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Quantum entanglement of identical particles by standard information-theoretic notions

Rosario Lo Franco, Giuseppe Compagno

arXiv:1511.03445v4quant-ph

TL;DR

Assessing entanglement for identical particles remains difficult because their indistinguishability and spatial overlap matter. The paper introduces an unlabeled state-based approach using localized measurements and the standard reduced-state entropy. It finds that entanglement can depend on overlap, particle type, and measurement location, with opposite-spin particles in the same site maximally entangled.

  • Problem

    The paper addresses how to determine entanglement for identical particles when indistinguishability and spatial overlap are relevant.

    Source: A. Description of the new approach · p. 3

  • Method

    The approach represents particles without labels and assesses entanglement using a localized-measurement partial trace and the reduced-state von Neumann entropy.

    Source: A. Description of the new approach · pp. 2–3, §II · p. 5

  • Results

    Entanglement depends on local measurements and, for partially overlapping opposite-pseudospin particles, on overlap and particle type; opposite pseudospins in the same site have E = 1.

    Source: B. Application · p. 4, §II · p. 5

  • Takeaways & Limitations

    The results provide theoretical grounds for setting and interpreting experiments that use quantum correlations when identical particles overlap.

    Source: §II · p. 6

Abstract

from arXiv · show

Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. We introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bosons or fermions) in assessing entanglement by notions commonly used in quantum information theory for distinguishable particles, like partial trace. Our approach furthermore shows that bringing identical particles into the same spatial location functions as an entangling gate, providing fundamental theoretical support to recent experimental observations with ultracold atoms. These results pave the way to set and interpret experiments for utilizing quantum correlations in realistic scenarios where overlap of particles can count, as in Bose-Einstein condensates, quantum dots and biological molecular aggregates.

A. Description of the new approach

The approach describes identical particles as an unlabeled two-particle state and derives their probability amplitudes from indistinguishable alternatives. It quantifies entanglement using localized one-particle measurements and the reduced-state entropy.

A. Description of the new approach

  • Identical particles are represented by an unlabeled state specified by its two one-particle states, with amplitudes formed by symmetrizing alternative transition paths.The exchange factor η enforces bosonic or fermionic symmetry.Source: A. Description of the new approach · p. 2
  • The reduced one-particle state is obtained by a physical partial trace based on projective measurements, then normalized to unit trace.The measurement outcomes are not registered, so the reduced state represents their statistical ensemble.Source: A. Description of the new approach · p. 3
  • Entanglement is quantified by the von Neumann entropy of the reduced state after measurements restricted to a localized region.Its value can depend on the measurement region, but is independent of that choice for spatially separated particles or particles in the same mode.Source: A. Description of the new approach · p. 3
  • A local one-particle measurement probes a property in a spatial region where the particle has nonzero probability of being found.For spatially separated particles, this reduces to measuring an addressable particle; overlap makes identity effects relevant.Source: A. Description of the new approach · p. 3
  • The method applies the standard partial-trace notion directly to identical-particle states and is presented as generalizable to many particles.It avoids requiring redefined entanglement measures and witnesses used by some name-labelled methods.Source: A. Description of the new approach · p. 3

B. Application

The application examines two identical qubits with partially overlapping spatial modes. It finds that overlap, particle statistics, internal-state configuration, and measurement location all matter for the entanglement assessed.

B. Application

  • For 0 < χ < 1, overlap and relative phase affect entanglement, and bosons and fermions can differ; at χ = 0, entanglement equals the nonidentical-particle value.For states of the same form, shifting θ by π maps fermionic entanglement to bosonic entanglement.Source: B. Application · p. 4, Figure 3 · p. 5
  • At complete overlap, |L ↑, L ↓⟩ has E = 1, with spin correlations matching Bell states for distinguishable particles measured along an arbitrary direction.The passage describes this entanglement as intrinsic to the system's spatial configuration, rather than induced by measurement.Source: B. Application · p. 4
  • For bosons, the same-site state |L ↑, L ↑⟩ has zero entanglement and leaves the remaining particle in |↑⟩ after a spin measurement.The contrast shows that internal-state configuration matters alongside spatial mode.Source: B. Application · pp. 4–5
  • Tracing out the spin in localized mode |R⟩ gives E_R(Ψ) = E_ni(Ψ), while entanglement can depend on the measurement location when particles overlap.That dependence disappears when particles are spatially separated or occupy the same mode.Source: B. Application · p. 5
  • Entanglement extraction from overlapping identical particles is described as drawing on entanglement already intrinsic to the state.The passage names Bose-Einstein condensates via tunneling and quantum-dot electrons via photon emission as example settings.Source: B. Application · p. 5

