Distinguishability-induced many-body decoherence
Christoph Dittel Andreas Buchleitner





christoph.dittel@physik.uni-freiburg.de
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstraße 19, 79104 Freiburg, Germany
Department of Physics, Lund University, Box 118, 221 00 Lund, Sweden


Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany








§ ABSTRACT 
We show that many-body interference (MBI) phenomena are exponentially suppressed in the particle number, if only the identical quantum objects brought to interference acquire a finite level of distinguishability through statistical mixing of some internal, unobserved degrees of freedom. We discuss consequences for cold atom and photonic circuitry experiments.


Distinguishability-induced many-body decoherence
    Andreas Buchleitner
    September 9, 2024
================================================







Wave-particle duality, the modulation of the statistics of particle-like detection events by wave-like interference patterns, is the essential feature which distinguishes the quantum from the classical realm. It hinges on a sufficient degree of purity of the interfering quantum object's state <cit.>, as the precondition to witness interference in a suitably chosen measurement set-up which probes the quantum  state's coherences in the associated basis. On the many-body level, coherent superpositions of many-body states give rise to multi-partite entanglement <cit.>, and to many-body interference (MBI) phenomena <cit.> when the involved elementary constituents are identical and at least partially indistinguishable. Entanglement (i.e., non-separability) between those degrees of freedom (dof) which are interrogated by the experimental measurement set-up and other, unobserved (“environmental", “bath",  “internal" or “ancilla") dof – which are traced over when sampling the measurement record – reduces the purity of the quantum object's state in its observed dof, and thus its ability to exhibit interference phenomena, by reducing the strength of the associated coherences <cit.>. It is intuitively plausible that the larger the number of involved dof and of constituents, it becomes ever more difficult to warrant separability of observed and unobserved dof (by preventing the former from any type of interactions with the latter). This is the fundamental impediment to push the quantum-classical demarcation line to meso- if not macroscopic scales. 

Notwithstanding, stunning progress has been achieved in preparing coherent superposition states of collective degrees of freedom of many-body compounds of ever increasing size, from bucky balls <cit.> to supercurrents <cit.>, micromechanical oscillators <cit.>, and Bose Einstein Condensates <cit.>. Furthermore, beyond such experiments, which probe effective single-body coherences, experimental progress in the manipulation of controlled many-body quantum systems on the level of single constituents <cit.> now allows to assess bona fide MBI phenomena. While on the single-body level the quantum object’s effective size or mass defines the scale on which interference phenomena are to be observed <cit.>, it is suggestive that the relevant scale is defined by the number N of interfering constituents on the MBI level. Our present purpose is to make this quantitative. We derive a scaling law which shows that distinguishability due to a finite degree of mixedness in the quantum objects' internal, ancilla degrees of freedom – easily brought about by some residual environment coupling – induces the exponential suppression of many-body coherences and hence of MBI phenomena with increasing N. We examine and discuss the consequences of our scaling law for MBI of cold atoms or of photons in optical lattices or in photonic circuits, respectively.










Consider a quantum many-body system composed of N identical bosons or fermions localized in mutually orthogonal external states (e.g., think of atoms in a Mott state <cit.>, or of photons in distinct optical modes <cit.>). To describe the particles' distribution across their individual external states, we make use of the first quantization formalism and denote the N-particle basis states by |E⃗⟩=|E_1⟩⊗⋯⊗|E_N⟩, where |E_α⟩ is the external state occupied by the αth particle, with E_αE_α'=δ_E_α,E_α' and E⃗E⃗'=∏_α=1^N δ_E_α,E'_α <cit.>. Further suppose that the particles are equipped with internal degrees of freedom (e.g., the arrival time and polarization state of photons, or the atoms' electronic energy levels), prepared in potentially mixed internal many-body states ρ, which are neither acted upon nor measured.


Given N identical bosons (fermions), we must (anti-) symmetrize |E⃗⟩⟨E⃗|⊗ρ with respect to all particle permutations of the symmetric group S_N of N elements. Since MBI is to be observed by interrogation of the external degrees of freedom alone, we subsequently trace over the internal degrees of freedom. This yields the external many-body state ρ_E= ∑_π,π' ∈N [ρ_E ]_π,π'|E⃗_π⟩⟨E⃗_π'|,

    [ρ_E ]_π,π'=(-1)_B(F)^ππ'1/N!Π_πρΠ^†_π'  ,

which has N!× N! matrix elements <cit.>. The many-particle basis states |E⃗_π⟩=|E_π(1)⟩⊗…⊗|E_π(N)⟩ of ρ_E result from |E⃗⟩ by permuting the particles according to π^-1∈N, much as the operator Π_π in (<ref>) performs a particle permutation π^-1 in the internal degrees of freedom. (-1)^ππ'_B=1 for bosons, with ππ' a composition of π and π', and (-1)^ππ'_F=(ππ') for fermions. 



