Distinguishing multipartite states by local measurements

(1)

Distinguishing multi-partite states by local measurements

Guillaume Aubrun / Andreas Winter

Université Claude Bernard Lyon 1 / Universitat Autònoma de Barcelona

Cécilia Lancien

UCBL1 / UAB

(2)

Outline

1 Introduction

2 Distinguishability norm associated with a single local measurement on a multi-partite system

3 Distinguishability norm associated with whole classes of locally restricted measurements on a multi-partite system

4 Conclusion and open questions

(3)

Outline

1 Introduction

(4)

Distinguishability norms (1)

Situation considered

System that can be in 2 quantum states,ρorσ, with respective prior probabilitiesqand 1

−

q.

→

Decide in which one it is most likely, based on accessible experimental data, i.e. on the outcomes of a POVMM

= (

M_x

)

_x∈X performed on it.

Probability of error :P_E

=

¹

2 1

− ∑

x∈X

^Tr

qρ

− (

1

−

q

)σ

M_x

!

Distinguishability (semi)-norm associated with the POVMM

= (

Mx

)

x∈X :

k∆k

_M

:= ∑

x∈X

^Tr

∆

M_x

SetMof POVMs :

k∆k

_M

:=

sup

M∈M

k∆k

_M

→

Holevo-Helstrom :

k∆k

_ALL

= k∆k

₁

:=

Tr

∆

(5)

Distinguishability norms (1)

Situation considered

−

q.

→

= (

M_x

)

=

¹

2 1

− ∑

x∈X

^Tr

qρ

− (

1

−

q

)σ

M_x

!

= (

Mx

)

x∈X :

k∆k

_M

:= ∑

x∈X

^Tr

∆

M_x

SetMof POVMs :

k∆k

_M

:=

sup

M∈M

k∆k

_M

→

k∆k

_ALL

= k∆k

₁

:=

Tr

∆

(6)

Distinguishability norms (1)

Situation considered

−

q.

→

= (

M_x

)

=

¹

2 1

− ∑

x∈X

^Tr

qρ

− (

1

−

q

)σ

M_x

!

= (

Mx

)

x∈X :

k∆k

_M

:= ∑

x∈X

^Tr

∆

M_x

SetMof POVMs :

k∆k

_M

:=

sup

M∈M

k∆k

_M

→

k∆k

_ALL

= k∆k

₁

:=

Tr

∆

(7)

Distinguishability norms (1)

Situation considered

−

q.

→

= (

M_x

)

=

¹

2 1

− ∑

x∈X

^Tr

qρ

− (

1

−

q

)σ

M_x

!

= (

Mx

)

x∈X :

k∆k

_M

:= ∑

x∈X

^Tr

∆

M_x

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Distinguishability norms (2)

Problem

When only POVMs from a restricted setMare allowed, how much smaller than

k · k

₁is

k · k

_M?

What kind of “restrictions” ? On a multi-partite system, experimenters are not able to implement any observable (measurements on their own sub-system).

→

Mis often defined by these locality constraints (e.g.LOCC

⊂

SEP

⊂

PPT).

Motivation :Existence of “Data-Hiding” states(DiVincenzo/Leung/Terhal) Orthogonal states (hence perfectly distinguishable by a suitable measurement) that are barely distinguishable by PPT (and even more so LOCC)

measurements.

→

Ex in the bipartite case :Completely symmetric and antisymmetric states onC^D

⊗

C^D^,σ

:=

_D2¹+D

(1 +

F)andα

:=

_D2¹−D

(1 −

F).

∆ :=

¹ 2σ

−

¹

2αis s.t.

k∆k

_PPT

=

²

D

+

1

= k∆k

₁.

(9)

Distinguishability norms (2)

Problem

When only POVMs from a restricted setMare allowed, how much smaller than

k · k

₁is

k · k

_M?

What kind of “restrictions” ? On a multi-partite system, experimenters are not able to implement any observable (measurements on their own sub-system).

→

Mis often defined by these locality constraints (e.g.LOCC

⊂

SEP

⊂

PPT).

Motivation :Existence of “Data-Hiding” states(DiVincenzo/Leung/Terhal) Orthogonal states (hence perfectly distinguishable by a suitable measurement) that are barely distinguishable by PPT (and even more so LOCC)

measurements.

