Entanglement between identical particles is a useful and consistent resource

The existence of fundamentally identical particles represents a foundational distinction between classical and quantum mechanics. Due to their exchange symmetry, identical particles can appear to be entangled -- another uniquely quantum phenomenon with far-reaching practical implications. However, a long-standing debate has questioned whether identical particle entanglement is physical or merely a mathematical artefact. In this work, we provide such entanglement with a consistent theoretical description as a quantum resource in processes frequently encountered in optical and cold-atom systems. Moreover, we demonstrate that identical particle entanglement is even a useful resource, being precisely the property resulting in directly usable entanglement from such systems when distributed to separated parties, with particle conservation laws in play. The utility of our results is demonstrated by a quantitative analysis of a recent experiment on Bose-Einstein condensates. This work is hoped to bring clarity to the debate with a unifying conceptual and practical understanding of entanglement between identical particles.


Introduction
Identical particles in quantum mechanics have a character quite distinct from those in classical mechanics. Classically, indistinguishability comes from limited abilities of the experimenter; in the quantum world, two particles of the same type, such as electrons, are fundamentally indistinguishable [1,2]. This feature applies not only to fundamental particles but is also crucial in describing identical composite particle systems such as Bose-Einstein condensates (BECs) [3,4]. Notably, exchanging two identical quantum particles results in an overall phase change in the wavefunction: no change for bosons and a minus sign for fermions.
In order to determine whether there is any meaningful interpretation of PE per se we follow the modern approach to entanglement within quantum information theory [26]. Here, entangled states are defined as those which cannot be prepared by two or more separated parties who are unable to send quantum information, and are as such limited to local operations (within their own laboratories) and classical communication -abbreviated as LOCC. Entanglement is then regarded as a resource for parties operating under such constraints, and can enable them to perform better at a vast range of tasks including quantum communication [27], computation [28], key distribution [29], and metrology [30], to name a few.
In systems of identical particles, the usable entanglement is that between modes [4,[31][32][33][34][35][36][37][38][39]. This is because (orthogonal) modes are by definition distinguishable systems and so can be addressed individually. Note that these modes need not be spatially separated; we only require that there exist some degree of freedom (such as momentum or internal spin) via which they can be separately addressed. Mode entanglement is distinct from entanglement between particles. For instance, a single particle existing in a superposition of two locations can be viewed as an entangled state of two spatial modes -but this state clearly contains no PE since there is only one particle. So if mode entanglement is the operationally useful quantity, and is not directly related to PE, why are we interested in the latter? There are strong reasons to believe that PE is a property worth quantifying and may be a resource in certain scenarios. For instance, many-body entangled states of cold atoms, such as spin-squeezed states, can increase precision in metrology thanks to their PE [40][41][42][43][44].
In order to justify PE as a resource, one needs to provide the appropriate setting -what is the analogue of LOCC for indistinguishable particles? In this work, we first answer that question by finding a physically relevant set of quantum operations in which PE cannot be created. These operations are constructed from combinations of appending vacuum states, performing passive linear unitaries and making either nondemolition measurements of total particle number, or else arbitrary but destructive measurements. We prove that each of these sets of elements is as general as possible while resulting in a consistent theory. In particular, the set of unitaries is physically motivated as "easy" in many settings, corresponding to beam splitters and phase shifters in optics, and to numberconserving non-interacting hamiltonians in condensed matter systems. These operations, which we call particle-separable, define the basis of a resource theory for PE. Such an approach has been widely employed recently to pin down a variety of quantum properties beyond entanglement, such as quantum thermodynamics [45], quantum coherence [46] and asymmetry [47]. With this structure in place, one can begin to rigorously quantify PE.
Next, we use our framework to find the complete setting in which PE is a resource for generating useful mode entanglement between parties. This fully generalises earlier observations by Yurke and Stoler [48] and more recently by Killoran et al. [38], the latter providing the starting impetus for this work. Specifically, by "useful" mode entanglement we mean that which is accessible to parties who are constrained not only by LOCC but also by a local particle-number superselection rule [49]. The latter constraint renders superpositions of different particle numbers unobservable, and applies when particle number is conserved and the two parties do not have access to a shared phase reference [50]. Under this limitation, less entanglement can be utilised [32,36]. We show that useful entanglement can be generated from an initial state by a particle-separable operation exactly when the initial state contains non-zero PE. Furthermore, we find quantitative relations between the amount of input PE and the output useful entanglement. This shows that PE mirrors other quantum resources which may be similarly "activated" into useful entanglement [51][52][53].
These results have direct applications to real systems of indistinguishable bosons, in particular entangled states of BECs [25,54]. We analyse one of a set of recent experimental advances witnessing mode entanglement in BECs [55][56][57]. We show that these fit into our framework and implement the above resource conversion. In particular, our results enable a quantitative determination of the PE content of the states produced in the experiment.
Finally, we find novel and surprising links between PE and non-classicality as employed in quantum optics. In that context, classical states are probabilistic mixtures of coherent states [58,59]. States lying outside this set are non-classical, and are essential in many quantum technological applications [60]. Aided by a recent resource theory formulation of nonclassicality [61][62][63], several connections can be formed between the two disparate topics. We find non-classicality to be a necessary but not sufficient prerequisite for PE -however, non-classicality can be "unlocked" by using multiple copies of a state. Thus we have a remarkable link between two uniquely quantum resources.

Results
Particle identity and superselection rules. We work with bosonic systems, for which m orthogonal modes have associated annihilation and creation operators a i , a † i , i = 0, . . . , m − 1, satisfying the canonical commutation relations [a i , For a particular choice of modes, the second quantised description is given in terms of the occupation numbers n i of each mode: |n 0 , . . . , n m−1 ∝ (a † m−1 ) n m−1 . . . (a † 0 ) n 0 |0, . . . , 0 . All bosonic states then live in the Fock space spanned by such vectors.
