k-extendibility of high dimensional bipartite quantum states

(1)

k-extendibility

of high dimensional bipartite quantum states

Université Claude Bernard Lyon 1 / Universitat Autònoma de Barcelona

Cécilia Lancien

Besançon - December 18^th2014

Cécilia Lancien k-extendibility Besançon - December 18^th2014 1 / 20

(2)

Outline

1 Introduction

2 Interlude : reminder about Gaussian and Wishart matrices

3 Mean-width of the sets of separable andk-extendible states

4 Separability andk-extendibility of random states

5 Summary, generalizations, open questions

(3)

Mathematical formalism of quantum physics in a nutshell

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

Finite numbernof free parameters

→

H

≡

Cⁿ.

Pure vs mixed :A quantum stateρonHispureif there exists a unit vector

|ψi

inHsuch thatρ

= |ψihψ|

. Otherwise, it ismixed.

Separability vs Entanglement :A bipartite quantum state onA

⊗

Bis separableif it may be written as a convex combination of product states, i.e.ρAB

=

∑ip_iσ⁽_Aⁱ⁾

⊗

τ⁽_Bⁱ⁾. Otherwise, it isentangled.

(4)

Mathematical formalism of quantum physics in a nutshell

→

H

≡

Cⁿ.

|

ψ

i

inHsuch thatρ

= |

ψ

ih

ψ

|

⊗

=

⊗

(5)

Mathematical formalism of quantum physics in a nutshell

→

H

≡

Cⁿ.

|

ψ

i

inHsuch thatρ

= |

ψ

ih

ψ

|

⊗

=

⊗

(6)

Separability vs Entanglement

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).

(7)

Separability vs Entanglement

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).

(8)

The k-extendibility criterion for separability (1)

Definition (k -extendibility)

Letk >^{2. A state}ρAB onA

⊗

Bisk-extendible with respect toBif there exists a stateρ_AB^k onA

⊗

B^⊗^k which is invariant under any permutation of theB subsystems and such thatρAB

=

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 is k-extendible w.r.t.Bfor allk >^2.

Proof idea :

•

“ρAB separable

⇒

ρAB k-extendible w.r.t.Bfor allk >2” is obvious since

⊗

τ⁽_Bⁱ⁾

=

Tr_B^k−1 h

⊗

τ⁽_Bⁱ^)⊗^k i

.

•

“ρAB k-extendible w.r.t.Bfor allk>²

⇒

ρAB separable” relies on the quantum De Finetti theorem(Christandl/König/Mitchison/Renner).

(9)

The k-extendibility criterion for separability (1)

Definition (k -extendibility)

⊗

=

Tr_B^k−1ρ_AB^k.

NSC for Separability

⊗

B, a state is separable if and only if it is k-extendible w.r.t.Bfor allk>^2.

Proof idea :

•

“ρAB separable

⇒

⊗

τ⁽_Bⁱ⁾

=

Tr_B^k−1 h

⊗

τ⁽_Bⁱ^)⊗^k i

.

• ⇒

(10)

The k-extendibility criterion for separability (1)

Definition (k -extendibility)

⊗

=

Tr_B^k−1ρ_AB^k.

NSC for Separability

⊗

B, a state is separable if and only if it is k-extendible w.r.t.Bfor allk>^2.

Proof idea :

•

“ρAB separable

⇒

⊗

τ⁽_Bⁱ⁾

=

Tr_B^k−1 h

⊗

τ⁽_Bⁱ^)⊗^k i

.

• ⇒

(11)

The k-extendibility criterion for separability (2)

Observation :ρABk-extendible w.r.t.B

⇒

ρAB k⁰-extendible w.r.t.Bfork⁰6^k^.

→

Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point.

Problem :For a givenk >2, how “close” to the set of separable states is the set ofk-extendible states ? how “powerful” is thek-extendibility NC for separability ?

(12)

The k-extendibility criterion for separability (2)

Observation :ρABk-extendible w.r.t.B

⇒

ρAB k⁰-extendible w.r.t.Bfork⁰6^k^.

→

Hierarchy of NC for separability, which an entangled state is guaranteed to stop passing at some point.

