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Entanglement in indistinguishable particle systems

F. Benatti, R. Floreanini, F. Franchini, U. Marzolino

arXiv:2007.06253v2quant-phcond-mat.quant-gashep-thphysics.optics

TL;DR

Indistinguishability prevents individual addressing and has led to discordant definitions of identical-particle entanglement. This review compares those approaches using bipartite correlation functions and three consistency criteria, finding that only the mode-based approach meets all three.

  • Problem

    Identical particles cannot be individually addressed, leaving multiple discordant definitions of entanglement rather than the agreed particle-based notion for distinguishable systems.

    Source: §1 · pp. 1, 4

  • Method

    The review compares existing approaches through bipartite correlation functions and tests them against criteria for locality, effective distinguishability, and information-processing resources.

    Source: §1 · pp. 4–5

  • Results

    Only the mode-based entanglement-V approach satisfies all three criteria; each particle-based approach violates at least one.

    Source: §5 · p. 28

  • Takeaways & Limitations

    Within the chosen criteria, correlations among second-quantized modes provide an adequate notion of identical-particle entanglement.

    Source: §5 · p. 28

    The review restricts its analysis to generalized bipartite settings using two-point correlations and excludes Bell-inequality non-locality and general resource theories.

    Source: §1 · pp. 3–4

Abstract

from arXiv · show

For systems consisting of distinguishable particles, there exists an agreed upon notion of entanglement which is fundamentally based on the possibility of addressing individually each one of the constituent parties. Instead, the indistinguishability of identical particles hinders their individual addressability and has prompted diverse, sometimes discordant definitions of entanglement. In the present review, we provide a comparative analysis of the relevant existing approaches, which is based on the characterization of bipartite entanglement in terms of the behaviour of correlation functions. Such a a point of view provides a fairly general setting where to discuss the presence of non-local effects; it is performed in the light of the following general consistency criteria: i) entanglement corresponds to non-local correlations and cannot be generated by local operations; ii) when, by "freezing" suitable degrees of freedom, identical particles can be effectively distinguished, their entanglement must reduce to the one that holds for distinguishable particles; iii) in absence of other quantum resources, only entanglement can outperform classical information protocols. These three requests provide a setting that allows to evaluate strengths and weaknesses of the existing approaches to indistinguishable particle entanglement and to contribute to the current understanding of such a crucial issue. Indeed, they can be classified into five different classes: four hinging on the notion of particle and one based on that of physical modes. We show that only the latter approach is consistent with all three criteria, each of the others indeed violating at least one of them.

1 Introduction

Indistinguishability removes the individual addressability that grounds the standard notion of entanglement, leaving competing definitions for identical particles. The review compares these definitions through bipartite correlation functions and three consistency criteria, finding that only the mode-based approach meets all three.

1 Introduction

  • Because identical particles cannot be individually addressed, their entanglement definitions remain discordant, unlike the established particle-based notion for distinguishable systems.The review asks whether symmetrized or antisymmetrized two-particle product-form states are entangled, and which locality notion underlies that judgment.Source: §1 · pp. 1–2
  • For distinguishable particles, entanglement supports quantum technologies and has been used to characterize many-body phenomena including phase transitions, area laws, and many-body localization.Individual addressing and manipulation are central to qubit-based quantum devices.Source: §1 · pp. 1–2
  • The review groups existing definitions into four particle-based classes and one mode-based class, reflecting different choices of what is correlated and considered local.The four particle-based classes are entanglement-I through -IV; entanglement-V concerns modes, which are singly addressable.Source: §1 · p. 4
  • When suitable degrees of freedom are frozen, identical particles can be labeled, and their entanglement should then reduce to the standard distinguishable-particle theory.Spatial confinement is one example of how frozen degrees of freedom can provide labels.Source: §1 · p. 4
  • The review tests entanglement definitions against non-local correlations that local operations cannot generate, recovery of standard entanglement after effective distinguishability, and classical limits on separable-state information tasks.The review defines non-locality through non-vanishing correlations of commuting observables in a bipartite scenario, not Bell-inequality violations.Source: §1 · pp. 4–5
  • Only entanglement-V, which concerns correlations among modes, satisfies all three criteria; each of the four particle-based approaches violates at least one.The criteria concern non-local correlations and local operations, effective distinguishability, and information-processing resources.Source: §1 · p. 5