In order to make statements about entanglement between particles, it is necessary to ensure that it is even sensible to talk about the particles comprising a state. Such statements are meaningless when a state contains a superposition of different particle numbers. Therefore we permit ourselves only to describe states of definite total particle number -or probabilistic mixtures of such states [4,31]. Mathematically, this is described by a particle-number superselection rule (SSR), which forces any state ρ under consideration to be block-diagonal with respect to the total number operatorN, also expressed as [ρ,N] = 0. (We distinguish between the operatorN and its eigenvalues N.) Similarly, all considered operations E (i.e., completely positive maps on the set of states) must respect the SSR. This is ensured by taking only covariant operations, defined by commutation [E, U θ ] = 0 with the phase rotation channel U θ (ρ) = e −iθN ρe iθN for all θ [50]. Equivalently, covariant operations can be performed via a dilation involving an initially number-diagonal environment and a global particle number conserving unitary interaction [64]. Any state of definite particle number N = i n i can be written in the first quantised picture, where each particle has an internal state in the single-particle space H 1 of dimension m (so that there is one degree of freedom for each mode). The overall state then lies in the symmetric subspace of the N-system space, denoted by H N = S[H ⊗ N 1 ]. A general mixture of particle numbers ρ = N p N ρ (N ) can be described as being a state on S[H ⊗ N 1 ] with probability p N . Where necessary, we distinguish between the first and second quantised forms of a pure state using the notation |ψ • and |ψ respectively, and similarly ρ • and ρ for a mixed state.
PE as a resource. A resource theory is defined by two components: the set of free states S, which possess no resource, and the set of free operations O, which do not add any new resource into the system. (One also tends to think of free operations as possible to perform without any resource, although this interpretation is not always clear.) The set of free states for PE is straightforward to define. For fixed particle number N, they must be non-entangled (separable) states in the first-quantised picture. Due to symmetry, a pure N-particle free state is thus of the form |Ψ • = |ψ ⊗ N , also known as a coherent spin state [41,65]. In secondquantised form, we have |Ψ ∝ (c † ψ ) N |0 , where c † ψ = i ψ i a † i creates a single particle in an arbitrary mode ψ. A mixed Nparticle free state is by definition symmetric and separable -it turns out (see Appendix A) that this is equivalent to the form Then the full set of free states -which we name particleseparable -consists of those ρ = N p N ρ (N ) such that each of these components in the first-quantised picture is of the form (2). We may then choose as free operations any set that preserves particle-separability. This is required in order to ensure a consistent notion of a resource. There is often tension between the desire for mathematical generality of these operations and wanting them to have a known physical implementation. In our approach, we do not take the largest set of quantum operations preserving particle-separability, but instead construct a physically transparent set from elementary types of operations. We prove that each of these elements is as general as possible.
In the spirit of the Stinespring dilation for quantum operations [66], we construct our free operations out of three basic steps: (i) appending ancilliary modes; (ii) global unitary operations; (iii) projective measurements. We investigate each of these in turn.
(i) Appending ancilliary modes: In mathematical terms, the action of appending to a state ρ another set of modes in a fixed state σ means ρ → ρ ⊗ σ in second quantisation. In order to consider this a free operation, we restrict σ ∈ S. In most resource theories this operation would preserve the set of free states [67]. However, the present theory is unusual in that this generally fails -the simplest example is appending the singleparticle state |1 to another copy of itself, as |1, 1 ≡ |1 |1 is not particle-separable. The reason for this is that appending particles in new modes requires symmetrisation in the first quantised picture, which creates PE. As we show in Appendix B, the only ancilliary state σ that guarantees preservation of free states is the vacuum.
(ii) Unitaries: The covariance condition for unitaries means that they preserve particle number: [U,N] = 0. Consider first the component U (N ) acting on the N-particle subspace. We see that U (N ) preserves S if and only if it has the first-quantised action U (N ) |ψ ⊗ N = |φ ⊗ N for every |ψ ∈ H 1 , where |φ can depend on |ψ . Perhaps unsurprisingly, this is equivalent to U (N ) = u ⊗N for any single-particle unitary u, although the argument is not immediate and invokes Wigner's theorem on inner-product-preserving transformations [68] (see Appendix C). In principle, this u could be different for each number N -however, the introduction of number measurements below implies that we lose no generality by taking a fixed u. Such unitaries have a simple second-quantised description via their action on ladder operators: U † a † i U = j u i j a † j , where u i j are the elements of a unitary matrix. They describe single-particle rotations without interaction, acting identically on all particles, and correspond to passive linear operations in optics, which are easily generated by beam splitters and phase shifters [69].
(iii) Projective measurements: A projective measurement is given by a set of projectors Π i which are orthogonal and complete: Π i Π j = δ i, j Π i , i Π i = 1. As for unitary operations, these must adhere to the SSR, [Π i ,N] = 0, and preserve the set of particle-separable states, Π •(N ) i |ψ ⊗N ∝ |φ ⊗ N . However, we find that these conditions are only met by a measurement of total particle number (see Appendix D). In order to enlarge the set of available measurements, we allow destructive measurements, in which the measured modes are subsequently discarded. In Appendix D we demonstrate that this relaxation allows any measurement adhering to the SSR to be performed on the system without introducing PE. Such destructive measurements correspond to the majority of experimental photonand atom-counting techniques.
The set O of particle-separable operations is defined as all possible protocols which result from combinations of the above elements, including possible conditioning of future operations on the results of measurement outcomes. We also allow for the use of classical randomness and coarse-graining -i.e., forgetting measurement outcomes. Mathematically, an element in O is represented as a quantum instrument, which is a set of CP maps E i where each i labels a single (possibly coarse-grained) measurement outcome and the sum i E i is deterministic (trace-preserving). Note that an instrument can equivalently be represented as a deterministic channel F (ρ) = i E i (ρ) ⊗ |i i| X , where the outcome is stored in a classical system X [70].