Many-body coherences [ρ_E ]_π,π', π≠π' in (<ref>), thus result from the (anti-) symmetrization and are associated with different orderings of the particles in the tensor product structure. By virtue of the trace in (<ref>), these coherences are governed by the particles' mutual indistinguishability <cit.> with respect to their internal degrees of freedom. In the limiting case of pure states of separable, perfectly indistinguishable bosons (B) or fermions (F), i.e., ρ=|ϕ⟩⟨ϕ| with |ϕ⟩=|φ⟩⊗…⊗|φ⟩, the trace in (<ref>) yields unity for all π,π'∈N, such that the corresponding reduced external state is fully coherent and described by a pure state, ρ_E^B(F)=|ψ_B(F)⟩⟨ψ_B(F)|, with |ψ_B(F)⟩=∑_π∈N(-1)_B(F)^π|E⃗_π⟩/√(N!) the usual Fock state of indistinguishable particles. On the other hand, pure states of separable, fully distinguishable (D) particles feature internal states with orthogonal support, i.e., ρ=|ϕ⟩⟨ϕ| with |ϕ⟩=|φ_1⟩⊗…⊗|φ_N⟩ and φ_jφ_k=δ_j,k, and thus give rise to a fully incoherent many-body state, ρ_E^D=∑_π∈N|E⃗_π⟩⟨E⃗_π|/N!. The normalized many-body coherence <cit.> of ρ_E is thus given by

    𝒲_C   = 1/N!-1∑_π,π' ∈N
    π≠π' [ρ_E ]_π,π'  ,  0≤𝒲_C≤ 1  ,

with pure states of separable, fully distinguishable (indistinguishable) particles saturating the lower (upper) bound. Since the modulus in Eq. (<ref>) erases the sign of the coherences (<ref>), the results presented hereafter apply for bosons as well as for fermions. 

Another source of distinguishability is due to mixedness of the particles' internal degrees of freedom, which typically arises through decoherence on the single-particle level <cit.>: Suppose that each particle is well described by the same single-particle state ρ_1p <cit.>, such that ρ=ρ_1p⊗⋯⊗ρ_1p in Eq. (<ref>). In Sec. II of [Supplemental Material] we show that in this case the normalized coherence 𝒲_C of many  bosons (fermions), N!≫ 1, can be identified with the expectation value of the projector Π_S(A) onto the N-particle (anti-) symmetric subspace, 𝒲_C≈Π_S(A)ρ_E 
[Note that for a bosonic state ρ_E, tight lower bounds of Π_Sρ_E can be efficiently measured <cit.>.], which, as we further show, is equivalent to the support of the unsymmetrized N-particle internal state ρ on the N-particle symmetric subspace, i.e., 𝒲_C≈Π_S(A)ρ_E=Π_Sρ. Although the internal states of all particles are described by the same density operator, the particles are indistinguishable if and only if ρ_1p is pure. To make this explicit, suppose that ρ_1p has a discrete spectrum of m eigenvalues λ_j≥ 0 with corresponding eigenvectors |j⟩, such that its eigendecomposition reads 
ρ_1p=∑_j=1^m λ_j |j⟩⟨j|. As we show in Sec. III of <cit.>, the normalized coherence (<ref>) can then be written, for N!≫ 1, as

    𝒲_C   ≈∑_J_1+J_2+⋯+J_m=Nλ_1^J_1λ_2^J_2⋯λ_m^J_m,

with the sum running over all non-negative integers J_1,J_2,…,J_m summing to N. Equation (<ref>) explicitly shows how the spectrum of ρ_1p controls many-body coherence, with 𝒲_C= 1 for pure ρ_1p, and 

    𝒲_C≈1/m^NN+m-1  m-1  ,

for ρ_1p maximally mixed. Since 𝒲_C≈Π_Sρ, the finite residual coherence quantified by (<ref>) is given by the relative dimension of the symmetric component S_N(ℋ_I) with respect to that of the total internal Hilbert space ℋ_I. In the limit of an infinite number m of single-particle internal states, S_N(ℋ_I) tends to zero, and so does the residual coherence in (<ref>). 



With 𝒲_C from Eq. (<ref>) at hand, we have a quantifier of many-body coherence, i.e., a quantifier of the very source of MBI, independently of the exact experimental protocol <cit.>. Any such experiment, however, can only unfold the complexity seeded by MBI for large system sizes. We therefore focus on the scaling behavior of 𝒲_C in the thermodynamic limit, N→∞ (at fixed particle density N/L, with L the dimension of the external single-particle Hilbert space, i.e., here, the number of external single-particle modes), and further examine the scaling behavior in the special case of faint particle distinguishability, i.e., for very weakly mixed internal states ρ_1p. 



