When Alice and Bob meet Banach: asymptotic geometric analysis and random matrix theory in quantum information

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When Alice & Bob meet Banach

Asymptotic geometric analysis and Random matrix theory in Quantum information

Université Claude Bernard Lyon 1 & Universitat Autònoma de Barcelona

Cécilia Lancien

WYRM 2015 - Madrid

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Outline

1 Introduction

2 Interlude : reminder about GUE and Wishart matrices

3 Mean-width of the set of states which are separablevssatisfying a separability criterion

4 EntanglementvsViolating a separability criterion for random states

5 Summary and broader perspectives

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Why Asymptotic geometric analysis and Random matrix theory in Quantum information ?

State of a quantum system :Positive and trace 1 operatorρ(density operator) on a Hilbert spaceH(state space).

Transformation on a quantum system :Completely Positive and Trace Preserving map φ:X∈

B

^(H)^7→^TrE(VXV^∗)∈

B

^(H⁰⁾for some isometryV:H,→E⊗H⁰(quantum channel).

→In finite dimension : Random states or transformations≡Random matrices.

Paradigm :Constructing explicitly an operator having certain properties may be harder than asserting that a suitably chosen random one should work with large probability.

→Usually, the curse of dimensionality then becomes an asset. A few (partial) examples :

•Additivity conjectures in QI : Both counterexamples(Aubrun/Szarek/Werner...)and “weak additivity” results(Montanaro, Collins/Fukuda/Nechita...)were provided by random matrix theory and local theory of Banach spaces.

•Generic properties of states, measurements, evolutions etc. under certain constraints, such as noise, energy, locality etc.(Hayden/Leung/Winter, Linden/Popescu/Short/Winter, Aubrun/Lancien...)

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Why Asymptotic geometric analysis and Random matrix theory in Quantum information ?

B

^(H)^7→^TrE(VXV^∗)∈

B

→Usually, the curse of dimensionality then becomes an asset. A few (partial) examples :

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Why Asymptotic geometric analysis and Random matrix theory in Quantum information ?

B

^(H)^7→^TrE(VXV^∗)∈

B

→Usually, the curse of dimensionality then becomes an asset.

A few (partial) examples :

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Why Asymptotic geometric analysis and Random matrix theory in Quantum information ?

B

^(H)^7→^TrE(VXV^∗)∈

B

→Usually, the curse of dimensionality then becomes an asset.

A few (partial) examples :

•Additivity conjectures in QI : Both counterexamples(Aubrun/Szarek/Werner...)and “weak additivity”

results(Montanaro, Collins/Fukuda/Nechita...)were provided by random matrix theory and local theory of Banach spaces.

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Separability vs Entanglement for bipartite quantum systems

Definition (SeparablevsEntangled)

A bipartite quantum stateρ_ABonA⊗Bisseparableif it may be written as a convex combination of product states, i.e.ρAB=∑ip_iσ⁽_Aⁱ⁾⊗τ⁽_Bⁱ⁾. Otherwise, it isentangled.

If a compound system is in a separable global state, there are no intrinsically quantum correlations between its local constituents.

→Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics.

Problem :It is known to be a hard task, both from a mathematical and a computational point of view(Gurvits).

Solution :Find set of states which are easier to characterize and which contain the set of separable states.

→Necessary conditions for separability that have a simple mathematical description and that may be checked efficiently on a computer (e.g. by a semi-definite programme).

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Separability vs Entanglement for bipartite quantum systems

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Separability vs Entanglement for bipartite quantum systems

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The PPT criterion for separability

Definition (Partial Transpose)

The partial transpose of a stateρABonA⊗Bis defined as Γ_AB(ρ_AB) =Id_A⊗T_B(ρ_AB), whereIddenotes the identity map andTdenotes the transpose map.

NC for Separability(Peres)

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

Remarks :

•This is obvious sinceΓ_AB(σ_A⊗τB) =σA⊗T_B(τ_B).

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

•Special instance in the class of separability relaxations built on :

ρABis separable iff for any positive but not completely positive mapΛ_B,Id_A⊗Λ_B(ρ_AB)is positive

(Horodecki).

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The PPT criterion for separability

Remarks :

ρABis separable iff for any positive but not completely positive mapΛ_B,Id_A⊗Λ_B(ρ_AB)is positive

(Horodecki).

