Correlation length in random MPS and PEPS

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Correlation length in random MPS and PEPS

Joint work with David Pérez-García (Universidad Complutense de Madrid) arXiv:1906.11682

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

Entangle This IV, Madrid - September 10 2019

Cécilia Lancien Correlation length in random MPS and PEPS Entangle This IV, Madrid - September 10 2019 1

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Outline

1 Background and motivations

2 Random MPS and PEPS : construction and statement of the main questions & results

3 Typical correlation length in a random TNS through typical spectral gap of its transfer operator

4 Typical correlation length in a random TNS through spectral gap of its parent Hamiltonian

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Tensor network states on many-body quantum systems

Known :In several contexts, ‘physically relevant’ states of many-body quantum systems are well approximated by states from a very small subset of the global, exponentially large, state space.

Tensor network state (TNS) on(C^d)^⊗^N:

Take a graphGwithNvertices andLedges. Put at each vertexva tensor|χvi ∈C^d⊗(C^q)^⊗δ(^v⁾ to get a tensor|˜χGi ∈(C^d)^⊗^N⊗(C^q)^⊗^2L. Contract together the indices of|˜χGicorresponding to a same edge to obtain a tensor|χGi ∈(C^d)^⊗^N.

−→Ifδ(v)6^r^{for all}v, then such state is described by at mostNq^rdparameters (linear rather than exponential inN).

• •

Gwith 6 vertices and 7 edges

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

|χGi ∈(C^d)^⊗⁶ d-dimensional indices :physicalindices.q-dimensional indices :bondindices.

In this talk :The underlying graphGis a regular lattice in dimension 1 or 2, so that|χ_Giis a matrix product state (MPS)or aprojected entangled pair state (PEPS).

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Tensor network states on many-body quantum systems

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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Tensor network states on many-body quantum systems

• •

|˜χGi ∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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Motivations behind TNS and related questions

Ideally, a TNS representation should be a (mathematically tractable) way of describing the state of a system composed of many sub-systems having a certain geometry and subject to interactions respecting this geometry.

Example :It is rigorously proven in some cases and conjectured in some others that being a TNS is approximately equivalent to being a ground state of a gapped local Hamiltonian (Hastings, Landau/Vazirani/Vidick...)

General question :Are statements of this kind at least true generically ?

−→What are the typical features of TNS ?

For instance : What is the typical amount of entanglement and correlations between two subsets of particles ? Is the parent Hamiltonian typically gapped or gapless ? etc.

Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability. (Regime we are interested in : larged,q, ideally anyN.)

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Motivations behind TNS and related questions

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Motivations behind TNS and related questions

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Motivations behind TNS and related questions

Idea :Sample a TNS at random and study the characteristics that it exhibits with high probability.

(Regime we are interested in : larged,q, ideally anyN.)

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Outline

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A simple model of random translation-invariant MPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗²whose entries are independent complex Gaussians with mean 0 and variance 1/dq. Repeat it onNsites disposed on a circle. Contract the neighbouring q-dimensional indices. The obtained tensor|χNi ∈(C^d)^⊗^Nis arandom translation-invariant MPS with periodic boundary conditions.

d q q

|χi= ∑^d

x=1 q

∑

i,j=1

g_xij|xiji

N

|χNi= ∑^d

x1,...,xN=1

q

∑

i1,...,iN=1

g_x₁_i_N_i₁· · ·g_x_N_i_N₋₁_i_N

|x₁· · ·x_Ni

Associatedtransfer operator:T:C^q⊗C^q→C^q⊗C^qobtained by contracting thed-dimensional indices of|χiand|¯χi.

T= ∑^d

x=1

q

∑

i,j,k,l=1

g_xij¯g_xkl|ikihjl|

=¹_d ∑^d

x=1

G_x⊗_G¯_x

TheG_x’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant MPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗²whose entries are independent complex Gaussians with mean 0 and variance 1/dq. Repeat it onNsites disposed on a circle. Contract the neighbouring q-dimensional indices. The obtained tensor|χNi ∈(C^d)^⊗^Nis arandom translation-invariant MPS with periodic boundary conditions.

d q q

|χi= ∑^d

x=1 q

∑

i,j=1

g_xij|xiji

N

|χNi= ∑^d

x1,...,xN=1

q

∑

i1,...,iN=1

g_x₁_i_N_i₁· · ·g_x_N_i_N₋₁_i_N

|x₁· · ·x_Ni

Associatedtransfer operator:T:C^q⊗C^q→C^q⊗C^qobtained by contracting thed-dimensional indices of|χiand|¯χi.