→

Ex in the bipartite case :Completely symmetric and antisymmetric states

D

⊗

^D ¹ ¹

−

(10)

Outline

1 Introduction

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Local measurements on a multi-partite quantum system

Finite-dimensional multi-partite quantum system :

H ⁼

C^d¹

⊗ · · · ⊗

C^d^K ^with^d

:=

dim

H ⁼

^d¹

^{× · · · ×}

^d^K ^<

^+∞

M

=

M⁽¹⁾

⊗ · · · ⊗

M⁽^K⁾a local POVM on

H

. Comparison of

k · k

_Mwith :

k · k

₁

→

most natural ?

k · k

₂

→

most relevant ! (norm equivalence with dimension-independent constants of domination)

Definition

K-partite generalization of the 2-norm :

k∆k

₂₍_K₎

:=

s

∑

I⊂[K]

Tr TrI

∆

2

Remark :On a single system,

k∆k

₂₍₁₎

=

p

|

Tr

∆|

²

+

Tr

|∆|

²reduces to

k∆k

₂

=

p

Tr

|∆|

²on traceless Hermitians

∆

.

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Local measurements on a multi-partite quantum system

H ⁼

C^d¹

⊗ · · · ⊗

C^d^K ^with^d

:=

dim

H ⁼

^d¹

^{× · · · ×}

^d^K ^<

^+∞

M

=

M⁽¹⁾

⊗ · · · ⊗

H

. Comparison of

k · k

_Mwith :

k · k

₁

→

most natural ?

k · k

₂

→

Definition

k∆k

₂₍_K₎

:=

s

∑

I⊂[K]

Tr TrI

∆

2

k∆k

₂₍₁₎

=

p

|

Tr

∆|

²

+

Tr

|∆|

²reduces to

k∆k

₂

=

p

Tr

|∆|

∆

.

(13)

Local measurements on a multi-partite quantum system

H ⁼

C^d¹

⊗ · · · ⊗

C^d^K ^with^d

:=

dim

H ⁼

^d¹

^{× · · · ×}

^d^K ^<

^+∞

M

=

M⁽¹⁾

⊗ · · · ⊗

H

. Comparison of

k · k

_Mwith :

k · k

₁

→

most natural ?

k · k

₂

→

Definition

k∆k

₂₍_K₎

:=

s

∑

I⊂[K]

Tr TrI

∆

2

k∆k

₂₍₁₎

=

p

|

Tr

∆|

²

+

Tr

|∆|

²reduces to

k∆k

₂

=

p

Tr

|∆|

∆

.

(14)

Local measurements on a multi-partite quantum system

H ⁼

C^d¹

⊗ · · · ⊗

C^d^K ^with^d

:=

dim

H ⁼

^d¹

^{× · · · ×}

^d^K ^<

^+∞

M

=

M⁽¹⁾

⊗ · · · ⊗

H

. Comparison of

k · k

_Mwith :

k · k

₁

→

most natural ?

k · k

₂

→

Definition

k∆k

₂₍_K₎

:=

s

∑

I⊂[K]

Tr TrI

∆

2

k∆k

₂₍₁₎

=

p

|

Tr

∆|

²

+

Tr

|∆|

²reduces to

k∆k

₂

=

p

Tr

|∆|

∆

.

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Tensor product of local 4-design POVMs (1)

M

:=

Dp_xP_x

x∈X is at-design POVM onC^D^if

(

p_x

)

x∈X is a probability distribution and

(

P_x

)

x∈X are rank-1 projectors onC^D^s.t.

∑

x∈X

pxP_x^⊗^t

=

Z

|ψi∈C^D,hψ|ψi=1

|ψihψ|

^⊗^tdψ

=

¹

D

× · · · × (

D

+

t

−

1

) ∑

σ∈St

Uσ.

Example :U

:= {

D

|ψihψ|dψ, |ψi ∈

C^D,

hψ|ψi =

1

}

∞-design POVM onC^D^.

Lemma :

“Moments’ method”

M

=

M⁽¹⁾

⊗ · · · ⊗

M⁽^K⁾withM⁽^j⁾

:=

djpxjPxj

xj∈Xj 4-design POVM onC^d^j^.

