isaac h. kim€¦ · isaac h. kim perimeter institute for theoretical physics waterloo, on n2l 2y5,...
TRANSCRIPT
![Page 1: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/1.jpg)
Conservation laws from entanglement
Isaac H. Kim
Perimeter Institute for Theoretical PhysicsWaterloo, ON N2L 2Y5, Canada
January 22th, 2015
*Powered by TikZ
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Conservation laws
• Charge conservation
• Energy-momentum conservation
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Noether’s theorem
Noether’s theorem asserts that a continuous symmetry gives rise toconservation laws.
• Conservation laws from symmetry
• It does not matter how complicated the action/Hamiltonianlooks like.
• However, the theorem only goes one way; symmetry impliesconservation laws, but the converse statement may not betrue.
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Emergent symmetry
For quantum mechanical many-body systems, physicists havedeveloped a notion of emergent symmetry over the past fewdecades.
• Topologically ordered system(e.g., toric code) is a canonicalexample.
• At this point, we do not have an analogue of Noether’stheorem for emergent conservation laws.
• One might argue that all we need to do is to identify theemergent symmetry and apply Noether’s theorem, but this isnot satisfactory.
• Any consistent fusion rules for boson/fermion must beequivalent to the multiplication rules of some irrep of somecompact group. (Doplicher, Roberts)
• For some anyon models, e.g., Fibonacci anyon model, theirfusion rules are not described by a group.
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Emergent symmetry
For quantum mechanical many-body systems, physicists havedeveloped a notion of emergent symmetry over the past fewdecades.
• Topologically ordered system(e.g., toric code) is a canonicalexample.
• At this point, we do not have an analogue of Noether’stheorem for emergent conservation laws.
• One might argue that all we need to do is to identify theemergent symmetry and apply Noether’s theorem, but this isnot satisfactory.
• Any consistent fusion rules for boson/fermion must beequivalent to the multiplication rules of some irrep of somecompact group. (Doplicher, Roberts)
• For some anyon models, e.g., Fibonacci anyon model, theirfusion rules are not described by a group.
![Page 6: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/6.jpg)
Emergent symmetry
For quantum mechanical many-body systems, physicists havedeveloped a notion of emergent symmetry over the past fewdecades.
• Topologically ordered system(e.g., toric code) is a canonicalexample.
• At this point, we do not have an analogue of Noether’stheorem for emergent conservation laws.
• One might argue that all we need to do is to identify theemergent symmetry and apply Noether’s theorem, but this isnot satisfactory.
• Any consistent fusion rules for boson/fermion must beequivalent to the multiplication rules of some irrep of somecompact group. (Doplicher, Roberts)
• For some anyon models, e.g., Fibonacci anyon model, theirfusion rules are not described by a group.
![Page 7: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/7.jpg)
Narrowing down the problem: algebraic theory of anyons
• The near-term goal of this program is to derive a generaltheory of anyons from plausible assumptions which do notinvoke any symmetries.
• No reference to the Hamiltonian either!• There exists a body of work in the algebraic quantum field
theory literature, but I do not want to assumetranslational/Lorentz invariance.
• Kitaev has ironed out such a theory, but he still starts withsome axioms.(cond-mat/0506438, Appendix E) Our goal is toderive
1. the axioms of this theory from a property of a single state,2. and identify the abstract objects in Kitaev’s theory to the
physical degrees of freedom.
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Narrowing down the problem: algebraic theory of anyons
• The near-term goal of this program is to derive a generaltheory of anyons from plausible assumptions which do notinvoke any symmetries.
• No reference to the Hamiltonian either!• There exists a body of work in the algebraic quantum field
theory literature, but I do not want to assumetranslational/Lorentz invariance.
• Kitaev has ironed out such a theory, but he still starts withsome axioms.(cond-mat/0506438, Appendix E) Our goal is toderive
1. the axioms of this theory from a property of a single state,2. and identify the abstract objects in Kitaev’s theory to the
physical degrees of freedom.
![Page 9: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/9.jpg)
Narrowing down the problem: algebraic theory of anyons
• The near-term goal of this program is to derive a generaltheory of anyons from plausible assumptions which do notinvoke any symmetries.
• No reference to the Hamiltonian either!• There exists a body of work in the algebraic quantum field
theory literature, but I do not want to assumetranslational/Lorentz invariance.
