k-extendibility
of high dimensional bipartite quantum states
Université Claude Bernard Lyon 1 / Universitat Autònoma de Barcelona
Cécilia Lancien
TUM - November 18th2014
Cécilia Lancien k-extendibility TUM - November 18th2014 1 / 21
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 and possible further developments
Separability vs Entanglement
A bipartite quantum stateρAB onA
⊗
Bisseparableif it may be written as a convex combination of product states, i.e.ρAB=
∑ipiσ(Ai)⊗
τ(Bi).Otherwise, it isentangled.
Problem :Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics. But 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 solving a semi-definite programme).Cécilia Lancien k-extendibility TUM - November 18th2014 3 / 21
Separability vs Entanglement
A bipartite quantum stateρAB onA
⊗
Bisseparableif it may be written as a convex combination of product states, i.e.ρAB=
∑ipiσ(Ai)⊗
τ(Bi).Otherwise, it isentangled.
Problem :Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics. But 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 solving a semi-definite programme).Separability vs Entanglement
A bipartite quantum stateρAB onA
⊗
Bisseparableif it may be written as a convex combination of product states, i.e.ρAB=
∑ipiσ(Ai)⊗
τ(Bi).Otherwise, it isentangled.
Problem :Deciding whether a given bipartite state is entangled or (close to) separable is an important issue in quantum physics. But 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 solving a semi-definite programme).Cécilia Lancien k-extendibility TUM - November 18th2014 3 / 21
The k-extendibility criterion for separability (1)
Definition (k -extendibility)
Letk >2. A stateρAB onA
⊗
Bisk-extendible with respect toBif there exists a stateρABk onA⊗
B⊗k which is invariant under any permutation of theB subsystems and such thatρAB=
TrBk−1ρABk.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∑ipiσ(Ai)
⊗
τ(Bi)=
TrBk−1 h∑ipiσ(Ai)
⊗
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).The k-extendibility criterion for separability (1)
Definition (k -extendibility)
Letk >2. A stateρAB onA
⊗
Bisk-extendible with respect toBif there exists a stateρABk onA⊗
B⊗k which is invariant under any permutation of theB subsystems and such thatρAB=
TrBk−1ρABk.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∑ipiσ(Ai)
⊗
τ(Bi)=
TrBk−1 h∑ipiσ(Ai)
⊗
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).Cécilia Lancien k-extendibility TUM - November 18th2014 4 / 21
The k-extendibility criterion for separability (1)
Definition (k -extendibility)
Letk >2. A stateρAB onA
⊗
Bisk-extendible with respect toBif there exists a stateρABk onA⊗
B⊗k which is invariant under any permutation of theB subsystems and such thatρAB=
TrBk−1ρABk.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∑ipiσ(Ai)
⊗
τ(Bi)=
TrBk−1 h∑ipiσ(Ai)
⊗
τ(Bi)⊗k i.
The k-extendibility criterion for separability (2)
Observation :ρABk-extendible w.r.t.B
⇒
ρAB k0-extendible w.r.t.Bfork06k.→
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 ?
Cécilia Lancien k-extendibility TUM - November 18th2014 5 / 21
The k-extendibility criterion for separability (2)
Observation :ρABk-extendible w.r.t.B
⇒
ρAB k0-extendible w.r.t.Bfork06k.→
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
Relaxing separability to k -extendibility : known results
Quantitative versions of the k-extendibility criterion :OnA
⊗
B, denote byS
the set of separable states and byE
k the set ofk-extendible states w.r.t.B. Ifρ∈ Ek, thenkρ − S k
162dB2/
k (Christandl/König/Mitchison/Renner)and
kρ − S k
LOCC→6p2 lndA
/
k(Brandão/Christandl/Yard).Problem :Interesting bounds only ifk
dB2orklndA→
What can be said in the case wheredA,
dBare “big” ? Since these bounds valid for anyk-extendible state are known to be close from optimal, one can only hope to make stronger statements about average behaviours...
