Typical amount of entanglement in multipartite quantum systems

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Typical amount of entanglement in multipartite quantum systems

Quantifying multipartite entanglement through tensor norms

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

Sakura Meeting – July 1 2021

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Outline

1 Tensor norms and entanglement

2 Quantifying the typical amount of entanglement/correlations in multipartite pure states

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Tensor norms in Banach spaces

LetA1, . . . ,AMbe Banach spaces. Givenx∈A1⊗ · · · ⊗AM, itsprojective tensor normis kxkA1⊗π···⊗πAM:= inf

( r

∑

k=1

|αk|:a^k_i ∈A_i,ka^k_ik6¹,x=

r

∑

k=1

αka₁^k⊗ · · · ⊗a^k_M,r∈N )

,

and itsinjective tensor normis

kxkA1⊗ε···⊗εAM:= sup{hb₁⊗ · · · ⊗b_M|xi:b_i∈A^∗_i,kb_ik6¹}.

These norms aredualto one another: for allx∈A₁⊗ · · · ⊗A_M, kxk_A₁_⊗_π_···⊗_π_A_M= sup

hy|xi:kyk_A^∗

1⊗ε···⊗εA^∗_M6¹ , kxk_A₁⊗ε···⊗εAM= sup

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗_M6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_M∈A_M, ka₁⊗ · · · ⊗a_MkA1⊗π···⊗πAM=ka₁⊗ · · · ⊗a_MkA1⊗ε···⊗εAM=ka₁k · · · ka_Mk. And they areextremalamong such norms: for any other tensor normk · konA1⊗ · · · ⊗AM,

k · k_A₁_⊗_ε_···⊗_ε_A_M6k · k6k · k_A₁_⊗_π_···⊗_π_A_M.

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Tensor norms in Banach spaces

( r

∑

k=1

r

∑

k=1

αka₁^k⊗ · · · ⊗a^k_M,r∈N )

,

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗_M6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_M∈A_M, ka₁⊗ · · · ⊗a_MkA1⊗π···⊗πAM=ka₁⊗ · · · ⊗a_MkA1⊗ε···⊗εAM=ka₁k · · · ka_Mk. And they areextremalamong such norms: for any other tensor normk · konA1⊗ · · · ⊗AM,

k · k_A₁_⊗_ε_···⊗_ε_A_M6k · k6k · k_A₁_⊗_π_···⊗_π_A_M.

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Tensor norms in Banach spaces

( r

∑

k=1

r

∑

k=1

αka₁^k⊗ · · · ⊗a^k_M,r∈N )

,

hy|xi:kyk_A^∗

1⊗π···⊗πA^∗_M6¹ .

The projective and injective norms are examples oftensor norms: for alla₁∈A₁, . . . ,a_M∈A_M, ka₁⊗ · · · ⊗a_MkA1⊗π···⊗πAM=ka₁⊗ · · · ⊗a_MkA1⊗ε···⊗εAM=ka₁k · · · ka_Mk.

And they areextremalamong such norms: for any other tensor normk · konA₁⊗ · · · ⊗A_M, k · k_A₁_⊗_ε_···⊗_ε_A_M6k · k6k · k_A₁_⊗_π_···⊗_π_A_M.

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Reminder: separability vs entanglement in multipartite quantum systems

H1, . . . ,HMcomplex Hilbert spaces. In this talk: of finite, but usually large, dimension.

−→H_i≡C^dⁱ withd_i1, for 16ⁱ6^M.

Definition [Separability and entanglement]

A stateρonH₁⊗ · · · ⊗H_Mis calledseparableif it is a convex combination of product states, i.e.

positive semidefinite operator with trace 1 onH1⊗ · · · ⊗HM

ρ=

r

∑

k=1

λkρ^k₁⊗ · · · ⊗ρ^k_M,with (

λk>⁰,16^k6^r,∑^r_k=1λk=1 ρ^k_i state onH_i,16^k6^r,16ⁱ6^M . Otherwise it is calledentangled.

