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
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
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
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)
x∈X performed on it.(Mx >0,x
∈ X
, and ∑x∈X
Mx
=
Id)Probability of error (applying “maximum likelihood rule”) :
PE
=
12 1
− ∑
x∈X
Tr
qρ
− (
1−
q)σ
Mx
!
:=
12 1
− k
qρ− (
1−
q)σ k
M→
Distinguishability (semi)-normk · k
Massociated with the POVMM. (Matthews/Wehner/Winter)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)
x∈X performed on it.(Mx >0,x
∈ X
, and ∑x∈X
Mx
=
Id)Probability of error (applying “maximum likelihood rule”) :
PE
=
12 1
− ∑
x∈X
Tr
qρ
− (
1−
q)σ
Mx
!
:=
12 1
− k
qρ− (
1−
q)σk
M→
Distinguishability (semi)-normk · k
Massociated with the POVMM. (Matthews/Wehner/Winter)Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 4 / 17
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 fork · k
M.SetMof POVMs onCd :
k · k
M:=
supM∈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 inH (
Cd)
s.t.±
Id∈
KM⊂ [−
Id,Id] MPOVM onCd⇔
KMzonoid inH (
Cd)
s.t.±Id ∈
KM⊂ [−Id,
Id]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 fork · k
M.SetMof POVMs onCd :
k · k
M:=
supM∈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 inH (
Cd)
s.t.±
Id∈
KM⊂ [−
Id,Id] MPOVM onCd⇔
KMzonoid inH (
Cd)
s.t.±Id ∈
KM⊂ [−Id,
Id]Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 5 / 17
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 fork · k
M.SetMof POVMs onCd :
k · k
M:=
supM∈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 inH (
Cd)
s.t.±Id ∈
KM⊂ [−Id,Id]
MPOVM onCd
⇔
KMzonoid inH (
Cd)
s.t.±Id ∈
KM⊂ [−Id,
Id]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
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=
dZ
|ψi∈S2(Cd)
|hψ|∆|ψi|dψ
.Property
(L/Winter)“Modified” 2-norm on
H (
Cd)
:k∆k
2(1):=
pTr
(∆
2) + (
Tr∆)
2. Dimension-independent norm equivalence : 1√
18
k · k
2(1)6k · k
U6k · 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...
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=
dZ
|ψi∈S2(Cd)
|hψ|∆|ψi|dψ
.Property
(L/Winter)“Modified” 2-norm on
H (
Cd)
:k∆k
2(1):=
pTr
(∆
2) + (
Tr∆)
2. Dimension-independent norm equivalence : 1√
18
k · k
2(1)6k · k
U6k · 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
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=
dZ
|ψi∈S2(Cd)
|hψ|∆|ψi|dψ
.Property
(L/Winter)“Modified” 2-norm on
H (
Cd)
:k∆k
2(1):=
pTr
(∆
2) + (
Tr∆)
2. Dimension-independent norm equivalence : 1√
18
k · k
2(1)6k · k
U6k · 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...
“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 probabilityk · k
M' k · k
2(1)?Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS
:=
∑nk=1
Pk andP
:= (e
Pk:=
S−1/2PkS−1/2)
16k6n. Pis a random POVM onCd withnoutcomes s.t.k∆k
P=
∑nk=1
TrPek
∆
.Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 8 / 17
“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 probabilityk · k
M' k · k
2(1)?Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS
:=
∑nk=1
Pk andP
:= (e
Pk:=
S−1/2PkS−1/2)
16k6n. Pis a random POVM onCd withnoutcomes s.t.k∆k
P=
∑nk=1
TrPek
∆
.“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 probabilityk · k
M' k · k
2(1)?Strategy :Draw independentlyP1, . . . ,Pnuniformly distributed rank-1 projectors onCd, setS
:=
∑nk=1
Pk andP
:= (e
Pk:=
S−1/2PkS−1/2)
16k6n. Pis a random POVM onCdwithnoutcomes s.t.k∆k
P=
∑nk=1
TrPek
∆
.Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 8 / 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)6k · k
P6(
1+
ε)k · k2(1)>1
−
e−cεd.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∆|
fromk∆k
2(1)?2 Large deviation probability of
n
∑
k=1
|
TrPek∆|
from dnn
∑
k=1
|
TrPk∆|
?“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)6k · k
P6(
1+
ε)k · k2(1)>1
−
e−cεd.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∆|
fromk∆k
2(1)?2 Large deviation probability of
n
∑
k=1
|
TrPek∆|
from dnn
∑
k=1
|
TrPk∆|
?Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 9 / 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)6k · k
P6(
1+
ε)k · k2(1)>1
−
e−cεd.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∆|
fromk∆k
2(1)?2 Large deviation probability of
n
∑
k=1
|
TrPek∆|
from dnn
∑
k=1
|
TrPk∆|
?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 · kM6k · k
M0 6k · 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 thanCεd2outcomes, 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
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 · kM6k · k
M0 6k · 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 thanCεd2outcomes, whered
=
d1× · · · ×
dk.Uniform POVMUonCd : explicit construction of an approximation ofUby a POVM with less thanCεd2outcomes.
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 · kM6k · k
M0 6k · 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 thanCεd2outcomes, 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
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
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
1ork · k
Mk · 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,σ:=
d21+d(Id+
F)andα:=
d21−d(Id −
F).∆ :=
1 2σ−
12αis s.t.
k∆k
LOCC6k∆k
SEP6k∆k
PPT=
2d
+
1 1= k∆k
1.Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 12 / 17
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
1ork · k
Mk · 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,σ:=
d21+d(Id+
F)andα:=
d21−d(Id −
F).∆ :=
1 2σ−
12αis s.t.
k∆k
LOCC6k∆k
SEP6k∆k
PPT=
2d
+
1 1= k∆k
1.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
1ork · k
Mk · 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,σ:=
d21+d(Id +
F)andα:=
d21−d(Id −
F).∆ :=
1 2σ−
12αis s.t.
k∆k
LOCC6k∆k
SEP6k∆k
PPT=
2d
+
1 1= k∆k
1.Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 12 / 17
“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−
e−c0d2,c6
kρ −
σkPPT6C and√
cd 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 valueEofkρ −
σkM.2 Probability that
kρ −
σkMdeviates fromE.“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−
e−c0d2,c6
kρ −
σkPPT6C and√
cd 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 valueEofkρ −
σkM.2 Probability that
kρ −
σkMdeviates fromE.Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 13 / 17
“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−
e−c0d2,c6
kρ −
σkPPT6C and√
cd 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 valueEofkρ −σk
M.2 Probability that
kρ −
σkMdeviates fromE.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
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1d.
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1d.
Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 15 / 17
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1d.
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1d.
Cécilia Lancien Distinguishability norms Erice nic@qs13 - October 8th2013 15 / 17
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1dk−1.
But what about the opposite high-dimensional setting, i.e.dfixed and k
→ +∞
?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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1dk−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
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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1dk−1.
But what about the opposite high-dimensional setting, i.e.dfixed and k
→ +∞
?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, askρ −
σkALL, whereaskρ −
σkSEPis of order √1dk−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
References
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