A useful observation
LetMABbe a Hermitian onA
⊗
B. Then, supσAB∈EkTr(
MABσAB) =
eMABk∞, whereMeABk
=
1k∑kj=1MABj⊗
IdBk\Bj.Cécilia Lancien k-extendibility TUM - November 18th2014 6 / 21
Relaxing separability to k -extendibility : known results
Quantitative versions of the k-extendibility criterion :OnA
⊗
B, denote byS
the set of separable states and byE
k the set ofk-extendible states w.r.t.B. Ifρ∈ Ek, thenkρ − S k
162dB2/
k (Christandl/König/Mitchison/Renner)and
kρ − S k
LOCC→6p2 lndA
/
k(Brandão/Christandl/Yard).Problem :Interesting bounds only ifk
dB2ork lndA→
What can be said in the case wheredA,
dBare “big” ? Since these bounds valid for anyk-extendible state are known to be close from optimal, one can only hope to make stronger statements about average behaviours...
A useful observation
LetMABbe a Hermitian onA
⊗
B. Then, supσAB∈EkTr(
MABσAB) =
eMABk∞, whereMeABk
=
1k∑kj=1MABj⊗
IdBk\Bj.Relaxing separability to k -extendibility : known results
Quantitative versions of the k-extendibility criterion :OnA
⊗
B, denote byS
the set of separable states and byE
k the set ofk-extendible states w.r.t.B. Ifρ∈ Ek, thenkρ − S k
162dB2/
k (Christandl/König/Mitchison/Renner)and
kρ − S k
LOCC→6p2 lndA
/
k(Brandão/Christandl/Yard).Problem :Interesting bounds only ifk
dB2ork lndA→
What can be said in the case wheredA,
dBare “big” ? Since these bounds valid for anyk-extendible state are known to be close from optimal, one can only hope to make stronger statements about average behaviours...
A useful observation
LetMABbe a Hermitian onA
⊗
B. Then, supσAB∈EkTr(
MABσAB) =
eMABk∞, whereMeABk
=
1k∑kj=1MABj⊗
IdBk\Bj.Cécilia Lancien k-extendibility TUM - November 18th2014 6 / 21
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 and possible further developments
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)
•
dµSC(σ2)(
x) =
21πσ2
√
4σ2
−
x21[−2σ,2σ](
x)d
x.•
dµMP(λ)(
x) =
(fλ
(
x)d
x ifλ>
1(
1−
λ)δ0+
λfλ(
x)d
x ifλ61 , wherefλis defined by fλ(
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 toµSC(1), and that of a
(
n,λ
n)
-Wishart matrix (rescaled byλn) converges toµMP(λ).Cécilia Lancien k-extendibility TUM - November 18th2014 8 / 21
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)
•
dµSC(σ2)(
x) =
21πσ2
√
4σ2
−
x21[−2σ,2σ](
x)d
x.•
dµMP(λ)(
x) =
(fλ
(
x)d
x ifλ>
1(
1−
λ)δ0+
λfλ(
x)d
x ifλ61 , wherefλis defined by fλ(
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 toµSC(1), and that of a
(
n,λ
n)
-Wishart matrix (rescaled byλn) converges toµMP(λ).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)
•
dµSC(σ2)(
x) =
21πσ2
√
4σ2
−
x21[−2σ,2σ](
x)d
x.•
dµMP(λ)(
x) =
(fλ
(
x)d
x ifλ>
1(
1−
λ)δ0+
λfλ(
x)d
x ifλ61 , wherefλis defined by fλ(
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 toµSC(1), and that of a
(
n,λ
n)
-Wishart matrix (rescaled byλn) converges toµMP(λ).Cécilia Lancien k-extendibility TUM - November 18th2014 8 / 21
Example : Density functions of a centered semicircular distribution of parameter 1 (on the left) and of a
Marˇcenko-Pastur distribution of parameter 4 (on the right)
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 and possible further developments
Cécilia Lancien k-extendibility TUM - November 18th2014 10 / 21
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,∆) =
supσ∈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=
Ek
Gk
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.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,∆) =
supσ∈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=
Ek
Gk
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.Cécilia Lancien k-extendibility TUM - November 18th2014 11 / 21
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,∆) =
supσ∈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=
