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(1)

Distinguishability norms

on high-dimensional quantum systems

joint work with Guillaume Aubrun arXiv:1309.6003

Université Claude Bernard Lyon 1 / Universitat Autònoma de Barcelona

Cécilia Lancien

Erice nic@qs13 - October 8th2013

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 1 / 17

(2)

Outline

1 Introduction

2 Approximating the uniform POVM by a POVM with “few” outcomes

3 “Typical” performance of PPT and separable measurements in distinguishing two bipartite states

4 Summary and open questions

(3)

Outline

1 Introduction

2 Approximating the uniform POVM by a POVM with “few” outcomes

3 “Typical” performance of PPT and separable measurements in distinguishing two bipartite states

4 Summary and open questions

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 3 / 17

(4)

Distinguishability norms and state discrimination

State discrimination problem

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

q.

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

= (

Mx

)

xX performed on it.

(Mx >0,x

∈ X

, and

xX

Mx

=

Id)

Probability of error (applying “maximum likelihood rule”) :

PE

=

1

2 1

− ∑

xX

Tr

− (

1

q

Mx

!

:=

1

2 1

− k

− (

1

q

)σ k

M

Distinguishability (semi)-norm

k · k

Massociated with the POVMM. (Matthews/Wehner/Winter)

(5)

Distinguishability norms and state discrimination

State discrimination problem

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

q.

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

= (

Mx

)

xX performed on it.

(Mx >0,x

∈ X

, and

xX

Mx

=

Id)

Probability of error (applying “maximum likelihood rule”) :

PE

=

1

2 1

− ∑

xX

Tr

− (

1

q

Mx

!

:=

1

2 1

− k

− (

1

q

)σk

M

Distinguishability (semi)-norm

k · k

Massociated with the POVMM. (Matthews/Wehner/Winter)

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 4 / 17

(6)

Distinguishability norms and convex geometry

Definitions

POVMMonCd : positive measureµon

H

+

(

Cd

)

s.t.

Z

H+(Cd)

M dµ(M

) =

Id. Associated distinguishability (semi-)norm :

k∆k

M

:=

Z

H+(Cd)

|

TrM

∆|dµ(

M

)

. Associated convex bodyKM: polar of the unit ball for

k · k

M.

SetMof POVMs onCd :

k · k

M

:=

sup

M∈M

k · k

M and KM

=

conv [

M∈M

KM

! .

Question :What “kind” of convex bodies are associated to POVMs ?

Statement

Mdiscrete POVM onCd

KMzonotope in

H (

Cd

)

s.t.

±

Id

KM

⊂ [−

Id,Id] MPOVM onCd

KMzonoid in

H (

Cd

)

s.t.

±Id

KM

⊂ [−Id,

Id]

(7)

Distinguishability norms and convex geometry

Definitions

POVMMonCd : positive measureµon

H

+

(

Cd

)

s.t.

Z

H+(Cd)

M dµ(M

) =

Id. Associated distinguishability (semi-)norm :

k∆k

M

:=

Z

H+(Cd)

|

TrM

∆|dµ(

M

)

. Associated convex bodyKM: polar of the unit ball for

k · k

M.

SetMof POVMs onCd :

k · k

M

:=

sup

M∈M

k · k

M and KM

=

conv [

M∈M

KM

! .

Question :What “kind” of convex bodies are associated to POVMs ?

Statement

Mdiscrete POVM onCd

KMzonotope in

H (

Cd

)

s.t.

±

Id

KM

⊂ [−

Id,Id] MPOVM onCd

KMzonoid in

H (

Cd

)

s.t.

±Id

KM

⊂ [−Id,

Id]

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 5 / 17

(8)

Distinguishability norms and convex geometry

Definitions

POVMMonCd : positive measureµon

H

+

(

Cd

)

s.t.

Z

H+(Cd)

M dµ(M

) =

Id. Associated distinguishability (semi-)norm :

k∆k

M

:=

Z

H+(Cd)

|

TrM

∆|dµ(

M

)

. Associated convex bodyKM: polar of the unit ball for

k · k

M.