T= ∑^d

x=1

q

∑

i,j,k,l=1

g_xij¯g_xkl|ikihjl|

=¹_d ∑^d

x=1

G_x⊗_G¯_x

TheG_x’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant PEPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗⁴whose entries are independent complex Gaussians with mean 0 and variance 1/dq². Repeat it onMNsites disposed on a torus. Contract the neighbouring q-dimensional indices. The obtained tensor|χMNi ∈(C^d)^⊗^MNis arandom translation-invariant PEPS with periodic boundary conditions.

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χ_Mi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:TM: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^Mobtained by contracting the d-dimensional indices of|χ_Miand|¯χMi.

T_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

G_x₁_i_M_i₁⊗_G¯_x

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM

TheG_xij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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A simple model of random translation-invariant PEPS

Pick a tensor|χi ∈C^d⊗(C^q)^⊗⁴whose entries are independent complex Gaussians with mean 0 and variance 1/dq². Repeat it onMNsites disposed on a torus. Contract the neighbouring q-dimensional indices. The obtained tensor|χMNi ∈(C^d)^⊗^MNis arandom translation-invariant PEPS with periodic boundary conditions.

q d q

q q

|χi= ∑^d

x=1 q

∑

i,j,i⁰,j⁰=1

g_xiji⁰_j⁰|xiji⁰j⁰i

M

|χ_Mi ∈(C^d⊗C^q⊗C^q)^⊗^M

Associatedtransfer operator:TM: (C^q⊗C^q)^⊗^M→(C^q⊗C^q)^⊗^Mobtained by contracting the d-dimensional indices of|χ_Miand|¯χMi.

T_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM

TheG_xij’s are independentq×qmatrices whose entries are independent complex Gaussians with mean 0 and variance 1/q.

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Correlations in a TNS

MPS :M=1. PEPS :M>1.

A,Bobservables on(C^d)^⊗^M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.

A A:C^d→C^d

M

A

A: (C^d)^⊗^M→(C^d)^⊗^M

Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e. v_χ(A,B,k) : = hχMN|A₁⊗Idk⊗B₁⊗IdN−k−2|χMNi

hχ_MN|χ_MNi .

hχ_MN|χ_MNi² . Correlationsin the TNS|χMNi:γ_χ(A,B,k) : =

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question :Do we haveγχ(A,B,k) −→

kN→∞0 ? And if so, at which speed ?

A B

N

k

A B

'?

Nk1 ×

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Correlations in a TNS

MPS :M=1. PEPS :M>1.

A,Bobservables on(C^d)^⊗^M, i.e. on 1 site for an MPS and on 1 row ofMsites for a PEPS.

A A:C^d→C^d

M

A

A: (C^d)^⊗^M→(C^d)^⊗^M

Compute the value on the TNS|χMNiof the observableA1⊗Idk⊗B1⊗IdN−k−2, i.e.

v_χ(A,B,k) : = hχMN|A₁⊗Idk⊗B₁⊗IdN−k−2|χMNi hχ_MN|χ_MNi .

Compare it to the product of the values on|χ_MNiofA₁⊗Id_N−1andId_k+1⊗B₁⊗Id_N−k−2, i.e.

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question :Do we haveγχ(A,B,k) −→

kN→∞0 ? And if so, at which speed ?

A B

N

k

A B

'?

Nk1 ×

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Main result : random TNS typically exhibit fast exponential decay of correlations

Intuition :Given a TNS|χMNi ∈(C^d)^⊗^MN, there existC,τ>0 such that, for anykNand any observablesA,Bon(C^d)^⊗^M,

γχ(A,B,k)6^Ce^−τ^kkAk∞kBk∞.

−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rateτ.

Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ.