Problem :For a givenk >2, how “close” to the set of separable states is the set ofk-extendible states ? how “powerful” is thek-extendibility NC for separability ?

(13)

Relaxing separability to k -extendibility : known results

Quantitative versions of the

k

-extendibility criterion

OnA

⊗

B, denote by

S

the set of separable states and by

E

k the set of k-extendible states w.r.t.B. Ifρ

∈ E

k, then

kρ − S ^k

16^2dB²

/

k (Christandl/König/Mitchison/Renner)

, kρ − S ^k

^LOCC^→6p

2 lndA

/

k (Brandão/Christandl/Yard)

.

Problem :Interesting bounds only ifk

d_B²ork

lndA

→

What can be said in the case whered_A

,

d_Bare “big” ? Since these bounds valid for any

k-extendible state are known to be close from optimal, one can only hope to make stronger statements about average behaviours...

(14)

Relaxing separability to k -extendibility : known results

Quantitative versions of the

k

-extendibility criterion

OnA

⊗

B, denote by

S

the set of separable states and by

E

k the set of k-extendible states w.r.t.B. Ifρ

∈ E

k, then

kρ − S ^k

16^2dB²

/

k (Christandl/König/Mitchison/Renner)

, kρ − S ^k

^LOCC^→6p

2 lndA

/

k (Brandão/Christandl/Yard)

.

Problem :Interesting bounds only ifk

d_B²ork

lndA

→

What can be said in the case whered_A

,

d_Bare “big” ? Since these bounds valid for any

k-extendible state are known to be close from optimal, one can only hope to make stronger statements about average behaviours...

(15)

Outline

1 Introduction

(16)

Asymptotic spectrum of Gaussian and Wishart matrices

Definitions (Gaussian Unitary and Wishart ensembles)

•

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 (Centered semicircular and Marˇcenko-Pastur distributions)

•

dµ_SC_(σ²₎

(

x

) =

₂¹

πσ²

√

4σ²

−

x²1_[−₂_σ,₂_σ]

(

x

)d

x.

•

dµ_MP_(λ)

(

x

) =

(

f_λ

(

x

)d

x ifλ

>

1

(

1

−

λ)δ0

+

λf_λ

(

x

)d

x ifλ6¹ ^{, where}^f^λis defined by f_λ

(

x

) =

√

(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_]

(

x

)

, withλ±

= ( √

λ

±

1

)

².

Link with random matrices :Whenn

→ +∞

, the spectral distribution of a n

×

nGUE matrix (rescaled by

√

n) converges toµ_SC₍₁₎, and that of a

(

n

,λ

n

)

-Wishart matrix (rescaled byλn) converges toµ_MP_(λ).

(17)

Asymptotic spectrum of Gaussian and Wishart matrices

Definitions (Gaussian Unitary and Wishart ensembles)

•

Gis an

×

nGUE matrix ifG

= (

H

+

H^∗

)/ √

2 withHan

×

•

W is a

(

n

,

s

)

-Wishart matrix ifW

=

HH^∗withHan

×

Definitions (Centered semicircular and Marˇcenko-Pastur distributions)

•

dµ_SC_(σ²₎

(

x

) =

₂¹

πσ²

√

4σ²

−

x²1_[−₂_σ,₂_σ]

(

x

)d

x.

•

dµ_MP_(λ)

(

x

) =

(f_λ

(

x

)d

x ifλ

>

1

(

1

−

λ)δ0

+

λf_λ

(

x

)d

(

x

) =

√

(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_]

(

x

)

, withλ±

= ( √

λ

±

1

)

².

→ +∞

×

√

(

n

,λ

n

)

-Wishart matrix (rescaled byλn) converges toµ_MP_(λ).

(18)

Asymptotic spectrum of Gaussian and Wishart matrices

Definitions (Gaussian Unitary and Wishart ensembles)

•

Gis an

×

nGUE matrix ifG

= (

H

+

H^∗

)/ √

2 withHan

×

•

W is a

(

n

,

s

)

-Wishart matrix ifW

=

HH^∗withHan

×

Definitions (Centered semicircular and Marˇcenko-Pastur distributions)

•

dµ_SC_(σ²₎

(

x

) =

₂¹

πσ²

√

4σ²

−

x²1_[−₂_σ,₂_σ]

(

x

)d

x.

