X. Zhou, author A. Tavakoli, author Z.-P. Xu, and author S. Yu, title title Bounding the amount of entanglement from witness operators, https://doi.org/10.1103/PhysRevLett.132.110204 journal journal Phys. Rev. Lett. volume 132, pages 110204 (year 2024)NoStop
[Sørensen et al.(2001)Sørensen, Duan, Cirac, and Zoller]S_rensen_2001
  author author A. Sørensen, author L.-M. Duan, author J. I. Cirac, and author P. Zoller, title title Many-particle entanglement with bose–einstein condensates, https://doi.org/10.1038/35051038 journal journal Nature volume 409, pages 63 (year 2001)NoStop
[Korbicz et al.(2005)Korbicz, Cirac, and Lewenstein]Korbicz_2005
  author author J. K. Korbicz, author J. I. Cirac, and author M. Lewenstein, title title Spin squeezing inequalities and entanglement of n qubit states, https://doi.org/10.1103/PhysRevLett.95.120502 journal journal Phys. Rev. Lett. volume 95, pages 120502 (year 2005)NoStop
[Tóth et al.(2007)Tóth, Knapp, Gühne, and Briegel]Toth_2007
  author author G. Tóth, author C. Knapp, author O. Gühne, and author H. J. Briegel, title title Optimal spin squeezing inequalities detect bound entanglement in spin models, https://doi.org/10.1103/PhysRevLett.99.250405 journal journal Phys. Rev. Lett. volume 99, pages 250405 (year 2007)NoStop
[Tóth et al.(2009)Tóth, Knapp, Gühne, and Briegel]Toth_2009
  author author G. Tóth, author C. Knapp, author O. Gühne, and author H. J. Briegel, title title Spin squeezing and entanglement, https://doi.org/10.1103/PhysRevA.79.042334 journal journal Phys. Rev. A volume 79, pages 042334 (year 2009)NoStop
[Pezzè et al.(2018)Pezzè, Smerzi, Oberthaler, Schmied, and Treutlein]2018_Pezze_Treutlein_rmp
  author author L. Pezzè, author A. Smerzi, author M. K. Oberthaler, author R. Schmied, and author P. Treutlein, title title Quantum metrology with nonclassical states of atomic ensembles, https://doi.org/10.1103/RevModPhys.90.035005 journal journal Rev. Mod. Phys. volume 90, pages 035005 (year 2018)NoStop
[Bornet et al.(2023)Bornet, Emperauger, Chen, Ye, Block, Bintz, Boyd, Barredo, Comparin, Mezzacapo, Roscilde, Lahaye, Yao, and Browaeys]Bornet_2023
  author author G. Bornet, author G. Emperauger, author C. Chen, author B. Ye, author M. Block, author M. Bintz, author J. A. Boyd, author D. Barredo, author T. Comparin, author F. Mezzacapo, author T. Roscilde, author T. Lahaye, author N. Y. Yao, and author A. Browaeys, title title Scalable spin squeezing in a dipolar rydberg atom array, https://doi.org/10.1038/s41586-023-06414-9 journal journal Nature volume 621, pages 728–733 (year 2023)NoStop
[Ma et al.(2011)Ma, Wang, Sun, and Nori]2011_Ma_Nori_physrep
  author author J. Ma, author X. Wang, author C. Sun, and author F. Nori, title title Quantum spin squeezing, https://doi.org/https://doi.org/10.1016/j.physrep.2011.08.003 journal journal Physics Reports volume 509, pages 89 (year 2011)NoStop
[Krammer et al.(2009)Krammer, Kampermann, Bruß, Bertlmann, Kwek, and Macchiavello]Krammer_2009
  author author P. Krammer, author H. Kampermann, author D. Bruß, author R. A. Bertlmann, author L. C. Kwek, and author C. Macchiavello, title title Multipartite entanglement detection via structure factors, https://doi.org/10.1103/PhysRevLett.103.100502 journal journal Phys. Rev. Lett. volume 103, pages 100502 (year 2009)NoStop