SetMof POVMs onCd :

k · k

M

:=

sup

M∈M

k · k

M and KM

=

conv [

M∈M

KM

! .

Question :What “kind” of convex bodies are associated to POVMs ?

Statement

Mdiscrete POVM onCd

KMzonotope in

H (

Cd

)

s.t.

±Id

KM

⊂ [−Id,Id]

MPOVM onCd

KMzonoid in

H (

Cd

)

s.t.

±Id

KM

⊂ [−Id,

Id]

(9)

Outline

1 Introduction

2 Approximating the uniform POVM by a POVM with “few” outcomes

3 “Typical” performance of PPT and separable measurements in distinguishing two bipartite states

4 Summary and open questions

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 6 / 17

(10)

The uniform POVM

Most “efficient” single POVM onCd

=

Most “symmetric” one.

Continuous POVMUin which all rank-1 projectors play the same role :

k∆k

U

=

d

Z

|ψi∈S2(Cd)

|hψ|∆|ψi|dψ

.

Property

(L/Winter)

“Modified” 2-norm on

H (

Cd

)

:

k∆k

2(1)

:=

p

Tr

(∆

2

) + (

Tr

∆)

2. Dimension-independent norm equivalence : 1

18

k · k

2(1)6

k · k

U6

k · k

2(1). Important theoretical result :derandomization of the state distinction problem (Ambainis/Emerson), bounds on the dimensionality reduction of quantum states (Harrow/Montanaro/Short)etc.

But of no practical use to describe an algorithm that would actually complete one of these tasks...

(11)

The uniform POVM

Most “efficient” single POVM onCd

=

Most “symmetric” one.

Continuous POVMUin which all rank-1 projectors play the same role :

k∆k

U

=

d

Z

|ψi∈S2(Cd)

|hψ|∆|ψi|dψ

.

Property

(L/Winter)

“Modified” 2-norm on

H (

Cd

)

:

k∆k

2(1)

:=

p

Tr

(∆

2

) + (

Tr

∆)

2. Dimension-independent norm equivalence : 1

18

k · k

2(1)6

k · k

U6

k · k

2(1).

Important theoretical result :derandomization of the state distinction problem (Ambainis/Emerson), bounds on the dimensionality reduction of quantum states (Harrow/Montanaro/Short)etc.

But of no practical use to describe an algorithm that would actually complete one of these tasks...

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 7 / 17

(12)

The uniform POVM

Most “efficient” single POVM onCd

=

Most “symmetric” one.

Continuous POVMUin which all rank-1 projectors play the same role :

k∆k

U

=

d

Z

|ψi∈S2(Cd)

|hψ|∆|ψi|dψ

.

Property

(L/Winter)

“Modified” 2-norm on

H (

Cd

)

:

k∆k

2(1)

:=

p

Tr

(∆

2

) + (

Tr

∆)

2. Dimension-independent norm equivalence : 1

18

k · k

2(1)6

k · k

U6

k · k

2(1). Important theoretical result :derandomization of the state distinction problem (Ambainis/Emerson), bounds on the dimensionality reduction of quantum states (Harrow/Montanaro/Short)etc.

But of no practical use to describe an algorithm that would actually complete one of these tasks...

(13)

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

Known :The “good” discriminating power of the uniform POVM is already achieved by certain discrete POVMs, e.g. 4-design POVMs.

However :

A 4-design POVM onCd must have

Ω(

d4

)

outcomes.

Neither the existence of a minimal 4-design POVM, nor the explicit construction of any 4-design POVM are known.

Minimal number of outcomes for a randomly chosen POVMMonCdso that with high probability

k · k

M

' k · k

2(1)?

Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS

:=

n

k=1

Pk andP

:= (e

Pk

:=

S1/2PkS1/2

)

16k6n. Pis a random POVM onCd withnoutcomes s.t.

k∆k

P

=

n

k=1

TrPek

.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 8 / 17

(14)

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

Known :The “good” discriminating power of the uniform POVM is already achieved by certain discrete POVMs, e.g. 4-design POVMs.