Let λ_max=max_jλ_j be the maximum eigenvalue of ρ_1p, and suppose that λ_max is non-degenerate. Unless all particles are perfectly indistinguishable (i.e., λ_max=1), it then follows from Eq. (<ref>) (see Sec. IV of <cit.>) that, in the thermodynamic limit, 

    𝒲_C≈λ_max^N ∏_j=1
     j≠max^m( 1-λ_j/λ_max)^-1  ,

i.e., 𝒲_C from Eq. (<ref>) vanishes exponentially in the number N of constituents [d-fold degeneracy of λ_max instead leads to 𝒲_C∝ (N+1)^d-1λ_max^N (Sec. IV of <cit.>)]. Alternatively, we decompose the internal single-particle state into a dominating pure and a faint mixed component, ρ_1p=(1-ϵ)|ϕ⟩⟨ϕ|+ϵρ̃_1p, with ϵ≪ 1/2, and ρ̃_1p a valid density operator. Again (Sec. V in <cit.>), for N!≫ 1, 𝒲_C exhibits exponential scaling: 

    𝒲_C≈ (1-ϵ)^N  .



To asses the scaling behavior of the entire hierarchy from two- to N-body coherences, we consider the reduced k-particle state ρ_E^(k)=N-kρ_E <cit.> obtained from ρ_E by tracing out all but k particles. As compared to ρ_E, the reduced k-particle state ρ_E^(k) carries no information about collective many-body properties of subsets of more than k particles <cit.>. The normalized coherence 𝒲_C^(k) of ρ_E^(k) (see Sec. VI of <cit.>) thus quantifies k-particle coherence of order k<N. Since we consider the particles to be localized in distinct external states, with equal, independent internal states ρ_1p, it is intuitively clear that, for k!≫ 1, 𝒲_C^(k) takes a similar form as Eq. (<ref>) (see Sec. VI of <cit.>): 𝒲_C^(k)≈∑_J_1+J_2+⋯+J_m=kλ_1^J_1λ_2^J_2⋯λ_m^J_m. Consequently, Eqs. (<ref>) and (<ref>) also apply to 𝒲_C^(k), but with an exponential scaling in k instead of N, such that k(<N)-body coherences fade away slower than those of order N. Note that 𝒲_C^(k) is, by the very purpose of its construction, undefined for k=1, and that the constituent particles interference with themselves is controlled by the coherence of the reduced single-particle state, as in standard single-body interference.






We now discuss the consequences of the above for specific experimental scenarios. First consider cold atoms in distinct optical lattice sites (external states) <cit.> which we model as harmonic potentials each with m=4 equidistant energy levels (internal states) with energy differences E_j+1-E_j=Δ E [see Fig <ref>(a)]. The atoms' population distribution over the oscillator levels be given by a Boltzmann distribution at equilibrium temperature T [see Fig. <ref>(a)]. Their single-particle internal states ρ_1p are then given as ρ_1p= ∑_j=1^m e^-β E_jZ(β)^-1|j⟩⟨j|, with β=1/ T the inverse temperature,  the Boltzmann constant, Z(β)=∑_j=1^m e^-β E_j the partition function, and {|j⟩}_j=1^m a set of m orthonormal oscillator energy states, with associated eigenvalues λ_j=e^-β E_j Z(β)^-1. For particle numbers up to N=100, Figs. <ref>(b,d) show a monotonous decrease of 𝒲_C with increasing T, from a fully coherent [k_BT/Δ E⪅ 0.1, see Fig. <ref>(c)] to an almost incoherent (k_BT/Δ E⪆ 1) external many-body state with finite residual coherence as described by Eq. (<ref>). The complementary, asymptotically exponential decrease of 𝒲_C as a function of the particle number N, at fixed temperature, is 
shown in Fig. <ref>(e) – in perfect agreement with (<ref>) [dashed lines in Fig. <ref>(e)].


Close to the critical temperature, 0.1⪅ k_BT/Δ⪅ 1, at which external decoherence sets in in Fig. <ref>(d), we can apply Eq. (<ref>): With e^-βΔ E≪ 1/2, i.e., T /Δ E≪ 1/ln(2), the internal ground state (i.e., lowest energy level of the local oscillator potential) dominates, leading to (see Sec. VII of <cit.>) 𝒲_C≈ (1-e^-βΔ E)^N [dashed lines in Fig. <ref>(d)], such that 

    T/Δ E≈-1/ln(1-𝒲_C^1/N)  .

If e^-βΔ E≪ 1/N, i.e., T/Δ E ≪ 1/ln(N), further approximation yields 𝒲_C≈ 1-N e^-βΔ E and T/Δ E ≈ 1/ln[N/(1- 𝒲_C)]. For our example illustrated in Fig. <ref>(a), Eq. (<ref>) is plotted  in Fig. <ref>(c), for fixed 
coherences 𝒲_C, as a function of N. We see that T/Δ E only gradually decreases for increasing N. This explains why the onset of decoherence, manifest in the drop of 𝒲_C in Figs. <ref>(b,d), only marginally shifts towards smaller critical temperatures with increasing N. In particular, this observation suggests that MBI remains observable, with high visibilities, in experiments with large particle numbers, at temperatures T not much below Δ E. 