•

dµ_MP_(λ)

(

x

) =

(f_λ

(

x

)d

x ifλ

>

1

(

1

−

λ)δ0

+

λf_λ

(

x

)d

(

x

) =

√

(λ+−x)(x−λ−)

2πλx 1_[λ₋_,λ₊_]

(

x

)

, withλ±

= ( √

λ

±

1

)

².

→ +∞

×

√

(

n

,λ

n

)

-Wishart matrix (rescaled byλn) converges toµ .

(19)

Outline

1 Introduction

(20)

Mean-width of a set of states

Definitions

LetK be a convex set of states onCⁿcontainingId

/

n.

•

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

×

nHermitians, equipped with the uniform probability measure.

It is equivalently defined asw

(

K

) =

Ew

(

K

,

G

)/γ

n, whereGis an

×

nGUE matrix andγn

=

E

k

G

k

_HS

∼

_n_→+∞n.

→

The mean-width of a set of states is a certain measure of its size.

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

Theorem (Wigner’s semicircle law)

OnCⁿ, the mean-width of the set of all states is asymptotically 2

/ √

n.

(21)

Mean-width of a set of states

Definitions

/

n.

•

For

∆

an

×

nHermitian s.t.

k∆k

_HS

=

∆

is w

(

K

,∆) =

sup_σ∈_KTr

(∆(σ −

Id

/

n

))

.

• (

K

,·)

×

(

K

) =

Ew

(

K

,

G

)/γ

n, whereGis an

×

nGUE matrix andγn

=

E

k

G

k

_HS

∼

_n_→+∞n.

→

Theorem (Wigner’s semicircle law)

/ √

n.

(22)

Mean-width of a set of states

Definitions

/

n.

•

For

∆

an

×

nHermitian s.t.

k∆k

_HS

=

∆

is w

(

K

,∆) =

sup_σ∈_KTr

(∆(σ −

Id

/

n

))

.

• (

K

,·)

×

(

K

) =

Ew

(

K

,

G

)/γ

n, whereGis an

×

nGUE matrix andγn

=

E

k

G

k

_HS

∼

_n_→+∞n.

→

Theorem (Wigner’s semicircle law)

/ √

n.

(23)

Mean-width of the set of separable states

Theorem

(Aubrun/Szarek)

Denote by

S

the set of separable states onC^d

⊗

C^d.

There exist universal constantsc

,

Csuch thatc

/

d³^/²6^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∈I a finite bounded Gaussian process.

•

Lower-bound : Estimate thevolume-radiusof

S

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

(24)

Mean-width of the set of separable states

Theorem

(Aubrun/Szarek)

Denote by

S

the set of separable states onC^d

⊗

C^d.

There exist universal constantsc

,

Csuch thatc

/

d³^/²6^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∈I a finite bounded Gaussian process.

•

Lower-bound : Estimate thevolume-radiusof

S

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

(25)

Mean-width of the set of k-extendible states

Theorem

Fixk >2 and denote by

E

k the 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 as

k

Ge

k

_∞for some suitableG. So one has to estimatee E

ke

G

k

_∞for the “modified” GUE matrix G. 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₍_k₎. The latter has 2

√

k as upper-edge.

(26)

Mean-width of the set of k-extendible states

Theorem

Fixk >2 and denote by

E

k the 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 as

k

Ge

k

_∞for some suitableG. So one has to estimatee E

ke

G

k

_∞for the “modified” GUE matrix G. 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₍_k₎. The latter has 2

√

k as upper-edge.

(27)

Outline

1 Introduction

(28)

Random induced states

System spaceH

≡

Cⁿ. Ancilla spaceH⁰

≡

C^s.

Random mixed state model onH:ρ

=

Tr_H⁰

|ψihψ|

with

|ψi

a uniformly distributed pure state onH

⊗

H⁰(quantum marginal).

Equivalent description :ρ

=

_TrW^W withW a

(

n

,

s

)

-Wishart matrix.

