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Symmetry protected topological phases under decoherence
Jong Yeon Lee, Yi-Zhuang You, Cenke Xu
TL;DR
The paper addresses how to identify topological information in SPT states after decoherence produces mixed states. It generalizes strange correlators and uses a doubled-Hilbert-space formulation to characterize these ensembles. Its calculations show that type-I and type-II correlators can retain distinct SPT information under decoherence.
Problem
A general definition and diagnosis of symmetry-protected topological ensembles, and their information-theoretic capacity under decoherence, remain lacking.
Source: §1 · p. 1Method
The paper generalizes type-I and type-II strange correlators and maps mixed-state density matrices to pure states in a doubled Hilbert space using the Choi-Jamiołkowski isomorphism.
Source: §4.1 · p. 15, §6 · p. 25Results
Exact lattice and field-theory calculations show that selective decoherence can preserve nontrivial SPT features in both strange correlators, with type-I linked to efficient probing and type-II encoding the mixed state’s full topological response.
Source: §1 · pp. 2–3Takeaways & Limitations
Type-I correlators provide a route to experimentally probing and utilizing SPT information, including preparation of long-range-entangled states, while type-II correlators capture information about the SPT state before decoherence.
Source: §1 · pp. 2–3Verifying the doubled-space ordering described can require exponentially large computational effort.
Source: §4.3 · p. 19
Abstract
from arXiv · showhide
We study ensembles described by density matrices with potentially nontrivial topological features. In particular, we study a class of symmetry protected topological (SPT) phases under various types of decoherence, which can drive a pure SPT state into a mixed state. We demonstrate that the system can still retain the nontrivial topological information from the SPT ground state even under decoherence. In the "doubled Hilbert space", we provide a general definition for symmetry protected topological ensemble (SPT ensemble), and the main quantity that we investigate is various types of (boundary) anomalies in the doubled Hilbert space. We show that the notion of the strange correlator, previously proposed to as a diagnosis for the SPT ground states, can be generalized to capture these anomalies in mixed-state density matrices. Using both exact calculations of the stabilizer Hamiltonians and field theory evaluations, we demonstrate that under decoherence the nontrivial features of the SPT state can persist in the two types of strange correlators: type-I and type-II. We show that the nontrivial type-I strange correlator corresponds to the presence of the SPT information that can be efficiently identified and utilized from experiments, such as for the purpose of preparing for long-range entangled states. The nontrivial type-II strange correlator encodes the full topological response of the decohered mixed state density matrix, i.e., the information about the presence of the SPT state before decoherence. Therefore, our work provides a unified framework to understand decohered SPT phases from the information-theoretic viewpoint.
1 Introduction
The paper asks how to identify and use SPT information when decoherence turns pure states into mixed ensembles. It generalizes strange correlators to mixed states and develops a doubled-Hilbert-space framework for their topological information.
1 Introduction
- The motivating gap is a lack of a general definition and diagnosis for SPT ensembles and limited understanding of the information capacity of underlying topological states under decoherence.The introduction identifies decoherence as prevalent in quantum devices and notes that prior information-theoretic studies focused mostly on pure SPT states.Source: §1 · p. 1
- Type-I strange correlators can indicate experimentally accessible SPT information, while type-II correlators encode the mixed state’s full topological response and ability to distinguish it from trivial states.The type-I correlator’s marginalized version can be probed with measurements and classical computation for a broad class of decorated-defect SPT states.Source: §1 · p. 3
- Selective decoherence can preserve nontrivial SPT features: type-II correlators may remain nontrivial when type-I correlators become short-ranged, while type-I can remain nontrivial at weak decoherence in the 2d cluster state.For the 2d cluster state, type-I becomes short-ranged beyond a critical decoherence strength.Source: §1 · p. 3
- The paper uses exact stabilizer-model calculations and field theory to study strange correlators, which reduce to boundary correlation functions with interactions coupling opposite boundaries.The field-theory treatment covers generic 1d and 2d SPT states.Source: §1 · p. 3
- The Choi-Jamiołkowski map sends mixed-state density matrices to pure states in a doubled Hilbert space, furnishing a definition of SPT ensembles and a decoding interpretation for type-II correlators.The type-II correlator is presented as a tool for decoding information transmitted through a noisy channel.Source: §1 · p. 3
- The paper relates type-I correlators to nonlocal order parameters, including Haldane-phase string order, and proposes an experimental probe.It establishes type-I correlators as an upper bound for nonlocal order parameters.Source: §1 · p. 3
2 Strange Correlators
This section examines strange correlators as diagnostics of SPT information in decohered density matrices, developing their symmetry framework and studying their behavior under different noise conditions and in cluster-state examples. It considers type-I and type-II correlators and whether they retain SPT information when conventional nonlocal order parameters may fail or become short-ranged.