As a second example, let us assess the case of N photons propagating along distinct, possibly coupled optical modes (external degrees of freedom) <cit.>. Different injection times (internal degrees of freedom) – a typical error source in photonic experiments – render the photons mutually partially distinguishable <cit.>. As above, we describe the internal state of every photon by the same mixed single-particle internal state ρ_1p= ∫_-∞^∞dt  P(t) |t⟩⟨t|, with arrival time probability distribution P(t), and |t⟩=(2 πΔ^2)^-1/4∫_-∞^∞dω e^ω t e^-(ω-Ω)^2/4Δ^2|ω⟩ a single photon's (internal) state with arrival time t, Gaussian frequency spectrum of spectral width Δ around the central frequency Ω, and ωω'=δ(ω-ω').

Figure <ref> provides an example of normally distributed arrival times, P(t)=exp(-(t-t_0)^2/(2σ^2))/√(2πσ^2), with mean ⟨t|=⟩t_0 and standard deviation σ=(⟨t^2|-⟩⟨t|^⟩2)^1/2, for which we calculate the behavior of 𝒲_C as a function of σΔ and N via Eqs. (<ref>) and (<ref>) in the following.

The finite width of the photonic arrival time distribution in terms of the temporal width of a photonic wave packet, σΔ, now reduces, similar to the finite temperature distribution over atomic energy bands above, the normalized coherence 𝒲_C of the N-photon state, plotted in Fig. <ref>(b). We observe a qualitatively similar transition  from a fully coherent to a fully incoherent many-body state, as a function of σΔ. However, since the here considered internal degrees of freedom – the arrival times – are continuous, 𝒲_C truly vanishes in the limit of large σΔ, since m→∞ in Eq. (<ref>). While a direct application of Eq. (<ref>) requires the spectral decomposition of ρ_1p, we here restrict to a qualitative observation, with Fig. <ref>(d) indicating a convergence into an exponential decay of 𝒲_C in the limit of large N. For N!≫ 1 and small distinguishabilities σΔ≪ 1/√(2), we show in Sec. VIII of <cit.> that Eq. (<ref>) can be reformulated as 𝒲_C≈ (1-σ^2Δ^2)^N (dashed lines in Fig. <ref>(b); compare to the average mutual fidelity in <cit.>). Accordingly, we have

    σΔ≈√(1-𝒲_C^1/N).

For very faint distinguishabilities σΔ≪ 1/√(N), this simplifies to σΔ≈√((1-𝒲_C)/N). In Fig. <ref>(c) we plot Eq. (<ref>) for our example of normally distributed arrival times. In particular, for σΔ≪ 1/√(N), it confirms the power law σΔ∝ N^-1/2 for a targeted level of coherence 𝒲_C, with σΔ sharply decreasing with increasing particle number N. Thus, the photons' arrival times must be increasingly well controlled (with respect to their inverse spectral width) in order to harvest the interference of an increasing number of particles. 







We have thus quantified the challenge to witness mutual interference of a large number of identical particles, given that, the larger this number, the more difficult to prevent the system constituents from interactions with environmental degrees of freedom. Our present analysis relies on the “static" description of a given many-body state with finite mixedness of its individual constituents, and thereby sets limits to the acceptable level of noise if a certain level of many-body coherence is to be guaranteed. What is not resolved in this analysis is the time dependence of the decoherence process, and a dynamical description of many-body decoherence appears an attractive topic for future theoretical research – much as the development of distillation <cit.> or error correction protocols to stabilize many-body coherences in actual experiments.











We thank Jonathan Brugger, Eric Brunner, Christian Haen, and Philipp Preiss for fruitful discussions. We are also thankful to Gabriel Dufour for fruitful discussions and comments at an early stage of the manuscript. C.D. acknowledges the Georg H. Endress Foundation for support and the Freiburg Institute for Advanced Studies for a FRIAS Junior Fellowship.



















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[Note1()]Note1
  note Supplemental MaterialNoStop
[Note2()]Note2
  note Note that for a bosonic state ρ _E,
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[Note3()]Note3
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Supplemental Material: Distinguishability-induced many-body decoherence
    Andreas Buchleitner
    September 9, 2024
=======================================================================


















§ SIMPLIFIED EXPRESSIONS OF 𝒲_C

In the following we provide a simplified expression for 𝒲_C from Eq. (<ref>) of the main text, assuming many-body internal states of the form ρ=ρ_1p⊗⋯⊗ρ_1p. To this end, we plug the matrix elements from Eq. (<ref>) of the main text into the definition of the many-body coherence from Eq. (<ref>) and use the cyclic property of the trace, 

    𝒲_C   =1/N!(N!-1)∑_π,π'∈N
    π≠π'Π_πρΠ^†_π'
       =1/N!-1( 1/N!∑_π,π'∈NΠ^†_π'Π_πρ -1).

