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Inaccessible entanglement in symmetry protected topological phases

Caroline de Groot, David T. Stephen, Andras Molnar, Norbert Schuch

arXiv:2003.06830v1quant-phcond-mat.str-el

TL;DR

The paper asks how enforcing symmetry changes entanglement distillation in SPT phases, a connection motivated by their dependence on global entanglement patterns. It defines symmetry-respecting G-LOCC and inaccessible entanglement, then shows that non-trivial phases retain inaccessible entanglement and that MNC phases attain log(|G|).

  • Problem

    The connection between entanglement under superselection rules and SPT phases, which are characterized by global entanglement patterns, had not yet been explored.

    Source: §I · p. 1

  • Method

    The paper defines G-LOCC as symmetry-respecting local operations and classical communication and studies inaccessible entanglement through distillation.

    Source: §I · p. 2, §VII · p. 14

  • Results

    In MNC phases, every state has Einacc = log(|G|), while random states across trivial and non-trivial phases typically have near-maximal inaccessible entanglement.

    Source: §VII · p. 14

  • Takeaways & Limitations

    Irrep probabilities distinguish trivial from non-trivial phases through an exact degeneracy in non-trivial phases, even though inaccessible entanglement alone is difficult to use for that distinction.

    Source: §VII · p. 14

Abstract

from arXiv · show

We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group $G$, which we call $G$-LOCC, and we define the inaccessible entanglement $E_{inacc}$ as the entanglement that cannot be used for distillation under $G$-LOCC. We derive a tight bound on $E_{inacc}$ which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely $\log(D_\omega^2) \leq E_{inacc} \leq \log(|G|)$, where $D_\omega$ is the topologically protected edge mode degeneracy of the SPT phase $\omega$ with symmetry $G$. For particular phases such as the Haldane phase, $D_\omega = \sqrt{|G|}$ so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of $E_{inacc}$ to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.

I. INTRODUCTION

The paper connects symmetry-restricted entanglement distillation to one-dimensional SPT order, showing that non-trivial phases retain entanglement inaccessible under symmetry-respecting operations. It introduces the tensor-network framework used to describe the symmetry and entanglement structure of these states.

I. INTRODUCTION

  • The motivation is an unexplored connection between entanglement under superselection rules and SPT order, whose properties depend on global entanglement patterns and symmetry.SPT order is distinguished from topological order here as short-range rather than long-range entanglement.Source: §I · pp. 1–2
  • Non-trivial one-dimensional SPT phases contain entanglement inaccessible under G-LOCC, and the paper relates its amount to the SPT phase.The work defines inaccessible entanglement operationally through distillation and derives a bound for it.Source: §I · p. 2
  • Random MPS states typically have near-maximal inaccessible entanglement, while irrep probabilities can distinguish trivial from non-trivial phases.The paper also studies effective reduced symmetry and reduced G-LOCC; accessible entanglement depends on the MPS junk-subspace dimension.Source: §I · p. 2

II. INTRODUCTION TO SPT PHASES WITH TENSOR NETWORKS

  • The analysis restricts to finite Abelian symmetries; the Haldane phase remains non-trivial under its Z2 × Z2 subgroup, also called the cluster phase.The paper notes that many phases protected by other groups remain non-trivial when a finite Abelian subgroup is enforced.Source: §II · p. 2

A. Introduction to tensor networks

  • Tensor networks encode global state properties through local tensors; for one-dimensional MPS, the projective symmetry action on virtual indices encodes entanglement and topological properties.The framework considers injective MPS, for which the transfer matrix has a unique largest-magnitude eigenvalue.Source: A. Introduction to tensor networks · pp. 2–3
  • The transfer operator is treated as a quantum channel whose right and left eigenvectors correspond to fixed points of the channel and its adjoint.Figure 1 also depicts the transfer-operator symmetries, the left fixed point, and a trace calculation for irrep probabilities.Source: A. Introduction to tensor networks · p. 3, Figure 1 · p. 3

B. Classification of SPT phases

One-dimensional SPT order is classified through projective symmetry actions, and finite Abelian groups give a particularly simple structure for their projective irreducible representations. Their shared dimension is denoted Dω.