2 Strange Correlators
- Because conventional nonlocal order parameters decay exponentially with their support under decoherence, the section introduces strange correlators as a way to quantify SPT features in mixed states.The correlators diagnose density matrices; the definition of an SPT ensemble is deferred to the doubled-Hilbert-space formalism.Source: §2 · p. 4
2.1 Basic formalism
- Local decoherence channels map pure states to mixed states, whose symmetry may be strong, acting separately on the density matrix’s left and right, or weak, acting by adjoint transformation.The section relates weak density-matrix symmetry to average symmetry of a state ensemble and distinguishes weakly from strongly symmetric channels.Source: §2.1 · p. 4
2.2 Review on Strange Correlator
- For pure SPT states, strange correlators of symmetry-charged operators are expected to approach a nonzero constant or decay as a power law, making them an SPT order parameter.They can be written as expectation values of an order-parameter operator in the SPT wave function.Source: §2.2 · p. 5
2.3 1d Cluster State
- In the 1d cluster-state example, the strange correlators take unit values, whereas they vanish for a trivial product state, illustrating their role as an SPT order parameter.The correlator is also connected to the Haldane-phase string order parameter.Source: §2.3 · p. 5
2.4 Decoherence
- Under even-site dephasing, type-I correlators decay exponentially for odd-sited operators while the even-sited correlator remains nontrivial; type-II correlators remain nontrivial for both sites.The type-I signal therefore does not capture the full mutual SPT response, motivating the type-II correlator as a more comprehensive diagnosis.Source: §2.4 · pp. 6–7
2.5 Away from the fixed point
- Away from the fixed point, decoherence makes type-I correlators decay exponentially with distance while type-II correlators remain robust, and both vanish at the transition h = 1.For even-sited operators, decoherence can increase the type-I correlator because it diffuses localized charged operators across the correlation length.Source: §2.5 · p. 7
2.6 Generic Noise Model
- With depolarization noise on all even sites, type-I and type-II correlators retain the behaviors found for the preceding noise model.The paper introduces this as a more generic decoherence channel.Source: §2.6 · pp. 7–8
- Under coherent multi-site noise, short-depth local unitaries can change the type-II strange-correlator magnitude; strong symmetry preserves SPT probes, while weak symmetry makes both correlators trivial unless higher-form symmetries yield emergent protection.The multi-site channel acts as stochastic short-depth local unitaries combined with local noise.Source: §2.6 · p. 8
2.7 2d Cluster State
- The passage reports B = 1 and a qualitative difference between CA and CB under decoherence; an expression involving ρ0, which acts trivially on vertices, is equivalent to the high-temperature expansion of an Ising correlation function.Source: §2.7 · p. 9
- A symmetric quantum circuit U commuting with a string of X acts nontrivially only on its endpoints, so within the same phase this order parameter is corrected by a constant amount; for the decohered mixed state, it is short-ranged.Source: §2.7 · p. 9
- With weakly symmetric vertex noise, the type-I strange correlator probing the 1-form symmetry decays by an area law and is short-ranged.This agrees with the stated 1d result after compactifying the system on a cylinder.Source: §2.7 · p. 10
3 Effective Field Theory Evaluation
The effective-field-theory treatment maps strange correlators to boundary correlations and examines how different decoherence channels affect them. Across 1D and 2D examples, type-I and type-II correlators can respond differently, and some decoherence settings leave both short-ranged.
3 Effective Field Theory Evaluation
- The field-theory construction treats decohered strange correlators as boundary correlation functions, with decoherence coupling the two boundaries.In the path-integral picture, decoherence couples the temporal boundaries; space-time rotation makes them opposite spatial boundaries.Source: Figure 3 · p. 10, §3 · p. 10
3.1 1d SPT states
- In 1D, decoherence favoring diagonal configurations cancels the two WZW terms, leaving type-I and type-II correlators short-ranged when symmetry-breaking terms make the boundary ground state nondegenerate.The analysis also suggests that symmetry-averaged decoherence of fermion bilinears can collapse fermionic TI and TSC classifications; fuller treatment is deferred.Source: §3.1 · p. 12
- With selective 1D decoherence, a boundary doublet can keep type-I correlators long-ranged while type-II correlators retain nontrivial information even when type-I vanishes on averaging.For each quantum trajectory, type-I fluctuates in sign while its magnitude stays robust; the squared order parameter, type-II, still captures SPT information.Source: §3.1 · pp. 12–13
3.2 2d SPT states
- Tuning K produces a transition marked by type-I becoming short-ranged, while θ+ remains gapless regardless of Hint’s renormalization-group fate, and type-II always decays as a power law, capturing pure-state SPT memory.Source: §3.2 · p. 14
3.3 Bosonic Integer Quantum Hall state
- For bosonic integer quantum Hall states, decoherence preserving only diagonal U(1) can make both type-I and type-II strange correlators short-ranged.The two interaction terms have scaling dimensions K and 1/K, so K can be chosen to gap both channels.Source: §3.3 · p. 14
4 Doubled System
The Choi-Jamiołkowski map represents a mixed-state density matrix as a pure state in a doubled Hilbert space, where SPT structure and strange correlators can be analyzed. Cluster-state examples show that topological features can persist under decoherence, with type-I and type-II correlators distinguishing different information-theoretic regimes.
4.1 Choi Representation of Decohered SPT States
- The Choi-Jamiołkowski isomorphism maps a density matrix to a pure state in the doubled Hilbert space, whose norm is set by the mixed-state purity.Applying the decoherence channel becomes an operator on the doubled state; for local commuting-projector Hamiltonians, the construction can yield a local parent Hamiltonian.Source: §4 · p. 15, §4.1 · pp. 15–16
4.1.1 1d cluster state
- In the 1D cluster-state example, the doubled parent Hamiltonian couples two SPT states through even-site charge tunneling, while ⟨Sodd(2r)⟩ remains unity for all r and decoherence strengths p.The coupling vanishes as p → 0 and diverges as p → 1/2.Source: §4.1.1 · pp. 16–17
4.1.2 2d cluster state
- For small decoherence, emergent one-form symmetries in the doubled 2D cluster state support anomaly inflow and a nontrivial type-I strange correlator.The authors connect robustness to extensively many conserved quantities, unlike the single globally conserved charge of a 0-form symmetry.Source: §4.1.2 · p. 17
4.2 SPT ensemble
- An SPT ensemble is a mixed state that maps to a nontrivial SPT state in the doubled Hilbert space, where generalized strange correlators diagnose its SPT features.The Choi representation relates type-II strange-correlator stability under selective decoherence to stability of the doubled-space SPT order.Source: §4.2 · pp. 17–18
4.3 Physical Implications
- Type-I and type-II strange correlators distinguish regimes in which SPT information is efficiently accessible, distinguishable in principle, or not discernible from the mixed state.A nontrivial type-I correlator means the physical boundary cannot be trivialized; a nontrivial type-II correlator implies the same for the doubled-space boundary, but requires boundary observables nonlinear in the density matrix.Source: §4.3 · pp. 18–19
- For any p > 0, verifying doubled-space ordering is exponentially hard, and type-II correlator measurement requires separate evaluation of exponentially small quantities.The purity of the decohered SPT state is also exponentially small.Source: §4.3 · p. 19
5 Experimentally probing the strange correlator
This section develops experimentally accessible protocols for probing strange-correlator signatures and related measurable order parameters, with and without post-selection. It examines how decoherence affects nonlocal order parameters and type-I strange correlators, including when decoding can recover useful SPT information and bounds on measurement-assisted long-range-state preparation.