II. DISCUSSION

The discussion presents a label-free approach for studying identical-particle entanglement with standard reduced-state measures, and finds that overlap and local measurement choices shape the result. It also identifies overlapping modes as an entangling operation and points to future studies of broader quantum correlations.

II. DISCUSSION

  • The label-free approach applies the standard reduced-state von Neumann entropy to bosons, fermions, and nonidentical particles under arbitrary wave-function overlap.The discussion says this treatment is technically comparable to that for nonidentical particles and facilitates generalization to many-particle systems.Source: §II · p. 5
  • For partially overlapping opposite-pseudospin particles measured in the shared site, entanglement depends on overlap and particle type and never exceeds the nonidentical-particle counterpart.The discussion also reports an ordering among particle types as overlap varies.Source: §II · p. 5
  • Local measurements cannot determine an absolute, measurement-independent entanglement amount unless identical particles are spatially separated or occupy the same spatial mode.Bringing identical particles into overlapping modes acts like an entangling gate whose effectiveness depends on overlap.Source: §II · p. 5
  • The analysis provides groundwork for studying entanglement dynamics and other quantum correlations in many-identical-particle systems, including mixed states.It also supplies theoretical grounds for setting and interpreting experiments in realistic overlap scenarios.Source: §II · pp. 5–6

Appendix A: Particle exchange

In the label-free description, physical exchange swaps the one-particle states and leaves the two-particle state unchanged up to the bosonic or fermionic phase. The exchange action therefore encodes the required spin-statistics symmetry without constructing a label-symmetrized state.

Appendix A: Particle exchange

  • Physical exchange maps |ϕ1, ϕ2⟩ to ±|ϕ1, ϕ2⟩, with + for bosons and − for fermions, while swapping the one-particle states in the state vector.The exchange operator is not needed to construct a label-symmetrized state in this approach.Source: Appendix A · pp. 6–7
  • The two-particle probability amplitude in Eq. (1) directly incorporates the required particle spin-statistics symmetry, or symmetrization postulate.After exchange, the state vector changes only by a global phase factor.Source: Appendix A · p. 7

Appendix B: Measurement-induced entanglement

For spatially separated identical qubits, a local projective measurement gives zero entanglement, whereas a nonlocal measurement basis yields maximal measurement-induced entanglement. The example demonstrates sensitivity to the measurement's nonlocal character.

Appendix B: Measurement-induced entanglement

  • With spatially separated orthogonal modes and local projective measurement, the identical particles behave like nonidentical ones and have E = Eni = 0.The example uses the state |L ↑, R ↓⟩.Source: Appendix B · p. 7
  • Measuring in a nonlocal one-particle basis gives Emi = 1, making the two qubits maximally measurement-induced entangled.The resulting reduced density matrix is (|L ↑⟩⟨L ↑| + |R ↓⟩⟨R ↓|)/2.Source: Appendix B · p. 7

Appendix C: Derivation of the identical particle entanglement

The appendix derives reduced states and entropies for two identical qubits as measurement location and spatial overlap vary. It shows how overlap, phase, particle statistics, and the choice of successive local measurements affect the resulting entanglement.