With this structure in place, we can now move naturally to define measures M PE of PE. As is standard in quantum resource theories [67], we require that any measure of PE fulfills the following three conditions. Condition (i)-It must not detect PE when there is none, meaning M PE (ρ) = 0 for all ρ ∈ S (and optionally the converse may be required). Condition (ii)-M PE must be a monotone, i.e. cannot increase under the action of any particle-separable operation. This reflects the idea that particle-separable operations cannot inject additional PE into the system. Monotonicity can be stated either deterministically, . Condition (iii)-Convexity, i.e., being non-increasing under probabilistically mixing different states, A straightforward class of PE measures are given by the minimal distance between a state and the set of particle-separable states: where D is any suitable measure of distinguishability between two quantum states. Conditions (i-iii) are met whenever D is contractive under quantum channels (so that D(E(ρ), E(σ)) ≤ D(ρ, σ) for any channel E) and jointly convex in its arguments [67].
Activating PE. Now that we have determined the set of protocols under which PE may abstractly be considered a resource, we are in a position to demonstrate a concrete task in which it is useful. The original seeds of the activation protocol that we study here are in work by Yurke and Stoler, who noted that two particles produced from separated, independent sources can in fact be used to violate a Bell inequality [48]. The protocol that we present is a full generalisation of [38]. Consider two separated parties, A and B, who want to perform some joint quantum information protocol but are con- The operation E converts a system of identical particles with PE into a bipartite state, whose SSR-entanglement can be extracted and utilised in quantum information tasks. The above diagram depicts the transformation b. An example of a particle-separable operation is the action of a beam-splitter with a vacuum, which can be used to activate the PE present in the state ρ C . strained to classical communication and additionally lack a shared phase reference (conjugate to the number observableN A orN B ). While each party may be unconstrained in their local operations, without sharing a phase reference, the amount of entanglement accessible to them is reduced by the application of an effective local SSR [50]. This SSR corresponds to both local particle numbersN A andN B . A third party C is tasked with providing A and B with a shared entangled state that they can use. To accomplish this, C has an initial resource state ρ C of m modes and can process it using any particle-separable operation E before distributing m A and m B modes to each of A and B. (Recall that the operation E may introduce new vacuum modes and trace out some modes; see Fig. 1). The question is: how much useful entanglement can be extracted in this way from ρ C ?
Let σ AB = E(ρ C ) be the output state sent to A and B, where E ∈ O is the distribution operation performed by C. (Without loss of generality, using classical flags, we can take this to be deterministic.) Due to the local SSR, from the perspective of A and B, this state is operationally as useful as the state [71], where Φ S is the dephasing channel local to subsystem S, removing quantum coherences between states of differing local numberN S [88].
For any measure E of bipartite entanglement, we can then define the corresponding measure of entanglement accessible to A and B [50]: We say that a state σ AB is SSR-separable whenever it has vanishing accessible entanglement -i.e., when is separable -and SSR-entangled otherwise. The inequality in (4) follows from the fact that Φ A ⊗ Φ B is a local operation -the local SSR generally reduces the amount of accessible entanglement. The aspect of the entanglement in σ AB that is inaccessible, sometimes referred to as "fluffy bunny entanglement" [72], is connected with superpositions of local number. Note that Wiseman and Vaccaro [31] proposed the same class of measures (4) and found such SSR-entanglement to require non-zero PE in the case of two particles. We prove that PE in the initial state ρ C is precisely the resource enabling the distribution of SSR-entanglement. Our first result is that the mapping between the two types of entanglement is faithful, in that SSR-entanglement can be extracted exactly when there is nonzero PE (see Appendix G for the proof): Moreover, almost any operation of the following type is sufficient to activate PE into non-zero SSR-entanglement: for each mode i in C, attach a new mode in the vacuum state, and perform a global passive-linear unitary coupling the modes (as in Fig. 1b). We say "almost all" because the unitary must not be trivial by failing to couple some of the modes. It was argued in [38] that activation is only meaningful for a specific class of unitary interactions, namely a set of beam-splitters with identical transmission coefficients. However, we see that a much more general statement is possible, expanding the scope to all particle-separable operations.
Beyond the faithful mapping between nonzero resources, we now quantitatively relate the input and output forms of entanglement. One approach uses measures of both PE and SSR-entanglement constructed in the same way. Recall the distance-based measure of PE M D PE ; by the same recipe, one can construct a measure of SSR-entanglement (see Appendix G): As shown in Appendix F, when ρ respects the local SSR, the minimisation can be equivalently performed over the smaller set of σ AB being separable and respecting the local SSR. Using this, we have: . This shows that the amount of accessible entanglement extracted never exceeds the initial amount of PE. Note, however, a subtlety: in general, this inequality is strict (apart from when both sides are zero), due to a necessary reduction in entanglement after applying the dephasing operation Φ A ⊗ Φ B and removing the "fluffy bunny entanglement".
Alternatively, we can take any measure of SSRentanglement and use it to construct a new measure of PE. This is given by the maximal amount of SSR-entanglement which can be created from a certain initial state: Theorem 3. For any (convex) entanglement measure E, the quantity defined as is a (convex) measure of PE.
In other words, for any entanglement measure E, the corresponding quantity M E PE satisfies criteria (i-iii). Theorem 3 gives a precise quantitative version of the statement that PE is the resource for producing SSR-entanglement.
Experimentally measuring PE. In this section we demonstrate that our resource theory for describing PE and its activation encompasses recent experimental investigations [55][56][57] converting PE into useful mode entanglement and analyse the experimental data from [55] to lower bound a measure of PE.
