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Entanglement in fermion systems
Quantum Physics · Submitted 20 Sep 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined in [J. Schliemann et al, Phys. Rev. A 64, 022303 (2001)], implying that "particle entanglement" may be seen as minimum "mode entanglement".
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We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. The "mode entanglement" between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space $\mathcal H$ is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of $\mathcal H$ is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined in [J. Schliemann et al, Phys. Rev. A 64, 022303 (2001)], implying that "particle entanglement" may be seen as minimum "mode entanglement". It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that of the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy are evaluated analytically, extending the results of previous ref. to general states.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. | contribution 0.67 | 0.85 | 0.07 | asserted 0.71 |
| s1 | The "mode entanglement" between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space $\mathcal H$ is defined. | contribution 0.60 | 0.91 | 0.05 | asserted 0.99 |
| s2 | The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of $\mathcal H$ is a function of the one-body density matrix. | result 0.60 | 0.94 | 0.19 | asserted 1.00 |
| s3 | Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. | result 0.96 | 0.95 | 0.12 | asserted 1.00 |
| s4 | These results are then used to quantify entanglement in fermion systems with four single-particle levels. | contribution 0.71 | 0.91 | 0.07 | asserted 0.99 |
| s5 | For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined in [J. Schliemann et al, Phys. Rev. A 64, 022303 (2001)], implying that "particle entanglement" may be seen as minimum "mode entanglement". | result 0.88 | 0.94 | 0.37 | hedged possibility 0.69 |
| s6 | It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that of the two-qubit case. | result 0.99 | 0.92 | 0.12 | asserted 1.00 |
| s7 | For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy are evaluated analytically, extending the results of previous ref. to general states. | result 0.52 | 0.93 | 0.12 | asserted 1.00 |
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Original abstract
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with fixed number parity yet not necessarily fixed particle number. The "mode entanglement" between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space $\mathcal H$ is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of $\mathcal H$ is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined in [J. Schliemann et al, Phys. Rev. A 64, 022303 (2001)], implying that "particle entanglement" may be seen as minimum "mode entanglement". It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that of the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy are evaluated analytically, extending the results of previous ref. to general states.
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We also use the single-orbital entanglement entropy, $S_i = -(1-γ_i) log_2 (1-γ_i) -γ_i log_2 γ_i$, with $γ_i= ⟨ψ(θ) |a^†_i a_i |ψ(θ) ⟩$ , bound between $0$ and $1$, to evaluate the importance of quantum correlations in the ansatz [ref] .
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