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The PPT criterion for separability

Remarks :

ρABis separable iff for any positive but not completely positive mapΛB,Id_A⊗ΛB(ρAB)is positive

(Horodecki).

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The k -extendibility criterion for separability

Definition (k-extendibility)

Letk>^{2. A state}ρABonA⊗Bisk-extendible with respect toBif there exists a stateρ_AB^k on A⊗B^⊗^kwhich is invariant under any permutation of theBsubsystems and such that

ρ_AB=Tr_B^k−1ρ_AB^k.

NSC for Separability(Doherty/Parrilo/Spedalieri)

On a bipartite Hilbert spaceA⊗B, a state is separable if and only if it isk-extendible w.r.t.Bfor allk>^2.

Remarks :

•“ρABseparable⇒ρABk-extendible w.r.t.Bfor allk>2” is obvious since σA⊗τB=Tr_Bk−1

h σA⊗τ^⊗_B^k

i .

•“ρABk-extendible w.r.t.Bfor allk>²⇒ρABseparable” relies on the quantum De Finetti theorem(Christandl/König/Mitchison/Renner).

•ρABk-extendible w.r.t.B⇒ρABk⁰-extendible w.r.t.Bfork⁰6^k.

→Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point (but one cannot tell whena priori).

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The k -extendibility criterion for separability

Remarks :

h σA⊗τ^⊗_B^k

i .

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The k -extendibility criterion for separability

Remarks :

h σA⊗τ^⊗_B^k

i .

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Quantifying the strength of separability relaxations

Problem :When relaxing the separability constraint to one which is easier to check, how “rough”

is the approximation ?

Known :There exist states which are PPT ork-extendible, and nevertheless “very” entangled (i.e. far away from the set of separable states in some standard or operational distance measure).

→Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours ?

Two possible quantitative strategies

•Estimate the size of the set of states, either satisfying a given separability criterion or being indeed separable.→Information on how much bigger than the separable set the relaxed set is.

•Characterize when certain random states are with high probability, either violating a given separability criterion or indeed entangled.→Information on how powerful the separability test is to detect entanglement.

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Quantifying the strength of separability relaxations

Problem :When relaxing the separability constraint to one which is easier to check, how “rough”

is the approximation ?

Known :There exist states which are PPT ork-extendible, and nevertheless “very” entangled (i.e. far away from the set of separable states in some standard or operational distance measure).

→Instead of looking at worst case scenarios, can we say something stronger about average/typical behaviours ?

Two possible quantitative strategies

•Estimate the size of the set of states, either satisfying a given separability criterion or being indeed separable.→Information on how much bigger than the separable set the relaxed set is.

•Characterize when certain random states are with high probability, either violating a given separability criterion or indeed entangled.→Information on how powerful the separability test is to detect entanglement.

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Outline

1 Introduction

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Asymptotic spectrum of GUE and Wishart matrices

Definitions (Gaussian Unitary Ensemble and Wishart matrices)

•Gis an×nGUE matrix ifG= (H+H^∗)/√

2 withHan×nmatrix having independent complex normal entries.

•W is a(n,s)-Wishart matrix ifW=HH^∗withHan×smatrix having independent complex normal entries.

Definitions (Semicircular and Marˇcenko-Pastur distributions)

•dµ_SC₍_m_,σ2)(x) =₂¹

πσ²

p4σ²−(x−m)²1_[_m₋₂_σ,_m₊₂_σ](x)dx.

•dµ_MP_(λ)(x) =

(f_λ(x)dxifλ>1

(1−λ)δ0+λf_λ(x)dxifλ6¹ ^{, where}^f^λis defined by f_λ(x) =

√(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_](x), withλ±= (√ λ±1)².

→For any HermitianMonCⁿ, denote byN_M=¹_n∑ⁿi=1δ_λ_i(M)its spectral distribution.

•(G_n)_n_∈_Nsequence ofn×nGUE matrices :(N_G

n/√

n)_n_∈_Nconverges toµ_SC₍₀_,₁₎.

•(W_n)_n_∈_Nsequence of(n,λn)-Wishart matrices :(N_W

n/λn)_n_∈_Nconverges toµ_MP_(λ). (convergence in distribution, but also in probability, and even almost surely).

