Post on 18-Jan-2016
A generalization of quantum Stein’s Lemma
Fernando G.S.L. Brandão and Martin B. Plenio
Tohoku University, 13/09/2008
Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state.
Measure two outcome POVM .
Error probabilities
- Type I error:
- Type II error:
(i.i.d.) Quantum Hypothesis Testing
nn AIA ,
))((:)( nn
nn AItrA
)(:)( nn
nn AtrA
Null hypothesis
Alternative hypothesis
Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state.
Measure two outcome POVM .
Error probabilities
- Type I error:
- Type II error:
(i.i.d.) Quantum Hypothesis Testing
nn AIA ,
))((:)( nn
nn AItrA
)(:)( nn
nn AtrA
The state is
The state is
Given n copies of a quantum state, with the promise that it is described either by or , determine the identity of the state
Measure two outcome POVM
Error probabilities
- Type I error:
- Type II error:
(i.i.d.) Quantum Hypothesis Testing
nn AIA ,
))((:)( nn
nn AItrA
)(:)( nn
nn AtrA
Several possible settings, depending on the constraints imposed on the probabilities of error
E.g. in symmetric hypothesis testing,
(i.i.d.) Quantum Hypothesis Testing
)()1()(min:0
nnnnIA
n ApAprn
)(infloglog
lim 1
]1,0[
ss
s
n
ntr
n
r
Quantum Chernoff bound (Audenaert, Nussbaum, Szkola, Verstraete 07)
)(),( nnnn AA
Several possible settings, depending on the constraints imposed on the probabilities of error
E.g. in symmetric hypothesis testing,
(i.i.d.) Quantum Hypothesis Testing
)()1()(min:0
nnnnIA
n ApAprn
)(infloglog
lim 1
]1,0[
ss
s
n
ntr
n
r
Quantum Chernoff bound (Audenaert, Nussbaum, Szkola, Verstraete 07)
)(),( nnnn AA
Several possible settings, depending on the constraints imposed on the probabilities of error
E.g. in symmetric hypothesis testing,
(i.i.d.) Quantum Hypothesis Testing
)()1()(min:0
nnnnIA
n ApAprn
)(infloglog
lim 1
]1,0[
ss
s
n
ntr
n
r
Quantum Chernoff bound (Audenaert, Nussbaum, Szkola, Verstraete 07)
)(),( nnnn AA
Asymmetric hypothesis testing
Quantum Stein’s Lemma
Quantum Stein’s Lemma
)(:)(min:)(0
nnnnIA
n AArn
)||()(log
lim,0 Sn
rnn
(Hiai and Petz 91; Ogawa and Nagaoka 00)
))log(log( tr
Asymmetric hypothesis testing
Quantum Stein’s Lemma
Quantum Stein’s Lemma
)(:)(min:)(0
nnnnIA
n AArn
)||()(log
lim,0 Sn
rnn
(Hiai and Petz 91; Ogawa and Nagaoka 00)
))log(log( tr
Most general setting
- Null hypothesis (null):
- Alternative hypothesis (alt.):
Quantum Stein’s Lemma
)( nnn HD
)( nnn HD
Known results
- (null) versus (alt.)
Quantum Stein’s Lemma
n )(: HDn
)(:)(min:)(0
nnnnIA
n AArn
)( nnAtr ))((sup n
n AItr
Known results
- (null) versus (alt.)
Quantum Stein’s Lemma
n )(: HDn
)(:)(min:)(0
nnnnIA
n AArn
)( nnAtr ))((sup n
n AItr
Known results
- (null) versus (alt.)
Quantum Stein’s Lemma
n )(: HDn
(Hayashi 00; Bjelakovic et al 04)
)||(inf)(log
lim,0
Sn
rnn
Known results
- (null) versus (alt.)
- Ergodic null hypothesis versus i.i.d. alternative hypothesis
Quantum Stein’s Lemma
n )(: HDn
(Hayashi 00; Bjelakovic et al 04)
)||(inf)(log
lim,0
Sn
rnn
(Hiai and Petz 91)
Known results
- (null) versus (alt.)