• Kitaev has ironed out such a theory, but he still starts withsome axioms.(cond-mat/0506438, Appendix E) Our goal is toderive
1. the axioms of this theory from a property of a single state,2. and identify the abstract objects in Kitaev’s theory to the
physical degrees of freedom.
![Page 10: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/10.jpg)
In relation to Jeongwan’s result
S(A|B) + S(A|C ) ≈ 0 everywhere
A
B
C→
D E
ΠaD ∼ Πa
E
where ΠaD(Πa
E ) is a fundamental projector for a topological chargea over an annulus D(E ).
Physical meaning: Superselection sectors are globally consistent.More concretely: There is an isomorphism between the logicalalgebras over different annuli.
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In relation to Jeongwan’s result
S(A|B) + S(A|C ) ≈ 0 everywhere
A
B
C→
D E
ΠaD ∼ Πa
E
where ΠaD(Πa
E ) is a fundamental projector for a topological chargea over an annulus D(E ).Physical meaning: Superselection sectors are globally consistent.
More concretely: There is an isomorphism between the logicalalgebras over different annuli.
![Page 12: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/12.jpg)
In relation to Jeongwan’s result
S(A|B) + S(A|C ) ≈ 0 everywhere
A
B
C→
D E
ΠaD ∼ Πa
E
where ΠaD(Πa
E ) is a fundamental projector for a topological chargea over an annulus D(E ).Physical meaning: Superselection sectors are globally consistent.More concretely: There is an isomorphism between the logicalalgebras over different annuli.
![Page 13: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/13.jpg)
Plan
1. Defining the charge sectors
2. Why charge sectors are globally well-defined
3. Consistency equations
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Charge sectors : sets of quantum statesLet’s suppose we have a state P = |ψ〉 〈ψ|. Consider the followingsets.
S = {ρ|Supp(ρ) = , ρ = Tr¯P}
S = {Tr ρ|ρ ∈ S }
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Charge sectors : convex set
S = {Tr ρ|ρ ∈ S }* S is a convex set, but convex sets come in different sizes andshapes.
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Charge sectors: simplex assumption
This is the main weakness of our work at this point. I will need toassume the following statement.
Assumption:
S = {ρ|ρ =∑i
piρi},
where∑
i pi = 1 and ρi ⊥ ρj ∀i 6= j .* We call ρi as the extreme points of the set.* This implies that there exists a set of orthogonal projectors {Πi}which project onto the support of ρi .
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Charge sectors: simplex assumption
This is the main weakness of our work at this point. I will need toassume the following statement.
Assumption:
S = {ρ|ρ =∑i
piρi},
where∑
i pi = 1 and ρi ⊥ ρj ∀i 6= j .* We call ρi as the extreme points of the set.* This implies that there exists a set of orthogonal projectors {Πi}which project onto the support of ρi .
![Page 18: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/18.jpg)
Charge sectors: justifications
Assumption:
S = {ρ|ρ =∑i
piρi},
where∑
i pi = 1 and ρi ⊥ ρj ∀i 6= j .* This implies that there exists a set of orthogonal projectors {Πi}which project onto the support of ρi .
• {Πi} coincides with the fundamental projectors in Jeongwan’swork for exactly solvable models.
• Holevo information of the set is bounded by 2γ (γ <∞ :topological entanglement entropy).
• Perhaps it is possible to derive this from some condition.
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Poke Ball condition
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Poke Ball condition
S(A|B) + S(A|C ) ≈ 0
A
B
C
* S(A|B) = S(AB)− S(B).
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Why we expect Poke Ball condition to hold
A
B
C
S(A) ≈ α|∂A| − γ
(Kitaev and Preskill 2006, Levin and Wen 2006) givesS(A|B) + S(A|C ) ≈ 0.
![Page 22: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/22.jpg)
A brief recap in the middle
All the things discussed in the remaining slides(starting from thenext one) follow from these three assumptions.1. ε, δ → 0.2. Simplex assumption
S = {ρ|ρ =∑i
piρi},
where∑
i pi = 1 and ρi ⊥ ρj∀i 6= j .
3. Poke Ball condition
S(A|B) + S(A|C ) ≈ 0
A
B
C
*I actually suspect that 3 implies 2, but I do not know how toshow that.