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Asymptotic spectrum of GUE and Wishart matrices

•dµ_SC₍_m_,σ2)(x) =₂¹

πσ²

p4σ²−(x−m)²1_[_m₋₂_σ,_m₊₂_σ](x)dx.

•dµ_MP_(λ)(x) = (

f_λ(x)dxifλ>1

√(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_](x), withλ±= (√ λ±1)².

n/√

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Asymptotic spectrum of GUE and Wishart matrices

•dµ_SC₍_m_,σ2)(x) =₂¹

πσ²

p4σ²−(x−m)²1_[_m₋₂_σ,_m₊₂_σ](x)dx.

•dµ_MP_(λ)(x) = (

f_λ(x)dxifλ>1

√(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_](x), withλ±= (√ λ±1)².

n/√

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Example : Density functions of a centered semicircular distribution of variance

parameter 1 (on the left) and of a Marˇcenko-Pastur distribution of parameter 4

(on the right)

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Outline

1 Introduction

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Mean-width of a set of states

Definitions

LetKbe a convex set of states onCⁿcontainingId/n(maximally mixed state).

•For∆an×nHermitian s.t.k∆k_HS=1, thewidth of K in the direction∆is w(K,∆) =sup_σ∈_KTr(∆(σ−Id/n)).

•Themean-width of K is the average ofw(K,·)over the Hilbert-Schmidt unit sphere ofn×n Hermitians, equipped with the uniform probability measure.

It is equivalently defined asw(K) =Ew(K,G)/γ_n, whereGis an×nGUE matrix and γn=EkGkHS∼n→+∞n.

→The mean-width of a set of states is a certain measure of its size (for any

“reasonable”K,w(K)'vrad(K), where vrad(K)is thevolume-radiusofK, i.e. the radius of the Euclidean ball with same volume asK).

Computing it amounts to estimating the supremum of some Gaussian process.

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Mean-width of a set of states

Definitions

LetKbe a convex set of states onCⁿcontainingId/n(maximally mixed state).

•For∆an×nHermitian s.t.k∆k_HS=1, thewidth of K in the direction∆is w(K,∆) =sup_σ∈_KTr(∆(σ−Id/n)).

•Themean-width of K is the average ofw(K,·)over the Hilbert-Schmidt unit sphere ofn×n Hermitians, equipped with the uniform probability measure.

It is equivalently defined asw(K) =Ew(K,G)/γ_n, whereGis an×nGUE matrix and γn=EkGkHS∼n→+∞n.

→The mean-width of a set of states is a certain measure of its size (for any

“reasonable”K,w(K)'vrad(K), where vrad(K)is thevolume-radiusofK, i.e. the radius of the Euclidean ball with same volume asK).

Computing it amounts to estimating the supremum of some Gaussian process.

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Mean-width of the set of separable states

Observation : Esup_σ_stateTr(G(σ−Id/n)) =EkGk_∞. So by Wigner’s semicircle law, the mean-width of the set of all states onCⁿis asymptotically 2√

n/γn, i.e. 2/√ n.

Theorem(Aubrun/Szarek)

Denote by

S

the set of separable states onC^d⊗C^d. There exist universal constantsc,C>0 such that c

d³^/² 6^w(

S

⁾6 ^C d³^/².

Remark :The mean-width of the set of separable states is of order 1/d³^/², hence much smaller than the mean-width of the set of all states (of order 1/d).

→On high dimensional bipartite systems, most states are entangled. Proof idea:

•Upper-bound : Approximate

S

by a polytope with “few” vertices, and use that Esup_i_∈_IZi6^Cp

log|I|for(Zi)i∈Ia finite bounded Gaussian process(Pisier).

•Lower-bound : Estimate the volume-radius of

S

by convex geometry considerations, and use thatvrad6^w^(Urysohn)^.

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Mean-width of the set of separable states

n/γn, i.e. 2/√ n.

Denote by

S

d³^/² 6^w(

S

)6 ^C d³^/².

→On high dimensional bipartite systems, most states are entangled.

Proof idea:

S

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Mean-width of the set of separable states

n/γn, i.e. 2/√ n.

Denote by

S

d³^/² 6^w(

S

)6 ^C d³^/².

→On high dimensional bipartite systems, most states are entangled.

Proof idea:

S

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Mean-width of the set of k -extendible states

Theorem

Fixk>2 and denote by

E

kthe set ofk-extendible states onC^d⊗C^d. Then,w(

E

k) ∼

d→+∞

√2 k d.