Our main result (informal)

For random translation-invariant MPS and PEPS with periodic boundary conditions this intuition is generically true, with a short correlation length.

(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow) Two ways of proving this :(suited to different dimensional regimes)

Show that the transfer operator associated to the random TNS generically has a large (upper) spectral gap.

Show that the parent Hamiltonian of the random TNS generically has a large (lower) spectral gap.

(‘large’=going to 1 asd,qgrow)

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Main result : random TNS typically exhibit fast exponential decay of correlations

Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)

(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow)

Two ways of proving this :(suited to different dimensional regimes)

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Main result : random TNS typically exhibit fast exponential decay of correlations

Correlation lengthin the TNS|χMNi:ξ(χ) : =1/τ. Our main result (informal)

(‘generically’=with probability going to 1 asd,qgrow, ‘short’=going to 0 asd,qgrow) Two ways of proving this :(suited to different dimensional regimes)

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Outline

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Correlation length in a TNS and spectrum of its transfer operator

|χMNi ∈(C^d)^⊗^MNan MPS (M=1) or a PEPS (M>1).

Tits associated transfer operator on(C^q⊗C^q)^⊗^M. Its correlation function can be re-written as : γχ(A,B,k) =

Tr _AT˜ ^k˜_BT^N⁻^k⁻²

Tr(T^N) −^Tr _AT˜ ^N⁻¹

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A A:C^d→C^d

A

A˜:C^q⊗C^q→C^q⊗C^q

Important consequence :

Denote byλ1(T),λ₂(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ1(T)| − |λ2(T)|andε(T) =|λ2(T)|/|λ1(T)|.

γ_χ(A,B,k)6^C _Tr

(T)

∆(T) 2

ε(T)^kkAk_∞kBk_∞.

−→Correlations between two 1-site or 1-row observables decay exponentially with the distance separating the two sites or rows, at a rate|logε(T)|, i.e.ξ(χ) =1/|logε(T)|.

Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?

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Correlation length in a TNS and spectrum of its transfer operator

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A A:C^d→C^d

A

Denote byλ1(T),λ₂(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ₁(T)| − |λ₂(T)|andε(T) =|λ₂(T)|/|λ₁(T)|.

γ_χ(A,B,k)6^C _Tr

(T)

∆(T) 2

Question :For a randomTwhat are typically the values of|λ1(T)|and|λ2(T)|?

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Correlation length in a TNS and spectrum of its transfer operator

Tr _BT˜ ^N⁻¹ (Tr(T^N))²

.

A A:C^d→C^d

A

Denote byλ1(T),λ₂(T), . . .the eigenvalues ofT, ordered such that|λ1(T)|>|λ2(T)|>· · ·. Set∆(T) =|λ₁(T)| − |λ₂(T)|andε(T) =|λ₂(T)|/|λ₁(T)|.

γ_χ(A,B,k)6^C _Tr

(T)

∆(T) 2

Question :For a randomTwhat are typically the values of|λ₁(T)|and|λ₂(T)|?

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Typical spectral gap of the transfer operator of a random MPS

Theorem (Typical spectral gap of the random MPS transfer operatorT) There exist universal constantsc,C>0 such that, for anyd,q∈N,

P

|λ1(T)|>¹−√^C

d and|λ2(T)|6√^C d

>¹−e⁻^cq.

Corollary (Typical correlation length in the random MPS|χ_Ni) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,

P

ξ(χ)6 ^C logd

>¹−e⁻^cmin⁽^q^,^d¹^/³⁾.

Remarks :

Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators. Other interpretation :

T

:X7→_d¹ ∑^d

x=1

G_xXG^∗_x, random completely positive map.

T

is with high probability a goodquantum expander(regularity≡high entropy approximate fixed point, low degree≡few Kraus operators, high expansion≡large spectral gap).

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Typical spectral gap of the transfer operator of a random MPS

P

|λ1(T)|>¹−√^C

>¹−e⁻^cq.

Corollary (Typical correlation length in the random MPS|χNi) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,

P

ξ(χ)6 ^C logd

>¹−e⁻^cmin⁽^q^,^d¹^/³⁾.

