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Mixed s-sourcery: Building many-body states using bubbles of Nothing

Brian Swingle, John McGreevy

arXiv:1607.05753v1cond-mat.str-elhep-thquant-ph

TL;DR

The paper extends s-sourcery from pure states to mixed states, focusing on thermal states of local Hamiltonians. It uses thermal-double purifications and approximate quantum conditional independence to establish mixed s-source fixed points for a wide variety of finite-temperature states, including results that make thermal expectation values efficiently computable. One scope boundary is that the proposed fixed-range unitaries remain unconstructed in most cases.

  • Problem

    The paper asks how the s-sourcery construction for pure states can be extended to mixed states, particularly thermal states of local Hamiltonians.

    Source: §1 · p. 3

  • Method

    The constructions use thermal-double purifications and approximate quantum conditional independence to treat thermal doubles as ground states of local Hamiltonians.

    Source: §7 · p. 38

  • Results

    A wide variety of thermal states have finite-depth-preparable purifications, making thermal expectation values efficiently computable.

    Source: §1 · pp. 4–5, §7 · p. 38

  • Takeaways & Limitations

    Ground-state techniques can address thermal states, so the ground-state problem is more general than it might first appear.

    Source: §1 · p. 3

    Fixed-range unitaries are expected on physical grounds but have yet to be constructed in most cases, a construction left beyond this work's scope.

    Source: §5 · p. 33

Abstract

from arXiv · show

In arXiv:1407.8203, we introduced the idea of s-sourcery, a general formalism for building many-body quantum ground states using renormalization-group-inspired quantum circuits. Here we define a generalized notion of s-sourcery that applies to mixed states, and study its properties and applicability. We prove a number of theorems establishing the prevalence of mixed s-source fixed points. For our examples we focus on thermal states of local Hamiltonians. Thermal double states (also called thermofield double states) and the machinery of approximate conditional independence are used heavily in the constructions.

1 Introduction and definitions

This section extends hierarchical s-sourcery to mixed states, defining mixed-state fixed points and introducing bubble-of-Nothing constructions and their properties. It also describes how a conjugate channel maps local operators to nearby local operators through Kraus-term cancellations.

1 Introduction and definitions

  • The original s-sourcery index counts ground-state copies used to double system size; states with s < 2d−1 obey an entanglement-entropy area law.The construction doubles the system hierarchically using local unitaries and product-state ancillas.Source: §1 · p. 3
  • For local-Hamiltonian thermal states, the paper uses approximate conditional independence and shows that many have finite-depth-preparable purifications, enabling efficient thermal expectation values.The paper also argues that purifications of thermal states can often be treated as ground states of local Hamiltonians.Source: §1 · pp. 3, 5

1.1 Definition of mixed s sourcery

  • The noisy, open, and purified definitions allow progressively more general constructions: noisy uses a quasi-local unitary, open permits tracing out an environment, and purified retains it.A noisy fixed point is also open, and an open fixed point is also purified.Source: §1.1 · pp. 6–8

1.2 Wormhole arrays and bubble-of-Nothing arrays

  • Bubble-of-Nothing arrays expand holes to decompose a state hierarchically; symmetry-protected states are open s = 0 fixed points, while inverse states can cancel protected edge obstructions.Long-range entanglement and gapless edges can obstruct the deformation; coupling a short-range-entangled state to its inverse removes protected edge states.Source: §1.2 · pp. 9–11

1.3.1 Equivalence of open and purified for s = 0

  • At s = 0, the open and purified definitions agree, although open fixed points need not be noisy fixed points.The noisy definition is more restrictive because it does not allow tracing out part of the environment.Source: §1.1 · p. 7, §1.3.1 · p. 11

1.3.2 Mutual information growth

  • A local channel can produce volume-law entropy but has limited capacity to generate mutual information across a region’s boundary.The mutual-information bound depends on region shape and local Hilbert-space dimension.Source: §1.3.2 · pp. 11–12

1.3.3 Induced map on local operators

  • Local channels map local operators to local operators, so expectation values can still be computed efficiently.In the Heisenberg picture, local Kraus operators preserve the locality of the operator map.Source: §1.3.3 · pp. 12–13
  • Under D⊗L†, most Kraus terms pair as M†αMα and sum to 1 away from a local operator O, while a few terms near O fail to cancel and produce a new local operator.Source: §1.3.3 · p. 13

2 Free fermions at finite temperature are s = 0

For quadratic local fermion Hamiltonians at any nonzero temperature, the thermal state is an s = 0 purified fixed point, with a required channel range set by its thermal correlation length. The construction uses a thermal-double Hamiltonian and a quasi-local continuation to the infinite-temperature state; the paper also extends the result to free bosons.