Since N forms a group, the summation can be reduced to

    𝒲_C   =1/N!-1( ∑_π∈NΠ_πρ -1).

As detailed in the main text, we now assume that all particles are in the same internal single-particle state ρ_1p such that ρ=ρ_1p⊗⋯⊗ρ_1p. Following our considerations from the main text, we assume that ρ_1p has a discrete spectrum with m eigenvalues λ_j ≥ 0 and corresponding eigenvectors |j⟩. Hence, its eigendecomposition reads

    ρ_1p=∑_j=1^m λ_j |j⟩⟨j|

and the many-body internal state ρ becomes

    ρ=∑_ℐ⃗∈{1,…,m}^Nλ_ℐ⃗ |ℐ⃗⟩⟨ℐ⃗|,

where ℐ⃗=(ℐ_1,…,ℐ_N) is the internal assignment list with N elements ℐ_α∈{1,…,m}, |ℐ⃗⟩=|ℐ_1⟩⊗⋯⊗|ℐ_N⟩, and λ_ℐ⃗=∏_α=1^N λ_ℐ_α. Using this, the traces in Eq. (<ref>) become

    Π_πρ   =∑_ℐ⃗∈{1,…,m}^Nλ_ℐ⃗ Π_π|ℐ⃗⟩⟨ℐ⃗|
       =∑_ℐ⃗∈{1,…,m}^Nλ_ℐ⃗ ℐ⃗ℐ⃗_π,

where Π_π|ℐ⃗⟩=|ℐ⃗_π⟩=|ℐ_π(1)⟩⊗…⊗|ℐ_π(N)⟩. Thus, since ℐ⃗ℐ⃗_π =∏_α=1^N δ_ℐ_α,ℐ_π(α)≥ 0 and λ_ℐ⃗≥ 0, we have Π_πρ≥ 0. Accordingly, we can drop the modulus in (<ref>), yielding the simplified expression

    𝒲_C   =1/N!-1( ∑_π∈NΠ_πρ -1).

With the help of the projector Π_S=1/N! ∑_π∈S_NΠ_π onto the symmetric N-particle subspace, this can further be written as

    𝒲_C   =N!/N!-1Π_Sρ -1/N!-1,

which, in the limit N!≫1, simplifies to 

    𝒲_C≈Π_Sρ.







§ RELATION BETWEEN 𝒲_C AND THE EXPECTATION VALUES Π_S(A)Ρ_E AND Π_SΡ


To calculate the expectation value of the projector Π_S(A)=1/N! ∑_τ∈S_N (-1)^τ _B(F)Π_τ onto the (anti)symmetric N-particle subspace with respect to the particles' external degrees of freedom, we use the matrix elements of ρ_E as provided in Eq. (<ref>) of the main text, 

    Π_S(A)ρ_E   =1/N!^2∑_τ,π,π' ∈S_N (-1)^τππ'_B(F)Π_πρΠ_π'^†⟨E⃗_π'|Π_τ|E⃗_π⟩
       =1/N!^2∑_τ,π,π' ∈S_N (-1)^τππ'_B(F)Π_π(π')^-1ρE⃗E⃗_πτ (π')^-1
       =1/N!^2∑_π,π' ∈S_NΠ_π(π')^-1ρ
       =1/N!∑_π∈S_NΠ_πρ
       =Π_Sρ.

That is, we just showed that Π_S(A)ρ_E=Π_Sρ. Using this in Eq. (<ref>) finally yields

    𝒲_C≈Π_S(A)=Π_Sρ,

as stated in the main text. 








§ PROOF OF EQ. (<REF>)

In the following we prove Eq. (<ref>) of the main text. We start wit plugging Eq. (<ref>) into Eq. (<ref>),

    𝒲_C   =1/N!-1(  ∑_ℐ⃗∈{1,…,m}^Nλ_ℐ⃗∑_π∈Nℐ⃗ℐ⃗_π  -1).