B. Classification of SPT phases

  • SPT phases are equivalence classes of short-range-entangled states connected by finite-depth circuits that respect the protecting symmetry.For one-dimensional systems, their classification is given by the second cohomology group H2(G, U(1)).Source: B. Classification of SPT phases · p. 3
  • The discussion focuses on bosonic SPT order at zero temperature, while noting the broader group-cohomology classification and models beyond it.The broader classification stated is by Hd+1(G, U(1)) for bosons in d dimensions.Source: B. Classification of SPT phases · p. 3

C. Projective irreps of Abelian groups

  • For a finite Abelian group, all projective irreps with a given cocycle have the same dimension, denoted Dω.The projective representation can be simplified to a tensor-product form by choosing an appropriate basis.Source: C. Projective irreps of Abelian groups · p. 4
  • The projective centre kω consists of group elements whose cocycle factors commute with those of every group element.The paper introduces kω to calculate Dω.Source: C. Projective irreps of Abelian groups · p. 4

1. Maximal non-commutativity

Maximally non-commutative phases have maximal protected edge degeneracy and saturate the upper bound on inaccessible entanglement. Edge degeneracy alone does not distinguish every such phase.

1. Maximal non-commutativity

  • A maximally non-commutative phase has trivial projective centre and the maximum protected edge degeneracy for its symmetry group.For these phases, Dω takes its maximum value, √|G|.Source: §1 · p. 4
  • The trivially transforming part of an MPS tensor is called the junk subspace, whose size determines bond dimension and bounds entanglement entropy.In non-MNC phases, multiple such subspaces can occur because there are multiple projective irreps.Source: §1 · p. 4
  • Some symmetry groups have multiple MNC phases that edge-mode degeneracy cannot distinguish; order parameters can distinguish them by probing the virtual action.The paper points to order parameters in Refs..Source: §1 · p. 4

III. ACCESSIBLE ENTANGLEMENT DISTILLATION

The paper defines symmetry-respecting G-LOCC and derives accessible entanglement through distillation. Because these operations do not mix symmetry sectors, only entanglement within individual sectors contributes.

III. ACCESSIBLE ENTANGLEMENT DISTILLATION

  • G-LOCC restricts local operations and classical communication to operations that commute with the global onsite symmetry.The paper uses this setting to ask how much entanglement can be extracted from a state obeying G.Source: §III · pp. 4–5
  • Under G-LOCC, accessible entanglement is the probability-weighted sum of entanglement within the individual symmetry sectors.The paper motivates and derives this expression through multi-shot entanglement distillation.Source: §III · pp. 4–5
  • Operations that cannot mix symmetry sectors erase entanglement between sectors, so accessible entanglement is no greater than total entanglement.The expression also ensures that product states have zero accessible entanglement.Source: §III · p. 5

A. Entanglement distillation under G-LOCC

The paper defines G-LOCC distillation by resolving the state into symmetry sectors and distilling within each sector. It then expresses the sector probabilities using a tensor-network formulation of the symmetry measurement.

A. Entanglement distillation under G-LOCC

  • In the asymptotic limit, the conversion ratio from many copies of a pure state to Bell pairs equals its entanglement entropy under LOCC.Distillation is described as converting N copies into M Bell pairs, with M/N determined by the entropy.Source: A. Entanglement distillation under… · p. 5
  • G-LOCC distillation measures the symmetry sectors and distills Bell pairs within each sector, weighting each branch’s entanglement by its probability.The operations are optimal within each sector, so the full distillation is optimal under the stated restriction.Source: A. Entanglement distillation under… · p. 5