[Lee et al.(2019)Lee, Najafabadi, Schumayer, Kwek, and Hutchinson]Lee_2019
  author author C. K. Lee, author M. S. Najafabadi, author D. Schumayer, author L. C. Kwek, and author D. A. W. Hutchinson, title title Environment mediated multipartite and multidimensional entanglement, https://doi.org/10.1038/s41598-019-45496-2 journal journal Scientific Reports volume 9,  (year 2019)NoStop
[Glauber(1963a)]Glauber1963
  author author R. J. Glauber, title title The quantum theory of optical coherence, https://doi.org/10.1103/PhysRev.130.2529 journal journal Phys. Rev. volume 130, pages 2529 (year 1963a)NoStop
[Glauber(1963b)]Glauber1963b
  author author R. J. Glauber, title title Coherent and incoherent states of the radiation field, https://doi.org/10.1103/PhysRev.131.2766 journal journal Phys. Rev. volume 131, pages 2766 (year 1963b)NoStop
[Loudon(1973)]Loudon:book
  author author R. Loudon, @noop  title The quantum theory of light (publisher Oxford Science Publications, year 1973)NoStop
[Werner(1989)]Werner_1989
  author author R. F. Werner, title title Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model, https://doi.org/10.1103/PhysRevA.40.4277 journal journal Phys. Rev. A volume 40, pages 4277 (year 1989)NoStop
[SM()]SM
  @noop  note See Supplemental Material at [URL will be inserted by publisher] for details on the derivation of the entanglement witness and application to single-excitation collective states.Stop
[Braginsky and Khalili(1996)]Braginsky1996
  author author V. B. Braginsky and author F. Y. Khalili, title title Quantum nondemolition measurements: the route from toys to tools, https://doi.org/10.1103/RevModPhys.68.1 journal journal Rev. Mod. Phys. volume 68, pages 1 (year 1996)NoStop
[Grangier et al.(1998)Grangier, Levenson, and Poizat]Grangier1998
  author author P. Grangier, author J. A. Levenson, and author J.-P. Poizat, title title Quantum non-demolition measurements in optics, https://doi.org/10.1038/25059 journal journal Nature volume 396, pages 537 (year 1998)NoStop
[Lehmberg(1970a)]Lehmberg_1970
  author author R. H. Lehmberg, title title Radiation from an n-atom system. i. general formalism, https://doi.org/10.1103/PhysRevA.2.883 journal journal Phys. Rev. A volume 2, pages 883 (year 1970a)NoStop
[Lehmberg(1970b)]Lehmberg_1970_2
  author author R. H. Lehmberg, title title Radiation from an n-atom system. ii. spontaneous emission from a pair of atoms, https://doi.org/10.1103/PhysRevA.2.889 journal journal Phys. Rev. A volume 2, pages 889 (year 1970b)NoStop
[Dicke(1954)]Dicke1954
  author author R. H. Dicke, title title Coherence in spontaneous radiation processes, https://doi.org/10.1103/physrev.93.99 journal journal Physical Review volume 93, pages 99 (year 1954)NoStop
[Arecchi et al.(1972)Arecchi, Courtens, Gilmore, and Thomas]Arecchi1972
  author author F. T. Arecchi, author E. Courtens, author R. Gilmore, and author H. Thomas, title title Atomic coherent states in quantum optics, https://doi.org/10.1103/PhysRevA.6.2211 journal journal Phys. Rev. A volume 6, pages 2211 (year 1972)NoStop
[MacGillivray and Feld(1976)]MacGillivray1976
  author author J. C. MacGillivray and author M. S. Feld, title title Theory of superradiance in an extended, optically thick medium, https://doi.org/10.1103/PhysRevA.14.1169 journal journal Phys. Rev. A volume 14, pages 1169 (year 1976)NoStop
