A generalization of quantum Stein’s Lemma Fernando G.S.L. Brandão and Martin B. Plenio Tohoku...

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:


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:


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,


)()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


)(:)(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.):


)( nnn HD

)( nnn HD

Known results

- (null) versus (alt.)


n )(: HDn

)(:)(min:)(0

nnnnIA

n AArn

)( nnAtr ))((sup n

n AItr

Known results



n )(: HDn

(Hayashi 00; Bjelakovic et al 04)

)||(inf)(log

lim,0

Sn

rnn

Known results


- Ergodic null hypothesis versus i.i.d. alternative hypothesis


n )(: HDn


)||(inf)(log

lim,0

Sn

rnn

(Hiai and Petz 91)

Known results


- Ergodic null hypothesis versus i.i.d. alternative hypothesis

- General sequence of states: Information spectrum


n )(: HDn


)||(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


)( 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


)( 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 )(




state , where satisfies



3. If , then

4. If and , then

5. If , then


n n

n)( n

nn HD nn

nnn HI )dim(/

1 n 1,...,1)( njtr nj

n m mn

n nnS )(




state , where satisfies



3. If , then

4. If and , then

5. If , then


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.


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.


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


)...( 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


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 ,


)(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.


(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.


(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 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


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


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


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


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


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


)(:),(: ,...,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


)(:),(: ,...,1,...,1 nnnnnn trtr

),(,2 ))(( nnn

nEn F

210

1

))(( ,2)( dnnn

nEn nXXd

)]log(11,)1(,[

)1()1()1(

12

,ndn

nnnEn tr


(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




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


(Proof sketch) Therefore

with

Finally

1

1

))((8' ,'22

nXXn nnn

nEdn

')''(..' Etrts

)(

)(suplim)()1(

'

En

EE n

n

n




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

A generalization of quantum Stein’s Lemma Fernando G.S.L. Brandão and Martin B. Plenio Tohoku...

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