Remarks :

Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators. Other interpretation :

T

:X7→_d¹ ∑^d

x=1

T

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Typical spectral gap of the transfer operator of a random MPS

P

|λ1(T)|>¹−√^C

>¹−e⁻^cq.

Corollary (Typical correlation length in the random MPS|χNi) There exist universal constantsc,C>0 such that, for anyd,q,N∈N,

P

ξ(χ)6 ^C logd

>¹−e⁻^cmin⁽^q^,^d¹^/³⁾.

Remarks :

Comparable results obtained on slightly different models : by Hastings and Pisier with unitary operators, by González-Guillén/Junge/Nechita with truncated unitary operators.

Other interpretation :

T

:X7→_d¹ ∑^d

x=1

T

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Proof strategy

Let’s do as ifTwere Hermitian and thus forget about all the subtleties eigen vs singular values...

By the variational characterization of eigenvalues : for any unit vector|ϕi ∈(C^q)^⊗²,

|λ₁(T)|>|hϕ|T|ϕi|and|λ₂(T)|6kT(Id− |ϕihϕ|)k_∞.

−→Goal :Find a|ϕiwhich gives good such lower and upper bounds.

Candidate :Maximally entangledunit vector|ψi: =^√¹_q

q

∑

i=1

|iii ∈C^q⊗C^q. Intuition :Recall thatT=¹_d ∑^d

x=1

G_x⊗_G¯_x_.

First,ET =|ψihψ|. SoTis expected to have largest eigenvalue close to 1 and corresponding eigenvector close to|ψi. Second, the singular value distribution of√

d(T− |ψihψ|)converges in moments to the quarter-circle distribution (Aubrun/Nechita). So asymptotically the singular values ofT− |ψihψ|are almost all of order at most 1/√

d(but what about potential isolated ones ?) Two steps in the proof :

Show that :E|hψ|T|ψi|=1 andEkT(Id− |ψihψ|)k∞6^C/√ d.

Show that :f: (G1, . . . ,Gd)7→ kTPk_∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).

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Proof strategy

q

∑

i=1

x=1

G_x⊗_G¯_x_.

d(but what about potential isolated ones ?)

Two steps in the proof :

Show that :f: (G1, . . . ,Gd)7→ kTPk_∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).

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Proof strategy

q

∑

i=1

x=1

G_x⊗_G¯_x_.

d(but what about potential isolated ones ?) Two steps in the proof :

Show that :f : (G₁, . . . ,G_d)7→ kTPk_∞, forPany fixed projector (e.g.P=|ψihψ|or P=Id− |ψihψ|), has a small probability of deviating from its average (local version of Gaussian concentration inequality by Aubrun/Szarek/Werner).

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Typical spectral gap of the transfer operator of a random PEPS

Theorem (Typical spectral gap of the random PEPS transfer operatorT_M)

There exist universal constantsc,C>0 such that, for anyd,q,M∈Nsatisfyingd'M^αand q'M^βwithα>8 and(α+1)/3<β<(α−2)/2,

P

|λ₁(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ₂(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

Corollary (Typical correlation length in the random PEPS|χMNi)

There exist universal constantsC₀,c,C>0 such that, for anyd,q,M,N∈NsatisfyingN>^C0M, d'M^αandq'M^βwithα>11 and(α+1)/3<β<(α−3)/2,

P

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM¹^/².

Main issue :Results are valid only in the regime whered,qgrow polynomially withM,N.

−→Enforce this scaling byblocking: Start from a lattice of_Mˆ×_Nˆsites, where Mˆ=M√

logM,_Nˆ=N√

logM, with physical and bond dimensionsˆ_d,ˆq= ˇq

√logM. Redefine 1 site as being√

logM×√

logMsites, to obtain a lattice ofM×Nsites, with physical and bond dimensionsd= ˆd^logM,q= ˆq

√logM= ˇq^logM(i.e.d=M^α,q=M^βforα=logˆ_d,β: =logˇq).

