Locality and entanglement of indistinguishable particles

Entanglement is one of the strongest quantum correlation, and is a key ingredient in fundamental aspects of quantum mechanics and a resource for quantum technologies. While entanglement theory is well settled for distinguishable particles, there are five inequivalent approaches to entanglement of indistinguishable particles. We analyse the different definitions of indistinguishable particle entanglement in the light of the locality notion. This notion is specified by two steps: (i) the identification of subsystems by means of their local operators; (ii) the requirement that entanglement represent correlations between the above subsets of operators. We prove that three of the aforementioned five entanglement definitions are incompatible with any locality notion defined as above.

Theorem 1. Given two commuting subsets A and B of operators that leave SEP I invariant, the factorisation condition (2) and Definition (1) of the main text imply either A or B consists only of operators propotional to the identity.
If |ψ is an eigenvector of z · σ, then while, due to the hermiticity of O and Q, Therefore, the factorisation condition (1) is fulfilled only if z = 0. Consider now the case z = 0, which implies y = γ x. The factorisation condition (1) is a forth order polynomial in ψ| x · σ|ψ . Furthermore, ψ| x · σ|ψ spans the real axis when |ψ varies: recall that we have relaxed the normalisation condition for states. Therefore, the coefficient of all powers of ψ| x · σ|ψ must vanish. In particular, the coefficient of ψ| x · σ|ψ 4 is γ 2 . In conclusion, γ = 0 and thus one of the subset, either A and B, is made only of operators proportional to the identity.

Examples
In this section, we provide some examples of the properties proved in the context of entanglement-I. Consider a generic separable-I state |Ψ = |ψ ⊗ |ψ , with |ψ = c 0 |0 + c 1 |1 as in Definition 1 of the main text, and impose the normalisation ψ|ψ = |c 0 | 2 + |c 1 | 2 = 1. First of all, the action of single-particle operators O ⊗ 1 do not preserve the particle permutation symmetry. Indeed, the resulting state is not invariant under the exchange of particles: On the other hand, symmetrised single-particle operators O ⊗ 1 + 1 ⊗ O generate entanglement-I: The state (6) is not separable-I. If, e.g., either c 0 = 1 or c 1 = 1 and O = σ 1 is the first Pauli matrix, the state (6) is |0 ⊗ |1 + |1 ⊗ |0 .

Entanglement-II
Definition 2 of the main text follows from the general frameworks in Refs. [8,9,1,2,10,3,11,12,13]. It is worthwhile to stress that separable-I states are also separable-II, but there are separable-II states that are entangled-I. Theorem 4 of the main text shows the incompatibility of entanglement-II with the locality notion, exploiting similar results for entanglement-I. Now, we provide a more detailed argument that relies on the explicit forms of operators that leave SEP II invariant.
Theorem 2. Any operator that leaves SEP II invariant either leaves SEP II \ SEP I invariant or sends SEP II in SEP I .
Proof. Consider an operator A that leaves SEP II invariant, and a state |Ψ ∈ SEP II \SEP I . Either A|Ψ ∈ SEP I or A|Ψ ∈ SEP II \ SEP I . A necessary and sufficient condition for A|Ψ ∈ SEP I is equation (8) of the main text, as in the proof of Theorem 3 of the main text, The crucial difference with the case |Ψ ∈ SEP I of Theorem 3 of the main text is that the polynomial P A now depends also on complex conjugates c 0 and c 1 . A necessary and sufficient condition for A|Ψ ∈ SEP II \ SEP I is equation (22) of the main text, exactly as in the proof of Theorem 3 of the main text. Now, also equation (11) can have no solutions. Therefore, fixing c 0 and Im c 1 , each of equation (11) and equation (22)  If equation (11) is the tautology, then A · SEP II \ SEP I ⊂ SEP I which implies, together with Theorem 3 of the main text, A · SEP II ⊂ SEP I . If equation (22) of the main text is the tautology, then A · SEP II \ SEP I ⊂ SEP II \ SEP I .
then O|ψ = O|ψ ⊥ which implies O = λ|+ +| with |+ = |ψ + |ψ ⊥ . This result must hold for any basis |ψ , |ψ ⊥ , according to Theorem 2. Therefore, λ must be zero, and there are no operators that transform SEP II \ SEP I to SEP I . The other possibility left by Theorem 2 is that for any basis |ψ , |ψ ⊥ . From the Definition 2 of the main text, The latter equation implies that O † O is diagonal in any basis, thus it must be proportional to the identity matrix. Therefore, O is proportional to a unitary matrix. In conclusion, it is not possible to identity an algebra of operators that do not generate entanglement-II, and this notion of entanglement is not compatible with the existence of local subsystems defined by subalgebras of their local operators.