However :

A 4-design POVM onCd must have

Ω(

d4

)

outcomes.

Neither the existence of a minimal 4-design POVM, nor the explicit construction of any 4-design POVM are known.

Minimal number of outcomes for a randomly chosen POVMMonCdso that with high probability

k · k

M

' k · k

2(1)?

Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS

:=

n

k=1

Pk andP

:= (e

Pk

:=

S1/2PkS1/2

)

16k6n. Pis a random POVM onCd withnoutcomes s.t.

k∆k

P

=

n

k=1

TrPek

.

(15)

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

Known :The “good” discriminating power of the uniform POVM is already achieved by certain discrete POVMs, e.g. 4-design POVMs.

However :

A 4-design POVM onCd must have

Ω(

d4

)

outcomes.

Neither the existence of a minimal 4-design POVM, nor the explicit construction of any 4-design POVM are known.

Minimal number of outcomes for a randomly chosen POVMMonCdso that with high probability

k · k

M

' k · k

2(1)?

Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS

:=

n

k=1

Pk andP

:= (e

Pk

:=

S1/2PkS1/2

)

16k6n. Pis a random POVM onCdwithnoutcomes s.t.

k∆k

P

=

n

k=1

TrPek

.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 8 / 17

(16)

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

Theorem

(Aubrun/L)

For any 0<ε<1, there existcε,Cε>0 s.t. ifn>Cεd2, then :

P 1

ε

18

k · k

2(1)6

k · k

P6

(

1

+

ε)k · k2(1)

>1

ecεd.

Remark :Optimal dimensional order of magnitude since a POVM onCd must have at leastd2outcomes to be informationally complete.

Main steps in the proof :

1 Large deviation probability of dn

n

k=1

|

TrPk

∆|

from

k∆k

2(1)?

2 Large deviation probability of

n

k=1

|

TrPek

∆|

from dn

n

k=1

|

TrPk

∆|

?

(17)

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

Theorem

(Aubrun/L)

For any 0<ε<1, there existcε,Cε>0 s.t. ifn>Cεd2, then :

P 1

ε

18

k · k

2(1)6

k · k

P6

(

1

+

ε)k · k2(1)

>1

ecεd.

Remark :Optimal dimensional order of magnitude since a POVM onCdmust have at leastd2outcomes to be informationally complete.

Main steps in the proof :

1 Large deviation probability of dn

n

k=1

|

TrPk

∆|

from

k∆k

2(1)?

2 Large deviation probability of

n

k=1

|

TrPek

∆|

from dn

n

k=1

|

TrPk

∆|

?

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 9 / 17

(18)

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

Theorem

(Aubrun/L)

For any 0<ε<1, there existcε,Cε>0 s.t. ifn>Cεd2, then :

P 1

ε

18

k · k

2(1)6

k · k

P6

(

1

+

ε)k · k2(1)

>1

ecεd.

Remark :Optimal dimensional order of magnitude since a POVM onCdmust have at leastd2outcomes to be informationally complete.

Main steps in the proof :

1 Large deviation probability of dn

n

k=1

|

TrPk

∆|

from

k∆k

2(1)?

2 Large deviation probability of

n

k=1

|

TrPek

∆|

from dn

n

k=1

|

TrPk

∆|

?

(19)

Existence of “short” approximations for any POVM

Correspondence “discrete POVM

zonotope” and “POVM

zonoid”.

Derive results on POVMs from results in local theory of Banach spaces.

(Bourgain/Lindenstrauss/Milman, Talagrand)

Theorem

(Aubrun/L)

For any POVMMonCdand any 0<ε<1, there exists a sub-POVMM0with less thanCεd2logd outcomes s.t.

(

1

ε)k · kM6

k · k

M0 6

k · k

M.

Refinements in some special cases :

Local uniform POVMLU

=

U1

⊗ · · · ⊗

Uk onCd1

⊗ · · · ⊗

Cdk : existence of an approximation ofLUby a local sub-POVM with less thanCεd2

outcomes, whered

=

d1

× · · · ×

dk.