Ek
Gk
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)
Mean-width of the set of separable states
Theorem
(Aubrun/Szarek)Denote by
S
the set of separable states onCd⊗
Cd.There exist universal constantsc
,
Csuch thatc/
d3/26w( S )
6C/
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 : ApproximateS
by a polytope with “few” vertices, and use that Esupi∈IZi6Cplog
|
I|
for(
Zi)
i∈I a finite bounded Gaussian process(Pisier).•
Lower-bound : Estimate thevolume-radiusofS
by “geometric” considerations, and use thatvrad6w(Urysohn).Cécilia Lancien k-extendibility TUM - November 18th2014 12 / 21
Mean-width of the set of separable states
Theorem
(Aubrun/Szarek)Denote by
S
the set of separable states onCd⊗
Cd.There exist universal constantsc
,
Csuch thatc/
d3/26w( S )
6C/
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 : ApproximateS
by a polytope with “few” vertices, and use that Esupi∈IZi6Cplog
|
I|
for(
Zi)
i∈I a finite bounded Gaussian process(Pisier).•
Lower-bound : Estimate thevolume-radiusofS
by “geometric” considerations, and use thatvrad6w(Urysohn).Mean-width of the set of separable states
Theorem
(Aubrun/Szarek)Denote by
S
the set of separable states onCd⊗
Cd.There exist universal constantsc
,
Csuch thatc/
d3/26w( S )
6C/
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 : ApproximateS
by a polytope with “few” vertices, and use that Esupi∈IZi6Cplog
|
I|
for(
Zi)
i∈I a finite bounded Gaussian process(Pisier).•
Lower-bound : Estimate thevolume-radiusofS
by “geometric”considerations, and use thatvrad6w(Urysohn).
Cécilia Lancien k-extendibility TUM - November 18th2014 12 / 21
Mean-width of the set of k-extendible states
Theorem
Fixk >2 and denote by
E
k the set ofk-extendible states onCd⊗
Cd. Asymptotically,w( Ek) =
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: supσk−extTr
(
G(σ −
Id/
d2)) = k
Gek
∞. So one has to estimate Ek
Gek
∞for the “modified” GUE matrixG. This is done by computing thee p-order momentsETrGep, and identifying the limiting spectral distribution (afterrescaling bykd) : a centered semicircular distributionµSC(k). The latter has 2
√
k as upper-edge.
Mean-width of the set of k-extendible states
Theorem
Fixk >2 and denote by
E
k the set ofk-extendible states onCd⊗
Cd. Asymptotically,w( Ek) =
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: supσk−extTr
(
G(σ −
Id/
d2)) = k
Gek
∞. So one has to estimate Ek
Gek
∞for the “modified” GUE matrixG. This is done by computing thee p-order momentsETrGep, and identifying the limiting spectral distribution (afterrescaling bykd) : a centered semicircular distributionµSC(k). The latter has 2
√
k as upper-edge.
Cécilia Lancien k-extendibility TUM - November 18th2014 13 / 21
Mean-width of the set of k-extendible states
Theorem
Fixk >2 and denote by
E
k the set ofk-extendible states onCd⊗
Cd. Asymptotically,w( Ek) =
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: supσk−extTr
(
G(σ −
Id/
d2)) = k
Gek
∞. So one has to estimate Ek
Gek
∞for the “modified” GUE matrixG. This is done by computing thee p-order momentsETrGep, and identifying the limiting spectral distribution (afterrescaling bykd) : a centered semicircular distributionµ . The latter has
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 and possible further developments
Cécilia Lancien k-extendibility TUM - November 18th2014 14 / 21
Random induced states
System spaceH
≡
Cn. Ancilla spaceH0≡
Cs.Random mixed state model onH:ρ
=
TrH0|ψihψ|
with|ψi
a uniformly distributed pure state onH⊗
H0(quantum marginal).Equivalent description :ρ
=
TrWW withW a(
n,
s)
-Wishart matrix.Question :Fixd
∈
Nand considerρa random state onCd⊗
Cd induced by some environmentCs.For which values ofsisρtypically separable ?k-extendible ?