[Gross and Haroche(1982)]Gross1982
  author author M. Gross and author S. Haroche, title title Superradiance: An essay on the theory of collective spontaneous emission, https://doi.org/10.1016/0370-1573(82)90102-8 journal journal Physics Reports volume 93, pages 301 (year 1982)NoStop
[Wolfe and Yelin(2014)]Wolfe2014
  author author E. Wolfe and author S. F. Yelin, title title Certifying separability in symmetric mixed states of n qubits, and superradiance, https://doi.org/10.1103/PhysRevLett.112.140402 journal journal Phys. Rev. Lett. volume 112, pages 140402 (year 2014)NoStop
[Guerin et al.(2016)Guerin, Araújo, and Kaiser]Guerin2016
  author author W. Guerin, author M. O. Araújo, and author R. Kaiser, title title Subradiance in a large cloud of cold atoms, https://doi.org/10.1103/PhysRevLett.116.083601 journal journal Phys. Rev. Lett. volume 116, pages 083601 (year 2016)NoStop
[Santos et al.(2022)Santos, Cidrim, Villas-Boas, Kaiser, and Bachelard]Santos2022
  author author A. C. Santos, author A. Cidrim, author C. J. Villas-Boas, author R. Kaiser, and author R. Bachelard, title title Generating long-lived entangled states with free-space collective spontaneous emission, https://doi.org/10.1103/PhysRevA.105.053715 journal journal Phys. Rev. A volume 105, pages 053715 (year 2022)NoStop
[Cipris et al.(2021)Cipris, Moreira, do Espirito Santo, Weiss, Villas-Boas, Kaiser, Guerin, and Bachelard]Cipris2021
  author author A. Cipris, author N. A. Moreira, author T. S. do Espirito Santo, author P. Weiss, author C. J. Villas-Boas, author R. Kaiser, author W. Guerin, and author R. Bachelard, title title Subradiance with saturated atoms: Population enhancement of the long-lived states, https://doi.org/10.1103/PhysRevLett.126.103604 journal journal Phys. Rev. Lett. volume 126, pages 103604 (year 2021)NoStop
[Osterloh et al.(2002)Osterloh, Amico, Falci, and Fazio]Osterloh_2002
  author author A. Osterloh, author L. Amico, author G. Falci, and author R. Fazio, title title Scaling of entanglement close to a quantum phase transition, https://doi.org/10.1038/416608a journal journal Nature volume 416, pages 608–610 (year 2002)NoStop
[Osborne and Nielsen(2002)]Osborne_2002
  author author T. J. Osborne and author M. A. Nielsen, title title Entanglement in a simple quantum phase transition, https://doi.org/10.1103/PhysRevA.66.032110 journal journal Phys. Rev. A volume 66, pages 032110 (year 2002)NoStop
[Hettich et al.(2002)Hettich, Schmitt, Zitzmann, Kuhn, Gerhardt, and Sandoghdar]Hettich2002
  author author C. Hettich, author C. Schmitt, author J. Zitzmann, author S. Kuhn, author I. Gerhardt, and author V. Sandoghdar, title title Nanometer resolution and coherent optical dipole coupling of two individual molecules, https://doi.org/10.1126/science.1075606 journal journal Science volume 298, pages 385–389 (year 2002)NoStop
[Wang and Schirmer(2010)]Wang2010
  author author X. Wang and author S. G. Schirmer, @noop  title Generating maximal entanglement between non-interacting atoms by collective decay and symmetry breaking (year 2010), https://arxiv.org/abs/1005.2114 arXiv:1005.2114 [quant-ph] NoStop
[Trebbia et al.(2022)Trebbia, Deplano, Tamarat, and Lounis]Trebbia2022