The experimental method is as follows -see [55] for more details. The BEC is initialised in a spin-squeezed state, which possesses PE. The BEC is then released from its trap and allowed to expand, and the spin components of the two spatially separated regions are measured. During the expansion, the effect of interactions between atoms on their spin state is negligible such that this step can be regarded as a beam-splitter operation and hence falls within our set of particle-separable operations , can be measured by applying appropriate spin rotations before detection, these local rotations also being allowed within SSR constraints.
In Ref. [55] the authors showed how these local spin measurements can violate the inequality [73] 4Var g zŜ in terms of variances and average values of spin observables. (7) is satisfied by all separable states and for any real constants g y,z , therefore certifying entanglement between system A and B whenever it is measured to be negative.
In Appendix H, we linearise (7) and use Theorem 3 to derive a lower bound on a measure of PE: where M Tr PE is defined according to (3) with the trace distance D Tr (ρ, σ) := 1 2 Tr | ρ − σ|. We show an evaluation of this bound using experimental results in Fig. 2. The parameters g y,z are optimised numerically so that the left-hand side of (7) is minimised, as this expression is more robust than (8) against experimental noise. This plot clearly shows a positive amount of PE has been activated from a spin squeezed BEC and none from a coherent spin BEC state, as predicted from our theory.
Connections to non-classicality. While coherent spin states are considered classical in cold atoms settings with fixed particle number, continuous-variable coherent states in quantum optics provide the model of classical light. Non-classical states display features such as photon anti-bunching, sub-poissonian statistics and squeezing [74], and form the basis of many quantum technological applications [60] As has been recently appreciated, [61][62][63] non-classicality can also be quantified with its own resource theory. In this section we demonstrate some remarkable connections between the resources theories for PE and non-classicality.
Recall that a single-mode coherent state |α is an eigenstate of the annihilation operator: a |α = α |α , and a multi-mode coherent state may be written as |α := |α 1 . . . |α m , where α = (α 1 , . . . , α m ) ∈ C m . A state is called classical if it can be written as a probabilistic mixture of coherent states: Due to the SSR employed here, we restrict to number-diagonal (ND) classical states -i.e, those satisfying [ρ,N] = 0. The operationally motivated free operations for nonclassicality, presented in Ref. [62], are very close to particleseparable operations. The only differences are that (i) rather than only the vacuum, any classical state may be prepared for free in a new mode and (ii) non-destructive measurements of total particle number can create non-classicality. Moreover, there is an entirely analogous protocol activating non-classicality into mode entanglement [75][76][77] (which in fact extends to more general notions of non-classicality [78]). Whereas PE can be activated under particle-separable operations into SSRentanglement, nonclassicality activates into entanglement accessible without local SSR constraints -equivalently, entanglement which can be accessed when a shared phase reference is available.
This observation immediately implies a relation between the free states of the two resource theories: all ND classical Based on the measurements [55] we are able to extract the lower bound given by the right-hand side of (8), on the PE measure M Tr PE . The two sets of points correspond to initialising the BEC either in a spin squeezed state (green), where Particle Entanglement is present, or in a coherent spin state (orange), which is particle-separable. Along the horizontal axis we vary the relative size of the two regions A and B from which we extract the spin values as explained in [55]. In the experiment, technical limitations in the resolution of assigning the atomic spins to the regions can lead to classical correlations, resulting in apparent entanglement. We give an upper bound for these correlations as the blue dashed line. For intermediate splitting ratios we find significant entanglement in the case of the spin squeezed state while the coherent spin state remains compatible with no particle entanglement within experimental error. On the right we show single-shot absorption images of the atomic densities for the two internal degrees of freedom, with an example of regions A and B used to define the collective spinsŜ A andŜ B entering in (8).
states are particle-separable. This follows from the fact that a classical state is always activated onto a separable state, which is always also SSR-separable, implying via Theorem 1 that the input is particle-separable. In fact, this can be shown by a more direct argument, with details in Appendix I. Essentially, any multi-mode coherent state |α can be regarded as a singlemode state -for any choice of mode decomposition, there is always a passive linear unitary U such that U |α = |ᾱ |0 . . . 0 , where |ᾱ| 2 = m i=1 |α i | 2 . So any classical state is a probabilistic mixture of terms in which all particles occupy the same mode.
Evidently, ND classical states form a strict subset of particleseparable states. Consequently, we may say that nonclassicality is lower-bounded by PE in the sense that, for any distance measure of nonclassicality M D NC constructed in the manner of (3), the inequality M D NC ≥ M D PE holds.
What distinguishes the two sets of free states? As noted earlier, a striking property of PE is that multiple copies of a free state ρ do not in general jointly form a free state. Viewed through the activation protocol, this is equivalent to saying that two copies of an SSR-separable state may be SSR-entangled. This is possible because of the way the SSR behaves for multiple copies of a system. If A and B share two pairs of entangled systems, (A 1 , B 1 ) and (A 2 , B 2 ), then the particle number local to A isN A =N A 1 +N A 2 and similarly for B. The local SSR is applied by The lack of factorisation is due to degeneracy in the eigenvalues of 2 is entangled but SSR-separable; the two copy state is SSR-entangled thanks to correlations in the block N A = N B = 1. This phenomenon is closely related to work-locking in quantum thermodynamics, whereby coherence in one copy of a state is useless for work extraction but becomes usable in two copies [79]. A tensor product of two classical states is always classical, hence multiple copies of an ND classical state always have zero PE. Are these the only states with this property? We first consider number-bounded states: those for which the expansion N p N ρ (N ) terminates at a finite maximum. In this case, the resource content of two copies is sufficient to distinguish the classical subset of particle-separable states (note that all classical states apart from the vacuum are necessarily unbounded in number): Theorem 4. Two copies ρ ⊗2 of a number-bounded state ρ are particle-separable if and only if ρ is the vacuum.