2 Identical and distinguishable particles

The section describes how identical-particle formalism differs from distinguishable-particle descriptions and how orthogonal single-particle states can make identical particles effectively distinguishable. Spatial localization provides the section’s practical example and yields benchmarks for testing entanglement definitions.

2 Identical and distinguishable particles

  • For identical particles, the Hilbert space is symmetrized for bosons or antisymmetrized for fermions, and single-particle operators must be distributed over all particles.The symmetrized single-particle operator is given as P(O, 1) = P(1, O) = O ⊗1 + 1 ⊗O.Source: §2 · pp. 5–6
  • Second quantization represents identical-particle states with creation and annihilation operators, whose commutation relations automatically enforce the required particle-exchange symmetry.The passage contrasts this with first quantization, where state vectors are explicitly symmetrized or antisymmetrized.Source: §2 · p. 6
  • Spatially orthogonal states, such as particles confined to separate regions, can label identical particles when the spatial degree of freedom is measured and frozen.This provides an effective particle label without changing the particles’ identity.Source: §2 · pp. 5, 7
  • With spatial localization, restricted observables can address internal degrees of freedom in the left and right regions, but those observables are not particle-local.The restricted observables cannot distinguish certain state descriptions, and their particle-non-locality is not witnessed by the states described in the passage.Source: §2 · pp. 7–8
  • Freezing orthogonal degrees of freedom recovers the standard distinguishable-particle description, making the Bell pair and separable state benchmarks for testing entanglement definitions.The passage identifies these as a prototypical entangled state and a separable state, respectively.Source: §2 · p. 8

3 Particle-partitioning

For distinguishable particles, particle partitioning identifies the relevant degrees of freedom with individual particles; indistinguishability makes that partitioning physically unavailable. The section instead considers algebraic partitions of selected degrees of freedom and compares particle-entanglement formulations through their implicit locality criteria.

3 Particle-partitioning

  • Individual addressability underpins particle partitioning, but identical particles cannot be partitioned into individual constituents, making particle-locality potentially untenable.The text motivates considering locality based on degrees of freedom beyond particles.Source: §3 · pp. 8–10
  • Algebraic partitioning selects degrees of freedom unrelated to specific particles, such as particle numbers or collective excitations, and considers their associated observables.The associated operators typically form algebras closed under summation and multiplication.Source: §3 · p. 9
  • For distinguishable particles, algebraic partitioning is tacitly identified with particle partitioning, and mixed-state separability is expressed through mixtures of particle-separable states.For pure states, separability is characterized by factorizing mean values of particle-local observables.Source: §3 · p. 9
  • A chosen algebraic bipartition defines particle-locality through commuting observables associated with its two selected subsets of single-particle degrees of freedom.The subsets cannot include all single-particle operators; otherwise both generated subalgebras coincide with M2(C)⊗M2(C).Source: §3 · p. 10
  • With a selected partition, pure-state particle-separability is defined by factorization for all observables in the two corresponding subalgebras.The text states that the equality condition extends to mixed states.Source: §3 · p. 11
  • The four reviewed particle-based formulations implicitly adopt this particle-locality criterion, and the review evaluates their compatibility with three physical criteria.The formulations are assessed through their separability and entanglement assignments and physical consequences.Source: §3 · pp. 10–11

3.1 Entanglement-I

Entanglement-I defines separability through particle-state structure, but its particle-based locality criterion conflicts with correlation factorization and does not recover distinguishable-particle entanglement under effective distinguishability.