Question :Fixd

∈

Nand considerρa random state onC^d

⊗

C^d induced by some environmentC^s.

For which values ofsisρtypically separable ?k-extendible ?

“typically”

=

“with probability going to 1 asdgrows”. Hence 2 steps : (i) Identify the range ofswhereρis, on average, separable/k-extendible. (ii) Show that the average behaviour is generic in high dimension

(concentration of measure).

(29)

Random induced states

System spaceH

≡

Cⁿ. Ancilla spaceH⁰

≡

C^s.

Random mixed state model onH:ρ

=

Tr_H⁰

|ψihψ|

with

|ψi

a uniformly distributed pure state onH

⊗

H⁰(quantum marginal).

Equivalent description :ρ

=

_TrW^W withW a

(

n

,

s

)

-Wishart matrix.

Question :Fixd

∈

Nand considerρa random state onC^d

⊗

C^d induced by some environmentC^s.

For which values ofsisρtypically separable ?k-extendible ?

“typically”

=

“with probability going to 1 asdgrows”. Hence 2 steps : (i) Identify the range ofswhereρis, on average, separable/k-extendible.

(ii) Show that the average behaviour is generic in high dimension (concentration of measure).

(30)

Separability of random induced states

Theorem

(Aubrun/Szarek/Ye)

Letρbe a random state onC^d

⊗

C^d induced byC^s. There exists a thresholds0

satisfyingcd³6^s06^Cd³^log²^d for some constantsc

,

Csuch that, ifs

<

s₀ thenρis 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. Ifs

d²thenρis expected to be close toId

/

d², therefore separable.

→

Phase transition between these two regimes ?

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

(31)

Separability of random induced states

Theorem

(Aubrun/Szarek/Ye)

⊗

C^d induced byC^s. There exists a thresholds0

satisfyingcd³6^s06^Cd³^log²^d for some constantsc

,

Csuch that, ifs

<

s₀ thenρis 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. Ifs

d²thenρis expected to be close toId

/

d², therefore separable.

→

Phase transition between these two regimes ?

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

(32)

k-extendibility of random induced states

Theorem

⊗

C^d induced byC^s. Ifs

<

⁽^k⁻_4k¹⁾²d²thenρis typically notk-extendible.

Proof strategy: If sup_σ_k₋_extTr

(ρσ) <

Tr

(ρ

²

)

, thenρis notk-extendible. To identify when such is the case, one should characterize when

Esup_σ_k₋_extTr

(

Wσ)

<

ETr

(

W²

)/

ETrW forW a

(

d²

,

s

)

-Wishart matrix. (+ Concentration around their average of the considered functions)

•

RHS : In the limitd

,

s

→ +∞

,ETr

(

W²

) =

d⁴s

+

d²s²andETrW

=

d²s.

•

LHS : Write sup_σ_k₋_extTr

(

Wσ) =

k

We

k

_∞, and estimateE

k

We

k

_∞for the

“modified” Wishart matrixWe. This may be done by computing thep-order momentsETrWe^p, and identifying the limiting spectral distribution (after rescaling bys

/

k) : a Marˇcenko-Pastur distributionµ_MP₍_ks_/_d²₎. The latter’s support has

(

p

ks

/

d²

+

1

)

²as upper-edge.

•

Fors

< (

k

−

1

)

²d²

/

4k,

(

p

ks

/

d²

+

1

)

²

< (

d²

+

s

)

k

/

s.

(33)

k-extendibility of random induced states

Theorem

⊗

C^d induced byC^s. Ifs

<

⁽^k⁻_4k¹⁾²d²thenρis typically notk-extendible.

Proof strategy: If sup_σ_k₋_extTr

(ρσ) <

Tr

(ρ

²

)

, thenρis notk-extendible. To identify when such is the case, one should characterize when

Esup_σ_k₋_extTr

(

Wσ)

<

ETr

(

W²

)/

ETrW forW a

(

d²

,

s

)

-Wishart matrix.

(+ Concentration around their average of the considered functions)

•

RHS : In the limitd

,

s

→ +∞

,ETr

(

W²

) =

d⁴s

+

d²s²andETrW

=

d²s.

