Generic aspects of entanglement in high-dimensional quantum systems

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Generic aspects of entanglement in high-dimensional quantum systems

Some applications of asymptotic geometric analysis in quantum information theory

Cécilia Lancien

Institut de Mathématiques de Toulouse & CNRS

EPIT, CIRM – May 28 2021

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Outline

1 Introduction

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

3 Typical strength of entanglement criteria for bipartite mixed states

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Overview of the lecture’s topic

Goal of this lecture:Understand howasymptotic geometric analysiscan be useful to tackle problems arising inquantum information theory.

[The reference:Alice and Bob meet Banach, by G. Aubrun and S. Szarek.] Question:What is asymptotic geometric analysis?

Use of probabilistic techniques in the study of Banach spaces of high but finite dimension.

Why “high dimension”?

Quantum system composed of 1 particle: described by a complex Hilbert spaceH. Quantum system composed ofMsuch particles: the associated space isH^⊗^M.

−→When studying multipartite quantum systems, one has to deal with high dimensions as soon as more than a few particles are involved.

Why “probabilistic techniques”?

1 Identify typical properties of high dimensional quantum systems.

2 Prove the existence of quantum systems having certain properties, using random constructions.

In this talk:We will focus on point (1), with a particular interest for multipartite quantum systems and properties related to entanglement.

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Overview of the lecture’s topic

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Overview of the lecture’s topic

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Background

Two seminal works from 2006:

1 Aspects of generic entanglement, by P. Hayden, D. Leung and A. Winter.

Goal: Quantify the typical amount of entanglement in bipartite quantum states.

2 Tensor products of convex sets and the volume of separable states onNqudits, by G. Aubrun and S. Szarek.

Goal: Study the typical performance of conditions which are checked in practice to guarantee the entanglement of bipartite quantum states.

In this talk:We will see various more recent results, following the two directions initiated by these papers.

Note:“typical”=“with probability going to 1 (exponentially) as the underlying dimension grows”. So there are usually two steps in the argument:

1 Identify the average behavior of the property under consideration.

2 Show that this average behavior is generic in high dimension.

Concentration of measure phenomenon: a sufficiently ‘well-behaved’ function has an exponentially small probability of deviating from its average as the dimension grows.

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Background

Note:“typical”=“with probability going to 1 (exponentially) as the underlying dimension grows”. So there are usually two steps in the argument:

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Background

Note:“typical”=“with probability going to 1 (exponentially) as the underlying dimension grows”.

So there are usually two steps in the argument:

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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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Outline

1 Introduction

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Outline

1 Introduction

2 Quantifying the typical amount of entanglement/correlations in multipartite pure states Typical amount of entanglement in uniformly distributed multipartite pure states What about more ‘physically relevant’ multipartite pure states?

3 Typical strength of entanglement criteria for bipartite mixed states Entanglement detection of bipartite states

Estimating the ‘size’ of sets of states in high-dimensional bipartite systems Typical properties of random high-dimensional bipartite states

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Characterizing and quantifying pure state entanglement through tensor norms

Letψ∈H₁⊗ · · · ⊗H_Mbe a pure state, i.e.kψk2=1.

Euclidean norm

Itsinjective normis:kψkε:= sup{|hϕ1⊗ · · · ⊗ϕM|ψi|:ϕi∈Hi,kϕik2=1}. Itsprojective normis:kψk_π:= inf

_r

∑

k=1

|αk|:χ^k_i ∈Hi,kχ^k_ik2=1,ψ=

r

∑

k=1

αkχ^k₁⊗ · · · ⊗χ^k_M

.

Facts:

Theεandπnorms aredual norms:kψk_ε= sup

kϕkπ6¹

|hϕ|ψi|andkψk_π= sup

kϕkε6¹

|hϕ|ψi|. Theεandπnorms are, respectively, minimal and maximal norms amongtensor norms (i.e. norms which factorize on product vectors:kψ₁⊗ · · · ⊗ψMk=kψ₁k · · · kψ_Mk). In particular:kψkε6kψk26kψkπ.

A pure stateψis separable iffkψk_ε=kψk_π=1.

−→Ifkψkε1 orkψkπ1, thenψis ‘very’ entangled.

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Characterizing and quantifying pure state entanglement through tensor norms

Letψ∈H₁⊗ · · · ⊗H_Mbe a pure state, i.e.kψk2=1.