Now, let us introduce the internal occupation list J⃗=(J_1,…,J_m), with J_j the number of particles in the internal state |j⟩ such that ∑_j=1^m J_j=N. Note that an internal occupation J⃗ can give rise to several assignment lists ℐ⃗ [see below Eq. (<ref>)], which differ by permutations of their elements. In particular, for the internal occupation J⃗, let I⃗ be the corresponding assignment list whose elements are listed in ascending order. Using the notation |I⃗_π⟩=|I_π(1)⟩⊗⋯⊗|I_π(N)⟩ for π∈N, we see that |I⃗_ξ⟩=|I⃗⟩ if and only if ξ∈J⃗ = J_1⊗⋯⊗J_m, which is a Young subgroup of N. Hence, for π∈N all permutations of the right coset J⃗π={ξπ|ξ∈J⃗} result in the same state, i.e., |I⃗_π⟩=|I⃗_π'⟩ for all π'∈J⃗π. Therefore, let us construct the transversal Σ(J⃗) of the set of right cosets of J⃗ in N containing one permutation of each distinct right coset such that I⃗_⃗μ⃗I⃗_⃗ν⃗=δ_μ,ν for μ,ν∈Σ(J⃗)  <cit.>. Note that Σ(J⃗) has cardinality J≡ |Σ(J⃗)| = N!/∏_j=1^m J_j!. Using this, we can rewrite the sum over all assignment lists ℐ⃗ in Eq. (<ref>), resulting in

    𝒲_C   =1/N!-1(   ∑_J⃗∑_μ∈Σ(J⃗)λ_I⃗_μ∑_π∈NI⃗_μI⃗_μπ  -1).

Using ∑_π∈NI⃗_μI⃗_μπ=N!/J, λ_I⃗_μ=λ_I⃗≡λ_J⃗, with λ_J⃗ =∏_j=1^m λ_j^J_j, this simplifies to 

    𝒲_C   =1/N!-1(   ∑_J⃗∑_μ∈Σ(J⃗)N!/Jλ_J⃗ -1)
       =1/N!-1(   N! ∑_J⃗λ_J⃗ -1)
       =N!/N!-1∑_J⃗λ_J⃗  - 1/N!-1.

Accordingly, with the help of Eq. (<ref>) we find Π_Sρ= ∑_J⃗λ_J⃗ such that for N!≫ 1, we get

    𝒲_C   ≈∑_J⃗λ_J⃗.

Using λ_J⃗ =∏_j=1^m λ_j^J_j and ∑_j=1^m J_j=N, this can also be written as

    𝒲_C   ≈∑_J_1+J_2+⋯+J_m=Nλ_1^J_1λ_2^J_2⋯λ_m^J_m,

which coincides with Eq. (<ref>) of the main text.









§ PROOF OF EQ. (<REF>)

We start our proof of Eq. (<ref>) of the main text by considering 𝒲_C from Eq. (<ref>) of the main text [see also Eq. (<ref>)]. Without loss of generality we can assume that the eigenvalues of ρ_1p satisfy λ_1 ≤λ_2 ≤⋯≤λ_m. That is, λ_max=λ_m is the maximal eigenvalue of ρ_1p. We further suppose that λ_m is d-fold degenerate, i.e., λ_m=λ_m-1=⋯=λ_m+1-d. With this in mind, let us rewrite 𝒲_C from Eq. (<ref>),

    𝒲_C   ≈∑_J_1+J_2+⋯+J_m=Nλ_1^J_1λ_2^J_2⋯λ_m^J_m
       = ∑_J_1=0^N λ_1^J_1∑_J_2=0^N λ_2^J_2⋯∑_J_m-1=0^N λ_m-1^J_m-1 λ_m^N-J_1-J_2-… -J_m-1 Θ(N-J_1-J_2-…-J_m-1) 
       = λ_m^N ∑_J_1=0^N ( λ_1/λ_m)^J_1∑_J_2=0^N ( λ_2/λ_m)^J_2⋯∑_J_m-1=0^N ( λ_m-1/λ_m)^J_m-1 Θ(N-J_1-J_2-…-J_m-1),

where 

    Θ(N-J_1-J_2-…-J_m-1)=  1  for  J_1+J_2+…+ J_m-1≤ N 
    
    0  otherwise

is the Heaviside function. By the degeneracy of the maximum eigenvalue, λ_m=λ_m-1=⋯=λ_m+1-d, this becomes 

    𝒲_C   ≈λ_m^N ∑_J_1=0^N ( λ_1/λ_m)^J_1⋯∑_J_m-d=0^N ( λ_m-d/λ_m)^J_m-d∑_J_m+1-d=0^N ⋯∑_J_m-1=0^N  Θ(N-J_1-J_2-…-J_m-1).

Now note that in the limit N→∞, the Heaviside function (<ref>) can be approximated by unity. Hence, in the limit N→∞ the many-body coherence is well approximated by

    𝒲_C   ≈λ_m^N ∑_J_1=0^N ( λ_1/λ_m)^J_1⋯∑_J_m-d=0^N ( λ_m-d/λ_m)^J_m-d∑_J_m+1-d=0^N ⋯∑_J_m-1=0^N

Next, we use the geometric series 

    ∑_J_j=0^N (λ_j/λ_m)^J_j=(1-λ_j/λ_m)^-1- (λ_j/λ_m)^N+1(1-λ_j/λ_m)^-1

such that 𝒲_C from Eq. (<ref>) becomes

    𝒲_C   ≈ (N+1)^d-1 λ_m^N  ∏_j=1^m-d[(1-λ_j/λ_m)^-1 - (λ_j/λ_m)^N+1(1-λ_j/λ_m)^-1].