[Note: Ifρis a pure state, i.e.ρ=|ψihψ|for some unit vectorψ∈H₁⊗ · · · ⊗H_M, thenρis separable iffψ=ψ1⊗ · · · ⊗ψMfor some unit vectorsψi∈Hi, 16ⁱ6^M.]

Fact:If a multipartite quantum system is in a separable state, there is no intrinsically quantum correlation between its subsystems. It thus does not provide any advantage over a classical system in information processing tasks.

−→Characterizing and quantifying the entanglement of multipartite quantum states is an important issue in practice.

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Reminder: separability vs entanglement in multipartite quantum systems

H1, . . . ,HMcomplex Hilbert spaces. In this talk: of finite, but usually large, dimension.

−→H_i≡C^dⁱ withd_i1, for 16ⁱ6^M.

Definition [Separability and entanglement]

A stateρonH₁⊗ · · · ⊗H_Mis calledseparableif it is a convex combination of product states, i.e.

positive semidefinite operator with trace 1 onH1⊗ · · · ⊗HM

ρ=

r

∑

k=1

λkρ^k₁⊗ · · · ⊗ρ^k_M,with (

λk>⁰,16^k6^r,∑^r_k=1λk=1 ρ^k_i state onH_i,16^k6^r,16ⁱ6^M . Otherwise it is calledentangled.

[Note: Ifρis a pure state, i.e.ρ=|ψihψ|for some unit vectorψ∈H₁⊗ · · · ⊗H_M, thenρis separable iffψ=ψ1⊗ · · · ⊗ψMfor some unit vectorsψi∈Hi, 16ⁱ6^M.]

Fact:If a multipartite quantum system is in a separable state, there is no intrinsically quantum correlation between its subsystems. It thus does not provide any advantage over a classical system in information processing tasks.

−→Characterizing and quantifying the entanglement of multipartite quantum states is an important issue in practice.

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Characterizing entanglement through tensor norms

•Pure state entanglement:

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^M.

Notation:∀x∈C^d,kxk_`d

2:= ∑^d_k₌₁|x_k|²1/2

vector 2-norm A pure stateψ∈C^d¹⊗ · · · ⊗C^d^Mis s.t.kψk_`^d1···dM

2

=1.

And it is separable iffkψk_`d1

2⊗ε···⊗ε`^dM₂ =kψk_`d1

2⊗π···⊗π`^dM₂ =1, where by definition:

kψk_`^d1

2⊗ε···⊗ε`^dM₂ := supn

hϕ1⊗ · · · ⊗ϕM|ψi:ϕi∈C^dⁱ,kϕik_`di 2

=1o , kψk_`^d1

2⊗π···⊗π`^dM₂ := inf _r

∑

k=1

|αk|:χ^k_i ∈C^dⁱ,kχ^k_ik_`di 2

=1,ψ= ∑^r

k=1

αkχ^k₁⊗ · · · ⊗χ^k_M

.

−→Ifkψk_ε1 orkψk_π1, thenψis ‘very’ entangled.

•Mixed state entanglement: Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^M.

Notation:∀X∈

M

d(C),kXk_Sd

1:= Tr|X|

matrix 1-norm A mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dM(C)is s.t.ρ>^{0 and}kρk

S1^d¹^···^dM

=1. And it is separable iffkρk_S^d1

1⊗π···⊗πS^dM₁ =1(Rudolph, Pérez-García), where by definition: kρkS1^d¹⊗π···⊗πS^dM1

:= inf _r

∑

k=1

|α_k|:τ^k_i ∈

M

di(C),kτ^k_ik

S1^di

=1,ρ= ∑^r

k=1

αkτ^k₁⊗ · · · ⊗τ^k_M

.

−→Ifkρk_π1, thenρis ‘very’ entangled.

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Characterizing entanglement through tensor norms

•Pure state entanglement:

Banach spaces

C^dⁱ,k · k_`di 2

, 16ⁱ6^M.