B. Symmetry measurement

  • The symmetry measurement is rewritten on the MPS virtual system, where cancellations leave an effective symmetry action at the boundaries.The derivation uses a ring in the infinite-site limit and the fixed point of the transfer matrix.Source: B. Symmetry measurement · pp. 5–6
  • In right-canonical form, the right fixed point is the identity, and the trace contribution used in the calculation is illustrated in Fig. 1.The symmetry projector acts on subsystem A, leaving the fixed-point action on the other half after tracing it out.Source: B. Symmetry measurement · p. 6

C. Inaccessible entanglement

Inaccessible entanglement is the entanglement that symmetry-restricted G-LOCC prevents from being accessed, and it measures entanglement protected by symmetry. The section gives it as the entropy of the symmetry-sector probabilities and establishes nonzero lower bounds for non-trivial SPT order.

C. Inaccessible entanglement

  • Inaccessible entanglement is the amount of entanglement that G-LOCC prevents access to.It measures entanglement protected by symmetry and stored in the state.Source: C. Inaccessible entanglement · p. 6
  • E_inacc is the base-2 entropy of the irrep probability distribution, E_inacc = S({pα}).The pα are the probabilities associated with the irreducible representations of the symmetry group.Source: C. Inaccessible entanglement · p. 6
  • Because E ≥ E_inacc, entanglement is a necessary condition for inaccessible entanglement.The derivation obtains inaccessible entanglement by subtracting accessible entanglement from the state’s entanglement entropy.Source: C. Inaccessible entanglement · p. 6

IV. BOUNDS FOR THE INACCESSIBLE ENTANGLEMENT

  • Product states and states without symmetry have zero inaccessible entanglement, while non-trivial SPT order has a nonzero lower bound.The bound is specified using the protecting group G and the projective centre k associated with the cocycle.Source: §IV · p. 6

A. Maximal inaccessible entanglement in the MNC phase

Maximally non-commutative (MNC) phases attain maximal inaccessible entanglement, while the generalization to non-MNC phases establishes a nonzero lower bound. The lower-bound argument uses degeneracies among irrep probabilities tied to the projective centre.

A. Maximal inaccessible entanglement in the MNC phase

B. Generalisation of bounds to non-MNC phases

  • The generalization to non-MNC phases establishes a nonzero lower bound on inaccessible entanglement.The proof simplifies the irrep probabilities using symmetry of the fixed point and then bounds entropy configurations.Source: B. Generalisation of bounds to non-MNC… · p. 7
  • For non-MNC phases, irrep probabilities depend on character values in the projective centre, producing degeneracies used to derive the lower bound.Each distinct probability occurs |G|/|k| times; the minimum-entropy configuration among these probabilities yields the bound.Source: B. Generalisation of bounds to non-MNC… · pp. 7–8

C. Comparison with string order parameter

The paper relates the irrep probabilities underlying inaccessible entanglement to string order parameters. It explains that inaccessible entanglement alone does not capture all phase information because string order includes additional end operators.

C. Comparison with string order parameter

  • The symmetry-sector probabilities are the Fourier transform of the string order parameter with identity end operators.String order uses carefully chosen end operators that detect a selection rule.Source: C. Comparison with string order… · p. 8
  • String order can distinguish different MNC phases because its selection rule changes at the phase transition.The operators are chosen for the symmetry and projective representation.Source: C. Comparison with string order… · p. 8
  • Inaccessible entanglement is not itself a full order parameter because it lacks the additional structure supplied by string-order end operators.The missing end operators mean it cannot capture the full information of the phase.Source: C. Comparison with string order… · p. 8

V. INVESTIGATION OF THE BOUNDS

The numerical investigation probes how states populate the inaccessible-entanglement bounds and constructs a symmetry-preserving path spanning them. Filtering selected symmetry sectors provides a controlled way to tune states along that path.