∆

an Hermitian on

H

^{, and}^Sthe random variable taking value Tr

∆

Px1

⊗ · · · ⊗

PxK

with probabilitypx1

× · · · ×

pxK, so that

k∆k

_M

=

d E

|

S

|

. By Jensen :E

|

S

| ≤

p

E

(

S²

)

, and by Hölder :E

|

S

| ≥

q

(E(S²))³ E(S⁴) . So :d

s

(

E

(

S²

))

³

E(S⁴

) ≤ k∆k

_M

≤

dp E

(

S²

)

.

(16)

Tensor product of local 4-design POVMs (1)

M

:=

Dp_xP_x

(

p_x

)

(

P_x

)

∑

x∈X

pxP_x^⊗^t

=

Z

|ψihψ|

^⊗^tdψ

=

¹

D

× · · · × (

D

+

t

−

1

) ∑

σ∈St

Uσ.

Example :U

:= {

D

|ψihψ|dψ, |ψi ∈

C^D,

hψ|ψi =

1

}

Lemma :

M

=

M⁽¹⁾

⊗ · · · ⊗

:=

djpxjPxj

∆

an Hermitian on

H

∆

Px1

⊗ · · · ⊗

PxK

with probabilitypx1

× · · · ×

pxK, so that

k∆k

_M

=

d E

|

S

|

. By Jensen :E

|

S

| ≤

p

E

(

S²

)

, and by Hölder :E

|

S

| ≥

q

(E(S²))³ E(S⁴) . So :d

s

(

E

(

S²

))

³

E(S⁴

) ≤ k∆k

_M

≤

dp E

(

S²

)

.

(17)

Tensor product of local 4-design POVMs (1)

M

:=

Dp_xP_x

(

p_x

)

(

P_x

)

∑

x∈X

pxP_x^⊗^t

=

Z

|ψihψ|

^⊗^tdψ

=

¹

D

× · · · × (

D

+

t

−

1

) ∑

σ∈St

Uσ.

Example :U

:= {

D

|ψihψ|dψ, |ψi ∈

C^D,

hψ|ψi =

1

}

Lemma :

M

=

M⁽¹⁾

⊗ · · · ⊗

:=

djpxjPxj

∆

an Hermitian on

H

∆

Px1

⊗ · · · ⊗

PxK

with probabilitypx1

× · · · ×

pxK, so that

k∆k

_M

=

d E

|

S

|

.

By Jensen :E

|

S

| ≤

p

E

(

S²

)

, and by Hölder :E

|

S

| ≥

q

(E(S²))³ E(S⁴) . So :d

s

(

E

(

S²

))

³

E(S⁴

) ≤ k∆k

_M

≤

dp E

(

S²

)

.

(18)

Tensor product of local 4-design POVMs (1)

M

:=

Dp_xP_x

(

p_x

)

(

P_x

)

∑

x∈X

pxP_x^⊗^t

=

Z

|ψihψ|

^⊗^tdψ

=

¹

D

× · · · × (

D

+

t

−

1

) ∑

σ∈St

Uσ.

Example :U

:= {

D

|ψihψ|dψ, |ψi ∈

C^D,

hψ|ψi =

1

}

Lemma :

M

=

M⁽¹⁾

⊗ · · · ⊗

:=

djpxjPxj

∆

an Hermitian on

H

∆

Px1

⊗ · · · ⊗

PxK

with probabilitypx1

× · · · ×

pxK, so that

k∆k

_M

=

d E

|

S

|

. By Jensen :E

|

S

| ≤

p

E

(

S²

)

, and by Hölder :E

|

S

| ≥

q

(E(S²))³ E(S⁴) .

So :d s

(

E

(

S²

))

³

E(S⁴

) ≤ k∆k

_M

≤

dp E

(

S²

)

.

(19)

Tensor product of local 4-design POVMs (1)

M

:=

Dp_xP_x

(

p_x

)

(

P_x

)

∑

x∈X

pxP_x^⊗^t

=

Z

|ψihψ|

^⊗^tdψ

=

¹

D

× · · · × (

D

+

t

−

1

) ∑

σ∈St

Uσ.