3.1 Entanglement-I

  • Bosonic states are separable-I when all particles occupy the same single-particle state; extending the definition to fermions makes every fermionic state entangled-I.Source: §3.1 · p. 11

3.1.1 Local operators

  • Separable-I states can fail to factorize on commuting particle-local observables, and single-particle operators P(O,1) can map them to entangled-I states.The approach also conflicts with treating particle-local operators as compatible with indistinguishability.Source: §3.1.1 · pp. 12–13

3.1.2 Effective distinguishability

  • With particles confined to disjoint L/R regions, some entangled-I states correspond to distinguishable-particle separable states, so entanglement-I does not recover distinguishable-particle entanglement.Source: §3.1.2 · p. 13
  • Applying U ⊗U can yield an entangled-I state even though it preserves the form of separable-I states, indicating that the entanglement was already present after spatial distinction.Source: §3.1.2 · p. 13
  • Preparations of entangled states for effectively distinguished particles rely on inter-particle interactions, suggesting entanglement stems more from interactions than from symmetrization.Source: §3.1.2 · pp. 13–14

3.1.3 Information resources

  • The reviewed evidence shows no conflict with the information-resource criterion, but operational applications of entanglement-I are restricted to interferometric phase estimation.Source: §3.1.3 · p. 14

3.2 Entanglement-II

Entanglement-II treats symmetrized or antisymmetrized products of single-particle states as separable when their constituents are pairwise equal or orthogonal. The review finds that this particle-based approach conflicts with locality and effective distinguishability, and does not account for some information-processing advantages.

3.2 Entanglement-II

  • Entanglement-II counts (anti-)symmetrized products as separable when single-particle states are pairwise equal or orthogonal.The approach was motivated partly by limits on using some symmetrization-entangled states for teleportation.Source: §3.2 · p. 14

3.2.1 Local operators

  • Entanglement-II is incompatible with identical-particle locality: product expectations factorize only when one of the two operators is proportional to the identity.The approach’s implicit locality criterion refers to particles.Source: §3.2.1 · p. 15

3.2.2 Effective distinguishability

  • Although initially appearing compatible with effective distinguishability in a one-particle-per-region setting, particle-local projection can create entanglement-II for some internal states.The operation leaves internal degrees of freedom untouched, and the review identifies their independence from spatial degrees of freedom as a further inconsistency.Source: §3.2.2 · pp. 16–18
  • Post-selecting double detections at two locations would yield entanglement-II solely because additional internal degrees of freedom exist.Without those internal degrees of freedom, the corresponding projector is onto a separable-II state and never generates entanglement-II.Source: §3.2.2 · p. 18

3.2.3 Information resources

  • Entanglement-II identifies teleportation resources but does not explain phase-estimation quantum advantage when particle non-locality is absent.The cited interferometric example is a separable-II state whose interferometer action cannot generate entanglement-II.Source: §3.2.3 · p. 18

3.3 Entanglement-III

Entanglement-III relates identical-particle correlations to distinguishable-particle entanglement by projecting onto sectors with fixed particle numbers in orthogonal subspaces. The review finds agreement with particle locality and effective distinguishability in those sectors, but says the approach can miss quantum effects outside them.

3.3 Entanglement-III

  • Entanglement-III relates identical-particle correlations to distinguishable-particle entanglement, motivated by transmitting information to registers of distinguishable qubits.Its locality criterion is tied to particles and to orthogonal sectors containing fixed numbers of particles.Source: §3.3 · pp. 18–19

3.3.1 Local operators

  • For fixed particle-number sectors, separability-III agrees with particle locality: local-operator expectations factorize exactly when the projected states are separable-III.The chosen commuting subalgebras support this agreement for generic states.Source: §3.3.1 · p. 20