“typically”
=
“with overwhelming probability asd grows”. 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).
Random induced states
System spaceH
≡
Cn. Ancilla spaceH0≡
Cs.Random mixed state model onH:ρ
=
TrH0|ψihψ|
with|ψi
a uniformly distributed pure state onH⊗
H0(quantum marginal).Equivalent description :ρ
=
TrWW withW a(
n,
s)
-Wishart matrix.Question :Fixd
∈
Nand considerρa random state onCd⊗
Cd induced by some environmentCs.For which values ofsisρtypically separable ?k-extendible ?
“typically”
=
“with overwhelming probability asd grows”. 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).
Cécilia Lancien k-extendibility TUM - November 18th2014 15 / 21
Random induced states
System spaceH
≡
Cn. Ancilla spaceH0≡
Cs.Random mixed state model onH:ρ
=
TrH0|ψihψ|
with|ψi
a uniformly distributed pure state onH⊗
H0(quantum marginal).Equivalent description :ρ
=
TrWW withW a(
n,
s)
-Wishart matrix.Question :Fixd
∈
Nand considerρa random state onCd⊗
Cd induced by some environmentCs.For which values ofsisρtypically separable ?k-extendible ?
“typically”
=
“with overwhelming probability asd grows”. 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).
Random induced states
System spaceH
≡
Cn. Ancilla spaceH0≡
Cs.Random mixed state model onH:ρ
=
TrH0|ψihψ|
with|ψi
a uniformly distributed pure state onH⊗
H0(quantum marginal).Equivalent description :ρ
=
TrWW withW a(
n,
s)
-Wishart matrix.Question :Fixd
∈
Nand considerρa random state onCd⊗
Cd induced by some environmentCs.For which values ofsisρtypically separable ?k-extendible ?
“typically”
=
“with overwhelming probability asd grows”. 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).
Cécilia Lancien k-extendibility TUM - November 18th2014 15 / 21
Separability of random induced states
Theorem
(Aubrun/Szarek/Ye)Letρbe a random state onCd
⊗
Cd induced byCs. There exists a thresholds0satisfyingcd36s06Cd3log2d for some constantsc
,
Csuch that, ifs<
s0 thenρis typically entangled, and ifs>
s0thenρis typically separable.Intuition :Ifs6d2thenρis 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.
Separability of random induced states
Theorem
(Aubrun/Szarek/Ye)Letρbe a random state onCd
⊗
Cd induced byCs. There exists a thresholds0satisfyingcd36s06Cd3log2d for some constantsc
,
Csuch that, ifs<
s0 thenρis typically entangled, and ifs>
s0thenρis typically separable.Intuition :Ifs6d2thenρis 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.
Cécilia Lancien k-extendibility TUM - November 18th2014 16 / 21
Separability of random induced states
Theorem
(Aubrun/Szarek/Ye)Letρbe a random state onCd
⊗
Cd induced byCs. There exists a thresholds0satisfyingcd36s06Cd3log2d for some constantsc
,
Csuch that, ifs<
s0 thenρis typically entangled, and ifs>
s0thenρis typically separable.Intuition :Ifs6d2thenρis 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.