V. INVESTIGATION OF THE BOUNDS

  • The investigation asks whether states can span the bounds, where they typically lie, and how bond and physical dimensions affect inaccessible entanglement.It also examines constructions on both bounds and their SPT-phase and entanglement properties.Source: §V · p. 8
  • Tuning λ filters support from a selected symmetry sector, smoothly changing the state's effective symmetry while each MPS remains a ground state of H(λ).The basis used for filtering diagonalises the physical symmetry action u(g).Source: §V · p. 8
  • Random MPS tensors are symmetrised using physical and projective symmetry representations to generate families of states in chosen SPT phases.The reported figures display samples of 10^6 states.Source: §V · p. 8

A. Tight bound for the inaccessible entanglement

  • A toy trivial-phase MPS interpolation connects the inaccessible-entanglement upper and lower bounds, demonstrating that the bound cannot be tightened for the trivial phase.The construction starts from a maximally entangled state with zero accessible entanglement and ends at a product state.Source: A. Tight bound for the inaccessible… · p. 9
  • For Z2 × Z2, filtering the trivial irrep connects a state with Einacc = 2 to a product state with Einacc = 0.The interpolation passes through equal irrep probabilities at the entangled endpoint and probabilities concentrated entirely on the trivial irrep at the product endpoint.Source: A. Tight bound for the inaccessible… · p. 9

B. Typical behaviour in the MNC and trivial phase

The cluster and MNC phases have fixed, maximal inaccessible entanglement, while trivial-phase states have less restricted irrep probabilities and can be tuned toward a product state. Irrep probabilities distinguish the phase structures even when their entanglement values are similar.

B. Typical behaviour in the MNC and trivial phase

  • The cluster state has pα = 1/4 for every irrep and Einacc = 2, the Z2 × Z2 upper bound.Its accessible entanglement is zero; the factor of two in its entanglement comes from cutting two bonds under periodic boundary conditions.Source: B. Typical behaviour in the MNC and… · p. 9
  • In the cluster phase, inaccessible entanglement is invariant across states, depending on the symmetry-action part rather than correlations in the junk subspace.A larger bond dimension can permit additional entanglement through a non-trivial junk subspace, including non-zero accessible entanglement.Source: B. Typical behaviour in the MNC and… · pp. 9–10
  • Trivial-phase states allow unrestricted irrep probabilities, whereas MNC states have a fixed degenerate distribution; random states in both phases are typically near the upper bound.Tuning toward a product state lowers inaccessible entanglement on average in the trivial phase, while the MNC value remains fixed.Source: B. Typical behaviour in the MNC and… · pp. 9–10
  • Irrep probabilities can distinguish typical trivial states from MNC states, even when their inaccessible entanglement is similar.The fixed value of inaccessible entanglement alone cannot distinguish mathematically inequivalent MNC phases protected by the same group.Source: B. Typical behaviour in the MNC and… · p. 10

C. Exploring a non-MNC phase

For the non-MNC phase of Z4 × Z2, inaccessible entanglement occupies a restricted interval and can be tuned toward its lower bound by reducing the effective symmetry. Irrep-probability structure distinguishes trivial and non-trivial phases more reliably than typical inaccessible-entanglement values.

C. Exploring a non-MNC phase

  • For random states near λ = 1, both trivial and non-MNC phases typically lie near the upper bound, with little difference in inaccessible entanglement.Increasing bond dimension increases the mean and variance of inaccessible entanglement, with substantial dependence on junk-subspace structure.Source: C. Exploring a non-MNC phase · p. 10, §1 · p. 11

1. Reduced effective symmetries

  • As λ approaches zero, filtering drives non-MNC states toward Einacc = 2 by enforcing an effective Z2 × Z2 symmetry.At λ = 0 the symmetry is actually enforced, but those non-injective states are excluded from the bound analysis.Source: §1 · pp. 10–11

2. Comparing trivial to non-trivial phase

  • Near-maximal inaccessible entanglement makes typical trivial states hard to distinguish from non-trivial states, but their irrep probabilities retain different structure.Trivial states may have eight distinct probabilities; non-MNC states show four-fold degeneracy, which becomes more apparent as filtering approaches the effective sub-symmetry.Source: §2 · pp. 11–12
  • Non-MNC states are bounded by 2 ≤ Einacc ≤ 3, while trivial states can approach zero inaccessible entanglement.As filtering increases, non-MNC states cluster above their lower bound, whereas trivial states approach it from below.Source: §2 · p. 11
  • Non-MNC irrep probabilities are four-fold degenerate, while each irrep probability can differ in the trivial phase.The degeneracy reflects that only two inequivalent irreps contribute to the measurement probabilities.Source: §2 · pp. 11–12