2 Free fermions at finite temperature are s = 0

  • At T > 0, any thermal state of a quadratic local fermion Hamiltonian is an s = 0 purified fixed point, with channel range given by its correlation length.The theorem assumes a local quadratic Hamiltonian and applies whether its ground-state spectrum is gapped or gapless.Source: §2 · pp. 13–14
  • The thermal double state is the ground state of a thermal-double Hamiltonian whose interactions decay exponentially over a range set by temperature and any energy gap.For a finite-temperature metal, the stated correlation length is ξ(T) = vF/T.Source: §2 · p. 16
  • A quasi-local unitary connects the thermal-double Hamiltonian's ground state to the infinite-temperature ground state, and tracing out the auxiliary system yields the thermal state.The interpolation uses a uniformly gapped family of Hamiltonians, and the quasi-adiabatic generator is explicitly computable once its filter is fixed.Source: §2 · pp. 16–17
  • The free-boson construction also works: its thermal double is a two-mode squeezed state and the ground state of a quadratic Hamiltonian.The paper says the remaining fermion analysis then applies.Source: §2 · p. 18

3 CFTs at finite temperature are s = 0

For CFT thermal states with the required entropy form, approximate conditional independence yields local reconstruction channels and an open s = 0 fixed point. The argument has dimensional and topological boundaries, and higher-dimensional corner terms require finite-temperature correlation decay.

3 CFTs at finite temperature are s = 0

  • Thermal states of CFTs smoothly connected to infinite temperature are open s = 0 fixed points, with channel range set by the thermal length.The construction uses approximate conditional independence of cellular regions to obtain local reconstruction channels.Source: §3 · pp. 18, 20

3.1 CFT1+1

  • In 1 + 1 dimensions, entropy terms cancel in the conditional mutual information, leaving exponentially suppressed corrections and an approximate s = 0 reconstruction.The recovery channel can be chosen local; the cited discussion notes a transpose-channel construction with error √ϵ rather than ϵ.Source: §3.1 · p. 22
  • Nonzero mutual information between buffer subregions can prevent a reconstruction channel from factoring into local channels, even when conditional mutual information vanishes.The paper's example has I(A : C|B) = 0 but I(b : b′) ≠ 0 for buffer subregions.Source: §3.1 · p. 22

3.2 CFTd+1

  • Finite-temperature topological terms can obstruct reconstruction in higher dimensions; the 2-form toric code in d = 4 remains s = 1 at low temperatures.The argument generalizes when the higher-dimensional entropy form holds; topological corrections can invalidate the needed cancellation.Source: §3 · p. 19, §3.2 · p. 23
  • At finite temperature, corner contributions are expected to depend on the thermal length, allowing their cancellation for sufficiently large regions.At zero temperature in a gapless system, corner logarithms can depend on subsystem sizes and may cancel only up to an order-one constant.Source: §3.2 · p. 23
  • In higher dimensions, successive local channels build the thermal state by adding 0-cells, 1-cells, and then 2-cells, yielding s = 0 with thermal-length range.The construction uses approximate conditional independence at each stage of the cellular decomposition.Source: §3.2 · p. 24
  • For CFTs obeying the entropy assumption, a local Hamiltonian exists whose ground state approximately equals the thermal double, but the general argument does not determine its detailed form.Its locality range is set by the thermal length or correlation length; the exact quasi-local version depends on a conjecture about exact reconstruction.Source: §3.2 · pp. 24–25

4 Holographic models at finite T are s = 0

Using approximate conditional independence and holographic entanglement entropies, the section argues that finite-temperature holographic states with an IR wall are s = 0 in the large-N limit. Subleading terms and the channel range qualify this result.