•

LHS : Write sup_σ_k₋_extTr

(

Wσ) =

k

We

k

_∞, and estimateE

k

We

k

_∞for the

“modified” Wishart matrixWe. This may be done by computing thep-order momentsETrWe^p, and identifying the limiting spectral distribution (after rescaling bys

/

k) : a Marˇcenko-Pastur distributionµ_MP₍_ks_/_d²₎. The latter’s support has

(

p

ks

/

d²

+

1

)

²as upper-edge.

•

Fors

< (

k

−

1

)

²d²

/

4k,

(

p

ks

/

d²

+

1

)

²

< (

d²

+

s

)

k

/

s.

(34)

Outline

1 Introduction

(35)

Summary, generalizations, open questions

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

→

Asymptotic weakness of thek-extendibility NC for separability. 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 thek-extendibility test ifs

<

Ckd².

→

What about the rangeCd²

<

s

<

cd³?ρis w.h.p.k-extendible if s

>

Ckd²log²d (geometric arguments).

Similar features are exhibited by all other known separability criteria, e.g. PPT^(Aubrun)or realignment(Aubrun/Nechita).

Possible generalization to the unbalanced caseA

≡

C^d^A,B

≡

C^d^B, d_A

6=

d_B(straightforward ifd_A

,

d_B

→ +∞

, more subtle ifd_Aord_Bis fixed). What happens ifk is not fixed, but grows withd? Work in progress...

(36)

Summary, generalizations, open questions

→

Asymptotic weakness of thek-extendibility NC for separability.

ρa random state onC^d

⊗

C^dinduced byC^s.

Whend

→ +∞

<

Ckd².

→

<

s

<

>

≡

C^d^A,B

≡

C^d^B, d_A

6=

,

d_B

→ +∞

(37)

Summary, generalizations, open questions

→

⊗

C^d induced byC^s.

Whend

→ +∞

<

Ckd².

→

<

s

<

>

≡

C^d^A,B

≡

C^d^B, d_A

6=

,

d_B

→ +∞

(38)

Summary, generalizations, open questions

→

⊗

C^d induced byC^s.

Whend

→ +∞

<

Ckd².

→

<

s

<

>

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

PPT^(Aubrun)or realignment(Aubrun/Nechita).

≡

C^d^A,B

≡

C^d^B, d_A

6=

,

d_B

→ +∞

(39)

Summary, generalizations, open questions

→

⊗

C^d induced byC^s.

Whend

→ +∞

<

Ckd².

→

<

s

<

>

≡

C^d^A,B

≡

C^d^B, d_A

6=

,

d_B

→ +∞

, more subtle ifd_Aord_Bis fixed).

What happens ifk is not fixed, but grows withd? Work in progress...

(40)

Summary, generalizations, open questions

→

⊗

C^d induced byC^s.

Whend

→ +∞

<

Ckd².

→

<

s

<

>

≡

C^d^A,B

≡

C^d^B, d_A

6=

,

d_B

→ +∞

, more subtle ifd_Aord_Bis fixed).

What happens ifk is not fixed, but grows withd? Work in progress...

(41)

A few references

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

G. Aubrun, Ion Nechita, “Realigning random states”,arXiv :1203.3974

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

G. Aubrun, Stanislaw Szarek, Deping Ye, “Entanglement threshold for random induced states”,arXiv :1106.2265

F.G.S.L. Brandão, M. Christandl, J.T. Yard, “Faithful Squashed Entanglement”, arXiv[quant-ph] :1010.1750

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

(42)

Combinatorics of permutations and moments of Gaussian and Wishart matrices

Notations :Letq

∈

N.S(q

)

/S₂

(

q

)

: permutations / pairings of

{

1

, . . . ,

q

}

, NC

(

q

)

/NC2

(

q

)

: non-crossing permutations / pairings of

{

1

, . . . ,

q

}

. γ: canonical full cycle,

](·)

: number of cycles in a permutation.

Lemma (Gaussian Wick formula)

LetX1

, . . . ,

Xq be centered Gaussian random variables. Ifq

=

2p

+

1, thenE

[

X₁

· · ·

X_q

] =

0.