3. Accessible entanglement in the junk subspace

The junk-subspace structure shapes how total entanglement is divided into accessible and inaccessible parts, while restricting the enforced symmetry can reduce inaccessible entanglement.

3. Accessible entanglement in the junk subspace

  • For MNC phases, a trivial junk subspace gives Eacc = 0, while a non-trivial one permits accessible entanglement to grow without changing Einacc.Increasing the junk subspace through higher bond dimension can allow more entanglement, but Einacc depends on the part transforming non-trivially under the symmetry.Source: §3 · p. 12
  • As filtering λ approaches zero, both phases’ irrep probabilities increasingly resemble the MNC Z2 × Z2 degeneracy, although the non-MNC phase remains four-fold degenerate at every filtering value.The trivial phase generally gives different probabilities to each irrep.Source: Figure 6 · p. 12
  • Unequal irrep multiplicities alter the junk-subspace parameter count and spread the possible inaccessible entanglement, while larger bond dimension raises total entanglement but leaves Einacc capped at 3.At equal total bond dimension, the [1, 5] and [3, 3] families have 26 and 18 junk-subspace parameters, respectively.Source: §3 · pp. 12–13

A. Restricting symmetry to a subgroup

  • For an MNC state, restricting operations from G to a subgroup H gives Einacc = log(|H|), so smaller enforced symmetry groups make more entanglement accessible.A state that is SPT-trivial under H can still have maximal inaccessible entanglement due to its MNC order under G.Source: A. Restricting symmetry to a subgroup · p. 13

B. Subsystem SPT order

The paper extends its results to subsystem SPT phases and examines connections between inaccessible entanglement, computational power, and hidden symmetry-protected order. Its conclusions also identify a scope boundary for the bounds.

B. Subsystem SPT order

  • For a 2D cluster state on a cylinder, dimensional reduction gives Einacc = N, equal to all its entanglement under the stated subsystem symmetries; the broader subsystem SPT family obeys an area law.The effective 1D symmetry is G = (Z2 × Z2)^N.Source: B. Subsystem SPT order · p. 13

C. Relation to computational power

  • In 1D MNC phases, maximal Einacc coincides with phases universal for MBQC, but measure-zero nonuniversal subsets remain under the analyzed scheme.The paper notes that a different MBQC scheme could have a different set of nonuniversal states.Source: C. Relation to computational power · p. 14
  • For non-MNC phases, random universal and non-universal states showed no clear differentiating Einacc behavior, leaving its relation to computational power unclear beyond MNC.Source: C. Relation to computational power · p. 14

VII. CONCLUSION AND OUTLOOK

  • For 1D systems with finite Abelian onsite symmetry, non-trivial SPT phases have inaccessible entanglement bounded below by a nonzero phase-dependent value and above by log(|G|); MNC phases reach the upper bound.The cluster states have zero accessible entanglement under suitable G-LOCC.Source: §VII · p. 14
  • Numerically, states across the full bound are attainable and typical states lie near maximal Einacc, even in the trivial phase; irrep probabilities distinguish phases through exact degeneracy.States near the lower bound have low weight in certain symmetry sectors and therefore an effective subgroup symmetry.Source: §VII · p. 14
  • Inaccessible entanglement can retain information about SPT order under an original larger symmetry even when restricting to a subgroup makes the state SPT-trivial.Source: §VII · pp. 14–15
  • The study covers 1D SPT phases and 2D subsystem SPT phases via dimensional reduction; bounds for 2D SPT phases with global symmetries remain unclear.Source: §VII · p. 15
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