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Poke Ball condition → Global consistency of charge sectors
First consider a state, |ψ〉 , which satisfies the Poke Ball conditioneverywhere:
S(A|B) + S(A|C ) ≈ 0
A
B
C
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Poke Ball condition → Global consistency of charge sectors
First consider a state, |ψ〉 , which satisfies the Poke Ball conditioneverywhere:
S(A|B) + S(A|C ) ≈ 0
A
B
C
![Page 25: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/25.jpg)
Poke Ball condition → Global consistency of charge sectors
First consider a state, |ψ〉 , which satisfies the Poke Ball conditioneverywhere:
S(A|B) + S(A|C ) ≈ 0
A
B
C
![Page 26: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/26.jpg)
Poke Ball condition → Global consistency of charge sectors
S = {ρ|Supp(ρ) = , ρ = Tr¯P}
S = {Tr ρ|ρ ∈ S }Poke Ball condition implies that S for all are isomorphic toeach other.
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Consistency of the charge sectors: contraction lemma
Lemma(Contraction lemma): If an annulus A is contained in A′,∀ρ, σ ∈ SA′ ,
‖ρ− σ‖1 ≥ ‖TrA′\A(ρ)− TrA′\A(σ)‖1.
Proof.Trivial
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Consistency of the charge sectors: expansion lemma
Lemma(Expansion lemma): If an annulus A is contained in A′, and thePoke Ball condition is satisfied near A′, ∀ρ, σ ∈ SA′ ,
‖TrA′\A(ρ)− TrA′\A(σ)‖1 ≥ ‖ρ− σ‖1 − o(1).
Proof.Nontrivial. Uses 1405.0137.
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More on expansion lemma, colloquially
Suppose we have two states, |ψ1〉 and |ψ2〉, such that
• They are close in trace distance over a subsystem A.
• They are close in trace distance over a set of (bounded) balls{Bi} in the neighborhood of A.
• In the balls, the Poke Ball condition is satisfied.
Expansion lemma says that, if A ∪ Bi has the same shape as A,then |ψ1〉 and |ψ2〉 are close in trace distance over A ∪ Bi .
A Bi & A Bi → A Bi
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Consistency of the charge sectors: an intermediateconclusion
S(A|B) + S(A|C ) ≈ 0
A
B
C
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Consistency of the charge sectors: an intermediateconclusion
S(A|B) + S(A|C ) ≈ 0
A
B
C
![Page 32: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/32.jpg)
Consistency of the charge sectors: an intermediateconclusion
S(A|B) + S(A|C ) ≈ 0
A
B
C
![Page 33: Isaac H. Kim€¦ · Isaac H. Kim Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada January 22th, 2015 *Powered by TikZ. Conservation laws Charge conservation](https://reader036.vdocuments.net/reader036/viewer/2022063007/5fbb2c563fd7e9587508f97c/html5/thumbnails/33.jpg)
Consistency of the charge sectors: an intermediateconclusion
What we want to show: ∀i , ∃j such that Π,i∼ Π
,j. Now let’s
see why.
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Consistency of the charge sectors: localized excitations
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Consistency of the charge sectors: localized excitations
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Consistency of the charge sectors: localized excitations
WLOG, choose ρ,i
to be one of the
extreme points of S . For
ρ,⊥ =
∑j 6=i
1N ρ ,j
,
‖ρ,i− ρ
,⊥‖1 ≤ ‖ρ ,i− ρ
,⊥‖1≤ ‖ρ
,i− ρ
,⊥‖1 + o(1).
This means that‖ρ
,i− ρ
,⊥‖1 ≈ ‖ρ ,i− ρ
,⊥‖1 ≈ 2.
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Consistency of the charge sectors: localized excitations
WLOG, choose ρ,i
to be one of the
extreme points of S . For
ρ,⊥ =
∑j 6=i
1N ρ ,j
,
‖ρ,i− ρ
,⊥‖1 ≤ ‖ρ ,i− ρ
,⊥‖1≤ ‖ρ
,i− ρ
,⊥‖1 + o(1).
This means that‖ρ
,i− ρ
,⊥‖1 ≈ ‖ρ ,i− ρ
,⊥‖1 ≈ 2.
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Consistency of the charge sectors: localized excitations
WLOG, choose ρ,i
to be one of the
extreme points of S . For
ρ,⊥ =
∑j 6=i
1N ρ ,j
,
‖ρ,i− ρ
,⊥‖1 = 2 = ‖ρ,i− ρ
,⊥‖1 + ε.