- Ergodic null hypothesis versus i.i.d. alternative hypothesis
- General sequence of states: Information spectrum
Quantum Stein’s Lemma
n )(: HDn
(Hayashi 00; Bjelakovic et al 04)
)||(inf)(log
lim,0
Sn
rnn
(Hiai and Petz 91)
(Han and Verdu 94; Nagaoka and Hayashi 07)
What about allowing the alternative hypothesis to
be non-i.i.d. and to vary over a family of states?
Only ergodicity and related concepts seems not
to be enough to define a rate for the decay of
Quantum Stein’s Lemma
)( nn A(Shields 93)
What about allowing the alternative hypothesis to
be non-i.i.d. and to vary over a family of states?
Only ergodicity and related concepts seems not
to be enough to define a rate for the decay of
Quantum Stein’s Lemma
)( nn A(Shields 93)
This talk: A setting where the optimal rate can be determined for varying correlated alternative hypothesis
Consider the following two hypothesis
- Null hypothesis: For every we have
- Alternative hypothesis: For every we have an unknown
state , where satisfies:
1. Each is closed and convex
2. Each contains the maximally mixed state
3. If , then
4. If and , then
5. If , then
A generalization of Quantum Stein’s Lemma
n n
n)( n
nn HD nn
nnn HI )dim(/
1 n nntr )(1
n m mn
n nnS )(
Consider the following two hypothesis
- Null hypothesis: For every we have
- Alternative hypothesis: For every we have an unknown
state , where satisfies
1. Each is closed and convex
2. Each contains the maximally mixed state
3. If , then
4. If and , then
5. If , then
A generalization of Quantum Stein’s Lemma
n n
n)( n
nn HD nn
nnn HI )dim(/
1 n 1,...,1)( njtr nj
n m mn
n nnS )(
Consider the following two hypothesis
- Null hypothesis: For every we have
- Alternative hypothesis: For every we have an unknown
state , where satisfies
1. Each is closed and convex
2. Each contains the maximally mixed state
3. If , then
4. If and , then
5. If , then
A generalization of Quantum Stein’s Lemma
n n
n)( n
nn HD nn
nnn HI )dim(/
1 n
n m mn
n nnS )(
)(
*(*)nSYM
n PPS
1,...,1)( njtr nj
theorem: Given satisfying properties 1-5 and
,
- (Direct Part) there is a s.t.
A generalization of Quantum Stein’s Lemma
nn
)(HD
nnn AIA ,
1)(lim
nn
nAtr
))((2)(,
En
nnnnn Atr
0
n
SE
n
n n
)||(minlim)(
theorem: Given satisfying properties 1-5 and
,
- (Strong Converse) s.t.
A generalization of Quantum Stein’s Lemma
nn
)(HD
0)(lim
nn
nAtr
))((2)(..
En
nnnnn Atrts
nnn AIA ,,0
We say is separable if
If it cannot be written in this form, it is entangled
The sets of separable sates over ,
satisfy properties 1-5
The rate function of the theorem is a well-known
entanglement measure, the regularized relative
entropy of entanglement
A motivation: Entanglement theory
)...( 1 kHHHD
kj
jjjp ...1
nH )( nHS
(Vedral and Plenio 98)
Given an entangled state
The theorem gives an operational interpretation to
this measure as the optimal rate of discrimination
of an entangled state to a arbitrary family of separable
states
More on the relative entropy of entanglement on Wednesday
Regularized relative entropy of entanglement
)...( 1 kHHHD
n
SE
n
HSnR n
)||(minlim)(
)(
(Vedral and Plenio 98)
Cor: For every entangled state
Regularized relative entropy of entanglement
)...( 1 kHHHD
0)( RE
Rate of conversion of two states by local operations and
classical communication:
The corollary implies that if is entangled,
The mathematical definition of entanglement is equal to
the operational: multipartite bound entanglement is real For bipartite systems see Yang, Horodecki, Horodecki, Synak-Radtke 05
Regularized relative entropy of entanglement
0||)(||minlim:supliminf)( 1
nk
LOCCn
n
nk
n
n n
kR
0)( R
Asymptotic continuity: Let
Non-lockability: Let
Some elements of the proofs
)(),||(min:)(
ESE
nn
)(,,)||(||)()( 1nHDfornfEE
(Horodecki and Synak-Radtke 05; Christandl 06)
j
jjp
j
jjj
jjj nnnEpEphEp )()()()(
(Horodecki3 and Oppenheim 05)
Lemma: Let and s.t.