Uniform POVMUonCd : explicit construction of an approximation ofUby a POVM with less thanCεd2outcomes.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 10 / 17

(20)

Existence of “short” approximations for any POVM

Correspondence “discrete POVM

zonotope” and “POVM

zonoid”.

Derive results on POVMs from results in local theory of Banach spaces.

(Bourgain/Lindenstrauss/Milman, Talagrand)

Theorem

(Aubrun/L)

For any POVMMonCdand any 0<ε<1, there exists a sub-POVMM0with less thanCεd2logd outcomes s.t.

(

1

ε)k · kM6

k · k

M0 6

k · k

M.

Refinements in some special cases :

Local uniform POVMLU

=

U1

⊗ · · · ⊗

Uk onCd1

⊗ · · · ⊗

Cdk : existence of an approximation ofLUby a local sub-POVM with less thanCεd2

outcomes, whered

=

d1

× · · · ×

dk.

Uniform POVMUonCd : explicit construction of an approximation ofUby a POVM with less thanCεd2outcomes.

(21)

Existence of “short” approximations for any POVM

Correspondence “discrete POVM

zonotope” and “POVM

zonoid”.

Derive results on POVMs from results in local theory of Banach spaces.

(Bourgain/Lindenstrauss/Milman, Talagrand)

Theorem

(Aubrun/L)

For any POVMMonCdand any 0<ε<1, there exists a sub-POVMM0with less thanCεd2logd outcomes s.t.

(

1

ε)k · kM6

k · k

M0 6

k · k

M.

Refinements in some special cases :

Local uniform POVMLU

=

U1

⊗ · · · ⊗

Uk onCd1

⊗ · · · ⊗

Cdk : existence of an approximation ofLUby a local sub-POVM with less thanCεd2

outcomes, whered

=

d1

× · · · ×

dk.

Uniform POVMUonCd : explicit construction of an approximation ofUby a POVM with less thanCεd2outcomes.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 10 / 17

(22)

Outline

1 Introduction

2 Approximating the uniform POVM by a POVM with “few” outcomes

3 “Typical” performance of PPT and separable measurements in distinguishing two bipartite states

4 Summary and open questions

(23)

Motivations

Seminal observation in state discrimination(Holevo/Helstrom):

k · k

ALL

= k · k

1.

Problem :On a multipartite system, locality constraints on the setMof POVMs that experimenters are able to implement :LOCC

SEP

PPT.

How do these restrictions affect their distinguishing power ?

k · k

M

' k · k

1or

k · k

M

k · k

1?

Example

Existence of “data-hiding” states on multipartite systems

(DiVincenzo/Leung/Terhal)i.e. states that would be well distinguished by a suitable global measurement but that are barely distinguishable by any local measurement.

In the bipartite case :Completely symmetric and antisymmetric states on Cd

Cd

:=

d21+d

(Id+

F)andα

:=

d21d

(Id −

F).

∆ :=

1

1

2αis s.t.

k∆k

LOCC6

k∆k

SEP6

k∆k

PPT

=

2

d

+

1

1

= k∆k

1.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 12 / 17

(24)

Motivations

Seminal observation in state discrimination(Holevo/Helstrom):

k · k

ALL

= k · k

1. Problem :On a multipartite system, locality constraints on the setMof POVMs that experimenters are able to implement :LOCC

SEP

PPT.

How do these restrictions affect their distinguishing power ?

k · k

M

' k · k

1or

k · k

M

k · k

1?

Example

Existence of “data-hiding” states on multipartite systems

(DiVincenzo/Leung/Terhal)i.e. states that would be well distinguished by a suitable global measurement but that are barely distinguishable by any local measurement.

In the bipartite case :Completely symmetric and antisymmetric states on Cd

Cd

:=

d21+d

(Id+

F)andα

:=

d21d

(Id −

F).

∆ :=

1

1

2αis s.t.

k∆k

LOCC6

k∆k

SEP6

k∆k

PPT

=

2

d

+

1

1

= k∆k

1.