(See the proof in Appendix I.) In the general unbounded case, let us first take pseudo-pure states, by which we mean those obtained by applying the SSR to a pure state: ρ = Φ(|ψ ψ|). It is known that in the limit n → ∞ of many copies |ψ ⊗n of a pure entangled state, the SSR is effectively lifted in that the full entanglement entropy is distillable [32]. One may then argue from the activation protocol as follows: a non-classical state at the input results in entanglement at the output; many copies of this state must therefore result in an SSR-entangled state. Hence any non-classical pseudo-pure state must fail to be particle-separable with sufficiently many copies. An even stronger statement is in fact possible: Theorem 5. Two copies Φ(|ψ ψ|) ⊗2 of a pseudo-pure state are particle-separable if and only if |ψ is classical.
Therefore we see that non-classicality of any pseudo-pure state, even if particle-separable, can always be unlocked into non-zero PE by taking only two copies.

Discussion
We have seen that entanglement between identical particles, despite its seemingly fictitious nature, is described by a consistent resource theory whose free operations are implementable in a wide range of physical systems. Far from just an abstract quantity, this entanglement can be activated, via the same types of operations, into directly accessible entanglement. This occurs in a setting where phase references are not easily shared between separated parties, enforcing a local SSR.
While we have found the most general form that this activation may take, some important questions remain open. Theorem 3 expresses the maximum activated SSR-entanglement from a given state as a measure of PE -however, the optimal operation that achieves this is unknown. It is plausible that this should be unitary; Lemma 5 in Appendix G proves a simplification from the full space of passive linear unitaries down to only one real parameter per mode, making the optimisation feasible. We conjecture that the optimal unitary is always a non-polarising beam-splitter in which transmission is independent of the internal state of the particle.
Our formulation reveals PE as fundamentally connected not only to entanglement under SSRs, but also to continuous variable non-classicality. In particular, we have shown that SSR-compliant classical states possess no PE. Consequently, PE is a stronger (rarer) resource than non-classicality. Nevertheless, by utilising multiple copies of a state, one may unlock its non-classicality into PE. Aside from our results, the only remaining case is that of mixed states with unbounded particle number. We conjecture that arbitrarily many copies ρ ⊗n are particle-separable if and only if ρ is classical. Thus it may be that non-classicality emerges as a many-copy limit of PE.
Finally, we would like to motivate the wider theoretical and experimental applicability of our framework for PE. In addition to describing the activation of entanglement from a BEC, the framework applies to any system of identical bosons, opening up the possibility of investigating PE beyond BECs and optics, to other condensed matter systems in which entanglement is of interest, such as superfluid Helium [80]. An extension to fermionic systems should also be pursued. It is hoped that the results presented here and future theoretical and experimental studies utilising this framework will provide insight into the genuinely quantum properties and behaviour of such systems. [87] Not to be confused with particle entanglement as named in [36].
[88] This may be written equivalently as a phase average Φ S (ρ) = ∫ 2π 0 dθ e −iθN S ρe iθN S /2π or as a "measure-and-forget" opera-tion of the local number: Φ S (ρ) = n P n,S ρP n,S , where P n,S is the projector onto the subspace of n particles in S.
[89] The interaction of ultracold 87 Rb atoms depends only very weakly on their spin state. During the expansion of the BEC, the interactions therefore do not affect the spin state and are furthermore quickly rendered small due to the decreasing density [81].
[90] Due to technical limitations a fraction of the atomic spins in a gap between the two regions is discarded in the measurement process.

Appendix A: Form of free states
Here we show that every particle-separable state of N particles is of the first-quantised form By assumption, ρ • is separable, so we can write ρ each term N k=1 ψ k i is in the symmetric subspace. It follows from this symmetry that all ψ k i are the same for a given i. Proof. It is sufficient to let ρ be the simplest free state, a single particle in a single mode: ρ = |1 1|. σ = N p N σ (N ) is arbitrary and may have unbounded particle number. Then The (N + 1)-particle component of this state is |1 1| ⊗ σ (N ) . In order to particle-separable, it must be possible to express as in terms of some set of m + 1 modes, with λ i ≥ 0 and the U i being free unitaries. The left-hand side has exactly one particle in the first mode and N in the remainder, so the same must be true of every term on the right-hand side. So for each i, U i |N + 1, 0, . . . = |1 |ψ i , which is impossible unless N = 0. To see this, note that we can write where b i is some linear combination of annihilation operators on the rightmost N modes. Expanding the bracket (a † 1 + b † i ) N +1 , we can never have a single term linear in a † 1 unless N = 0. Therefore p N = 0 for N ≥ 0, so σ = |0 0|. Conversely, it is trivially seen that appending vacuum modes always preserves the set of free states.

Appendix C: Free unitaries
In the following section, we work with states of N particles and always in the first-quantised picture, so we drop the additional notation for convenience.

Theorem 7. A unitary U on H N maps free states into free states if and only if U
Proof. Note that we only specify the restriction of U to H N rather than the "full" Hilbert space H ⊗N 1 . For example, permutations between particles are not of the given form but have trivial action on the symmetric subspace.