Then s.t.
and
Lemma: Let ,
Some elements of the proofs
)(HD 0, Y Y
)(' HD )(1),'( trF
(Datta and Renner 08)Y'
)(, HD 0
0)2(lim ))||((
nnSn
ntr
(Ogawa and Nagaoka 00)
Almost power states:
Exponential de Finetti theorem: For any permutation-
symmetric state there exists a measure
over and states s.t.
Some elements of the proofs
(Renner 05)
rrr
rnrnn HnSYMPHV ),(::),(
)),(()(],,[ rnnnrn
HVspanHSym
)( knkn HD
kn
rkH
nnEknk ntrdtr
)1(
)dim(
1,...,1 2)()(
2
EHH rn
n
,,
Almost power states:
Exponential de Finetti theorem: For any permutation-
symmetric state there exists a measure
over and states s.t.
Some elements of the proofs
(Renner 05)
rrr
rnrnn HnSYMPHV ),(::),(
)),(()(],,[ rnnnrn
HVspanHSym
)( knkn HD
kn
rkH
nnEknk ntrdtr
)1(
)dim(
1,...,1 2)()(
2
EHH ],,[ rn
n
(Proof sketch) We can write the statement of the theorem as
The dual formulation of the convex optimization above reads
It is then clear that it suffices to prove
Elements of the proof
)(,0
)(,1)2,(lim
Ey
Eyynnn
n
nynn
IA
ynnn AtrAtr
2)(:)(max)2,(0
)(
,2)2(min)2,( ybnbnn
b
ynnn tr
n
)(,0
)(,1)2(minlim
Ey
Eytr ynn
n n
(Proof sketch) We can write the statement of the theorem as
The dual formulation of the convex optimization above reads
It is then clear that it suffices to prove
Elements of the proof
)(,0
)(,1)2,(lim
Ey
Eyynnn
n
nynn
IA
ynnn AtrAtr
2)(:)(max)2,(0
)(
,2)2(min)2,( ybnbnn
b
ynnn tr
n
)(,0
)(,1)2(minlim
Ey
Eytr ynn
n n
(Proof sketch) We first show that for every
Take sufficiently large such that
Let be such that
By the strong converse of quantum Stein’s Lemma
As we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
n 2//)()(
nEE n
n
nn )||()( SEn
0)2(lim )2/)((
m
nnEnm
ntr
nmm
n
0)2(mininflim ))((
nEn
ntr
n
(Proof sketch) We first show that for every
Take sufficiently large such that
Let be such that
By the strong converse of quantum Stein’s Lemma
As we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
n 2//)()(
nEE n
n
nn )||()( nn SE
n
0)2(lim )2/)((
m
nnmEnm
ntr
nmm
n
0)2(mininflim ))((
nEn
ntr
n
(Proof sketch) We first show that for every
Take sufficiently large such that
Let be such that
By the strong converse of quantum Stein’s Lemma
As we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
n 2//)()(
nEE n
n
nn )||()( nn SE
n
0)2(lim )2/)((
m
nnmEnm
ntr
nmm
n
0)2(mininflim ))((
nEn
ntr
n
(Proof sketch) We now show
Let be an optimal sequence in the eq. above
We can write
Assuming conversely that the limit is zero, we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
nn
)2(2 ))(())((
nnEn
nnEn
0,21
))(( n
nnnE
n
(Proof sketch) We now show