(25)

Motivations

Seminal observation in state discrimination(Holevo/Helstrom):

k · k

ALL

= k · k

1. Problem :On a multipartite system, locality constraints on the setMof POVMs that experimenters are able to implement :LOCC

SEP

PPT.

How do these restrictions affect their distinguishing power ?

k · k

M

' k · k

1or

k · k

M

k · k

1?

Example

Existence of “data-hiding” states on multipartite systems

(DiVincenzo/Leung/Terhal)i.e. states that would be well distinguished by a suitable global measurement but that are barely distinguishable by any local measurement.

In the bipartite case :Completely symmetric and antisymmetric states on Cd

Cd

:=

d21+d

(Id +

F)andα

:=

d21d

(Id −

F).

∆ :=

1

1

2αis s.t.

k∆k

LOCC6

k∆k

SEP6

k∆k

PPT

=

2

d

+

1

1

= k∆k

1.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 12 / 17

(26)

“Typical” behaviour of PPT and separable POVMs on a bipartite system

Theorem

(Aubrun/L)

There existc0,c,C>0 s.t. forρ,σrandom states onCd

Cd(picked independently and uniformly), with probability greater than 1

ec0d2,

c6

kρ −

σkPPT6C and

c

d 6

kρ −

σkSEP6

C d.

Conclusion :In comparison, typically,

kρ −

σkALL

= kρ −

σk1

'

1.

The PPT constraint only affects the distinguishability by a constant factor, whereas the separability one implies a dimensional loss (“generic” data-hiding). Main steps in the proof :

1 Mean-widthofKM

Expected valueEof

kρ −

σkM.

2 Probability that

kρ −

σkMdeviates fromE.

(27)

“Typical” behaviour of PPT and separable POVMs on a bipartite system

Theorem

(Aubrun/L)

There existc0,c,C>0 s.t. forρ,σrandom states onCd

Cd(picked independently and uniformly), with probability greater than 1

ec0d2,

c6

kρ −

σkPPT6C and

c

d 6

kρ −

σkSEP6

C d. Conclusion :In comparison, typically,

kρ −

σkALL

= kρ −

σk1

'

1.

The PPT constraint only affects the distinguishability by a constant factor, whereas the separability one implies a dimensional loss (“generic” data-hiding).

Main steps in the proof :

1 Mean-widthofKM

Expected valueEof

kρ −

σkM.

2 Probability that

kρ −

σkMdeviates fromE.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 13 / 17

(28)

“Typical” behaviour of PPT and separable POVMs on a bipartite system

Theorem

(Aubrun/L)

There existc0,c,C>0 s.t. forρ,σrandom states onCd

Cd(picked independently and uniformly), with probability greater than 1

ec0d2,

c6

kρ −

σkPPT6C and

c

d 6

kρ −

σkSEP6

C d. Conclusion :In comparison, typically,

kρ −

σkALL

= kρ −

σk1

'

1.

The PPT constraint only affects the distinguishability by a constant factor, whereas the separability one implies a dimensional loss (“generic” data-hiding).

Main steps in the proof :

1 Mean-widthofKM

Expected valueEof

kρ −σk

M.

2 Probability that

kρ −

σkMdeviates fromE.

(29)

Outline

1 Introduction

2 Approximating the uniform POVM by a POVM with “few” outcomes

3 “Typical” performance of PPT and separable measurements in distinguishing two bipartite states

4 Summary and open questions

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 14 / 17

(30)

Summary

1 OnCd, a random POVM made of

Θ(

d2

)

independent uniformly distributed rank-1 projectors (appropriately renormalized) emulates the uniform POVM with high probability.

More generally, any POVM onCd may be approximated by a sub-POVM with

Θ(

d2logd

)

outcomes.

2 For generic statesρ,σonCd

Cd,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

d.

(31)

Summary

1 OnCd, a random POVM made of

Θ(

d2

)

independent uniformly distributed rank-1 projectors (appropriately renormalized) emulates the uniform POVM with high probability.