* The projector onto the positiveeigenvalue subspace of ρ
,i− ρ
,⊥ picks
out ρ,i
and annihilates all ρ,j 6=i
with
error O(Nε). (Same for .)
* We declare these projectors to beΠ
,i∼ Π
,j.
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Consistency of the charge sectors: localized excitations
WLOG, choose ρ,i
to be one of the
extreme points of S . For
ρ,⊥ =
∑j 6=i
1N ρ ,j
,
‖ρ,i− ρ
,⊥‖1 = 2 = ‖ρ,i− ρ
,⊥‖1 + ε.
* The projector onto the positiveeigenvalue subspace of ρ
,i− ρ
,⊥ picks
out ρ,i
and annihilates all ρ,j 6=i
with
error O(Nε). (Same for .)
* We declare these projectors to beΠ
,i∼ Π
,j.
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Poke Ball condition → Global consistency of charge sectors
* We can now unambiguously label all the charges lying inside anannulus by some fixed universal set, as long as the annuli can bedeformed into one another without encountering .
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Existence of the trivial charge
We learned that all the annuli have a canonical set of (almost)orthogonal projectors {Πi |i = 1, · · · ,N}. The index set,{i = 1, · · · ,N}, is universal for all annuli. These indices representthe charge sectors.
* Let’s first recall the expansion lemma.
A Bi & A Bi → A Bi
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Existence of the trivial charge
We learned that all the annuli have a canonical set of (almost)orthogonal projectors {Πi |i = 1, · · · ,N}. The index set,{i = 1, · · · ,N}, is universal for all annuli. These indices representthe charge sectors.* Let’s first recall the expansion lemma.
A Bi & A Bi → A Bi
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Existence of the trivial charge
We learned that all the annuli have a canonical set of (almost)orthogonal projectors {Πi |i = 1, · · · ,N}. The index set,{i = 1, · · · ,N}, is universal for all annuli. These indices representthe charge sectors.
For , pick Πi such that Πiρ Πi ≈ ρ .* You are guaranteed to have only one such Πi !* We denote this charge as 1.
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Charge of a single localized excitation is trivial (on asphere)
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Consistency equations
Playing the same game, one can show that
• All the charges have their own unique antiparticles.
• Charges can be transported unitarily.
We are almost at the cusp of deriving the anyon theory!
We onlyneed to prove the following list.
• Pentagon equation
• Hexagon equation
• Triangle equation
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Consistency equations
Playing the same game, one can show that
• All the charges have their own unique antiparticles.
• Charges can be transported unitarily.
We are almost at the cusp of deriving the anyon theory! We onlyneed to prove the following list.
• Pentagon equation
• Hexagon equation
• Triangle equation
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Consistency equations(flavor of the argument)
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Consistency equations(flavor of the argument)
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Consistency equations(flavor of the argument)
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Consistency equations(flavor of the argument)
The projectors on and may not commute with each other.* This is reminiscent to an observation that the eigenvalues of(L2, Lz) as well as (L2, Lx) forms a set of good quantum numbers,but the eigenvalues of (L2, Lz , Lx) do not.
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Summary
We are at the cusp of deriving all the axioms of cond-mat/0506438(Appendix E) from the following set of assumptions:1. ε, δ → 0.2. Simplex assumption
S = {ρ|ρ =∑i
piρi},
where∑
i pi = 1 and ρi ⊥ ρj∀i 6= j .
3. Poke Ball condition
S(A|B) + S(A|C ) ≈ 0
A
B
C
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Future directions
• We are still forcefully injecting the notion of charge from thevery beginning.(Simplex assumption) I would like to be able toprove the simplex assumption from the Poke Ball condition.
• Now we can mechanically derive the axioms of the effectivetheory that describe interesting classes of quantummany-body systems at low energy.
• I expect this machinery to be applicable in the presence ofopen boundary condition/higher dimensions.
• Classification of phases?
• A more general framework to derive conservation laws fromentanglement analysis?
Thank you!
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Future directions
• We are still forcefully injecting the notion of charge from thevery beginning.(Simplex assumption) I would like to be able toprove the simplex assumption from the Poke Ball condition.
• Now we can mechanically derive the axioms of the effectivetheory that describe interesting classes of quantummany-body systems at low energy.
• I expect this machinery to be applicable in the presence ofopen boundary condition/higher dimensions.
• Classification of phases?
• A more general framework to derive conservation laws fromentanglement analysis?
Thank you!