Remark :The mean-width of the set ofk-extendible states is of order 1/d, hence much bigger than the mean-width of the set of separable states.

→On high dimensional bipartite systems, the set ofk-extendible states is a very rough approximation of the set of separable states.

Proof strategy: sup_σ_k₋_extTr(G(σ−Id/d²))may be expressed askeGk∞for some suitableG. Soe one has to estimateEkeGk_∞for the “modified” GUE matrixG. This is done by computing thee p-order momentsETrGe^p, and identifying the limiting spectral distribution (after rescaling byd/k) : a centered semicircular distributionµ_SC₍₀_,_k₎. The latter has 2√

kas upper-edge.

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Mean-width of the set of k -extendible states

Theorem

Fixk>2 and denote by

E

kthe set ofk-extendible states onC^d⊗C^d. Then,w(

E

k) ∼

d→+∞

√2 k d.

Remark :The mean-width of the set ofk-extendible states is of order 1/d, hence much bigger than the mean-width of the set of separable states.

→On high dimensional bipartite systems, the set ofk-extendible states is a very rough approximation of the set of separable states.

Proof strategy: sup_σ_k₋_extTr(G(σ−Id/d²))may be expressed askeGk∞for some suitableG. Soe one has to estimateEkeGk_∞for the “modified” GUE matrixG. This is done by computing thee p-order momentsETrGe^p, and identifying the limiting spectral distribution (after rescaling byd/k) : a centered semicircular distributionµ_SC₍₀_,_k₎. The latter has 2√

kas upper-edge.

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Outline

1 Introduction

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Random induced states

Definition (Purevsmixed)

A quantum stateρonHispureif there exists a unit vector|ψiinHsuch thatρ=|ψihψ|. Otherwise, it ismixed.

System spaceH≡Cⁿ. Ancilla spaceH⁰≡C^s.

Random mixed state model onH:ρ=Tr_H⁰|ψihψ|with|ψia uniformly distributed pure state onH⊗H⁰(quantum marginal).

Equivalent description :ρ=_TrW^W withWa(n,s)-Wishart matrix.

Question :Fixd∈Nand considerρa random state onC^d⊗C^dinduced by some environment C^s. For which values ofsisρtypically separable ? PPT ?k-extendible ?

“typically”=“with probability going to 1 asdgrows”. Hence 2 steps :

•Identify the range ofswhereρis, on average, separable/PPT/k-extendible.

•Show that the average behaviour 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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Random induced states

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Random induced states

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Separability of random induced states

Theorem(Aubrun/Szarek/Ye)

Letρbe a random state onC^d⊗C^dinduced byC^s. There exists a thresholds₀satisfying cd³6^s06^Cd³^log²^dfor some universal constantsc,C>0 such that, ifs<s₀thenρis typically entangled, and ifs>s₀thenρis typically separable.

Intuition :Ifs6^d²^thenρis uniformly distributed on the set of states of rank at mosts, therefore generically entangled. Ifsd²thenρis expected to be close toId/d², therefore separable.

→Phase transition between these two regimes ?

Proof idea: Convex geometry + Comparison of random matrix ensembles (majorization) + Concentration of measure in high dimension.

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Separability of random induced states

Theorem(Aubrun/Szarek/Ye)

Letρbe a random state onC^d⊗C^dinduced byC^s. There exists a thresholds₀satisfying cd³6^s06^Cd³^log²^dfor some universal constantsc,C>0 such that, ifs<s₀thenρis typically entangled, and ifs>s₀thenρis typically separable.

Intuition :Ifs6^d²^thenρis uniformly distributed on the set of states of rank at mosts, therefore generically entangled. Ifsd²thenρis expected to be close toId/d², therefore separable.

→Phase transition between these two regimes ?

Proof idea: Convex geometry + Comparison of random matrix ensembles (majorization) + Concentration of measure in high dimension.

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PPT of random induced states

Theorem(Aubrun)

Letρbe a random state onC^d⊗C^dinduced byC^s. Ifs<4d²thenρis typically not PPT, and if s>4d²thenρis typically PPT.

Remark :The threshold environment dimension at which random induced states are generically either PPT or NPT is of orderd², hence much smaller than the one at which they are generically either separable or entangled (of orderd³).