Example :U

:= {

D

|ψihψ|dψ, |ψi ∈

C^D,

hψ|ψi =

1

}

Lemma :

M

=

M⁽¹⁾

⊗ · · · ⊗

:=

djpxjPxj

∆

an Hermitian on

H

∆

Px1

⊗ · · · ⊗

PxK

with probabilitypx1

× · · · ×

pxK, so that

k∆k

_M

=

d E

|

S

|

. By Jensen :E

|

S

| ≤

p

E

(

S²

)

, and by Hölder :E

|

S

| ≥

q

(E(S²))³ E(S⁴) .

(20)

Tensor product of local 4-design POVMs (2)

Theorem

(L/Winter)

M

=

M⁽¹⁾

⊗ · · · ⊗

M⁽^K⁾withM⁽^j⁾4-design POVM onC^d^j^. E(S²

) =

K

∏

j=1

1

dj

(

dj

+

1

) ∑

I⊂[K]

Tr TrI

∆

2

E

(

S⁴

) ≤

K

∏

j=1

1

d_j

× · · · × (

d_j

+

3

)

¹⁸

K

"

∑

I⊂[K]

Tr TrI

∆

2

#2

So : 1

18^K^/²

k∆k

₂₍_K₎

≤ k∆k

_M

≤ k∆k

₂₍_K₎.

Conclusion :IfM is a “sufficiently symmetric” local POVM on

H

^(tensor

product of local 4-design POVMs), then

k · k

_Mis “essentially” equivalent to

k · k

₂₍_K₎(dimension-independent constants of domination).

(21)

Tensor product of local 4-design POVMs (2)

Theorem

(L/Winter)

M

=

M⁽¹⁾

⊗ · · · ⊗

) =

K

∏

j=1

1

dj

(

dj

+

1

) ∑

I⊂[K]

Tr TrI

∆

2

E

(

S⁴

) ≤

K

∏

j=1

1

d_j

× · · · × (

d_j

+

3

)

¹⁸

K

"

∑

I⊂[K]

Tr TrI

∆

2

#2

So : 1

18^K^/²

k∆k

₂₍_K₎

≤ k∆k

_M

≤ k∆k

₂₍_K₎.

H

^(tensor

k · k

(22)

Tensor product of local 4-design POVMs (2)

Theorem

(L/Winter)

M

=

M⁽¹⁾

⊗ · · · ⊗

) =

K

∏

j=1

1

dj

(

dj

+

1

) ∑

I⊂[K]

Tr TrI

∆

2

E

(

S⁴

) ≤

K

∏

j=1

1

d_j

× · · · × (

d_j

+

3

)

¹⁸

K

"

∑

I⊂[K]

Tr TrI

∆

2

#2

So : 1

18^K^/²

k∆k

₂₍_K₎

≤ k∆k

_M

≤ k∆k

₂₍_K₎.

H

^(tensor

k · k

(23)

Related work and applications

Remark :Other 4^thvs2^ndorder moment inequalities may be obtained from hypercontractive inequalities(Montanaro).

Previously known results :Let

∆

be a traceless Hermitian on

H

^.

K

=

1: ¹₃

k∆k

₂

≤ k∆k

_M

≤ k∆k

₂forM a 4-design POVM.

→

Applications in quantum algorithms(Ambainis/Emerson)and quantum dimensionality reduction(Harrow/Montanaro/Short).

K

=

2: ^√¹

153

k∆k

₂

≤ k∆k

_MforM a tensor product of two 4-design POVMs(Matthews/Wehner/Winter).

→

Applications in entanglement theory(Brandão/Christandl/Yard).

→

Results for anyK and non-necessarily traceless

∆

useful too ?

(24)

Related work and applications

Remark :Other 4^thvs2^ndorder moment inequalities may be obtained from hypercontractive inequalities(Montanaro).

Previously known results :Let

∆

be a traceless Hermitian on

H

^.

K

=

1: ¹₃

k∆k

₂

≤ k∆k

_M

≤ k∆k

₂forM a 4-design POVM.

→

Applications in quantum algorithms(Ambainis/Emerson)and quantum dimensionality reduction(Harrow/Montanaro/Short).

K

=

2: ^√¹

153

k∆k

₂

≤ k∆k

_MforM a tensor product of two 4-design POVMs(Matthews/Wehner/Winter).