  author author J.-B. Trebbia, author Q. Deplano, author P. Tamarat, and author B. Lounis, title title Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters, https://doi.org/10.1038/s41467-022-30672-2 journal journal Nature Communications volume 13,  (year 2022)NoStop
[Oliveira et al.(2023)Oliveira, Higgins, Zhang, Predojevi ćć, Hennrich, Bachelard, and Villas-Boas]Oliveira2023
  author author M. H. Oliveira, author G. Higgins, author C. Zhang, author A. Predojevi ćć, author M. Hennrich, author R. Bachelard, and author C. J. Villas-Boas, title title Steady-state entanglement generation for nondegenerate qubits, https://doi.org/10.1103/PhysRevA.107.023706 journal journal Phys. Rev. A volume 107, pages 023706 (year 2023)NoStop
[Sherson et al.(2010)Sherson, Weitenberg, Endres, Cheneau, Bloch, and Kuhr]Sherson2010
  author author J. F. Sherson, author C. Weitenberg, author M. Endres, author M. Cheneau, author I. Bloch, and author S. Kuhr, title title Single-atom-resolved fluorescence imaging of an atomic mott insulator, https://doi.org/10.1038/nature09378 journal journal Nature volume 467, pages 68–72 (year 2010)NoStop
[Christakis et al.(2023)Christakis, Rosenberg, Raj, Chi, Morningstar, Huse, Yan, and Bakr]Christakis2023
  author author L. Christakis, author J. S. Rosenberg, author R. Raj, author S. Chi, author A. Morningstar, author D. A. Huse, author Z. Z. Yan, and author W. S. Bakr, title title Probing site-resolved correlations in a spin system of ultracold molecules, https://doi.org/10.1038/s41586-022-05558-4 journal journal Nature volume 614, pages 64–69 (year 2023)NoStop
[Krämer and Ritsch(2015)]Helmut_2015
  author author S. Krämer and author H. Ritsch, title title Generalized mean-field approach to simulate the dynamics of large open spin ensembles with long range interactions, https://doi.org/10.1140/epjd/e2015-60266-5 journal journal The European Physical Journal D volume 69,  (year 2015)NoStop
[Pezzagna and Meijer(2021)]2021_Pezzagna_Meijer_apr
  author author S. Pezzagna and author J. Meijer, title title Quantum computer based on color centers in diamond, https://doi.org/10.1063/5.0007444 journal journal Applied Physics Reviews volume 8, pages 011308 (year 2021)NoStop
[Vitagliano et al.(2014)Vitagliano, Apellaniz, Egusquiza, and Tóth]2014_Vitagliano_Toth_pra
  author author G. Vitagliano, author I. Apellaniz, author I. n. L. Egusquiza, and author G. Tóth, title title Spin squeezing and entanglement for an arbitrary spin, https://doi.org/10.1103/PhysRevA.89.032307 journal journal Phys. Rev. A volume 89, pages 032307 (year 2014)NoStop
[Kraft et al.(2018)Kraft, Ritz, Brunner, Huber, and Gühne]2018_Kraft_Guhne_prl
  author author T. Kraft, author C. Ritz, author N. Brunner, author M. Huber, and author O. Gühne, title title Characterizing genuine multilevel entanglement, https://doi.org/10.1103/PhysRevLett.120.060502 journal journal Phys. Rev. Lett. volume 120, pages 060502 (year 2018)NoStop
[Lewenstein et al.(2000)Lewenstein, Kraus, Cirac, and Horodecki]Lewenstein_2000
  author author M. Lewenstein, author B. Kraus, author J. I. Cirac, and author P. Horodecki, title title Optimization of entanglement witnesses, https://doi.org/10.1103/PhysRevA.62.052310 journal journal Phys. Rev. A volume 62, pages 052310 (year 2000)NoStop