Appendix C: Derivation of the identical particle entanglement

  • For partial overlap, the reduced-state calculation for |Ψ⟩ depends on the spatial overlap, relative phase θ, and particle statistics η.The appendix gives the state as |Ψ⟩ = a|L ↑, B ↓⟩ + be^{iθ}|L ↓, B ↑⟩ and derives its entropy from the nonzero eigenvalues of the reduced density matrix.Source: Appendix C · p. 7
  • Tracing via localized R measurements leaves a reduced state localized in L, with entropy ER(Ψ) = −a² log2 a² − (1 − a²) log2(1 − a²).The appendix notes that the overlap does not affect this case because the relevant modes are orthogonal.Source: Appendix C · p. 7
  • Successive local measurements can be made in the same or different localized modes, and tracing first on L then on R gives ELR(Ψ) = ER(Ψ).The appendix describes projecting the remaining particle onto R after the first partial trace on L.Source: Appendix C · p. 7
  • Tracing first on L and measuring again on L gives ELL(Ψ) = 1.This result follows the projection onto |L⟩ and renormalization.Source: Appendix C · p. 8
  • The derived entanglement varies quantitatively with how the local measurement is performed, a behavior also seen with detection-level concurrence Cd.The appendix notes that Cd is unsuitable for particles in the same site and for generalization to many particles.Source: Appendix C · p. 8

Appendix D: Entanglement extraction

Splitting identical particles from one mode creates a component in distinct modes whose entanglement is maximally entangled and exploitable by LOCC. The approach also extracts entanglement for three identical bosons without particle name labels.

Appendix D: Entanglement extraction

  • The components with all particles remaining in a single mode retain the input state’s entanglement, but it remains unexploitable by LOCC.The input is two identical qubits with opposite spins in the same mode.Source: Appendix D · p. 8
  • The split-state component in modes C and D is maximally entangled and becomes an exploitable LOCC resource between accessible, distinguishable spatial modes.The component is established probabilistically by the splitting operation.Source: Appendix D · p. 8
  • Entanglement extraction for three identical bosons in the same mode can be carried out analogously without particle name labels.The cited procedure in Ref. required labels, which this approach demonstrates are unnecessary.Source: Appendix D · p. 8

Appendix E: Comparing our particle-based approach with the usual name-labelled one

The particle-based approach treats an identical-particle state as a whole without name labels and defines measurement and partial trace as physical operations on the system. It differs from name-labelled criteria that can miss entanglement in overlapping modes and impede extension to many particles.

Appendix E: Comparing our particle-based approach with the usual name-labelled one

  • The approach treats the two-particle state as a whole without labels, whereas the standard labelled description writes the corresponding state as a symmetrized or antisymmetrized form.For separated particles, an entangled state has the same two-term form used for distinguishable particles in this framework; the labelled description has four ket states.Source: Appendix E · p. 8
  • The approach’s partial trace is a physical operation on the two-particle system, unlike a name-labelled trace that addresses a fictitious individual particle.The authors say their physical trace avoids the boson–fermion contradictions associated with the name-labelled reduced-state entropy.Source: Appendix E · p. 8
  • A state of identical particles in orthogonal one-particle states can still be entangled when their spatial modes overlap.This includes two particles in the same mode with orthogonal internal states, contrary to the name-labelled criterion described in the passage.Source: Appendix E · p. 9
  • Detection-level concurrence misses same-site intrinsic entanglement and requires overall-state tomography at spatially separated detectors, limiting its suitability for many-particle systems.The physical partial-trace approach is described as naturally extendable to many particles.Source: Appendix E · p. 9

Appendix F: Spin correlations for two identical particles in the same site

For two identical particles in the same site, opposite-spin states yield spin correlations that depend on particle nature, while the same-spin state is unentangled. The opposite-spin correlations behave like those of a maximally entangled pair.

Appendix F: Spin correlations for two identical particles in the same site

  • After a one-particle spin measurement on the opposite-spin same-site state, the other particle is left in the corresponding opposite spin direction for fermions, but in a direction u′ for bosons.For bosons, u′ has angles θ′ = π − θ and ϕ′ = ϕ; the results match those for distinguishable particles in Bell states.Source: Appendix F · p. 9
  • For the same-site state |L ↑, L ↑⟩, the remaining particle is |↑⟩ independently of the first measurement, so the state is unentangled.The passage identifies the state as separable or uncorrelated.Source: Appendix F · p. 9
  • The opposite-spin particles in the same site exhibit perfect spin correlations and behave exactly like a maximally entangled pair.The passage describes the two particles as a whole object in a perfectly correlated, non-separable state.Source: Appendix F · p. 9
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