→In the ranged².^s.^d³, typical entanglement of random induced states is typically not detected by the PPT test.

Proof strategy: Everything boils down to characterizing when a partially transposed

(d²,s)-Wishart matrixWis positive. This is done by computing thep-order momentsETrΓ(W)^p, and identifying the limiting spectral distribution (after rescaling bys) : a non-centered semicircular distributionµ_SC₍₁_,_d2/s). The latter has positive support iff 1−2p

d²/s>^{0 i.e. iff}^s>^4d²^.

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PPT of random induced states

Theorem(Aubrun)

Letρbe a random state onC^d⊗C^dinduced byC^s. Ifs<4d²thenρis typically not PPT, and if s>4d²thenρis typically PPT.

Remark :The threshold environment dimension at which random induced states are generically either PPT or NPT is of orderd², hence much smaller than the one at which they are generically either separable or entangled (of orderd³).

→In the ranged².^s.^d³, typical entanglement of random induced states is typically not detected by the PPT test.

Proof strategy: Everything boils down to characterizing when a partially transposed

(d²,s)-Wishart matrixWis positive. This is done by computing thep-order momentsETrΓ(W)^p, and identifying the limiting spectral distribution (after rescaling bys) : a non-centered semicircular distributionµ_SC(1,d²/s). The latter has positive support iff 1−2p

d²/s>^{0 i.e. iff}^s>^4d²^.

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Outline

1 Introduction

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Summary and broader perspectives

On high dimensional bipartite systems, the volume of PPT ork-extendible states is more like the one of all states than like the one of separable states.

→Asymptotic weakness of these NC for separability : most PPT ork-extendible states are entangled in high dimension.

ρa random state onC^d⊗C^dinduced byC^s.

Whend→+∞,ρis w.h.p. entangled ifs<cd³, and this entanglement is w.h.p. detected by the PPT or thek-extendibility test ifs<Cd². But in the ranged²sd³, PPT or k-extendible entanglement is generic.

Similar features are exhibited by all other known separability criteria, e.g. realignment

(Aubrun/Nechita)or reduction(Jivulescu/Lupa/Nechita).

Possible generalization to the unbalanced caseA≡C^d^A,B≡C^d^B,d_A6=d_B: often straightforward ifd_A,d_B→+∞, but more subtle ifd_Aord_Bis fixed (free probability approach usually more relevant and powerful in this latter setting).

Generalizations to the multi-partite case ? The problem becomes much richer because checking separability across every bi-partite cut is not even enough to assert full separability...

→“Small” number of “big” subsystems can generally be dealt with by the same techniques, but the picture changes completely in the opposite high-dimensional setting, i.e. a “big” number of “small” subsystems.

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Summary and broader perspectives

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Summary and broader perspectives

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Summary and broader perspectives

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Summary and broader perspectives

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Summary and broader perspectives

→“Small” number of “big” subsystems can generally be dealt with by the same techniques, but the picture changes completely in the opposite high-dimensional setting, i.e. a “big”

number of “small” subsystems.

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A few references

G. Aubrun, “Partial transposition of random states and non-centered semicircular distributions”,arXiv :1011.0275 G. Aubrun, I. Nechita, “Realigning random states”,arXiv :1203.3974

G. Aubrun, S. Szarek, “Tensor products of convex sets and the volume of separable states on N qudits”, arXiv :quant-ph/0503221

G. Aubrun, S. Szarek, D. Ye, “Entanglement threshold for random induced states”,arXiv :1106.2265 M. Christandl, R. König, G. Mitchison, R. Renner, “One-and-a-half quantum De Finetti theorems”, arXiv :quant-ph/0602130

A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, “A complete family of separability criteria”,arXiv :quant-ph/0308032 P. Hayden, D. Leung, A. Winter, “Aspects of generic entanglement”,arXiv :quant-ph/0407049

M. Horodecki, P. Horodecki, R. Horodecki, “Separability of mixed states : necessary and sufficient conditions”, arXiv :quant-ph/9605038

M.A Jivulescu, N. Lupa, I. Nechita, “On the reduction criterion for random quantum states”,arXiv :1402.4292 C. Lancien, “k-extendibility of high-dimensional bipartite quantum states”,arXiv :1504.06459

A. Peres, “Separability criterion for density matrices”,arXiv :quant-ph/9604005