→

Applications in entanglement theory(Brandão/Christandl/Yard).

→

Results for anyK and non-necessarily traceless

∆

useful too ?

(25)

“Implementable” POVM with “good” discriminating power (1)

Problem

A 4-design POVM onC^d must have at least

Ω(

d⁴

)

outcomes.

No explicit constructions of 4-design POVMs are known.

→

Minimal number of outcomes for a randomly chosen POVMMonC^d^{so that} with high probability

k · k

_M

' k · k

₂₍₁₎(dimension-independently) ?

Strategy :Draw independentlyP₁, . . . ,P_nuniformly distributed rank-1 projectors onC^d^{, set}^S

:=

∑ⁿ

k=1

Pk andP

:= (e

Pk

:=

S⁻¹^/²PkS⁻¹^/²

)

1≤k≤n. Pis a random POVM onC^d^withⁿoutcomes s.t.

k∆k

_P

=

n

∑

k=1

Tr(∆eP_k

)

.

Theorem

(Aubrun/L)

∀α

>0,

∃

Cα>0

:

n

≥

Cαd²

⇒

P 1

8√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵₈

k · k

₂₍₁₎

≥

1

−

α

Remark :“Optimal result” sincePinformationally complete

⇒

n

≥

d².

(26)

“Implementable” POVM with “good” discriminating power (1)

Problem

Ω(

d⁴

)

outcomes.

→

k · k

_M

' k · k

:=

∑ⁿ

k=1

P_k andP

:= (e

P_k

:=

S⁻¹^/²P_kS⁻¹^/²

)

₁≤k≤n. Pis a random POVM onC^d^withⁿoutcomes s.t.

k∆k

_P

=

n

∑

k=1

Tr(∆eP_k

)

.

Theorem

(Aubrun/L)

∀α

>0,

∃

Cα>0

:

n

≥

Cαd²

⇒

P 1

8√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵₈

k · k

₂₍₁₎

≥

1

−

α

⇒

n

≥

d².

(27)

“Implementable” POVM with “good” discriminating power (1)

Problem

Ω(

d⁴

)

outcomes.

→

k · k

_M

' k · k

:=

∑ⁿ

k=1

P_k andP

:= (e

P_k

:=

S⁻¹^/²P_kS⁻¹^/²

)

k∆k

_P

=

n

∑

k=1

Tr(∆eP_k

)

.

Theorem

(Aubrun/L)

⇒

n

≥

d².

(28)

“Implementable” POVM with “good” discriminating power (1)

Problem

Ω(

d⁴

)

outcomes.

→

k · k

_M

' k · k

:=

∑ⁿ

k=1

P_k andP

:= (e

P_k

:=

S⁻¹^/²P_kS⁻¹^/²

)

k∆k

_P

=

n

∑

k=1

Tr(∆eP_k

)

.

Theorem

(Aubrun/L)

∀α

>0,

∃

Cα>0

:

n

≥

Cαd²

⇒

P 1

8√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵₈

k · k

₂₍₁₎

≥

1

−

α

⇒

n

≥

d².

(29)

“Implementable” POVM with “good” discriminating power (2)

Idea of the proof :

1 Large deviation probability of ^d_n

n

∑

k=1

|Tr(

P_k

∆)|

from

k∆k

₂₍₁₎?

2 Large deviation probability of

n

∑

k=1

|Tr(e

Pk

∆)|

from ^d_n

n

∑

k=1

|Tr(

Pk

∆)|

?

→

Use twice :

(i) Moments’ estimate :

∀

p

≥

1,E

|

X

|

^p

≤ (

e^cp

)

^p

⇒ k

X

k

_ψ₁ .e^c (ii) Bernstein-type tail bound for sums of i.i.d centeredψ1-r.v. (iii) Net argument : individual to global error term

1

{

X_k

:=

d

Tr P_k

∆

− k∆k

_U

}

₁_≤_k_≤_nfor a given

∆ ∈

S₂₍₁₎

2

{

Xk

:= hψ|

dPk

−

1|ψi}₁_≤_k_≤_nfor a given unit vector

|ψi

P

₁ 8

√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵

8

k · k

₂₍₁₎

≥

1

−

C^d²e⁻^cn

−

C⁰^de⁻^c⁰ⁿ^/^d

(30)

“Implementable” POVM with “good” discriminating power (2)

n

∑

k=1

|Tr(

P_k

∆)|

from

k∆k

₂₍₁₎?

n

∑

k=1

|Tr(e

Pk

∆)|

from ^d_n

n

∑

k=1

|Tr(

Pk

∆)|

?