k-extendibility of random induced states
Theorem
Letρbe a random state onCd
⊗
Cd induced byCs. Ifs<
(k−4k1)2d2thenρis typically notk-extendible.Proof strategy: If supσk−extTr
(ρσ) <
Tr(ρ
2)
, thenρis notk-extendible. To identify when such is the case, one should characterize whenEsupσk−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 supσk−extTr(
Wσ) =k
Wek
∞, and estimateEk
Wek
∞for the“modified” Wishart matrixWe. This may be done by computing thep-order momentsETrWep, and identifying the limiting spectral distribution (after rescaling bys
/
k) : a Marˇcenko-Pastur distributionµMP(ks/d2). The latter’s support has(
pks
/
d2+
1)
2as upper-edge.•
Fors< (
k−
1)
2d2/
4k,(
pks
/
d2+
1)
2< (
d2+
s)
k/
s.Cécilia Lancien k-extendibility TUM - November 18th2014 17 / 21
k-extendibility of random induced states
Theorem
Letρbe a random state onCd
⊗
Cd induced byCs. Ifs<
(k−4k1)2d2thenρis typically notk-extendible.Proof strategy: If supσk−extTr
(ρσ) <
Tr(ρ
2)
, thenρis notk-extendible. To identify when such is the case, one should characterize whenEsupσk−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 supσk−extTr(
Wσ) =k
Wek
∞, and estimateEk
Wek
∞for the“modified” Wishart matrixWe. This may be done by computing thep-order momentsETrWep, and identifying the limiting spectral distribution (after rescaling bys
/
k) : a Marˇcenko-Pastur distributionµ . The latter’sOutline
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 and possible further developments
Cécilia Lancien k-extendibility TUM - November 18th2014 18 / 21
Some generalizations and refinements
Possible generalizations to the unbalanced caseA
≡
CdA andB≡
CdB withdA6=
dB : straightforward in the casedA,
dB→ +∞
, slightly more subtle in the case wheredAordBis fixed.Adding the constraint that the symmetric extension is PPT
→
Other complete hierarchy of separability NC where the SDP to be solved is bigger but faster(Navascués/Owari/Plenio).Result :Imposing that the symmetric extension is PPT across one cut reduces the mean-width of the set ofk-extendible states by a factor
√
2.Some generalizations and refinements
Possible generalizations to the unbalanced caseA
≡
CdA andB≡
CdB withdA6=
dB : straightforward in the casedA,
dB→ +∞
, slightly more subtle in the case wheredAordBis fixed.Adding the constraint that the symmetric extension is PPT
→
Other complete hierarchy of separability NC where the SDP to be solved is bigger but faster(Navascués/Owari/Plenio).Result :Imposing that the symmetric extension is PPT across one cut reduces the mean-width of the set ofk-extendible states by a factor
√
2.Cécilia Lancien k-extendibility TUM - November 18th2014 19 / 21
Some generalizations and refinements
Possible generalizations to the unbalanced caseA
≡
CdA andB≡
CdB withdA6=
dB : straightforward in the casedA,
dB→ +∞
, slightly more subtle in the case wheredAordBis fixed.Adding the constraint that the symmetric extension is PPT
→
Other complete hierarchy of separability NC where the SDP to be solved is bigger but faster(Navascués/Owari/Plenio).Result :Imposing that the symmetric extension is PPT across one cut reduces the mean-width of the set ofk-extendible states by a factor
√
2.Summary and 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. Whend→ +∞
, a random state onCd⊗
Cd induced byCs is w.h.p. entangled ifs<
cd3, and this entanglement is w.h.p. detected by the k-extendibility test ifs<
ckd2.→
What about the rangeCd2<
s<
cd3? By “geometric” arguments, if s>
Ckd2log2d then a random state onCd⊗
Cd induced byCs is w.h.p. k-extendible.Similar features are exhibited by all other known separability criteria, e.g. PPT(Aubrun)or realignment(Aubrun/Nechita).
What happens whenk is not fixed, but instead grows withd? Work in progress...
Cécilia Lancien k-extendibility TUM - November 18th2014 20 / 21
Summary and 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.Whend
→ +∞
, a random state onCd⊗
Cd induced byCs is w.h.p. entangled ifs<
cd3, and this entanglement is w.h.p. detected by the k-extendibility test ifs<
ckd2.→
What about the rangeCd2<
s<
cd3? By “geometric” arguments, if s>
Ckd2log2d then a random state onCd⊗
Cd induced byCs is w.h.p. k-extendible.Similar features are exhibited by all other known separability criteria, e.g. PPT(Aubrun)or realignment(Aubrun/Nechita).