	
	
	
	
 






 Supplemental Material for: 
 Detecting entanglement from macroscopic measurements of the electric field and its fluctuations


0.5Pedro Rosario 0000-0002-7628-7373,^1 Alan C. Santos 0000-0002-6989-7958,^1,2 Nicola Piovella,^3 Robin Kaiser,^4 André Cidrim 0000-0003-0007-2330,^1 and Romain Bachelard 0000-0002-6026-509X^1


0.5 ^1Departamento de Física, Universidade Federal de São Carlos,
 Rodovia Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brazil

^2Instituto de Física Fundamental (IFF), Consejo Superior de Investigaciones Científicas (CSIC), 
 Calle Serrano 113b, 28006 Madrid, Spain

^3Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy & INFN Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

^4Université Côte d'Azur, CNRS, Institut de Physique de Nice, 06560 Valbonne, France









§ DERIVATION  OF THE ENTANGLEMENT WITNESSES





 §.§ Definitions and phase-dependent basis


Let us introduce the following basis of Pauli operators, which incorporate optical-path terms:

    σ̂^𝐤,x_j=e^-i𝐤.𝐫_jσ̂^-_j+e^i𝐤.𝐫_jσ̂^+_j,   σ̂^𝐤,y_j=i(e^i𝐤.𝐫_jσ̂^+_j-e^-i𝐤.𝐫_jσ̂^-_j),   σ̂^z_j=|↑⟩⟨↑|_j-|↓⟩⟨↓|_j,

where σ̂^-_j=|↓⟩⟨↑|_j and σ̂^+_j=|↑⟩⟨↓|_j. Then, a one-qubit state can be represented by the density matrix

    ρ̂_j=1/2(+𝐯^𝐤_j·σ^𝐤_j),

where 𝐯^𝐤_j=(v^𝐤,x_j,v^𝐤,y_j,v^z_j) is the coherence vector for atom j and σ^𝐤_j=(σ̂^𝐤,x_j,σ̂^𝐤,y_j,σ̂^z_j). Any density matrix ρ̂  satisfies Tr(ρ̂^2)≤ 1, which translates into

    (v^𝐤,x_j)^2+ (v^𝐤,y_j)^2+(v^z_j)^2≤ 1.

The Bloch vector elements are defined as

    v^𝐤,η_j=Tr[σ̂^𝐤,η_jρ̂_j],   with  η ∈{x,y} and  v^z_j=Tr[σ̂^z_jρ̂_j].

Using the phase-dependent basis σ^𝐤_j, an arbitrary one-qubit state can be written as

    ρ̂_j =[    a b+ic; b-ic    d ],
       with  |a|^2+|b|^2=1.


Inserting this expression into Eq. (<ref>) leads to

    v^𝐤,x_j=2[bcos(𝐤.𝐫_j)-csin(𝐤.𝐫_j)],   v^𝐤,y_j=-2[bsin(𝐤.𝐫_j)+cos(𝐤.𝐫_j)c],    v^z_j=a-d,

so that 𝐯^𝐤_j∈ℝ^3. 

We now consider the electric field radiated by the two-level atoms in the far field, Ê_𝐤^±=∑_j=1^Ne^∓ i𝐤.𝐫_jσ̂^∓_j, and express its quadrature, X̂_𝐤 and Ŷ_𝐤, and population imbalance, Ẑ, in this basis:

    X̂_𝐤=Ê^+_𝐤+Ê^-_𝐤=∑_j^N(e^ i𝐤.𝐫_jσ̂^+_j+e^- i𝐤.𝐫_jσ̂^-_j)=∑_j=1^Nσ̂^𝐤,x_j,
       Ŷ_𝐤=i(Ê^+_𝐤-Ê^-_𝐤)=∑_j^Ni(e^ i𝐤.𝐫_jσ̂^+_j-e^- i𝐤.𝐫_jσ̂^-_j)=∑_j=1^Nσ̂^𝐤,y_j,
       Ẑ=∑_j=1^Nσ̂^z_j.






 §.§ First witness



Let us first evaluate the quantity  ⟨X̂^2_𝐤⟩ + ⟨Ŷ^2_𝐤⟩ + ⟨Ẑ^2⟩ for separable states, in order to determine an upper or lower bound in terms of the number of particles. A separable state with N particles reads

    ρ̂=∑_l=1^Lp_lρ̂^(l)_1⊗ρ̂^(l)_2⊗⊗ρ̂^(l)_N   with ∑_l=1^Lp_l=1.