Appendix D: Free measurements
As in Appendix C, we temporarily drop the first-quantised notation. As a first step in the investigation of non-destructive measurements, we need the following Lemma: Lemma 1. Let Π be a projector with support on the symmetric subspace of N particles, i.e. Π = P N ΠP N , where P N projects onto H N . Then Π is non-entangling if and only if there exists a projector π on H 1 such that Proof. It is immediate that any Π of the form (D1) preserves symmetric product states; so we need only prove the converse. We start from the observation that for any |ψ ∈ H 1 , there is a (normalised) |φ ∈ H 1 such that Π|ψ ⊗ N = c|φ ⊗ N , where either c = 0 or else c 0 and |φ ⊗N ∈ supp Π. If c = 0 ∀ |ψ , then Π = 0 since states of the form |ψ ⊗ N span H N [82]. Otherwise, there must exist some |0 such that |0 ⊗ N ∈ supp Π. If rank Π = 1, then Π = |0 0| ⊗N and we are done. If rank Π > 1, then consider any |ψ orthogonal to |0 . Again, we must have Π|ψ ⊗N = c|φ ⊗ N . Note that having used Π|0 ⊗ N = |0 ⊗ N . So either c = 0, or else c 0 and |φ is orthogonal to |0 . Considering all |ψ orthogonal to |0 , it follows that either Π|ψ ⊗ N = 0 for all such |ψ , or else there exists |1 orthogonal to |0 , with |1 ⊗ N ∈ supp Π. Continuing this procedure, we are able to construct a complete basis {|k } of H 1 such that for some r. Now take an arbitrary |ψ ∈ H 1 , written in terms of the chosen basis as |ψ = d−1 k=0 ψ k |k . Given the properties of this basis, it follows that But since Π preserves product states, Π|ψ ⊗N = |φ ⊗ N (where |φ need not be normalised).
where n k ∈ {0, . . . , N − 1}. In principle, n k may be a function of |ψ ; however, the continuity of the mapping under Π ensures that n k is continuous and hence constant. Furthermore, since |φ ⊗N is invariant under this mapping, we must have n k ≡ 0, so that φ k = ψ k ∀k ≤ r − 1. The action of Π on an arbitrary product |ψ ⊗ N is therefore identical to the action of π ⊗N , where Again, since such product states span H N , this gives (D1).
be a set of non-zero orthogonal projectors onto subspaces of H N (where N > 1) such that k i=1 Π i = P N and each Π i preserves the set of particle-separable states. Then k = 1 and Proof. From Lemma 1, there exist projectors π i such that Π i = P N π ⊗ N i P N ∀i. It follows from this that the orthogonality relation Π i Π j = δ i, j Π i implies π i π j = δ i, j π i . Hence there exist orthogonal |ψ i such that |ψ i ∈ supp π i . From these, we construct from which the completeness relation gives Using the form of |ψ , the right-hand side evaluates to Hence there is a contradiction unless k = 1, which forces the single projector to be Π 1 = P N .
Theorem 8 says that any non-destructive free projective measurement in the N-particle subspace must be trivial. Extending this to measurements over the whole Fock space, respecting the SSR, shows that only a measurement of the number observableN is permissible.
Theorem 9. Any destructive measurement respecting the SSR preserves the set of particle-separable states S.
Proof. It is sufficient to prove this for a single projector. Let the measurement be performed on m B modes of an (m A + m B )-mode system, having the action where Π B is a projector such that [Π B ,N B ] = 0. Any particle-separable pure state has the form |ψ ∝ (c † ) N |0 , where c is a single-particle annihilation operator. Choosing some orthogonal mode set {a i }, where i = 1, . . . , m A for the unmeasured modes and i = m A + 1, . . . , m A + m B for the measured modes, we can write c = a + b, where a and b are linear combinations of the unmeasured and measured a i , respectively. Thus we can effectively treat |ψ as a two-mode state: where the r N A are coefficients. Then the post-measurement (unnormalised) state is where we have used the fact that Π B is diagonal in particle number, Hence σ A ∈ S; the extension to mixed initial states ρ A follows by linearity.

Appendix E: Measures of PE
The following results are used to show that if D satisfies a few straightforward properties, then the resulting measure of PE can be expressed as an average over different particle numbers. We write this in a more abstract form which shows a generalisation to arbitrary resource theories with a block-diagonal structure.
for any sets of states ρ i , σ i and probabilities p i ; Then it also satisfies a. (direct sum linearity) Proof. To show (a): where, in the last line, we have used the fact that adding and removing an uncorrelated system are both reversible channels which must therefore leave D unchanged. The left-and right-hand sides are equal, thus the initial inequality must actually be an equality.
To show (b), we construct from the instrument a channel E(ρ) = i E i (ρ) ⊗ |i i|, so that D(ρ, σ).
where M D N is defined similarly to M D , but minimising over states in F N . Proof. For the first part, we take τ to be the closest state to ρ in F. For any instrument For the second part, which shows that the closest state can be chosen to have q N = p N . Finally, we use (a).
The relative entropy S(ρ||σ) := tr[ρ log ρ − ρ log σ] satisfies all three assumptions of Lemma 2 -in particular, (3) follows from where the last term is the classical relative entropy (or Kullback-Leibler divergence). Hence the relative entropy measure of PE is The same property also holds for distances defined by Schatten p-norms, D p (ρ, σ) = ρ − σ p [67].

Appendix F: SSR-entanglement
The activation protocol converts particle entanglement into entanglement that is of use to two parties A, B who are limited to local covariant operations that respect the SSR and classical communication.
Definition 1. [32,71] An operation between two or more parties is said to be covariant-LOCC when it is composed of local operations respecting the local superselection rule, and classical communication.
Although not spelled out explicitly by [32,71], the free states of this resource theory (in a bipartite setting; easily generalised) are the following: Definition 2. A bipartite state ρ AB is free in the resource theory of SSR-entanglement when it can be written in the form such that each ρ i A , ρ i B respects the SSR, i.e., Φ S (ρ i S ) = ρ i S , S = A, B. Such a free state is said to be invariant-separable (since it is invariant under local phase rotations).
Of course every invariant-separable state is separable, but not vice-versa. This set of free states may be motivated as being those accessible from a given primitive state, such as the vacuum |0 |0 under covariant-LOCC.
Lemma 3. The following statements are equivalent: is pure and contains a definite number of particles. 3. ρ AB is separable and satisfies the local SSR constraint Proof. The equivalence of (1) and (2) is easily seen from the fact that every local-SSR-respecting state ρ i A = Φ A (ρ i A ) can be written as a mixture of pure states of definite number. (1) ⇒ (3) is also straightforward. Conversely, suppose (3) holds, then we so that if each term in the RHS is separable, then the LHS also is. Finally, we show that (1) ⇒ (4). We have which is separable.