What happens whenk is not fixed, but instead grows withd? Work in progress...
Summary and 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.Whend
→ +∞
, a random state onCd⊗
Cd induced byCs is w.h.p.entangled ifs
<
cd3, and this entanglement is w.h.p. detected by the k-extendibility test ifs<
ckd2.→
What about the rangeCd2<
s<
cd3? By “geometric” arguments, if s>
Ckd2log2d then a random state onCd⊗
Cd induced byCs is w.h.p.k-extendible.
Similar features are exhibited by all other known separability criteria, e.g. PPT(Aubrun)or realignment(Aubrun/Nechita).
What happens whenk is not fixed, but instead grows withd? Work in progress...
Cécilia Lancien k-extendibility TUM - November 18th2014 20 / 21
Summary and 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.Whend
→ +∞
, a random state onCd⊗
Cd induced byCs is w.h.p.entangled ifs
<
cd3, and this entanglement is w.h.p. detected by the k-extendibility test ifs<
ckd2.→
What about the rangeCd2<
s<
cd3? By “geometric” arguments, if s>
Ckd2log2d then a random state onCd⊗
Cd induced byCs is w.h.p.k-extendible.
Similar features are exhibited by all other known separability criteria, e.g.
PPT(Aubrun)or realignment(Aubrun/Nechita).
What happens whenk is not fixed, but instead grows withd? Work in progress...
Summary and 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.Whend
→ +∞
, a random state onCd⊗
Cd induced byCs is w.h.p.entangled ifs
<
cd3, and this entanglement is w.h.p. detected by the k-extendibility test ifs<
ckd2.→
What about the rangeCd2<
s<
cd3? By “geometric” arguments, if s>
Ckd2log2d then a random state onCd⊗
Cd induced byCs is w.h.p.k-extendible.
Similar features are exhibited by all other known separability criteria, e.g.
PPT(Aubrun)or realignment(Aubrun/Nechita).
What happens whenk is not fixed, but instead grows withd? Work in progress...
Cécilia Lancien k-extendibility TUM - November 18th2014 20 / 21
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
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· · ·
Xq] =
0.Ifq
=
2p, thenE[
X1· · ·
Xq] =
∑(i1,j1)···(ip,jp)∈S2(2p)∏pm=1E[
XimXjm]
.Consequence :Gan
×
nGUE matrix,W a(
n,
λn)
-Wishart matrix. ETr G2p=
∑α∈S2(2p)n](γ−1α)∼
n→+∞|
NC2(
2p)|
np+1.ETr
(
Wp) =
∑α∈S(p)λ](α)n](γ−1α)+](α)∼
n→+∞∑α∈NC(p)λ](α)np+1.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· · ·
Xq] =
0.Ifq
=
2p, thenE[
X1· · ·
Xq] =
∑(i1,j1)···(ip,jp)∈S2(2p)∏pm=1E[
XimXjm]
.Consequence :Gan
×
nGUE matrix,W a(
n,
λn)
-Wishart matrix. ETr G2p=
∑α∈S2(2p)n](γ−1α)∼
n→+∞|
NC2(
2p)|
np+1.ETr
(
Wp) =
∑α∈S(p)λ](α)n](γ−1α)+](α)∼
n→+∞∑α∈NC(p)λ](α)np+1.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· · ·
Xq] =
0.Ifq
=
2p, thenE[
X1· · ·
Xq] =
∑(i1,j1)···(ip,jp)∈S2(2p)∏pm=1E[
XimXjm]
.Consequence :Gan
×
nGUE matrix,W a(
n,
λn)
-Wishart matrix.ETr G2p
=
∑α∈S2(2p)n](γ−1α)∼
n→+∞|
NC2(
2p)|
np+1.ETr