→

Use twice :

∀

p

≥

1,E

|

X

|

^p

≤ (

e^cp

)

^p

⇒ k

X

k

_ψ₁ .e^c (ii) Bernstein-type tail bound for sums of i.i.d centeredψ1-r.v.

(iii) Net argument : individual to global error term

1

{

X_k

:=

d

Tr P_k

∆

− k∆k

_U

}

∆ ∈

S₂₍₁₎

2

{

Xk

:= hψ|

dPk

−

|ψi

P

₁ 8

√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵

8

k · k

₂₍₁₎

≥

1

−

C^d²e⁻^cn

−

(31)

“Implementable” POVM with “good” discriminating power (2)

n

∑

k=1

|Tr(

P_k

∆)|

from

k∆k

₂₍₁₎?

n

∑

k=1

|Tr(e

Pk

∆)|

from ^d_n

n

∑

k=1

|Tr(

Pk

∆)|

?

→

Use twice :

∀

p

≥

1,E

|

X

|

^p

≤ (

e^cp

)

^p

⇒ k

X

k

1

{

X_k

:=

d

Tr P_k

∆

− k∆k

_U

}

∆ ∈

S₂₍₁₎

2

{

Xk

:= hψ|

dPk

−

|ψi

P ₁

8

√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵

8

k · k

₂₍₁₎

≥

1

−

C^d²e⁻^cn

−

(32)

“Implementable” POVM with “good” discriminating power (2)

n

∑

k=1

|Tr(

P_k

∆)|

from

k∆k

₂₍₁₎?

n

∑

k=1

|Tr(e

Pk

∆)|

from ^d_n

n

∑

k=1

|Tr(

Pk

∆)|

?

→

Use twice :

∀

p

≥

1,E

|

X

|

^p

≤ (

e^cp

)

^p

⇒ k

X

k

1

{

X_k

:=

d

Tr P_k

∆

− k∆k

_U

}

∆ ∈

S₂₍₁₎

2

{

Xk

:= hψ|

dPk

−

|ψi

P

₁ 8

√

18

k · k

₂₍₁₎

≤ k · k

_P

≤

¹⁵

8

k · k

₂₍₁₎

≥

1

−

C^d²e⁻^cn

−

(33)

Outline

1 Introduction

(34)

Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements

A tensor product of 4-design POVMs is a particular LOCC strategy. Consequently : 1

18^K^/²

k · k

₂

≤

¹

18^K^/²

k · k

₂₍_K₎

≤ k · k

_LOCC.

On am-partite Hilbert space, the ball of radius 2¹⁻^m^/²for

k · k

₂(centered at1) is fully separable(Barnum/Gurvits).

Consequently : 2

2^K^/²

k · k

₂

≤ k · k

SEPand

k · k

₂

≤ k · k

PPT.

Comparison of k · k

LOCC

, k · k

SEP

and k · k

PPT

with k · k

ALL

1 18^K^/²

√

1

d

k · k

₁

≤ k · k

_LOCC

≤ k · k

₁ 2

2^K^/²

√

1

d

k · k

₁

≤ k · k

_SEP

≤ k · k

₁

√

1

d

k · k

₁

≤ k · k

_PPT

≤ k · k

₁

(35)

Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements

A tensor product of 4-design POVMs is a particular LOCC strategy.

Consequently : 1

18^K^/²

k · k

₂

≤

¹

18^K^/²

k · k

₂₍_K₎

≤ k · k

_LOCC.

k · k

Consequently : 2

2^K^/²

k · k

₂

≤ k · k

SEPand

k · k

₂

≤ k · k

PPT.