We then take α̂∈{X̂,Ŷ}, which leads to

    α̂^2_𝐤=∑_j=1^N∑_s=1^Nσ̂^𝐤,α_jσ̂^𝐤,α_s=∑_j=1^N +∑_j=1^N∑_s≠ j^Nσ̂^𝐤,α_jσ̂^𝐤,α_s=N +∑_j=1^N∑_s≠ j^Nσ̂^𝐤,α_jσ̂^𝐤,α_s.

Its expectation value then reads

    ⟨α̂^2_𝐤⟩   = N+Tr[(∑_l=1^Lp_lρ̂^(l)_1⊗ρ̂^(l)_2⊗⊗ρ̂^(l)_N)(∑_j=1^N∑_s≠ j^Nσ̂^𝐤,α_jσ̂^𝐤,α_s)]
       =N +∑_j=1^N∑_s≠ j^N∑_l=1^Lp_lTr[ρ̂^(l)_jσ̂^𝐤,α_j]Tr[ρ̂^(l)_sσ̂^𝐤,α_s]
       =N+∑_j=1^N∑_s≠ j^N∑_l=1^Lp_lv^𝐤,α,l_jv^𝐤,α,l_s.

This allows us to write

    ⟨X̂^2_𝐤⟩ + ⟨Ŷ^2_𝐤⟩ + ⟨Ẑ^2⟩ =3N+∑_j=1^N∑_s≠ j^N∑_l=1^Lp_l(v^𝐤,x,l_jv^𝐤,x,l_s+v^𝐤,y,l_jv^𝐤,y,l_s+v^z,l_jv^z,l_s)=3N+∑_j=1^N∑_s≠ j^N∑_l=1^Lp_l𝐯_j^𝐤,l·𝐯_s^𝐤,l.

Here 𝐯_j^𝐤,l·𝐯_s^𝐤,l stands for the dot product between two Bloch vectors. Therefore using Eq. (<ref>) leads to 𝐯_j^𝐤,l·𝐯_s^𝐤,l≤ 1 and eventually to the first inequality:

    ⟨X̂^2_𝐤⟩ + ⟨Ŷ^2_𝐤⟩ + ⟨Ẑ^2⟩≤  N(2+N) .

This results in the definition of the first witness (<ref>).





 §.§ Second witness




Let us now evaluate the sum of the variances, (ΔX̂_𝐤)^2+(ΔŶ_𝐤)^2+(ΔẐ)^2. 
The concavity of the variances makes that, for separable states:

    (Δα̂_𝐤)^2≥∑_l=1^L∑_j=1^Np_l[Δσ̂^𝐤,α_j]^2_l,   with  α̂∈{X̂,Ŷ},

where [Δσ̂^𝐤,α_j]^2_l is the (local) variance associated with particle j. This leads to

    (ΔX̂_𝐤)^2+(ΔŶ_𝐤)^2+(ΔẐ)^2   ≥∑_l=1^L∑_j=1^Np_l([Δ (σ̂^𝐤,x_j)]^2_l+[Δ (σ̂^𝐤,y_j)]^2_l+[Δ (σ̂^z_j)]^2_l)
       =∑_l=1^L∑_j=1^Np_l(3-[⟨σ̂^𝐤,x_j⟩^2]_l -[⟨σ̂^𝐤,y_j⟩^2]_l -[⟨σ̂^z_j⟩^2]_l)
       =∑_l=1^L∑_j=1^Np_l(3-[ (v^𝐤,x_j)^2]_l -[ (v^𝐤,y_j)^2]_l -[(v^z_j)^2]_l)
       =∑_l=1^L∑_j=1^Np_l(3-[ (v^𝐤,x_j)^2+(v^𝐤,y_j)^2+(v^z_j)^2]_l)
       =2∑_l=1^L∑_j=1^Np_l=2N,

where we have used that (v^𝐤,x_j)^2+(v^𝐤,y_j)^2+(v^z_j)^2=1 for pure states to set the lower bound. Hence, we obtain the inequality from which the second witness, Eq. (<ref>), is derived:

    (ΔX̂_𝐤)^2+(ΔŶ_𝐤)^2+(ΔẐ)^2≥ 2N.