After performing the product, we see that there are factors of the form λ_m^N, λ_m^N (λ_j/λ_m)^N+1, λ_m^N (λ_j/λ_m)^N+1 (λ_k/λ_m)^N+1, etc. However, since λ_j/λ_m<1 for all j=1,…,m-d, in the limit N→∞ the dominant factors are those of the form λ_m^N. That is, the approximation of 𝒲_C simplifies to

    𝒲_C   ≈ (N+1)^d-1 λ_m^N  ∏_j=1^m-d(1-λ_j/λ_m)^-1.

Note that one arrives at the same result faster by setting the upper limit of the first m-d sums in (<ref>) to infinity. Equation (<ref>) describes the behaviour of 𝒲_C in the thermodynamic limit. In the case of a non-degenerate maximum eigenvalue λ_m, i.e., d=1, the approximation simplifies to

    𝒲_C   ≈λ_m^N ∏_j=1^m-1(1-λ_j/λ_m)^-1

as stated in Eq. (<ref>) of the main text. 









§ PROOF OF EQ. (<REF>)

In the following we prove Eq. (<ref>) of the main text. As stated in the main text, in the case of small deviations from perfectly indistinguishable particles, ρ_1p can be written as ρ_1p =(1-ϵ)|ϕ⟩⟨ϕ| + ϵρ̃_1p, with ϵ≪ 1/2. Note that ρ̃_1p is Hermitian and has unit trace. Using this decomposition of ρ_1p, the many-body internal state ρ becomes

    ρ   =[(1-ϵ)|ϕ⟩⟨ϕ| + ϵρ̃_1p] ⊗⋯⊗[(1-ϵ)|ϕ⟩⟨ϕ| + ϵρ̃_1p]
       =(1-ϵ)^N  |ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ|
       +(1-ϵ)^N-1ϵ( ρ̃_1p⊗|ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ| +… + |ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ|⊗ρ̃_1p)
       +(1-ϵ)^N-2ϵ^2 ( ρ̃_1p⊗ρ̃_1p⊗|ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ| +… + |ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ|⊗ρ̃_1p⊗ρ̃_1p)
       +…

Now note that for ϵ≪ 1/2, we have (1-ϵ)^N-jϵ^j ≫ (1-ϵ)^N-j-1ϵ^j+1. Accordingly, the first term of the sum in (<ref>) dominates. Therefore we can approximate 𝒲_C from Eq. (<ref>) as

    𝒲_C   ≈1/N!-1( (1-ϵ)^N ∑_π∈NΠ_π|ϕ⟩⟨ϕ|⊗…⊗|ϕ⟩⟨ϕ| -1)
       = N!/N!-1 (1-ϵ)^N -1/N!-1.

In the limit  N!≫ 1 this simplifies to

    𝒲_C≈ (1-ϵ)^N,

 as stated in Eq. (<ref>) in the main text. 






§ LOWER ORDER COHERENCES


Let us consider the reduced external state [see Eq. (<ref>) in the main text]

    ρ_E= ∑_π,π' ∈N [ρ_E ]_π,π'|E⃗_π⟩⟨E⃗_π'|

with 

    [ρ_E ]_π,π'=(-1)_B(F)^ππ'1/N!Π_πρΠ^†_π'.

By tracing out a particle, we obtain the reduced external N-1-particle state

    ρ_E^(N-1)   = ∑_π,π' ∈N [ρ_E ]_π,π'E_π'(N)E_π(N) |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|
       =∑_π,π' ∈N
    π(N)=π'(N) [ρ_E ]_π,π' |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|,

where |E⃗^(N-1)_π⟩ = |E_π(1)⟩⊗…⊗|E_π(N-1)⟩. With the help of the Young subgroup S_N-1;α=S_{1,…,N}∖{α}⊗S_{α}, this can be written as

    ρ_E^(N-1)   =∑_α=1^N ∑_π,π' ∈S_N-1;α [ρ_E ]_π,π' |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|
       =1/N∑_α=1^N ∑_π,π' ∈S_N-1;α [ρ_E^(N-1) ]_π,π' |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|
       =1/N∑_α=1^N ρ_E^(N-1;α),

where [ρ_E^(N-1)]_π,π'=N [ρ_E]_π,π', and 

    ρ_E^(N-1;α)=∑_π,π' ∈S_N-1;α [ρ_E^(N-1) ]_π,π' |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|.