Examples
In this section, we discuss some examples of the above theorems for entanglement-II. We remind that separable-I states are also separable-II (see Definition 2 in the main text), and that operators that do not generate entanglement-II are special cases of those that do not gneerate entanglement-I (see Theorem 3). In particular, the examples discussed for entanglement-I applies also to the framework of entanglement-II. The only difference is that it is not enough to compare states (6), that result from the action of symmetrised single-particle operators O ⊗ 1 + 1 ⊗ O, with separable-I states, but rather with the larger class of separable-II states. Nevertheless, the condition that the state (6) is separable-II, although not separable-I, reads ψ|O|ψ = 0 (see Definition 2 in the main text). This condition, together with the arbitrariness of the state |ψ , implies that O is the zero operator. Therefore, symmetrised single-particle operators We now complement the examples in the previous section with others that exploit separable-II states that are not separable-I, namely where with the normalisation condition |c 0 | 2 + |c 1 | 2 = 1. Consider, as in section , the operators O ⊗ O and Q ⊗ Q with O = σ 1 and Q = σ 2 , that do not generate entanglement-II. The expectation values of these operators are and Therefore, the factorisation condition (2) in the main text is violated in general, as in equation (10).

Entanglement-III
Note that Definition 3 of the main text is called entanglement-IV in [6] whereas entanglement-III therein is the definition called SSR-entanglement in the main text.
In order to define separable-III states in Definition 3 of the main text, we need to introduce some preliminary notions. First of all, define reductions from the two-particle Hilbert space to the single-particle Hilbert space.
where η = +1(−1) for bosons (fermions). Within this approach, a so-called reduced single-particle density matrix is defined relative to a single-particle subspace K: where |ψ k k is any orthonormal basis (ONB) of K. In second quantisation, ρ (1) reads where a † ψ k (a ψ k ) creates (annihilates) a particle in the state |ψ k , and a ψ k , a † ψ k = δ k,k . Note that the density matrix ρ (1) does not reproduce expectations of single-particle observables [14], and, yet, is used to define entanglement.

Examples
In this section, we present some examples for the impossibility to reconcile the definition of entanglement-III with the Werner's formulation. First of all, consider separable-III states with two particles in the left location, namely separable-III states that lie in the support of the projector P LL defined in equations (25) of the main text. The support of P LL is spanned by states where only internal degrees of freedom varies. Therefore, this subspace is isomorphic to C 2 ⊗ C 2 , and entanglement-III in this subspace is the same as entanglement-I. Therefore, all the examples shown in section applies also here.
Consider now more general examples of separable-III states: in first quantization, or in second quantization, with the normalisation Ψ|Ψ = |c LL | 2 + |c LR | 2 + |c RR | 2 = 1. Note that thess state are separable-III for both choices of the single-particle subspace K = span{|L, 0 , |L, 1 } and K = span{|R, 0 , |R, 1 } (see the comments before Definition 3 in the main text and equations (23) and (24)). Consider also the operators where 1 is the identity matrix on the single-particle Hilbert space with spatial and internal degrees of freedom. These operators can be written in second quantization as A = a † L,0 a L,0 − a † L,1 a L,1 , and are considered local within the theory of entanglement-III [15,16]. Expectation values are Therefore, the factorisation condition is violated: