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Classification of phases for mixed states via fast dissipative evolution

Andrea Coser, David Perez-Garcia

arXiv:1810.05092v3quant-phcond-mat.stat-mechmath-phmath.MP

TL;DR

How fast dissipative evolution classifies mixed-state phases, and how that classification relates to Hamiltonian phases for pure states, remains an open question. The paper defines phases through fast local Lindbladian evolution and finds that all 1D SPT states are trivial under this definition, while equivalence with Hamiltonian phases is established in both directions only for 2D Zn quantum doubles.

  • Problem

    The paper asks how the fast-dissipative classification of phases relates to the established Hamiltonian classification when restricted to pure states.

    Source: §1 · p. 4

  • Method

    The paper defines two states as belonging to the same phase when a time-independent, local Lindbladian drives one to the other in fast evolution.

    Source: §1 · p. 4

  • Results

    Mixed states form equivalence classes partially ordered by topological complexity; for pure states, all 1D SPT states are trivial, while both-way equivalence with Hamiltonian phases is shown for 2D Zn quantum doubles.

    Source: §1 · p. 5, §5 · p. 44

  • Takeaways & Limitations

    When two states are connected by the defined fast dissipative dynamics, their local observables behave smoothly along the evolution.

    Source: §2.2 · p. 11

    The paper’s approach for the 2D example fails on genus-zero systems, such as a sphere, because there are no logical operators and the ground state is unique.

    Source: §4.2 · p. 41

Abstract

from arXiv · show

We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by K\"onig and Pastawski in 2013, is that it takes time to create new topological correlations, even with the use of dissipation. We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of "topological complexity". (2) It provides a path between any two states belonging to the same phase where observables behave smoothly. We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here. We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group. We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for the $\mathbb{Z}_n$ quantum double models.

1 Introduction

The introduction proposes classifying mixed-state phases by fast local Lindbladian evolution, which cannot create new long-range correlations. It outlines how this classification relates to Hamiltonian phases of pure states and differs for 1D SPT and 2D topological order.

1 Introduction

  • The introduction distinguishes this proposal from open-system phase definitions by Diehl et al. and their generalization, whose exact connection to the present definition remains open.Those earlier approaches concern Gaussian states and subsequent generalizations.Source: §1 · p. 3
  • Two mixed states are in the same phase when a fast local Lindbladian can drive one to the other, preserving their kind of long-range correlations.Fast evolution is defined by reaching an ε-close final state in sublinear time in the system’s linear dimension, with ε vanishing in the thermodynamic limit.Source: §1 · pp. 3–4
  • The definition induces a partial order: destroying long-range correlations may be fast, while creating correlations absent from the initial state may require at least linear time.Ref. shows that any state can be driven to a product state very quickly, whereas some states cannot be prepared from a product state in sublinear time; the authors conjecture a stronger asymmetry between nontrivial phases.Source: §1 · p. 4
  • Any two pure states connected by a gapped Hamiltonian path are also connected by a fast Lindbladian evolution.The introduction also previews a rigorous intermediate result: such a path implies a finite-depth unitary circuit with gates acting on regions of poly-logarithmic size.Source: §1 · pp. 4–5
  • For 1D symmetry-protected topological phases, the Lindbladian classification has a unique phase regardless of the symmetry group.The introduction contrasts this with the Hamiltonian classification, which is stated to depend on the second cohomology group of the symmetry group.Source: §1 · p. 5
  • In 2D, certain topologically ordered states cannot be obtained by fast dissipation when the initial state’s topological content is a subset of the final state’s.The introduction says this contrasts with 1D symmetry protection, which can be overcome by a symmetry-preserving Lindbladian evolution.Source: §1 · p. 5
  • The authors also discuss fast evolution from a state with topological order to one with a subset of that order, illustrating the possibility with 1D GHZ states.They conjecture that the corresponding mechanism for 2D topological states is related to anyon condensation, but do not provide a 2D example.Source: §1 · p. 5

2 Definition of phases for mixed states

This section defines mixed-state phases through fast, local, time-independent dissipative evolutions that connect states in both directions. It establishes transitivity by showing how local timers can switch between successive fast evolutions while preserving a fast connection.

2.1 The formal definition

  • Locality limits how quickly correlations can spread, while local observables along a fast evolution can reach their final values after a finite time independent of system size and remain smooth.Nonlocal evolution could connect states with different long-range order in short times, making the phase diagram trivial.Source: §2.1 · pp. 6, 9
  • Two states are in the same phase when each can be driven to the other in polylogarithmic time by a local, time-independent Lindbladian, with locality-preserving ancillas returned decoupled and error vanishing in the thermodynamic limit.Fast convergence is required for the specified initial state, not for every state under the Lindbladian; the analysis tracks system-size scaling without formally taking the thermodynamic limit.Source: §2.1 · pp. 6, 9

2.2 Connections with previous work

  • Unlike Gaussian-state classifications based on purity and dissipative gaps, this approach defines phases dynamically; whether it coincides with Diehl et al.’s classification remains open, and their method is limited to Gaussian states.A prior quasi-thermal approach found a trivial phase when requiring globally fast Lindbladians, whereas allowing polylogarithmic convergence times can retain a richer phase diagram.Source: §2.2 · pp. 9–10
  • The analyticity of local observables during these evolutions does not conflict with dynamical quantum phase transitions, whose nonanalyticity concerns the nonlocal Loschmidt echo after a quench.The definition fixes initial and final states and admits fast connecting dynamics only within a phase; a quench has no final global state and may show nonanalyticities in the echo.Source: §2.2 · p. 11

2.3 Transitivity

  • Fast reachability is transitive despite the time-independent Lindbladian requirement: locality-preserving timers and switches can compose two evolutions while leaving ancillas decoupled at the end.The construction implements the first Lindbladian before switching to the second, with the switching time chosen so the intermediate state is sufficiently close to the target.Source: §2.3 · pp. 11–12
  • Timers make the system evolve under L1 before τ and L2 after τ, with the resulting evolution approximating e^(t−τ)L2e^τL1(ρ0) up to vanishing errors.The switch is implemented locally, with timer probabilities concentrated on the appropriate switch states before and after the transition window.Source: §2.3 · pp. 13, 16
  • The timer construction requires its size and coupling rate to scale with system size, with T polynomial in N and γ also scaling polynomially in N.The authors note that this tuning is needed to trigger the switch in a shrinking time window while keeping the construction local.Source: §2.3 · p. 14

Transitivity.

  • The constructed Lindbladian proves transitivity: if successive evolutions drive ρ0 to ρ1 and ρ1 to ρ2 fast, one evolution drives ρ0 to ρ2 fast.The proof uses the triangle inequality and contractivity of Lindbladian evolution; the resulting error vanishes as N grows.Source: Transitivity · p. 17

3 The pure state case, a path of gapped Hamiltonians implies a fast Lindbladian evolution

This section examines how a uniformly gapped path of local Hamiltonians connecting pure states can be implemented as a fast Lindbladian evolution. It approximates quasi-adiabatic evolution by a finite-depth circuit and shows how to implement that circuit with a time-independent, log-local Lindbladian in polylogarithmic time.

3 The pure state case, a path of gapped Hamiltonians implies a fast Lindbladian evolution

  • A gapped Hamiltonian path between two pure states implies a log-local Lindbladian evolution between them in time poly log N.The Lindbladian is a sum of terms supported on subsets of size poly log N; the construction uses a finite-depth circuit and ancillary timers.Source: §3 · pp. 17–18

3.1 Adiabatic evolution as a poly log-local circuit of constant depth

  • The quasi-adiabatic evolution can be approximated by a finite-depth circuit with gates acting on regions of size poly log N.The quasi-adiabatic Hamiltonian is quasi-local, so the construction adapts circuit-approximation methods rather than applying strictly local-Hamiltonian results directly.Source: §3.1 · p. 18

3.1.1 Quasi-locality of the quasi-adiabatic Hamiltonian

  • The quasi-adiabatic Hamiltonian is quasi-local, and the authors derive bounds on its terms sufficient for the later circuit approximation.The derivation follows Ref. while using filter-function results from Ref. to improve the bounds.Source: §3.1.1 · pp. 20, 23

3.1.2 Approximating the quasi-adiabatic evolution with a finite depth quantum circuit

  • For gate diameter Ω ∼ log^(1+δ) N with δ > 0, the circuit-approximation error vanishes faster than any power as N grows.The construction uses local evolutions within a region and its complement, together with a boundary patch operator approximated by a finite-support gate.Source: §3.1.2 · pp. 23–24
  • Quasi-adiabatic evolution can be approximated by a finite-depth circuit of log-local gates with error vanishing faster than any polynomial in N.Choosing the boundary-patch width Ω to grow as log^(1+δ) N makes the truncation error vanish; the circuit depth is 2d in d dimensions.Source: §3.1.2 · pp. 27–28

3.2 Fast Lindbladian evolution from a finite depth circuit

  • Any finite-depth circuit mapping one pure state to another can be implemented by a time-independent Lindbladian in polylogarithmic time, with error vanishing as N grows.The construction uses a locality-preserving ancillary system; its timers switch the system through the circuit layers, after which the ancillas reach a discardable product fixed point.Source: §3.2 · pp. 28, 31

4 The other implication: Comments and examples

The section examines examples of the reverse implication for fast Lindbladian evolution, spanning symmetry-protected and GHZ states in 1D and quantum-double ground states in 2D. It discusses directional constraints on reachability and the difficulty of obtaining pure final states under dissipative evolution.

4.1 Unique SPT phase in one dimension: a difference with the Hamiltonian classification

  • In 1D, any two SPT phases can be connected by a fast Lindbladian covariant under the symmetry, so symmetry does not protect distinct phases in this setting.The construction connects representatives of different cohomology classes; states within the same Hamiltonian phase also remain connected by a covariant Lindbladian.Source: §4.1 · pp. 33–34

4.2 An example of two-dimensional topological states that are not in the same phase

  • For m<n, no local Lindbladian drives a pure D(Zm) ground state to a pure D(Zn) ground state in time shorter than the order of the system size.The result is stated for pure states and is limited to geometries with ground-state degeneracy.Source: §4.2 · p. 35
  • The 2D obstruction follows by evolving logical-operator-related initial states and deriving a full-rank matrix in the smaller ground-state space, contradicting its lower dimension.The argument also establishes that, after poly-logarithmic time, evolved states from the smaller ground space are close to states supported in the larger model’s ground space.Source: §4.2 · p. 37
  • The two-dimensional no-go result also applies to mixed states supported on H0(m), but establishes closeness to H0(n) only at a time t ≳ poly log N, not at infinite time.If G(ℓ) is purely exponential, the argument can instead use a time proportional to the linear system size L; the authors leave the genus-zero case for future work.Source: §4.2 · pp. 38, 41
  • For m < n, no fast Lindbladian evolution can drive D(Zm) to D(Zn): it would yield n² independent vectors arbitrarily close to the m²-dimensional ground space.The proof uses fattened logical operators to construct the vectors and show their overlap matrix has full rank; the approximation errors vanish in the thermodynamic limit.Source: §4.2 · pp. 40–41

4.3 Fast dissipative evolution between 1D ‘topological states’

  • The authors conjecture that fast dissipative dynamics could selectively destroy part of a two-dimensional state's anyon content, implementing condensation and confinement, but leave this fine-tuned construction for future work.They give GHZ states as a simpler one-dimensional example of the proposed reverse process.Source: §4.3 · pp. 41–42
  • The reverse GHZm-to-GHZn evolution is impossible by a fast Lindbladian when m < n, by the same dimension-based argument used for the two-dimensional models.The proof constructs n linearly independent states that are approximate ground states of a Hamiltonian whose ground space has dimension m.Source: §4.3 · pp. 42–43
  • For n = 4 and m = 2, a strictly local Lindbladian implements the fast conversion GHZ2 → GHZ4, with convergence to its limiting state after t ≳ log N.It sums local generators Li = Ti − id, where each single-site CPTP map is idempotent; for any initial state, the evolution converges to TΛ(ρ).Source: §4.3 · p. 43
  • Finite-time evolution is mixed, but fine-tuned dissipative dynamics reaches pure |GHZ2⟩; constructing pure final states is harder for 2D topological systems.The passage raises defining topological order for generic mixed states as an open question; Gibbs states were considered in Refs., while a broader approach is left for future work.Source: §4.3 · p. 43

5 Conclusions and outlook

The paper defines mixed-state phases through time-independent local Lindbladian evolution, yielding smooth paths of observables and equivalence classes partially ordered by topological complexity. For pure states, it establishes broad agreement with Hamiltonian phases for standard topological order, while finding a distinct result for 1D SPT phases and proposing anyon condensation as a possible ordering mechanism.

Definition and motivation

  • The proposed definition connects states by irreversible evolution under a time-independent local Lindbladian, giving a path along which observables behave smoothly.The paper presents this as an operational link between phases and the presence or absence of divergences in observable behavior.Source: §5 · pp. 43–44

Mixed-state classification

  • The definition partitions mixed states into equivalence classes and partially orders them by topological complexity.The authors describe these results as first steps toward establishing the definition as a reasonable choice.Source: §5 · p. 44

Pure-state phases

  • For pure states with standard topological order, the definitions give the same classes: one implication holds generally, while the converse is proven only for 2D Zn quantum double phases.The paper separately reports that all 1D SPT states belong to the trivial Lindbladian phase, unlike in the Hamiltonian case.Source: §5 · p. 44

Outlook

  • The authors conjecture that fast Lindbladian evolution can implement anyon condensation, which would make condensation the mechanism ordering phases under their definition.They describe the supporting results as very preliminary.Source: §5 · p. 44
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