Euclidean norm

Itsinjective normis:kψkε:= sup{|hϕ1⊗ · · · ⊗ϕM|ψi|:ϕi∈Hi,kϕik2=1}. Itsprojective normis:kψk_π:= inf

_r

∑

k=1

|αk|:χ^k_i ∈Hi,kχ^k_ik2=1,ψ=

r

∑

k=1

αkχ^k₁⊗ · · · ⊗χ^k_M

. Facts:

Theεandπnorms aredual norms:kψk_ε= sup

kϕkπ6¹

|hϕ|ψi|andkψk_π= sup

kϕkε6¹

|hϕ|ψi|. Theεandπnorms are, respectively, minimal and maximal norms amongtensor norms (i.e. norms which factorize on product vectors:kψ₁⊗ · · · ⊗ψMk=kψ₁k · · · kψ_Mk).

In particular:kψk_ε6kψk26kψk_π.

A pure stateψis separable iffkψk_ε=kψk_π=1.

−→Ifkψkε1 orkψkπ1, thenψis ‘very’ entangled.

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Particular case of bipartite pure states

Let us look at the previous definitions in the caseM=2 andH₁≡H₂≡C^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

Let us look at the previous definitions in the caseM=2 andH₁≡H₂≡C^d. We can identify|ψi=

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

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Particular case of bipartite pure states

Let us look at the previous definitions in the caseM=2 andH₁≡H₂≡C^d. We can identify|ψi=

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

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Geometric measure of entanglement

Definition [Geometric measure of entanglement(Wei/Goldbart)]

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

=−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

=−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

=−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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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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Outline

1 Introduction

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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.

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

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

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

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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

hχ_M|χ_Mi .

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

hχM|χMi² . Correlationsin the MPSχ_M:γ_χ(A,B,k) :=

v_χ(A,B,k)−v_χ(A)v_χ(B) . Question:Do we haveγ_χ(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 hχ_M|χ_Mi .

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

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

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Exponential decay of correlations in 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

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Exponential decay of correlations in 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 and|λ2(T)|6 ^√^C

>¹−e⁻^c⁰^q.

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Perspectives

The amount of correlations in a random MPS is generically small. Is it also the case for the amount of multipartite entanglement?

−→Can we estimate the GME of a random MPS? Work in progress with I. Nechita...

What about more complicated models of random MPS, where the random 1-site tensor has some symmetries?

Can the generic amount of correlations and multipartite entanglement be computed in TNS with a more complicated geometry?

Typically small correlation length can be proven for random TNS on a 2-dimensional regular lattice(Lancien/Pérez-García), but everything else remains essentially open.

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Outline

1 Introduction

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Outline

1 Introduction

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Entanglement criteria

Known:The problem of deciding whether a given multipartite quantum state is entangled or separable (and even just approximate versions of it) is in general computationally hard(Gharibian).

−→Solution in practice: Look for necessary conditions to separability, which are easier to check than separability itself, akaentanglement criteria.

Here, we focus on the bipartite case. And we look at two such necessary conditions to separability, which can be efficiently checked and are thus widely used in practice:

beingpositive under partial transposition (PPT), beingk -extendible.

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Entanglement criteria

Known:The problem of deciding whether a given multipartite quantum state is entangled or separable (and even just approximate versions of it) is in general computationally hard(Gharibian).

−→Solution in practice: Look for necessary conditions to separability, which are easier to check than separability itself, akaentanglement criteria.

Here, we focus on the bipartite case. And we look at two such necessary conditions to separability, which can be efficiently checked and are thus widely used in practice:

beingpositive under partial transposition (PPT), beingk -extendible.

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The PPT criterion

Definition [Partial transposition]

Thepartial transposition(onB) of a stateρABonA⊗Bis defined as Γ_AB(ρ_AB) :=I_A⊗T_B(ρ_AB), whereIdenotes the identity map andTdenotes the transposition map.

Theorem [Necessary condition to separability(Peres)]

On a bipartite Hilbert spaceA⊗B, if a state is separable, then it is positive under partial transposition (PPT).

Remarks:

This is obvious sinceΓAB(σA⊗τB) =σA⊗T_B(τB).

NSC for separability onC²⊗C²orC²⊗C³(Horodecki’s). In higher dimensions, there exist PPT entangled states.

Special instance in the class of separability relaxations built on:

ρABis separable iff for any positive mapΛ_B,Id_A⊗Λ_B(ρ_AB)is positive(Horodecki’s).