We recognize that ρ_E^(N-1;α) corresponds to the reduced external state with the αth particle excluded. Since we consider the internal product state ρ=ρ_1p⊗…⊗ρ_1p in Eq. (<ref>), for different α, the states ρ_E^(N-1;α) only differ by the labeling of the external states, and, thus, must have equal many-body coherences. Accordingly, by Eq. (<ref>), and the linearity of 𝒲_C^(N-1) with respect to ρ_E^(N-1), the states ρ_E^(N-1) and ρ_E^(N-1;N) have equal many-body coherences. Thus, since 

    ρ_E^(N-1;N)=∑_π,π' ∈S_N-1 [ρ_E^(N-1) ]_π,π' |E⃗^(N-1)_π⟩⟨E⃗^(N-1)_π'|

coincides with the reduced external state ρ_E from Eq. (<ref>) with N-1 instead of N particles, we can conclude that the normalized many-body coherence 𝒲_C^(N-1) of the reduced N-1-particle state ρ_E^(N-1) coincides with the normalized many-body coherence 𝒲_C [see Eq. (<ref>) in the main text] in the case of N-1 instead of N particles. Similarly, by tracing out further particles, the same reasoning lets us conclude that the many-body coherence 𝒲_C^(k) of the reduced k-particle state ρ_E^(k)=N-kρ_E coincides with 𝒲_C in the case of k instead of N particles. 


The same conclusion can be drawn faster by considering the expression of 𝒲_C from Eq. (<ref>): Since each term of the reduced k-particle state ρ_E^(k) is associated with the unsymmetrized internal state ρ^(k)=ρ_1p^⊗ k, by Eq. (<ref>), we must have

    𝒲_C^(k)   =k!/k!-1Π_S^(k)ρ^(k) -1/k!-1,

with Π_S^(k) = 1/k! ∑_π∈S_kΠ_π. 






§ ATOMS IN THE SMALL TEMPERATURE LIMIT

In the small temperature limit T ≪Δ E/ln(2) the atoms are with hight probability (e^-β E_1 Z(β)^-1) in the internal ground state |1⟩. Hence, by rewriting the single-particle internal state ρ_1p= ∑_j=1^m e^-β E_jZ(β)^-1|j⟩⟨j| as

    ρ_1p   =e^-β E_1/Z(β)|1⟩⟨1|+ ∑_j=2^m e^-β E_j/Z(β)|j⟩⟨j|
       =(1-ϵ) |1⟩⟨1|+ϵ ρ̃_1p,
 
we can identify 

    1-ϵ=e^-β E_1/Z(β)≈e^-β E_1/e^-β E_1+e^-β E_2 = 1/1+e^-βΔ E≈ 1- e^-βΔ E,

i.e., we have ϵ≈ e^-βΔ E. For particle numbers N! ≫ 1 we can then apply Eq. (<ref>) [see also Eq. (<ref>)], resulting in 

    𝒲_C≈ (1- e^-βΔ E)^N

as stated in the main text. 









§ FAINT DISTINGUISHABILITIES OF PHOTONS WITH RANDOM ARRIVAL TIMES

Let us consider the single-particle internal state ρ_1p=∫dt P(t) |t⟩⟨t| for any probability distribution P(t), i.e., P(t) is not necessarily a normal distribution. Without loss of generality we suppose that ⟨t|=⟩∫dt t P(t) =0 such that, in the case of faint distingiushabilities σΔ≪ 1/√(2), the photons arrive at time t≈⟨t|=⟩0 with high probability. Hence, we can write the single-particle internal state as

    ρ_1p   = ⟨0|ρ_1p|0⟩|0⟩⟨0| + (1-⟨0|ρ_1p|0⟩) ρ̃_1p
       =(1-ϵ) |0⟩⟨0| + ϵ ρ̃_1p,

and identify 1-ϵ =⟨0|ρ_1p|0⟩. Calculating the expectation value yields

    ⟨0|ρ_1p|0⟩   =∫_-∞^∞dt P(t) 0t^2
       =∫_-∞^∞dt P(t) e^-Δ^2 t^2,

where we used |t⟩=(2 πΔ^2)^-1/4∫_-∞^∞dω e^ω t e^-(ω-Ω)^2/4Δ^2|ω⟩ to obtain the overlap 0t^2=e^-Δ^2 t^2. Since P(t) in (<ref>) must be small for times |t|>1/Δ [recall that we consider faint disinguishabilities σΔ≪ 1/√(2), i.e., σ≪ 1/√(2)Δ], we can expand the exponential function in (<ref>), resulting in

    ⟨0|ρ_1p|0⟩   ≈∫_-∞^∞dt P(t) ( 1-Δ^2 t^2)
       =1-Δ^2 σ^2,

where we used ⟨t^2|=⟩σ^2, since ⟨t|=⟩0. Hence, we find 1-ϵ≈ 1-Δ^2 σ^2 (compare to the average mutual fidelity in <cit.>). By Eq. (<ref>) [see also Eq. (<ref>)], for N!≫ 1, we then have 𝒲_C≈ (1-Δ^2 σ^2)^N as provided in the main text. 