Notation:∀x∈C^d,kxk_`d

2:= ∑^d_k₌₁|x_k|²1/2

vector 2-norm A pure stateψ∈C^d¹⊗ · · · ⊗C^d^Mis s.t.kψk_`^d1···dM

2

=1.

And it is separable iffkψk_`d1

2⊗ε···⊗ε`^dM₂ =kψk_`d1

2⊗π···⊗π`^dM₂ =1, where by definition:

kψk_`^d1

2⊗ε···⊗ε`^dM₂ := supn

hϕ1⊗ · · · ⊗ϕM|ψi:ϕi∈C^dⁱ,kϕik_`di 2

=1o , kψk_`^d1

2⊗π···⊗π`^dM₂ := inf _r

∑

k=1

|αk|:χ^k_i ∈C^dⁱ,kχ^k_ik_`di 2

=1,ψ= ∑^r

k=1

αkχ^k₁⊗ · · · ⊗χ^k_M

.

−→Ifkψk_ε1 orkψk_π1, thenψis ‘very’ entangled.

•Mixed state entanglement:

Banach spaces

M

di(C),k · k

S^di1

, 16ⁱ6^M.

Notation:∀X∈

M

d(C),kXk_Sd

1:= Tr|X|

matrix 1-norm A mixed stateρ∈

M

d1(C)⊗ · · · ⊗

M

dM(C)is s.t.ρ>^{0 and}kρk_S^d1···_dM

1

=1.

And it is separable iffkρk_S^d1

1⊗π···⊗πS^dM₁ =1(Rudolph, Pérez-García), where by definition:

kρkS1^d¹⊗π···⊗πS^dM1

:= inf _r

∑

k=1

|α_k|:τ^k_i ∈

M

di(C),kτ^k_ik

S1^di

=1,ρ= ∑^r

k=1

αkτ^k₁⊗ · · · ⊗τ^k_M

.

−→Ifkρk_π1, thenρis ‘very’ entangled.

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Particular case of bipartite pure states

Let us look at the previous definitions in the case of pure states whenM=2 andd₁=d₂=:d.

We can identify|ψi=

d

∑

k,l=1

ψkl|kli ∈C^d⊗C^d withM_ψ=

d

∑

k,l=1

ψkl|kihl| ∈

M

d(C). Then clearly,kψk2=kM_ψk2.

And theSchmidt decompositionofψcorresponds to thesingular value decompositionofM_ψ:

|ψi=

r

∑

k=1

p

λk|e_kf_ki ←→M_ψ=

r

∑

k=1

p

λk|e_kihf_k|,

withr6^d^theSchmidt rankofψ,∑^r_k₌₁λ_k=1,{e_k}^r_k₌₁,{f_k}^r_k₌₁orthonormal sets inC^d. Sokψkε= max₁₆_k₆_r√

λk=kM_ψk∞andkψkπ=∑^r_k=1

√

λk=kM_ψk1.

−→Checking bipartite pure state separability is easy. Quantitatively, for allψ∈C^d⊗C^ds.t.kψk2=1,^√¹

d 6kψkε6^{1 and 1}6kψkπ6√ d. But no such simple characterization forM>2 (no equivalent of the Schmidt decomposition).

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Particular case of bipartite pure states

d

∑

k,l=1

d

∑

k,l=1

ψkl|kihl| ∈

M

|ψi=

r

∑

k=1

p

r

∑

k=1

p

λk|e_kihf_k|,

withr6^d^theSchmidt rankofψ,∑^r_k₌₁λ_k=1,{e_k}^r_k₌₁,{f_k}^r_k₌₁orthonormal sets inC^d. Sokψk_ε= max₁₆_k₆_r√

λk=kM_ψk_∞andkψk_π=∑^r_k=1

√

λk=kM_ψk1.