A state can fail to be invariant-separable in two different (but not mutually exclusive) ways: it may break the local SSR, or it may be entangled. The measures of SSR-entanglement defined here capture the amount of entanglement accessible from a single copy of the state under the local SSR. However, there are states which have E SSR = 0 yet are not invariant-separable -for example, product states which break the local SSR.
Lemma 4. The distance-based measure of SSR-entanglement can be calculated by a restricted optimisation over SSR-separable states: Equivalently, the closest separable state to Proof. Let E D SSR be the quantity defined by the right-hand side of (F4). We prove an inequality in both directions. Since invariant-separable states form a subset of separable states, it is clear that where we have used the monotonicity of D under Φ A ⊗ Φ B and the fact that A useful consequence of Theorem 10 is that the relative entropy measure of SSR-entanglement can be written as where p N A , N B = Tr (P N A ⊗ P N B )ρ AB . This measure is seen to provide an extension of the pure-state measure defined by Wiseman and Vaccaro [31].
Proof. Lemma 2 of [62] shows that U can be decomposed as It is worth noting that the number of vacuum modes introduced can always be assumed to be no greater than m -again, as a consequence of Lemma 2 in [62].
The faithfulness of the activation is proven below for almost all such unitaries (apart from those with vanishing beam-splitter parameters). Theorem 1. There exists an activation operation E C→AB ∈ O creating an SSR-entangled state σ AB from ρ C if and only if ρ C S.
Moreover, E can be taken to be any of the unitary operations described in Lemma 5, as long as all of the parameters r i , t i are non-vanishing.
Proof. We first prove that any particle-separable initial state results in no SSR-entanglement. This follows from a more general observation: any bipartite particle-separable state ρ AB also SSR-separable. (This was stated in the two-particle case in Ref. [31].) As in the proof of Theorem 9, a particle-separable bipartite state |ψ AB can be regarded as an effective two-mode state -taking a and b as linear combinations of the modes in A and B respectively, we have where the r N A are unimportant coefficients. It is immediate from this expression that P N A ⊗ P N −N A |ψ AB is separable for all N A . Since every particle-separable state is a convex combination of pure particle-separable states, the result follows for all mixed free states. So if ρ C is a particle-separable state, then for any E C→AB ∈ O, E C→AB (ρ C ) is also particle-separable, and hence SSR-separable in the A/B partition. Conversely, we prove that any unitary operation as in Lemma 5 with r i , t i 0 ∀i is sufficient to activate SSR-entanglement from PE. The simplest case -with a pure state and a "non-polarising beam-splitter", r i = r ∀i -was proven in Ref. [38]. Let us first argue that this extends to mixed states.
Suppose that the output state σ AB is SSR-separable, so that each (P N A ⊗ P N B )σ AB (P N A ⊗ P N B ) is separable. As shown in Ref. [38], the entanglement structure of (P , in which the first-quantised form of the input state is partitioned into N A versus N B particles. Hence ρ •(N ) (with N = N A + N B ) is bi-separable with respect to this partition, i.e., where , λ i ≥ 0. Since ρ •(N ) has support in the symmetric subspace H N , we must have |φ i N A | χ i N B ∈ H N ∀i. But any bi-separable symmetric pure state must also be fully separable. Therefore |φ i N A | χ i N B = |ψ i ⊗N , so ρ •(N ) is particle-separable. Finally, we extend to the case of general r i . Via a straightforward generalisation of the argument from Ref. [38], we find the output of the activation taking a Fock state |n as input -the details are in Appendix J. Denote by |ξ AB the output of activating |n with beam-splitter parameters r i = 1/ √ 2 ∀i, and similarly denote by |η AB the output obtained with some arbitrary set of r i . From (J5) with two parties and α Ai = r i , α Bi = t i , we have It is clear from this expression that |η can be obtained from |ξ by application of the local operators L A ⊗ L B , where Since these operators are independent of the choice of initial Fock state, the same relationship holds for any input state -that is, the output from an arbitrary set of beam-splitters can be obtained by applying L A ⊗ L B to the output from a set of balanced beam-splitters. As long as r i , t i 0 ∀i, these operators are invertible. The application of invertible local operators to a bipartite state does not change its Schmidt number [83]. This proves that the faithfulness of activation from a set of arbitrary non-trivial beam-splitters is equivalent to activation from balanced beam-splitters.

Theorem 2. For any activation
. Proof. Let τ be the closest particle-separable state to ρ according to the measure D, then The first two inequalities use the contractivity of D under channels. The final inequality uses the fact that τ is free, so that Φ A ⊗ Φ B • E C→AB (τ C ) is separable, but not in general the closest separable state to σ AB .
Theorem 3. For any (convex) entanglement measure E, the quantity defined as where the supremum is over all deterministic particle-separable operations, is a (convex) measure of PE.
Proof. The faithfulness of the measure is the content of Theorem 1. Deterministic monotonicity follows immediately from the definition and the fact that the set of operations O is closed under composition. Non-deterministic (strong) monotonicity states that M E PE (ρ) does not increase on average, where Λ i (ρ) = p i σ i and {Λ i } i ∈ O. From the definition (G12), we have, for every activating channel E C→AB ∈ O, We now continue to prove strong monotonicity by contradiction, showing that a violation of strong monotonicity (G13), implies a violation of (G14). If strong monotonicity (G13) is violated, then there must exist a set of operations E i,C→AB ∈ O such that the following is true: We now invoke a general property of entanglement measures (and SSR-entanglement measures), namely monotonicity under the partial trace over a subsystem. We split B into two subsystems B 1 , B 2 , in which B 2 contains a classical flag. Then, for any ensemble of state ρ i, AB 1 with probabilities p i , Applying this to (G15), we obtain Note that the operations appearing on the right-hand side above can be combined into a single operation F C→AB 1 B 2 ∈ O, which is performed by first applying {Λ i } i , storing the outcome i in a classical flag, and then conditionally applying E i . Thus, The above is a direct contradiction of (G14), thus establishing that M PE is a strong monotone for any entanglement monotone E SSR . We now continue by showing convexity: From the definition of M PE , we have where we have made use of the fact that taking the supremum over each term in the sum individually cannot give less than a single supremum.