4 Holographic models at finite T are s = 0

  • Holographic black-hole states obey approximate conditional independence up to O(1/N) terms, supporting an s = 0 construction in the large-N limit.The argument controls the first quantum correction using bulk-field entanglement; the analysis is extended to other dimensions, with a fuller treatment deferred to.Source: §4 · pp. 25, 26, 29, 30
  • For two intervals at large separation scale ℓ, mutual information drops to zero at the critical separation x_c and remains zero thereafter.The sharp transition is an artifact of the large-N classical-gravity approximation.Source: §4 · p. 28
  • At large central charge c, the required range ℓ grows like log(c), a blowup the authors conjecture is artificial.They attribute the factor to demanding small total trace-norm error across c copies, which requires each copy to be accurate to error ϵ/c.Source: §4 · p. 29
  • The conditional-information argument must control subleading entropy terms, since even an O(1) term can obstruct reconstruction.In d = 1, the O(1) contribution is also exponentially small; the holographic account uses the first quantum correction to entropy.Source: §4 · pp. 26, 29

5 RG decomposition of s = 0 fixed points

The section interprets s = 0 thermal fixed points through an RG flow that retains ground-state scaling behavior until the thermal length is reached. It also distinguishes approximate recovery from locality of the recovery channel and notes that fixed microscopic-range constructions remain largely open.

5 RG decomposition of s = 0 fixed points

  • Approximate recovery does not guarantee a channel local within the buffer; the authors propose that small mutual information among buffer subsets could enable approximate locality.A classical thermal state can be exactly Markov yet have long-range correlations and a recovery channel that is not local on the buffer.Source: §5 · p. 30
  • Although channel range grows as T → 0, the channels can be viewed as products of fixed-range channels, with their number bounded by log(IR cutoff).The authors argue that this structure keeps calculation of local observables formally efficient.Source: §5 · p. 31
  • For scale-invariant theories, the entropy comparison gives s = 2θ for ground states, while finite-temperature s = 0 requires range of order ξ.At microscopic temperatures the thermal length reaches the cutoff, and the state crosses over to s = 0.Source: §5 · pp. 31–32
  • The RG picture keeps the ground-state value s = 2θ until coarse-graining reaches the thermal scale, while the finite-temperature state has s = 0 with channel range of order ξ.For scale-invariant theories, a range-ξ unitary can be decomposed into log(ξ/a) fixed-range unitaries.Source: §5 · pp. 31–32
  • In the ideal scaling limit, Rényi entropies also obey a size-and-temperature relation; for a cutoff-free mathematical CFT, the density matrix depends only on LT.An exact equality would imply matching spectra for the size-L state and 2^θ copies of the size-L/2 state.Source: §5 · p. 32
  • Fixed microscopic-range unitaries are expected on physical grounds but have not been constructed in most cases, so that construction remains beyond this work’s scope.The proposed finite-temperature crossover with fixed-range unitaries runs from s = 2θ to s = 0.Source: §5 · pp. 32–33

6 Bubble-of-Nothing analysis of topological gauge theories

The section uses conditional mutual information to identify where topological gauge theories obstruct bubble-of-Nothing reconstruction, then shows how those obstructions behave at nonzero temperature. One-form theories lose the obstruction at sufficiently large scales, whereas 2-form theories in four dimensions retain finite-temperature topological order and self-correcting memory.

6 Bubble-of-Nothing analysis of topological gauge theories

  • For p = 2 in d = 4, the 1→2 reconstruction step has I(A : C|B) ≃ 7 and an obstruction, whereas the 2→3 step has I(A : C|B) = 0 and no obstruction.At small but non-zero temperature, 2-form gauge theory in d = 4 remains s = 1; its topological order survives, making it a robust self-correcting quantum memory.Source: §6 · p. 33, §6.1 · pp. 36–37

6.1 Obstruction theory at T = 0

  • At T = 0, p-form gauge theories have at most two reconstruction obstructions, at the steps that add topological data detectable as new charges.The two obstructions are related by duality and coincide when d = 2p; the authors anticipate, but have not completed, a general Mayer–Vietoris proof.Source: §6.1 · pp. 34–35
  • The reconstruction examples diagnose an obstruction by nonzero conditional mutual information, with conventional local area-law terms canceling from that quantity.The calculations focus on topological entropy terms; for example, the 0→1 step in the 0-form case has I(A : C|B) = 2 and an obstruction, while 1→2 has zero and none.Source: §6.1 · p. 35