−→Checking bipartite pure state separability is easy. Quantitatively, for allψ∈C^d⊗C^ds.t.kψk2=1,^√¹

d 6kψkε6^{1 and 1}6kψkπ6√ d. But no such simple characterization forM>2 (no equivalent of the Schmidt decomposition).

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Particular case of bipartite pure states

d

∑

k,l=1

d

∑

k,l=1

ψkl|kihl| ∈

M

|ψi=

r

∑

k=1

p

r

∑

k=1

p

λk|e_kihf_k|,

withr6^d^theSchmidt rankofψ,∑^r_k₌₁λ_k=1,{e_k}^r_k₌₁,{f_k}^r_k₌₁orthonormal sets inC^d. Sokψk_ε= max₁₆_k₆_r√

λk=kM_ψk_∞andkψk_π=∑^r_k=1

√

λk=kM_ψk1.

−→Checking bipartite pure state separability is easy.

Quantitatively, for allψ∈C^d⊗C^ds.t.kψk2=1,^√¹

d 6kψkε6^{1 and 1}6kψkπ6√ d.

But no such simple characterization forM>2 (no equivalent of the Schmidt decomposition).

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Geometric measure of entanglement

Definition [Geometric measure of entanglement(Shimony, Wei/Goldbart, Zhu/Chen/Hayashi)] Letψ∈H1⊗ · · · ⊗HMbe a pure state. Itsgeometric measure of entanglement (GME)is

E(ψ) :=−log sup

|hϕ1⊗ · · · ⊗ϕM|ψi|²:ϕi∈Hi,kϕik2=1 . By definition,E(ψ) =−2logkψkε. SoE(ψ) =0 iffψis separable.

−→Eis afaithful entanglement measurefor multipartite pure states.

Remark:The definition of the GME can be extended to mixed states onH₁⊗ · · · ⊗H_M(but it is not an entanglement measure anymore), as

E(ρ) :=−log sup{hϕ1⊗ · · · ⊗ϕM|ρ|ϕ1⊗ · · · ⊗ϕMi:ϕi∈Hi,kϕik2=1}

=−log sup{Tr(ρσ) :σseparable state onH1⊗ · · · ⊗HM}. Fact:For any unit vectorψ∈(C^d)^⊗^M,kψk_ε> ^√_d¹M−1, i.e.E(ψ)6(M−1) logd. This can be checked recursively, starting from the bipartite case:

For any unit vectorψ∈C^d¹⊗C^d²,kψk_ε> ^√¹_d^{, where}^d:= min(d1,d2).

maximal possible Schmidt rank ofψ Question:Are multipartite pure states generically ‘very’ or ‘little’ entangled?

−→What is the typical value of the GME for a unit vectorψ∈(C^d)^⊗^Msampled at random?

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Geometric measure of entanglement

E(ψ) :=−log sup

=−log sup{Tr(ρσ) :σseparable state onH1⊗ · · · ⊗HM}.

Fact:For any unit vectorψ∈(C^d)^⊗^M,kψk_ε> ^√_d¹M−1, i.e.E(ψ)6(M−1) logd. This can be checked recursively, starting from the bipartite case:

For any unit vectorψ∈C^d¹⊗C^d²,kψk_ε> ^√¹_d^{, where}^d:= min(d1,d2).

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Geometric measure of entanglement

E(ψ) :=−log sup

=−log sup{Tr(ρσ) :σseparable state onH1⊗ · · · ⊗HM}. Fact:For any unit vectorψ∈(C^d)^⊗^M,kψk_ε> ^√_d¹M−1, i.e.E(ψ)6(M−1) logd.

This can be checked recursively, starting from the bipartite case:

For any unit vectorψ∈C^d¹⊗C^d²,kψk_ε> ^√¹_d^{, where}^d:= min(d₁,d₂).

maximal possible Schmidt rank ofψ

Question:Are multipartite pure states generically ‘very’ or ‘little’ entangled?