More generally, any POVM onCd may be approximated by a sub-POVM with

Θ(

d2logd

)

outcomes.

2 For generic statesρ,σonCd

Cd,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

d.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 15 / 17

(32)

Summary

1 OnCd, a random POVM made of

Θ(

d2

)

independent uniformly distributed rank-1 projectors (appropriately renormalized) emulates the uniform POVM with high probability.

More generally, any POVM onCd may be approximated by a sub-POVM with

Θ(

d2logd

)

outcomes.

2 For generic statesρ,σonCd

Cd,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

d.

(33)

Summary

1 OnCd, a random POVM made of

Θ(

d2

)

independent uniformly distributed rank-1 projectors (appropriately renormalized) emulates the uniform POVM with high probability.

More generally, any POVM onCd may be approximated by a sub-POVM with

Θ(

d2logd

)

outcomes.

2 For generic statesρ,σonCd

Cd,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

d.

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 15 / 17

(34)

Open questions

Approximation of any POVM onCd by a POVM with

Θ(

d2logd

)

outcomes ?

Explicit construction of such POVM ? Typical behaviour of

k · k

LOCC?

Generalization to the multipartite case :

On

(

Cd

)

k, withk fixed andd

→ +∞

,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

dk−1.

But what about the opposite high-dimensional setting, i.e.dfixed and k

→ +∞

?

(35)

Open questions

Approximation of any POVM onCd by a POVM with

Θ(

d2logd

)

outcomes ?

Explicit construction of such POVM ?

Typical behaviour of

k · k

LOCC?

Generalization to the multipartite case :

On

(

Cd

)

k, withk fixed andd

→ +∞

,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

dk−1.

But what about the opposite high-dimensional setting, i.e.dfixed and k

→ +∞

?

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 16 / 17

(36)

Open questions

Approximation of any POVM onCd by a POVM with

Θ(

d2logd

)

outcomes ?

Explicit construction of such POVM ? Typical behaviour of

k · k

LOCC?

Generalization to the multipartite case :

On

(

Cd

)

k, withk fixed andd

→ +∞

,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

dk−1.

But what about the opposite high-dimensional setting, i.e.dfixed and k

→ +∞

?

(37)

Open questions

Approximation of any POVM onCd by a POVM with

Θ(

d2logd

)

outcomes ?

Explicit construction of such POVM ? Typical behaviour of

k · k

LOCC?

Generalization to the multipartite case :

On

(

Cd

)

k, withk fixed andd

→ +∞

,

kρ −

σkPPTis of order 1, as

kρ −

σkALL, whereas

kρ −

σkSEPis of order 1

dk−1.

But what about the opposite high-dimensional setting, i.e.dfixed and k

→ +∞

?

Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 16 / 17

(38)

References

M. Talagrand, “Embedding subspaces ofL1intol1N”.

D. Chafaï, O. Guédon, G. Lecué, A. Pajor, “Interactions between compressed sensing, random matrices and high dimensional geometry”.

G. Aubrun, S.J. Szarek, “Tensor product of convex sets and the volume of separable states onNqudits”,arXiv :quant-ph/0503221.

K. ˙Zyczkowski, H-J. Sommers, “Induced measures in the space of mixed quantum states”,arXiv :quant-ph/0012101.

D.P. DiVincenzo, D. Leung, B.M. Terhal, “Quantum Data Hiding”,arXiv :quant-ph/0103098.

A.W. Harrow, A. Montanaro, A.J. Short, “Limitations on quantum dimensionality reduction”,arXiv[quant-ph] :1012.2262.

A. Ambainis, J. Emerson, “Quantum t-designs : t-wise independence in the quantum world”,arXiv :quant-ph/0701126.

W. Matthews, S. Wehner, A. Winter, “Distinguishability of quantum states under restricted families of measurements with an application to data hiding”,arXiv :0810.2327[quant-ph].

C. Lancien, A. Winter, “Distinguishing multi-partite states by local measurements”, arXiv[quant-ph] :1206.2884.

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