6.2 Fate of obstructions at T > 0

  • At nonzero temperature, 1-form gauge theory is s = 1 below the defect spacing lq(T) and s = 0 at larger scales.Thermally excited particles have density n(T) ∝ e^−∆/T, so the obstruction persists only for R ≪ lq(T).Source: §6.2 · p. 37
  • For 2-form gauge theory in d > 3, large closed-string defects are suppressed below a Hagedorn temperature, allowing topological order to persist.Above that temperature, string condensation destroys the order; numerical evidence for a finite-temperature transition is reported for the four-dimensional theory.Source: §6.2 · pp. 37–38

7 Discussion, conjectures and questions

The discussion presents mixed s-sourcery as a framework for a broad range of finite-temperature states and develops approximate conditional independence as a route to recovery channels and thermal-double constructions. It also states practical bounds and limitations, alongside applications and open conjectures.

7 Discussion, conjectures and questions

  • The paper extends s-sourcery to mixed states and reports examples across free particles, conformal field theories, topological phases, and holographic states.It also proposes a further renormalization-group-inspired decomposition of the local unitaries.Source: §7 · p. 38

7.1 Form of local channel arising from approximate conditional independence

  • Approximate conditional independence yields a recovery channel that can successively “untrace out” regions in the cellular decomposition and supports constructing a local thermal-double Hamiltonian.The construction purifies the recovery channel into a unitary with an environment, then conjugates a trivial Hamiltonian whose ground state is the initial product state.Source: §7.1 · p. 40

7.2 Thermal double Hamiltonians

  • The general thermal-double Hamiltonian has range of order the thermal length ξ and terms containing ∼ξ^d or fewer operators.The Hamiltonian has an approximately thermal-double ground state; additional physical input can sharpen the construction, as in the free-particle case with two-body terms of range ξ.Source: §7.2 · p. 40
  • For classical states, a local few-body thermal-double Hamiltonian can exist even with long-range correlations, but it need not have an energy gap.The authors describe these classical states as special and speculate that such local Hamiltonians may exist more broadly.Source: §7.2 · p. 42

7.3 Some further applications

  • Suitable thermal states are approximate fixed points of a local Lindblad flow, while uniqueness and mixing-time bounds remain possible further results.The discussion also conjectures that thermal states in the same phase can be related by a local channel, but proves this only for free fermion states.Source: §7.3 · pp. 42–43, §7.4 · p. 45
  • The recovery-channel construction gives a thermal-double state-complexity upper bound that is polynomial in L for d = 1 and quasi-polynomial in L for d > 1.The estimate is temperature-dependent and therefore does not imply that complexity decreases monotonically as temperature increases.Source: §7.3 · pp. 43–44

A.1 Entropy bounds

The appendix gives a lower bound relating the mutual information in a mixed state to entropies in its purification.

A.1 Entropy bounds

  • The mutual information I(A, Ac) is bounded above by the entropy S(A ˜A) of A together with its corresponding purifying region.The derivation uses purity, swap invariance, and strong subadditivity.Source: §A.1 · p. 46
  • The bound follows by exchanging systems with complements using overall purity, then applying strong subadditivity.The appendix identifies swap invariance as an additional premise.Source: §A.1 · p. 46
  • The authors note obstacles to obtaining a useful upper bound, owing to freedom to act with an arbitrary unitary on the purifying system.They add that a gapped local parent Hamiltonian for the thermal double would make an area-law bound on entanglement entropy highly likely.Source: §A.1 · pp. 46–47

A.2 Review of reversibility

The appendix reviews recovery channels, including the transpose (Petz) channel, and illustrates recovery with a toric-code example.