Appendix H: Lower bound on PE measure from an an entanglement criterion In order to witness the entanglement present in the system a criterion of separability from [73] is used, which is satisfied for all separable states, where λ max [A], λ min [A] are the maximum and minimum eigenvalues of the operator A, respectively, and in last line we have again used the fact that the value is maximised when all the particles are in the same internal mode. Substituting the above into the last line of equation (H2) and maximising over each term individually results in, (H20) Now we have bounded both the maximum and minimum values the witness can take, we can bound the normalisation N from equation (H14) and therefore bound the entanglement measure with a normalised witness, Appendix I: Non-classicality Theorem 11. Every number-diagonal (ND) classical state is particle-separable.
Proof. If ρ is classical and ND, then with P(α) ≥ 0. Hence it is sufficient to prove the claim for all Φ(|α α|). For any multi-mode coherent state |α , there exists a passive linear unitary U that brings all the particles into a single mode: U |α = |ᾱ |0 ⊗(n−1) , where |ᾱ| 2 = n i=1 |α i | 2 . Since this unitary is number-conserving, it commutes with Φ, so which is particle-separable.
Theorem 4. Two copies ρ ⊗2 of a number-bounded state ρ are particle-separable if and only if ρ is the vacuum.
Proof. Let both ρ and ρ ⊗2 be free with bounded particle number, and we decompose ρ = N 0 N =0 p N ρ (N ) . Then The maximal number component of this state is p 2 N 0 ρ (N 0 ) ⊗ ρ (N 0 ) , where p N 0 0 by assumption. This component must be particle-separable, thus must be obtainable by mixtures of the form i p i U i |2N 0 , 0, 0, . . . 2N 0 , 0, 0, . . .|U † i , where the U i are passive linear. Now this state has exactly N 0 particles on each of the two parties, and so the same must be true for every term in the sum. In other words, for each i, U i |2N 0 , 0 = (V i |N 0 ) (W i |N 0 ) with pair of additional passive linear unitaries V i , W i acting on each subsystem. It is easily seen that this is impossible unless N 0 = 0.
Proof. We first show that the activation of an arbitrary pure state |ψ into SSR-entanglement is exactly the same as for the pseudo-pure state Φ(|ψ ψ|). Let Φ AB be the joint dephasing operator with respect to the total number over two parties A, B. This operation is already implemented by dephasing with respect to local number, so that (Φ A ⊗ Φ B ) = (Φ A ⊗ Φ B ) • Φ AB . We use this to connect the SSR-entanglement activated by a unitary U ∈ O from |ψ ψ| to that activated from Φ(|ψ ψ|): where we have used the fact that U is number-conserving, so [U, Φ AB ] = 0, and the last line holds because B contains no particles. Now let |ψ be activated by U consisting of a set of non-trivial beam-splitters into |φ AB . Then we can write |φ AB = k,l φ k,l AB := k,l P k, A P l, B |φ AB . If two copies of |ψ are activated in the same way in parallel, then the output state is |φ ⊗2 = |φ A 1 B 1 |φ A 2 B 2 . Given that Φ(|ψ ψ|) ⊗2 is particle-separable, Theorem 1 says that the projection of the activated state onto local particle number must be unentangled -so there exist (unnormalised) |a n,m A 1 A 2 , |b n,m B 1 B 2 such that, for each n, m, Applying the projector P k, A 1 P l, B 1 onto local numbers in the first copy, we find φ k,l A 1 B 1 φ n−k,m−l A 2 B 2 = P k, A 1 |a n,m A 1 A 2 P l, B 1 |b n,m B 1 B 2 .
Both sides of the above equation must be separable with respect to both the A 1 A 2 /B 1 B 2 and A 1 B 1 /A 2 B 2 partitions. Therefore there must exist (unnormalised) states a n,m The left-hand side of the above is independent of n and m, so the same must be true of the states on the right -removing these labels, we obtain Summing over k and l, we see that φ k,l A 1 B 1 = ( k |a k A 1 )( l |b l B 1 ) is separable. From the result in quantum optics saying that all non-classical states are activated into entangled states, it follows that |ψ must be classical.

Appendix J: Unitary activation of Fock states
Here we generalise the main result of Ref. [38] to multiple modes and to general beam-splitters. We also present the results without much additional effort for arbitrary numbers of parties, although the rest of our work uses only the bipartite case. Let us first find the first-quantised form of an m-mode Fock state |n , partitioned into sets of N A , N B , . . . , N Z particles, where K=A,B,..., Z N K = N := i n i . We have where N n is a multinomial coefficient and the sum runs over distinct permutations Π of m−1 i=0 |i ⊗n i . Dividing initially into N A versus NĀ = N − N A particles, it may be verified that where nĀ i = n i − n Ai . Recursively continuing the subdivision ofĀ in this way, we obtain Next, we find show how a Fock state on A is activated into a multipartite SSR-entangled state by mixing with vacuum modes on B, . . . , Z at a generalised beam splitter. Specifically, we take the beam-splitter U to have the action a † Ai → K α Ki a † Ki -a generalisation of Ref. [38], in which α Ki was independent of i. Then Conditioning on local particle number, which is of the same form as (J3), up to the coefficients N N A ,..., N Z 1/2 K,i α n K i Ki .