3.3.2 Effective distinguishability

  • When particles are effectively distinguished by spatial localization, entanglement-III recovers distinguishable-particle separability and analogous entanglement-monotone results.The review describes this as consistency with effective distinguishability through freezing suitable number degrees of freedom.Source: §3.3.2 · pp. 20–21

3.3.3 Information resources

  • Projection loses quantum coherence, and two-level protocols can show quantum-enhanced performance while their states remain separable-III.The cited protocols include phase estimation, teleportation, entanglement distillation, data hiding, and Bell inequalities; the review says entanglement-III is inadequate for protocols that do not leave the projected subspaces invariant.Source: §3.3.3 · p. 21

3.4 Entanglement-IV

Entanglement-IV defines correlations relative to a chosen single-particle subspace, but it fails the local-operations criterion: particle-local measurements can generate states it calls entangled. Its effective-distinguishability behavior is compatible with standard entanglement, while its information-resource implications expose a contradiction.

3.4 Entanglement-IV

  • The entanglement-IV properties depend on the choice of K, although they are invariant under a change of orthonormal basis within K.The reduced single-particle density matrix X1 is not a standard one-particle reduced density matrix: its single-particle observable expectations need not reproduce those of the original two-particle pure state.Source: §3.4 · p. 22
  • Entanglement-IV is defined relative to correlations between a chosen single-particle subspace K and its orthogonal complement.The construction uses operators that map the two-particle (anti-)symmetrized space to the single-particle Hilbert space; their adjoints are creation and annihilation operators and do not depend on particle labels.Source: §3.4 · p. 22

3.4.1 Local operators

  • For any K, the bosonic state |φ⟩⊗|φ⟩ is separable-IV, yet some commuting observables in an algebraic bipartition have nonfactorizing correlations in that state.Thus the locality criterion fails for this definition.Source: §3.4.1 · p. 23

3.4.2 Effective distinguishability

  • When spatial labels effectively distinguish particles, separability-IV includes the effectively distinguished separable states; the argument generalizes to other forms of effective distinguishability.This is the section’s stated compatibility with effective distinguishability.Source: §3.4.2 · p. 23

3.4.3 Information resources

  • Particle-local measurements can transform a separable-IV state into an entangled-IV state with probability 1/2, violating consistency with particle-local operations.The resulting resource would let an inherently quantum information protocol proceed using only local operations, classical communication, and a separable-IV state.Source: §3.4.3 · p. 24
  • The four particle-based definitions use first quantization and focus on correlations among particles; the review next turns to second quantization for a physically consistent definition.The section’s information-resource discussion also reports an entanglement-swapping application of entanglement-IV.Source: §3.4.3 · pp. 23–24

4 Mode-partitioning

Mode-partitioning defines entanglement through correlations between groups of orthogonal modes rather than between particles. The approach extends local operations and classical communication to modes, recovers standard separability under effective distinguishability, and supports quantum-enhanced information tasks.

4 Mode-partitioning

  • Mode-local operators are products of operators from the two mode subsets, and mode separability is defined by factorization of correlations across those subsets.For bosons, orthogonal mode subsets commute; fermionic locality must account for CAR, with choices involving even operators or microcausality and a parity superselection rule.Source: §4 · pp. 24–25
  • Entanglement-V concerns quantum correlations between occupancies of orthogonal modes, which can be individually addressed in experiments, but it depends on the chosen mode partition.The framework extends to multipartitions; it can also be extended from discrete to continuous sets of modes.Source: §4 · pp. 25–26

4.1 Local operators

  • Entanglement-V does not increase under mode-local operations and classical communication, extending the distinguishable-particle result by replacing particle-local operators with mode-local ones.The framework also generalizes the Schmidt decomposition to pure states.Source: §4.1 · p. 26

4.2 Effective distinguishability

  • Under effective distinguishability, mode separability reduces to standard separability for distinguishable particles, and corresponding entanglement monotones reduce to standard expressions.The mode bipartition also extends beyond states representable as distinguishable-particle states when local operators can change particle number.Source: §4.2 · p. 27