Let be an optimal sequence in the eq. above
We can write
Assuming conversely that the limit is zero, we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
nn
)2(2 ))(())((
nnEn
nnEn
0,21
))(( n
nnnE
n
(Proof sketch) We now show
Let be an optimal sequence in the eq. above
We can write
Assuming conversely that the limit is zero, we find
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
nn
)2(2 ))(())((
nnEn
nnEn
0,21
))(( n
nnnE
n
(Proof sketch) We now show
Let be an optimal sequence in the eq. above
We can write
Assuming conversely that the limit is zero, we find
Then
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
nn
)2(2 ))(())((
nnEn
nnEn
0,21
))(( n
nnnE
n
)(
)(lim)( E
n
EE n
n
n
(Proof sketch) We now show
Let be an optimal sequence in the eq. above
We can write
Assuming conversely that the limit is zero, we find
Then
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
nn
)2(2 ))(())((
nnEn
nnEn
0,21
))(( n
nnnE
n
)(
)(lim)( E
n
EE n
n
n
(Proof sketch) Finally we now show
with . Suppose conversely that .From
we can write
Note that we can take to be permutation-symmetric
Elements of the proof
0)2(minlim ))((
nEn
ntr
n
1 1
)2(2 ))(())((
nnEn
nnEn
),(,2 ))(( n
nnnE
n F
nn ,
(Proof sketch) Define
We have
We can write
Elements of the proof
)(:),(: ,...,1,...,1 nnnnnn trtr
),(,2 )1())(( nnn
nEn F
210
1
))(( ,2)( dnnn
nEn nXXd
)]log(11,)1(,[
)1()1()1(
12
,ndn
nnnEn tr
(Proof sketch) Define
We have
We can write
Elements of the proof
)(:),(: ,...,1,...,1 nnnnnn trtr
),(,2 ))(( nnn
nEn F
210
1
))(( ,2)( dnnn
nEn nXXd
)]log(11,)1(,[
)1()1()1(
12
,ndn
nnnEn tr
Elements of the proof
(Proof sketch) Because
Therefore we can write
and
with
),( )1( nnF
)1()()(8/1
nB
d
2
8/1
10
1
))((
)(
,)1(2)1()(' dnnn
nE
B
n nXXOOd
n
222
4
10
1
8))((8
)'(
,2)('' dnn
dn
nEd
B
n nXXnnd
n
)(' 8/1 nB
Elements of the proof
(Proof sketch) Because
Therefore we can write
and
with
),( )1( nnF
)1()()(8/1
nB
d
2
8/1
10
1
))((
)(
,)1(2)1()(' dnnn
nE
B
n nXXOOd
n
222
4
10
1
8))((8
)'(
,2)('' dnn
dn
nEd
B
n nXXnnd
n
)(' 8/1 nB
Elements of the proof
(Proof sketch) Because
Therefore we can write
and
with
),( nnF
)1()()(8/1
nB
d
2
8/1
10
1
))((
)(
,)1(2)1()(' dnnn
nE
B
n nXXOOd
n
222
4
10
1
8))((8
)'(
,2)('' dnn
dn
nEd
B
n nXXnnd
n
)(' 8/1 nB
Elements of the proof
(Proof sketch) Therefore
with
Finally
1
1
))((8' ,'22
nXXn nnn
nEdn
')''(..' Etrts
)(
)(suplim)()1(
'
En
EE n
n
n
Elements of the proof
(Proof sketch) Because
Therefore we can write
and
with
),( nnF
)1()()(8/1
nB
d
2
8/1
10
1
))((
)(
,)1(2)1()(' dnnn
nE
B
n nXXOOd
n
222
4
10
1
8))((8
)'(
,2)('' dnn
dn
nEd
B
n nXXnnd
n
)(' 8/1 nB