ENTANGLEMENT OF FERMIONIC STATES

Entanglement measures

The entanglement of a pure state of a bipartite quantum system can be quantified uniquely by the entropy of one of its subsystems. Namely, if the global pure state Ψ is entangled, i.e., it cannot be factorized into individual pure states of the subsystems, each of the subsystems (1 and 2) is in a mixed state. The degree of mixing can be determined by means of the von Neumann entropy

where ρ₁ = Tr₂|Ψ⟩⟨Ψ| and ρ₂ = Tr₁|Ψ⟩⟨Ψ| are the reduced density matrices, describing the mixed states of subsystems 1 and 2, respectively. Alternatively, one can calculate the linear entropy

that can be obtained from the von Neumann entropy by expanding the logarithm of the reduced density matrix and retaining the leading term. Both entropies meet the necessary requirements for an entanglement measure. The linear entropy, however, is more convenient to compute.

Entanglement measure for states of two identical fermions

For the systems consisting of two identical fermions (e.g. electrons) the measure must be modified in order to exclude the contribution related to the antisymmetric character of fermionic states. The correlations due to the Pauli principle do not contribute to the state’s entanglement. For instance, a two-fermion state of a Slater rank 1 (i.e., a state represented by one Slater determinant) must be regarded as nonentangled and its measure has to be zero. In order to satisfy this requirement, the entanglement measure based on the linear entropy in the case of two identical fermions has the form

Here ρ_r is the reduced single-particle density matrix obtained by tracing the two-particle density matrix ρ = |Ψ⟩⟨Ψ| (describing the pure state Ψ) over one of the two particles. As a result, for a factorized wave function Ψ = ψ (r₁,r₂) χ (σ₁,σ₂) (ψ and χ are the orbital and spin parts of Ψ) measure ℰ transforms to the following form

where ρ_r^(orb) and ρ_r^(spin) are the reduced single-particle density matrices in the orbital and spin spaces, respectively. In principle, the trace of [ρ_r^(orb)]² has to be calculated in some single-particle basis. It is shown in Ref. [1] how it can be determined in a convenient way.

The trace of [ρ_r^(spin)]² in the two-electron spin states with a definite symmetry χ_S,Ms has values 1/2 if M_S = 0 (antiparallel spins of two electrons) and 1 if M_S = ±1 (parallel spins), i.e.,

Two-electron quantum dots

The Hamiltonian of a circular (2D) two-electron quantum dot (QD) reads

where m* is the effective electron mass and λ is the parameter describing the strength of electron-electron (Coulomb) interaction.

Entanglement of the lowest two-electron states in the limit of noninteracting electrons

Linear entropy of the lowest two-electron states with different values of the magnetic quantum number for relative kretanje (m) is in the limit of noninteracting electrons (λ → 0) given by analytical expression [1]

It follows that the lowest states with m = 0 and 1 are not entangled (in this limit), but those with m > 1 are.

Dependence of ℰ on the electron-electron interaction strength

References