 §.§ Third witness



The third inequality is obtained by analyzing (N-1)(ΔÂ)^2-⟨B̂^2⟩ -⟨Ĉ^2⟩, where (Â,B̂,Ĉ) is any cyclic permutation of (X̂_𝐤,Ŷ_𝐤,Ẑ). Using the above relations for α̂,β̂∈{X̂,Ŷ} we get

    (Δα̂_𝐤)^2=N +∑_j=1^N∑_s≠ j^N⟨σ̂^𝐤,α_jσ̂^𝐤,α_s⟩-∑_j=1^N∑_s=1^N⟨σ̂^𝐤,α_j⟩⟨σ̂^𝐤,α_s⟩, 
       ⟨β̂^2_𝐤⟩=N +∑_j=1^N∑_s≠ j^N⟨σ̂^𝐤,β_jσ̂^𝐤,β_s⟩.

Taking into account the concavity relation in Eq.(<ref>) and the inequality 

    -(∑_j=1^Nv^𝐤,α_j)^2≥ -N∑_j=1^N(v^𝐤,α_j)^2,

we obtain

    (N-1)(ΔÂ)^2-⟨B̂^2⟩ -⟨Ĉ^2⟩≥ -2N,

from which entanglement witness Eq. (<ref>) is obtained.





 §.§ Fourth witness



The last set of witnesses rely on the quantity (N-1)[(ΔÂ)^2+(ΔB̂)^2]-⟨Ĉ^2⟩, where (Â,B̂,Ĉ) are cyclic permutations of (X̂_𝐤,Ŷ_𝐤,Ẑ). Using the concavity argument in Eq.(<ref>), combined with Eq.(<ref>), we get

    (N-1)[(ΔÂ)^2+(ΔB̂)^2]-⟨Ĉ^2⟩≥ N (N-2).





§ SINGLE EXCITATION COLLECTIVE STATES





 §.§ Dicke state



We here derive the properties for single excitation Dicke state ρ̂, written under the form 

    ρ̂=|𝒟(N)⟩⟨𝒟(N)|,   with 𝒟(N)=∑_ne^iϕ_n/√(N)|↑_n⟩.

The expectation value ⟨Ŷ^2_𝐤⟩ takes the form

    ⟨Ŷ^2_𝐤⟩   =∑_j=1^N∑_s=1^N⟨σ̂^𝐤,y_jσ̂^𝐤,y_s⟩
       =N+∑_j=1^N∑_s≠ j^N⟨σ̂^𝐤,y_jσ̂^𝐤,y_s⟩
       =N+Tr[|𝒟(N)⟩⟨𝒟(N)|(∑_j=1^N∑_s≠ j^Nσ̂^𝐤,y_jσ̂^𝐤,y_s)]
       =N+∑_j=1^N∑_s≠ j^N⟨𝒟(N)|σ̂^𝐤,y_jσ̂^𝐤,y_s|𝒟(N)⟩.

Using the properties σ̂^𝐤,y_s|↓⟩=ie^i𝐤·𝐫_s|↑⟩, σ̂^𝐤,y_s|↑⟩=-ie^-i𝐤·𝐫_s|↓⟩, we obtain