A.2 Review of reversibility

  • The transpose or Petz channel R is trace preserving and reproduces the reference state σ after the channel E acts on it.The appendix notes that recovery may also be sought for high-probability pure states from the ensemble represented by σ.Source: §A.2 · p. 47
  • For a toric code on a torus, single-link dephasing mixes the normalized four-ground-state projector with its Z-conjugated state.The example sets the dephasing Kraus operators to M1 = √(1 − p) and M2 = √p Z.Source: §A.2 · p. 47
  • For any particular pure ground state, the recovery channel reverses the mixture of that state and its Z-conjugated state shown in the toy construction.The authors describe this as a toy version of the toric-code error-recovery construction.Source: §A.2 · p. 48

A.3 Review of quantum Markov chains

Quantum Markov chains have vanishing conditional mutual information, and their state can be reconstructed with a Petz recovery channel. The recovery map's dependence only on marginals makes it local, while approximate versions extend the construction to small conditional mutual information.

A.3 Review of quantum Markov chains

  • A quantum Markov chain is a tripartite state with I(A : C|B) = 0, where I(A : C|B) is the conditional mutual information.Such states saturate strong subadditivity; the passage contrasts this with classical probability factorization.Source: §A.3 · p. 48
  • When I(A : C|B) = 0, the Petz recovery channel reconstructs ρABC from ρAB and is given by a local map.Its dependence only on ρBC, rather than the whole-system state, is essential to the constructions in the main text.Source: §A.3 · pp. 49–50
  • For I(A : C|B) ≈ 0, an approximate recovery map acts from B to BC, and the later map is independent of ρA.The paper uses this recovery machinery in its construction.Source: §A.3 · p. 50

B Another construction of thermal double Hamiltonians

This section constructs Hamiltonians with thermal double states as exact ground states, including a Hermitian, frustration-free option. The non-Hermitian constraints can be non-local, and the resulting Hermitian Hamiltonian's spectrum is not immediately controlled.

B Another construction of thermal double Hamiltonians

  • The earlier Klich and Feiguin construction gives a thermal-double ground-state Hamiltonian that may be non-local and is non-Hermitian.The section introduces freedom in choosing the operators annihilating the state as potentially useful for gapping the spectrum.Source: B Another construction of thermal… · pp. 50–51
  • The thermal double state is the unique ground state of the infinite-temperature construction, and tracing out its auxiliary degrees of freedom reproduces ρ(T) after applying the square root of ρ(T).The infinite-temperature state is formed from two copies of the system.Source: B Another construction of thermal… · pp. 50–51
  • The thermal double is an exact ground state of non-Hermitian constraints, which may be non-local; short-ranged thermal correlations suggest their range may be set by the correlation length.The constraints have integer spectrum and exactly annihilate the thermal double state.Source: B Another construction of thermal… · p. 51
  • A Hermitian, frustration-free Hamiltonian also has the thermal double as a ground state, but its spectrum is not under immediate control.The section identifies analyzing whether the resulting Hamiltonian is gapped as an open question.Source: B Another construction of thermal… · pp. 51–52
  • For the toric-code construction, the constraints commute with one another, while a constraint's conjugate fails to commute with constraints sharing a vertex or plaquette.The section notes that the spectrum of the associated Hermitian Hamiltonian still needs analysis.Source: B Another construction of thermal… · pp. 51–52

C Cellular decomposition of R4

The cellular-decomposition analysis tracks cell counts and Betti numbers, with the first two steps treated in general dimension and the 2→3 step specified for d = 4. In the displayed 0→1 calculation, the obstruction vanishes.

C Cellular decomposition of R4

  • The analysis treats the first two decomposition steps for general Rd, then specifies d = 4 for the 2→3 step.The table uses d for spatial dimension and L for unit cells; Σk denotes the k-skeleton assembled from 0-cells through k-cells.Source: C Cellular decomposition of R4 · p. 52
  • For p = 1 and p = 2-form gauge theory, the rightmost table column gives the obstructions.At the 2→3 step, the analysis determines y1 ≡ b1(∂Σ2) by imposing that the p=2 obstruction vanish.Source: C Cellular decomposition of R4 · p. 53
  • The 0→1 calculation gives ABC0→1 = −2d + (d + 1) − (d − 1) = 0.The table identifies the obstruction as vanishing at this step.Source: C Cellular decomposition of R4 · p. 54
  • The displayed 4 × Sd−3 × S2 entry is 4 × S1 × S2, with z0 = 4 and z1 = 4.Source: C Cellular decomposition of R4 · p. 54
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