Ifq

=

2p, thenE

[

X1

· · ·

Xq

] =

∑₍i1,j1)···(ip,jp)∈S2(2p)∏^p_m₌₁E

[

XimXjm

]

.

Consequence :Gan

×

nGUE matrix,W a

(

n

,

λn

)

-Wishart matrix. ETr G^2p

=

∑α∈S2(2p)n^](γ⁻¹^α)

∼

_n→+∞

|

NC₂

(

2p

)|

n^p⁺¹.

ETr

(

W^p

) =

∑α∈S(p)λ^](α)n^](γ⁻¹^α)+](α)

∼

_n→+∞∑α∈NC(p)λ^](α)n^p⁺¹.

(43)

Combinatorics of permutations and moments of Gaussian and Wishart matrices

Notations :Letq

∈

N.S(q

)

/S₂

(

q

)

{

1

, . . . ,

q

}

, NC

(

q

)

/NC2

(

q

)

{

1

, . . . ,

q

}

](·)

Lemma (Gaussian Wick formula)

LetX1

, . . . ,

Xq be centered Gaussian random variables.

Ifq

=

2p

+

1, thenE

[

X1

· · ·

Xq

] =

0.

Ifq

=

2p, thenE

[

X1

· · ·

Xq

] =

∑₍i1,j1)···(ip,jp)∈S2(2p)∏^p_m₌₁E

[

XimXjm

]

.

×

nGUE matrix,W a

(

n

,

λn

)

-Wishart matrix. ETr G^2p

=

∑α∈S2(2p)n^](γ⁻¹^α)

∼

_n→+∞

|

NC₂

(

2p

)|

n^p⁺¹.

ETr

(

W^p

) =

∑α∈S(p)λ^](α)n^](γ⁻¹^α)+](α)

∼

_n→+∞∑α∈NC(p)λ^](α)n^p⁺¹.

(44)

Combinatorics of permutations and moments of Gaussian and Wishart matrices

Notations :Letq

∈

N.S(q

)

/S₂

(

q

)

{

1

, . . . ,

q

}

, NC

(

q

)

/NC2

(

q

)

{

1

, . . . ,

q

}

](·)

Lemma (Gaussian Wick formula)

LetX1

, . . . ,

Ifq

=

2p

+

1, thenE

[

X1

· · ·

Xq

] =

0.

Ifq

=

2p, thenE

[

X1

· · ·

Xq

] =

∑₍i1,j1)···(ip,jp)∈S2(2p)∏^p_m₌₁E

[

XimXjm

]

.

×

nGUE matrix,W a

(

n

,

λn

)

-Wishart matrix.

ETr G^2p

=

∑α∈S2(2p)n^](γ⁻¹^α)

∼

_n→+∞

|

NC₂

(

2p

)|

n^p⁺¹.

ETr

(

W^p

) =

∑α∈S(p)λ^](α)n^](γ⁻¹^α)+](α)

∼

_n→+∞∑α∈NC(p)λ^](α)n^p⁺¹.

(45)

Combinatorics of permutations and moments of Gaussian and Wishart matrices

Notations :Letq

∈

N.S(q

)

/S₂

(

q

)

{

1

, . . . ,

q

}

, NC

(

q

)

/NC2

(

q

)

{

1

, . . . ,

q

}

](·)

Lemma (Gaussian Wick formula)

LetX1

, . . . ,

Ifq

=

2p

+

1, thenE

[

X1

· · ·

Xq

] =

0.

Ifq

=

2p, thenE

[

X1

· · ·

Xq

] =

∑₍i1,j1)···(ip,jp)∈S2(2p)∏^p_m₌₁E

[

XimXjm

]

.

×

nGUE matrix,W a

(

n

,

λn

)

-Wishart matrix.

ETr G^2p

=

∑α∈S2(2p)n^](γ⁻¹^α)

∼

_n→+∞

|

NC₂

(

2p

)|

n^p⁺¹.

ETr

(

W^p

) =

∑α∈S(p)λ^](α)n^](γ⁻¹^α)+](α)

∼

_n→+∞∑α∈NC(p)λ^](α)n^p⁺¹.