Comparison of k · k

LOCC

, k · k

SEP

and k · k

PPT

with k · k

ALL

1 18^K^/²

√

1

d

k · k

₁

≤ k · k

_LOCC

≤ k · k

₁ 2

2^K^/²

√

1

d

k · k

₁

≤ k · k

_SEP

≤ k · k

₁

√

1

d

k · k

₁

≤ k · k

_PPT

≤ k · k

₁

(36)

Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements

Consequently : 1

18^K^/²

k · k

₂

≤

¹

18^K^/²

k · k

₂₍_K₎

≤ k · k

_LOCC.

k · k

Consequently : 2

2^K^/²

k · k

₂

≤ k · k

_SEPand

k · k

₂

≤ k · k

_PPT.

Comparison of k · k

LOCC

, k · k

SEP

and k · k

PPT

with k · k

ALL 1

18^K^/²

√

1

d

k · k

₁

≤ k · k

_LOCC

≤ k · k

₁ 2

2^K^/²

√

1

d

k · k

₁

≤ k · k

_SEP

≤ k · k

₁

√

1

d

k · k

₁

≤ k · k

_PPT

≤ k · k

₁

(37)

Bounds on the distinguishability of multi-partite states under LOCC, SEP and PPT measurements

Consequently : 1

18^K^/²

k · k

₂

≤

¹

18^K^/²

k · k

₂₍_K₎

≤ k · k

_LOCC.

k · k

Consequently : 2

2^K^/²

k · k

₂

≤ k · k

_SEPand

k · k

₂

≤ k · k

_PPT.

Comparison of k · k

LOCC

, k · k

SEP

and k · k

PPT

with k · k

ALL

1 18^K^/²

√

1

d

k · k

₁

≤ k · k

_LOCC

≤ k · k

₁

2 1

√ k · k ≤ k · k ≤ k · k

(38)

Data-Hiding and optimality of the lower bounds

M

:=

U_C^d1

⊗ · · · ⊗

U_CdK tensor product of the local uniform POVMs. Tightness of ¹

18^K/2

√1

d

k · k

₁

≤ k · k

_M?

∃ ∆ 6=

0

: k∆k

_M

≤

^δ^√^K/2

d

k∆k

₁with ²

π<δ<1

→

Dependence onK anddof the constant relating the norms is “real”. Tightness of ^√¹

d

k · k

₁

≤ k · k

_PPT? If

∃

I

⊂ [

K

] :

∏

i∈I

di

=

∏

i∈/I

di

= √

d, then

∃ ∆ 6=

0

: k∆k

_PPT

≤

^√²

d+1

k∆k

₁

→

Dependence ond of the constant relating the norms is “real”. Remark :In the special case

H ^{= (}

C^D

)

^⊗^K

(

d

=

D^K

)

withK fixed and D

→ +∞

, it may be shown by volumic considerations that “typically” : (

k · k

_SEP

'

√ ¹

d^1−1/K

k · k

₁

k · k

_PPT

' k · k

₁ , so that :

k · k

_PPT

k · k

_SEP

^√¹

d

k · k

₁(Aubrun/L).

→

Data-Hiding Hermitians are “exceptionnal”.

(39)

Data-Hiding and optimality of the lower bounds

M

:=

U_C^d1

⊗ · · · ⊗

U_CdK tensor product of the local uniform POVMs.

Tightness of ¹

18^K/2

√1

d

k · k

₁

≤ k · k

_M?

∃ ∆ 6=

0

: k∆k

_M

≤

^δ^√^K/2

d

k∆k

₁with ²

π<δ<1

→

Dependence onK anddof the constant relating the norms is “real”.

Tightness of ^√¹

d

k · k

₁

≤ k · k

_PPT? If

∃

I

⊂ [

K

] :

∏

i∈I

di

=

∏

i∈/I

di

= √

d, then

∃ ∆ 6=

0

: k∆k

_PPT

≤

^√²

d+1

k∆k

₁

→

Dependence ond of the constant relating the norms is “real”. Remark :In the special case

H ^{= (}

C^D

)

^⊗^K

(

d

=

D^K

)

withK fixed and D

→ +∞

, it may be shown by volumic considerations that “typically” : (

k · k

_SEP

'

√ ¹

d^1−1/K

k · k

₁

k · k

_PPT

' k · k

₁ , so that :

k · k

_PPT

k · k

_SEP

^√¹

d

k · k

₁(Aubrun/L).

→

Data-Hiding Hermitians are “exceptionnal”.