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Geometric measure of entanglement

E(ψ) :=−log sup

=−log sup{Tr(ρσ) :σseparable state onH1⊗ · · · ⊗HM}. Fact:For any unit vectorψ∈(C^d)^⊗^M,kψk_ε> ^√_d¹M−1, i.e.E(ψ)6(M−1) logd.

This can be checked recursively, starting from the bipartite case:

For any unit vectorψ∈C^d¹⊗C^d²,kψk_ε> ^√¹_d^{, where}^d:= min(d₁,d₂).

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Outline

1 Tensor norms and entanglement

2 Quantifying the typical amount of entanglement/correlations in multipartite pure states

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GME of uniformly distributed multipartite pure states

Theorem [Typicalεnorm of a random unit vector(Aubrun/Szarek)]

There exist constantsc,C,c₀>0 s.t., forψ∈(C^d)^⊗^Ma uniformly distributed unit vector,

P c

rMlogM

d^M⁻¹ 6kψk_ε6^C

rMlogM d^M⁻¹

!

>¹−e⁻^c⁰^dM^log^M.

Consequence:Forψ∈(C^d)^⊗^Ma uniformly distributed unit vector, whendorMis large, E(ψ) = (M−1) logd−log(MlogM) +O(1)with high probability.

−→A random multipartite pure state is typically close to maximally entangled.

Proof idea:Observe thatψ∼g/kgk2, whereg∈(C^d)^⊗^Mhas independent complex Gaussian entries with mean 0 and variance 1.

•By the standard Gaussian concentration inequality:P

kgk2≷√

d^M(1±ε)

6^e⁻^d^M^ε²^.

•Set

V

^:=ϕ1⊗ · · · ⊗ϕM :ϕi∈C^d,kϕik2=1 , so thatEkgkε=Esup_ϕ∈_V|hϕ|gi|. To estimate the latter quantity, use results about suprema of Gaussian processes. Upper bound: ‘small’ covering subset of

V

. Lower bound: ‘large’ separated subset of

V

^.

Conclusion:Ekgkεis of order√

dMlogM.

Then show thatg7→ kgk_εalso concentrates around its average.

Question:Are ‘interesting’ multipartite pure states really captured by the uniform distribution?

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GME of uniformly distributed multipartite pure states

P c

rMlogM

rMlogM d^M⁻¹

!

kgk2≷√

d^M(1±ε)

6^e⁻^d^M^ε²^.

•Set

V

^:=ϕ1⊗ · · · ⊗ϕM :ϕi∈C^d,kϕik2=1 , so thatEkgkε=Esup_ϕ∈_V|hϕ|gi|. To estimate the latter quantity, use results about suprema of Gaussian processes.

Upper bound: ‘small’ covering subset of

V

^.

Conclusion:Ekgk_εis of order√

dMlogM.

Question:Are ‘interesting’ multipartite pure states really captured by the uniform distribution?

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GME of uniformly distributed multipartite pure states

P c

rMlogM

rMlogM d^M⁻¹

!

kgk2≷√

d^M(1±ε)

6^e⁻^d^M^ε²^.

•Set

V

^:=ϕ1⊗ · · · ⊗ϕM :ϕi∈C^d,kϕik2=1 , so thatEkgkε=Esup_ϕ∈_V|hϕ|gi|. To estimate the latter quantity, use results about suprema of Gaussian processes.

Upper bound: ‘small’ covering subset of

V

^.

Conclusion:Ekgk_εis of order√

dMlogM.

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‘Physical’ states of many-body quantum systems and tensor network states

Curse of dimensionalityin many-body quantum systems: A system composed ofM d-dimensional subsystems has dimensiond^M, which is exponential inM.

However, ‘physically relevant’ states of many-body quantum systems, such asground states of gapped local Hamiltonians, are (conjectured to be) well approximated by so-calledtensor network states (TNS), which form a small subset of the global state space(Hastings, Landau/Vazirani/Vidick).

Tensor network state on(C^d)^⊗^M:Take a graphGwithMvertices andLedges.