5.1 Strange correlator against a generic trivial state
- Under Zeven decoherence, the 1D even type-I correlator remains unchanged while the odd correlator becomes short-ranged, and its sign structure persists beyond the stabilizer limit.A generic trivial reference state must have nonzero overlap with the SPT state, requiring matching total symmetry charge; its locally charged background does not change qualitative behavior.Source: §5.1 · p. 20
- Under 0-form decoherence, the type-I correlator’s sign structure remains robust; its magnitude can change with decoherence strength and is an RBIM correlation at inverse temperature β = tanh−1(1 − 2p), depending only on bond frustration.Source: §5.1 · p. 20
- A weighted average of type-I correlators, with weights set by reference-state overlaps, remains long-ranged at small decoherence because maximally frustrated references have extremely small weight.The type-I correlator can therefore depend on the chosen trivial reference state, motivating the overlap-weighted average as a diagnostic.Source: §5.1 · p. 20
5.2 With post-selection
- Post-selection makes type-II correlators estimable from relative frequencies of paired charge outcomes, while measuring even sites in the X basis yields an accessible marginalized type-I correlator.The marginalized type-I quantity is a weighted average of type-I correlators; without post-selection, estimating type-II from all possible outcomes is exponentially challenging. Classical-shadow fidelity estimation also requires exponentially many measurements because the fidelities are exponentially small.Source: §5.2 · pp. 21–22
5.3 Without post-selection
- Post-selection becomes experimentally impractical as system size grows because the number of samples required increases exponentially, motivating a protocol without post-selection.Source: §5.3 · p. 22
5.3.1 Without decoherence
- In the pure SPT state, a measurable string order parameter equals the marginalized type-I correlator and can reveal its nontrivial behavior without post-selection using polynomially many measurement snapshots.The protocol measures even sites in the X basis and odd sites in the Z basis; in higher dimensions, the corresponding non-local order parameter exhibits a perimeter law in the SPT phase.Source: §5.3.1 · p. 23
- For GB of form degree d−1, the associated nonlocal order parameter is finite in the SPT phase; for degree d−2, it decays with boundary length rather than area, extending earlier results to these SPT states.Source: §5.3.1 · p. 23
5.3.2 With decoherence
- With weak decoherence, the nonlocal order parameter becomes short-ranged while the type-I strange correlator can remain nontrivial, because outcome-dependent signs cancel in its weighted average.Computational assistance can decode those signs; an efficient decoder extracting a nontrivial observable guarantees a nontrivial SPT ensemble, while short-ranged type-I correlators bound efficiently extractable topological information.Source: §5.3.2 · p. 24
- With an identity decoding channel, a nonlocal SPT order parameter lower-bounds the marginalized type-I strange correlator, but the correlator can remain nontrivial when that order parameter is short-ranged.The proposed procedure measures GB charges, decodes the sign based on outcome sB, and transforms the remaining qubits before measuring the associated observable.Source: §5.3.2 · p. 24
5.4 Stability of measurement-assisted state preparation under decoherence
- Under decoherence, efficient SPT diagnosis and measurement-assisted long-range-state preparation are possible only if the marginalized type-I strange correlator is nontrivial.For the 2d cluster-state SPT under weak 1-form-symmetry decoherence, this correlator can remain nontrivial at small p, though the theorem does not give a decoding protocol; Ref. [24] demonstrates an efficient one.Source: §5.4 · p. 25
6 Conclusion and Outlook
The paper frames type-I and type-II strange correlators and the doubled Hilbert space as a unified way to diagnose topological information in selectively decohered SPT ensembles. It identifies further directions in classifying mixed states and studying decohered fermionic and gapless systems.
Conclusion
- Under selective decoherence, type-I and type-II strange correlators diagnose persistent SPT information, while the doubled-space mapping explains type-II robustness and distinguishes experimentally accessible information from the full topological response.The mapping treats the mixed state as a pure state in the doubled Hilbert space and places it under a shallow-depth symmetric circuit.Source: §6 · p. 25
Outlook
- The Choi-Jamiołkowski mapping may support classifying density matrices by the quantum phases of their Choi states, subject to Hermiticity, positive semi-definiteness, and average symmetry.The authors describe this as a speculative direction for quantifying density-matrix information.Source: §6 · p. 25
- A systematic study of decohered fermionic topological insulators and superconductors is left for future work, where different decoherence channels may produce rich physics.The authors also suggest decoherence could lead to symmetric mass generation or collapsed classifications.Source: §6 · p. 26
- Decohered gapless quantum states, including quantum critical points, are identified as another direction for exploration.The passage highlights critical points whose bulk is an SPT state as a setting involving two boundary effects.Source: §6 · p. 26
A More example
This example extends strange-correlator analysis beyond Z2 symmetry to a one-dimensional Z3 × Z3 SPT, then describes its phases, string order, and possible decoherence behavior.
A More example
- Strange correlators are demonstrated as detectors of a one-dimensional SPT with Z3 × Z3 symmetry, extending their use beyond Z2 symmetries.The local Hilbert space has dimension three, with Z and X obeying ZX = ωXZ.Source: A More example · p. 34
- The Z3 × Z3 classification has three SPT phases labeled m = 0, 1, 2, with m = 0 denoting the trivial paramagnet.The passage identifies the second group cohomology as H2(Z3 × Z3, U(1)) = Z3.Source: A More example · p. 34
- The fixed-point Hamiltonian has commuting terms, and its ground state has unit expectation values for the stated non-local string order parameters at every r.The cited passage describes these as string order parameters characterizing the ground state.Source: A More example · p. 34
- A decoherence channel respecting doubled symmetry preserves the string-order expectation values, and the construction is stated to generalize to Zn × Zn.For the generalization, the passage specifies ω = e2πi/n and m ∈ {0, 1, ..., n − 1}.Source: A More example · pp. 34–35
B Lemma
The lemma constrains when operators on either sublattice of a cluster SPT can have nonzero expectation values, linking them to the corresponding symmetry.