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Typical spectral gap of the transfer operator of a random PEPS

P

|λ₁(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ₂(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

P

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM¹^/².

logM,_Nˆ=N√

logM×√

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Typical spectral gap of the transfer operator of a random PEPS

P

|λ₁(T_M)|>¹− ^C

M^α/²⁻¹^−β and|λ₂(T_M)|6 ^C M^α/²⁻¹^−β

>¹−e⁻^cM³^β−α.

P

ξ(χ)6 ^C log(M^α/²⁻¹^−β)

>¹−e⁻^cM¹^/².

logM,_Nˆ=N√

logM×√

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Idea of the proof and extension of the result

Recall thatT_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM. Proof idea :Some kind of recursion procedure that uses the MPS results as building blocks.

Approximate first eigenvector :|ψi^⊗^M∈(C^q⊗C^q)^⊗^M.

Extension :The result on the typical spectral gap ofT_Mremains unchanged in the non translation-invariant case (1-site random tensors picked independently identically distributed).

−→Potential application : Let

T

Mbe the corresponding random completely positive map, i.e.

T

M:X7→ ¹ d^Mq^M

d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

G_x¹

1iMi1⊗ · · · ⊗G^M_x

MiM−1iM

X G¹_x^∗

1jMj1⊗ · · · ⊗G_x^M^∗

MjM−1jM

.

It is a model of random local dissipative evolution, in analogy to random local circuits in the reversible setting.

Main question : How close to the maximally mixed state is the approximate fixed point of

T

M? And what is the speed of convergence towards it ? (cf : After which depth does the action of a circuit composed of local Haar unitaries ‘resembles’ that of a global Haar unitary ?)

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Idea of the proof and extension of the result

Recall thatT_M=_dM¹q^M d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

1jMj1⊗ · · · ⊗G_x_M_i_M₋₁_i_M⊗_G¯_x

MjM−1jM. Proof idea :Some kind of recursion procedure that uses the MPS results as building blocks.

Approximate first eigenvector :|ψi^⊗^M∈(C^q⊗C^q)^⊗^M.

Extension :The result on the typical spectral gap ofT_Mremains unchanged in the non translation-invariant case (1-site random tensors picked independently identically distributed).

−→Potential application : Let

T

Mbe the corresponding random completely positive map, i.e.

T

M:X7→ ¹ d^Mq^M

d

∑

x1,...,xM=1 q

∑

i1,j1,...,iM,jM=1

G_x¹

1iMi1⊗ · · · ⊗G^M_x

MiM−1iM

X G¹_x^∗

1jMj1⊗ · · · ⊗G_x^M^∗

MjM−1jM

.

It is a model of random local dissipative evolution, in analogy to random local circuits in the reversible setting.

Main question : How close to the maximally mixed state is the approximate fixed point of

T

M? And what is the speed of convergence towards it ? (cf : After which depth does the action of a circuit composed of local Haar unitaries ‘resembles’ that of a global Haar unitary ?)

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Outline

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Parent Hamiltonian of a TNS

Parent Hamiltonianof a TNS : local Hamiltonian which has the TNS as ground state (unique ground state in theinjectivity regime, by Cirac/Pérez-García/Verstraete/Wolf).

Fact :By the Lieb-Robinson (LR) bound (Hastings/Koma...) or the detectability lemma (Aharonov/Arad/Landau/Vazirani...), if the parent Hamiltonian of a TNS is gapped above its ground state energy then the correlations between two observables on sub-regions decay exponentially with the distance separating the two sub-regions.

Question :Is a random parent Hamiltonian typically gapped or gapless ?

Construction for our random MPS :Assumed>q², so that|χ_Niis almost surely injective. DefineV: =Span

( d

∑

x1,x2=1

Tr(G_x₁G_x₂M)|x₁x₂i:M q×qmatrix )

⊂C^d⊗C^d, where theG_x’s are theq×qmatrices appearing in the transfer operatorT.

Denoting byΠthe projector onV, the parent HamiltonianHof|χNiis then defined as H: =

N

∑

i=1

Π^⊥_i_,_i₊₁⊗Id1,...,i−1,i+2,...,N.

By constructionHis a 2-local frustration-free Hamiltonian, that has|χNias unique ground state. Construction for our random PEPS :Assumed>q⁴, so that|χ_MNiis almost surely injective.