Put at each vertexva tensorχv∈C^d⊗(C^q)^⊗δ(^v⁾to get a tensorˆχG∈(C^d)^⊗^M⊗(C^q)^⊗^2L. Contract together the indices ofˆχGassociated to a same edge to get a tensorχG∈(C^d)^⊗^M.

−→Ifδ(v)6δfor allv, thenχ_Gis described by at mostMq^δdparameters, which is linear inM.

• •

• • Gwith 6 vertices and 7 edges

• •

ˆχG∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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

If the underlying graphGis 1-dimensional (line or circle),χGis amatrix product state (MPS).

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‘Physical’ states of many-body quantum systems and tensor network states

However, ‘physically relevant’ states of many-body quantum systems, such asground states of gapped local Hamiltonians, are (conjectured to be) well approximated by so-calledtensor network states (TNS), which form a small subset of the global state space(Hastings, Landau/Vazirani/Vidick). Tensor network state on(C^d)^⊗^M:Take a graphGwithMvertices andLedges.

• •

ˆχG∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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‘Physical’ states of many-body quantum systems and tensor network states

However, ‘physically relevant’ states of many-body quantum systems, such asground states of gapped local Hamiltonians, are (conjectured to be) well approximated by so-calledtensor network states (TNS), which form a small subset of the global state space(Hastings, Landau/Vazirani/Vidick). Tensor network state on(C^d)^⊗^M:Take a graphGwithMvertices andLedges.

• •

ˆχG∈(C^d)^⊗⁶⊗(C^q)^⊗¹⁴

• •

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A simple model of random translation-invariant MPS

Mparticles on a circle

• • • • • M

Pick a tensorχ∈C^d⊗(C^q)^⊗²whose entries are independent complex Gaussians with mean 0 and variance 1/dq.

Repeat it on all sites and contract neighboringq-dimensional indices.

−→Obtained tensorχM∈(C^d)^⊗^M:random translation-invariant MPS with periodic boundary conditions(typically almost normalized).

d q q

|χi= ∑^d

i=1 q

∑

a,a⁰=1

g_iaa0|iaa⁰i

M

|χ_Mi= ∑^d

i1,...,iM=1

q

∑

a1,...,aM=1

g_i₁_a_M_a₁· · ·g_i_M_a_M₋₁_a_M

|i₁· · ·i_Mi

Associatedtransfer operator: T:C^q⊗C^q→C^q⊗C^q, obtained by contracting the

d-dimensional indices ofχandχ¯. T= ∑^d

i=1

q

∑

a,a⁰,b,b⁰=1

g_iaa⁰¯g_ibb⁰|abiha⁰b⁰|

=: ∑^d

i=1

G_i⊗_G¯_i

Remark:The parameterqquantifies the amount of bipartite entanglement: Across any bipartite cut preserving the ordering of subsystems,χMhas Schmidt rank at mostq²d^M^/².

area vs volume law Now what about genuinely multipartite entanglement?

−→Ifq=1,χM=χ^⊗^Mis separable. But what can we say forq1?

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A simple model of random translation-invariant MPS

• • • • • M

d q q

|χi= ∑^d

i=1 q

∑

a,a⁰=1

g_iaa0|iaa⁰i

M

|χ_Mi= ∑^d

i1,...,iM=1

q

∑

a1,...,aM=1

g_i₁_a_M_a₁· · ·g_i_M_a_M₋₁_a_M

|i₁· · ·i_Mi

Associatedtransfer operator:

T:C^q⊗C^q→C^q⊗C^q, obtained by contracting the

i=1

q

∑

a,a⁰,b,b⁰=1

=: ∑^d

i=1

G_i⊗_G¯_i

−→Ifq=1,χM=χ^⊗^Mis separable. But what can we say forq1?