Lemma
- For the stated cluster SPT with mixed topological response, an operator restricted to one sublattice can have nonzero expectation only if it contains a symmetry action on that sublattice.The two symmetries act on different sublattices.Source: B Lemma · p. 35
Proof
- The proof decomposes an operator into terms carrying stabilizer quantum numbers and rules out terms with nontrivial quantum numbers because their expectation values vanish.The reasoning uses the ability to pull a stabilizer out of the ground state and absorb it back.Source: B Lemma · p. 35
- The surviving operator on the first sublattice is a symmetry in G1, with the analogous result applying to the second sublattice.The passage says the result is generic for SPT states with a stabilizer structure.Source: B Lemma · p. 35
C Choi-Jamiołkowski isomorphism
The appendix defines Choi states in a doubled Hilbert space and describes structural and symmetry constraints that distinguish them from arbitrary doubled-space states.
Choi-Jamiołkowski isomorphism
- A density matrix on H is represented as a Choi state in the doubled space Hd = Hu ⊗ Hl.The passage introduces this as the Choi-state construction for a given density matrix.Source: C Choi-Jamiołkowski isomorphism · p. 36
- SWAP exchanges the upper and lower spaces, while complex conjugation corresponds to Hermitian conjugation in matrix language.The associated symmetry is described as SWAP* symmetry.Source: C Choi-Jamiołkowski isomorphism · p. 36
- A valid Choi state must have a Schmidt decomposition across Hu and Hl with nonnegative coefficients, corresponding to a Hermitian matrix with nonnegative eigenvalues.The passage identifies this as a strong condition on valid Choi states.Source: C Choi-Jamiołkowski isomorphism · p. 36
- Choi states from density matrices generally respect Gd × Z2*, whereas those from pure-state density matrices respect a larger symmetry group.The upper-space representation of g is the complex conjugate U*(g) of the original representation.Source: C Choi-Jamiołkowski isomorphism · p. 36
D A review for strange correlator and the NLSM description of SPT phases
This section reviews the strange correlator and the nonlinear sigma model (NLSM) description of pure SPT states. It relates strange-correlator behavior to boundary anomalies and outlines NLSM accounts of SPT boundary physics.
D A review for strange correlator and the NLSM description of SPT phases
- The strange correlator is a correlation function at the temporal interface between an SPT state and a trivial disordered state, and inherits the bulk topological action’s ’t Hooft anomaly.The path-integral description places the interface at Euclidean time τ = 0.Source: D A review for strange correlator and… · p. 37
- For one- and two-dimensional SPTs, the boundary anomaly excludes all symmetry-charged operator correlations being short-ranged at either temporal or spatial boundaries.If every such correlation were short-ranged, integrating out boundary matter would produce only local gauge-invariant response.Source: D A review for strange correlator and… · p. 37
- In three spatial dimensions, anomalous topological order can coexist with short-ranged local correlations, so a strange correlator can in principle be short-ranged.The authors know of no explicit example of this scenario at the temporal boundary.Source: D A review for strange correlator and… · p. 37
- For the Haldane phase, the O(3) NLSM has Θ = 2π; its boundary WZW term has level k = 1 and describes a dangling spin S = 1/2.The bulk Θ-term leaves a boundary imprint even though it produces no special bulk dynamics in the stated disordered regime.Source: D A review for strange correlator and… · pp. 37–38
- A (2 + 1)d bosonic SPT NLSM reduces to a boundary NLSM with a WZW term; at Θ = 2π its level is k = 1, and SO(4) symmetry yields an SU(2)k CFT fixed point.When SO(4) is reduced to U(1) × U(1), Abelian bosonization relates scaling dimensions to the Luttinger parameter.Source: D A review for strange correlator and… · p. 38
- The NLSM also describes a temporal-boundary SPT wavefunction and makes decorated defects explicit, while strange-correlator calculations can use boundary CFT methods such as Abelian bosonization.A vortex of (n1, n2) reduces the Θ-term to a (0 + 1)d term describing a U(1)-symmetric SPT.Source: D A review for strange correlator and… · p. 38
- Comparing strange correlators against representative wavefunctions for each SPT class can identify which class the state belongs to: only the matching reference has an anomaly-free temporal interface.The correlator for that matching class is the one expected to be trivial.Source: D A review for strange correlator and… · p. 39