4.3 Information resources

  • Entanglement-V supports quantum-enhanced information protocols using mode-local operations and classical communication, while mode-nonlocal interferometers can generate the entanglement needed to exceed classical performance.Examples listed include teleportation, distillation, data hiding, Bell-inequality checks, and interferometric phase estimation.Source: §4.3 · pp. 27–28

5 Outlook

The review tests definitions of identical-particle entanglement against three consistency criteria. Particle-based definitions each fail at least one, whereas mode-based entanglement satisfies all three within the review’s setting.

5 Outlook

  • All formulations treating entanglement as a property of particles fail at least one of the three consistency criteria, while entanglement-V conforms to all three.The criteria concern non-local correlations, agreement with standard entanglement under effective distinguishability, and identifying entanglement as necessary to outperform classical performance absent other quantum resources.Source: §5 · p. 28
  • The review uses nonfactorization of correlation functions involving commuting observables as its guiding characterization of non-locality.The framework is presented as applicable to both distinguishable and indistinguishable particles, yielding the standard notion for distinguishable particles.Source: §5 · p. 28

A Identical particle effective distinguishability

The appendix sketches an isomorphism between suitable Fock subspaces of identical particles and tensor-product Hilbert spaces. It uses orthogonal external states to distinguish particles and restricts accessible observables to preserve that distinction.

A Identical particle effective distinguishability

  • For N ≥ 3 identical particles, the appendix sketches an isomorphism between suitable Fock subspaces and tensor-product Hilbert spaces.It introduces permutation operators and symmetrization or antisymmetrization as part of the construction.Source: A Identical particle effective… · pp. 28–29
  • Orthogonal external states can effectively distinguish identical particles by projecting onto the subspace associated with those states, which is mapped unitarily to a tensor-product Hilbert space.The setup assumes an external/internal tensor-product structure for each single-particle Hilbert space.Source: A Identical particle effective… · pp. 29–30
  • To preserve effective distinguishability, accessible observables must commute with the distinguishing projector; under the isomorphism, symmetrized operators act as tensor-product local operators.The appendix’s Table 1 summarizes the consistency test, with symbols indicating compliance, noncompliance, or no known evidence of violation.Source: Table 1 · p. 29, A Identical particle effective… · p. 30

B Entanglement-IV

Entanglement-IV is evaluated relative to a chosen single-particle subspace, with the examples considering spin within a spatial region. Its relation to other entanglement definitions depends on whether particles occupy the same or non-overlapping regions.

B Entanglement-IV

  • For the product state |L, ↑⟩⊗|L, ↑⟩, X1 is pure and S(X1)=0, so the state is separable-IV.Source: B Entanglement-IV · p. 30
  • For particles in non-overlapping regions, entanglement-IV coincides with entanglement-III, and the displayed state is entangled unless its coefficients factorize.The entropy S(X1) is −μ↑log μ↑−μ↓log μ↓; coefficient factorization also distinguishes separable-III from entangled-III states.Source: B Entanglement-IV · p. 30
  • Changing the orthonormal basis in K does not change X1.Source: B Entanglement-IV · p. 31

B.1 Local operators

The locality condition in Definition 6 fails for separable-IV states. Factorization tests can also depend on the chosen bipartition and internal state.

B.1 Local operators

  • A separable-I pure state is also separable-IV, yet fails local-expectation factorization, so the locality condition fails for Definition 6.Source: §B.1 · p. 31
  • For |L, ↑⟩⊗|L, ↑⟩, factorization holds only when α=↑, whereas for |L, +⟩⊗|L, +⟩ it holds only when α=+; neither is simultaneously separable.Source: §B.1 · p. 31
  • For a spatially localized bipartition and an arbitrary internal state |α⟩, the factorization condition is not generally satisfied.Source: §B.1 · p. 32
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