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A simple model of random translation-invariant MPS

• • • • • M

d q q

|χi= ∑^d

i=1 q

∑

a,a⁰=1

g_iaa0|iaa⁰i

M

|χ_Mi= ∑^d

i1,...,iM=1

q

∑

a1,...,aM=1

g_i₁_a_M_a₁· · ·g_i_M_a_M₋₁_a_M

|i₁· · ·i_Mi

Associatedtransfer operator:

T:C^q⊗C^q→C^q⊗C^q, obtained by contracting the

i=1

q

∑

a,a⁰,b,b⁰=1

=: ∑^d

i=1

G_i⊗_G¯_i

−→Ifq=1, = ^⊗^Mis separable. But what can we say forq1?

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Correlations in an MPS

LetA,Bbe 1-site observables, i.e. observables onC^d. Goal:Quantify the correlations between the outcomes of AandB, when performed on ‘distant’ sites.

A A:C^d→C^d

B B:C^d→C^d

Compute the value on the MPSχMof the observableA₁⊗I_k⊗B₁⊗I_M₋_k₋₂, i.e. v_χ(A,B,k) :=hχM|A₁⊗I_k⊗B₁⊗I_M₋_k₋₂|χMi.

Compare it to the product of the values onχ_MofA₁⊗I_M−1andI_k+1⊗B₁⊗I_M−k−2, i.e. v_χ(A)v_χ(B) :=hχ_M|A₁⊗I_M₋₁|χ_Mihχ_M|I_k+1⊗B₁⊗I_M₋_k₋₂|χ_Mi. Correlationsin the MPSχM:γ_χ(A,B,k) :=

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

kM→∞0? And if so, at which speed?

A B

M

k

A B

'?

Mk1 ×

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Correlations in an MPS

LetA,Bbe 1-site observables, i.e. observables onC^d. Goal:Quantify the correlations between the outcomes of AandB, when performed on ‘distant’ sites.

A A:C^d→C^d

B

B:C^d→C^d Compute the value on the MPSχMof the observableA₁⊗I_k⊗B₁⊗I_M₋_k₋₂, i.e.

v_χ(A,B,k) :=hχM|A₁⊗I_k⊗B₁⊗I_M₋_k₋₂|χMi.

Compare it to the product of the values onχMofA₁⊗I_M−1andI_k+1⊗B₁⊗I_M−k−2, i.e.

v_χ(A)v_χ(B) :=hχM|A₁⊗I_M₋₁|χMihχM|I_k+1⊗B₁⊗I_M₋_k₋₂|χMi.

Correlationsin the MPSχM:γ_χ(A,B,k) :=

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

kM→∞0? And if so, at which speed?

A B

M

k

A B

'?

Mk1 ×

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Exponential decay of correlations in random translation-invariant MPS

Clearly, separability implies no correlation between 1-site observables:

IfχM=χ^⊗^M, thenγ_χ(A,B,k) =0 for anyk6^Mand any observablesA,BonC^d.

Intuition:In an MPSχM, the correlations between 1-site observables decay exponentially with the distance separating the sites, i.e. there existC(χ),τ(χ)>0 s.t., for anykMand any observablesA,BonC^d,

γχ(A,B,k)6^C(χ)e^−τ(χ)^kkAk∞kBk∞. Correlation lengthin the MPSχM:ξ(χ) :=1/τ(χ).

Theorem [Typical correlation length of a random MPS(Lancien/Pérez-García)] There exist constantsC,c₀>0 s.t., forχM∈(C^d)^⊗^Ma random translation-invariant MPS,

P

ξ(χ)6 ^C logd

>¹−e⁻^c⁰^q.

Proof idea:Letλ1(T),λ₂(T)be the two largest eigenvalues of the transfer operatorT and set ε(T) :=|λ2(T)|/|λ1(T)|. Then,γχ(A,B,k)6^C(T)ε(T)^kkAk∞kBk∞. Soξ(χ) =1/|logε(T)|. We can then prove thatP

|λ1(T)|>¹−^√^C

d and|λ2(T)|6 ^√^C_d

>¹−e⁻^c⁰^q. spectral analysis for a non-normal random matrix with tensor product structure