Quantum entanglement of identical particles

Abstract

We consider entanglement in a system of fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement. How does one characterize entanglement in a fixed number of identical particles? Obviously, a correct characterization must exclude the nonfactorization due to (anti)symmetrization. Here, we clarify that it can be done by using the (anti)symmetrized basis, which is equivalent to the particle number representation. This naturally leads to the use of occupation numbers of different single-particle basis states as the (distinguishable) degrees of freedom in quantifying identical-particle entanglement even when the number of particles is conserved. The occupation-numbers of different modes have already been used in quantum computing [1]. The use of modes was made in a previous study of identicalparticle entanglement, based on formally mapping the Fock space to the state space of qubits or harmonic oscillators [2], but it was under the unphysical presumption of full access to the Fock space. We shall elaborate that the concept of entanglement in a system of identical particles is fundamentally different from that of distinguishable particles, for which entanglement is invariant under local unitary transformations. There is no local operation that acts only on one of the identical particles. The single-particle basis transformation is made on each particle and chooses a different set of particles in representing the many-particle system. Thus, the entanglement property of a system of identical particles depends on the single-particle basis used. The particle number basis state for a fixed number of particles is just the normalized (anti)symmetrized basis in the configuration space, i.e., Slater determinants or permanents. Therefore, the occupation-number entanglement in a fixed number of particles is nothing but the situation that the state is a superposition of different Slater de- * Email address:ys219@phy.cam.ac.uk terminants or permanents. Another consequence is that the two-identical-particle counterpart of the Schmidt decomposition, which we call Y...

Compositeness effects, Pauli's principle and entanglement

“…This is so because in the second case the effects of antisymmetrization are generated by the anticommutation relations (see Ref. [18] for similar considerations in a different context).…”

Entanglement of indistinguishable particles in condensed-matter physics

“…entanglement of formation, entanglement of distillation, negativity), A and B each control some subset of the total modes, and ρ AB is the total state shared by A and B. This approach is advocated in, for example, [16,26,27]. The entanglement of modes depends on which modes A and B control, as discussed in [28,29], but not on the local mode decomposition that they choose.…”

“…Examples of such systems span many fields of physics: quantum optics experiments, ultracold atomic gases, itinerant electrons and superconductors. Even the question of which states are entangled is the subject much recent debate [7,8,9,10,11,12,13,14,15,16,17,18]. The difficulty arises from the lack of individual identity of the particles that are supposed to be entangled, which is manifest as the necessary symmetrization or antisymmetrization of the quantum wavefunction.…”

Spectral properties of reduced fermionic density operators and parity superselection rule

“…For instance, bipartitions with respect to different spin components were considered recently [51]. Mathematically, we deal with the algebraic bipartition in the second quantized description [52][53][54][55][56], which is used in the study of entanglement [45,[57][58][59][60].…”