    ⟨𝒟(N)|σ̂^𝐤,y_jσ̂^𝐤,y_s|𝒟(N)⟩   =1/N∑_m∑_ne^i(ϕ_n-ϕ_m)(δ_snδ_jnδ_mne^i𝐤· (𝐫_j-𝐫_s)+δ_sn(1-δ_jn)δ_mje^i𝐤· (𝐫_j-𝐫_s)),
       +1/N∑_m∑_ne^i(ϕ_n-ϕ_m)((1-δ_sn)δ_jnδ_mse^-i𝐤· (𝐫_j-𝐫_s)+(1-δ_sn)δ_jsδ_mne^-i𝐤· (𝐫_j-𝐫_s)),

which then leads to

    ∑_j=1^N∑_s≠ j^N⟨𝒟(N)|σ̂^𝐤,y_jσ̂^𝐤,y_s|𝒟(N)⟩   =1/N∑_j=1^N∑_s≠ j^N∑_m∑_ne^i(ϕ_n-ϕ_m)(δ_snδ_mje^i𝐤· (𝐫_j-𝐫_s)+δ_jnδ_mse^-i𝐤· (𝐫_j-𝐫_s))
       =1/N∑_j=1^N∑_s≠ j^N(e^i(ϕ_s-ϕ_j)e^i𝐤· (𝐫_j-𝐫_s)+e^i(ϕ_j-ϕ_s)e^-i𝐤· (𝐫_j-𝐫_s))
       =2/N∑_j=1^N∑_s≠ j^Ncos[(ϕ_s-ϕ_j)+𝐤·(𝐫_j-𝐫_s)].

It is straightforward to verify that ⟨Ŷ^2_𝐤⟩=⟨X̂^2_𝐤⟩, so  we finally obtain

    ⟨X̂^2_𝐤⟩=N+2/N∑_j=1^N∑_s≠ j^Ncos[(ϕ_s-ϕ_j)+𝐤·(𝐫_j-𝐫_s)],
       ⟨Ŷ^2_𝐤⟩=N+2/N∑_j=1^N∑_s≠ j^Ncos[(ϕ_s-ϕ_j)+𝐤·(𝐫_j-𝐫_s)],
       ⟨Ẑ^2⟩ = N+(N-4)(N-1),
       ⟨Ẑ⟩ = (N-2),
       ⟨Ŷ_𝐤⟩ =0,
       ⟨X̂_𝐤⟩ = 0.

Note that if one omits the phases related to the field (or sets them to zero), the second and third inequalities derived previously take the form

    0  ≤4/N∑_j=1^N∑_s≠ j^Ncos(ϕ_j-ϕ_s),
      
        0  ≤ -4/N∑_j=1^N∑_s≠ j^Ncos(ϕ_j-ϕ_s).

Hence, unless the single-excitation Dicke state has phases which satisfy ∑_j=1^N∑_s≠ j^Ncos(ϕ_j-ϕ_s)=0, either (<ref>) or (<ref>) is violated, and the entanglement is detected.

If the optical path terms are considered, one obtains the full inequalities

    0≤4/N∑_j=1^N∑_s≠ j^Ncos[(ϕ_s-ϕ_j)+𝐤·(𝐫_j-𝐫_s)],
       0≤ -4/N∑_j=1^N∑_s≠ j^Ncos[(ϕ_s-ϕ_j)+𝐤·(𝐫_j-𝐫_s)].

In particular, changing the direction of observation alters the phase terms in the cosine, so while the inequality may not be violated for a specific angle, it should then be violated for a different one.


If we now look for a specific set of phases of the form {ϕ_n=nν}_n=1^N which satisfy the relation

    ∑_j=1^N∑_s≠ j^Ncos(ϕ_j-ϕ_s)=cos(Nν)-1/cos(ν)-1-N=0,

we obtain

    cos(Nν)-1/cos(ν)-1-N=Nδ -T_N(δ)+(1-N)=0,

where
T_N(δ) are the Chebyshev polynomials of the first kind, and where δ should be different from -1 and 1. Then the set of phases takes the form 

    {ϕ_n}_n=1^N={narccosδ}_n=1^N,

with δ a solution of (<ref>). For the specific case of N=2 qubits, the solution is δ=0, which leads to the phases

    {ϕ_n}_n=1^2={0,π/2},

up to a global phase.

