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A generalization of quantum Stein’s Lemma
Fernando G.S.L. Brandão and Martin B. Plenio
Tohoku University, 13/09/2008
(i.i.d.) Quantum Hypothesis Testing

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
An , I  An  .
Null
 Error probabilities
hypothesis
- Type I error:
- Type II error:
Alternative
hypothesis
n
 n ( An ) : tr(  ( I  An ))
 n ( An ) : tr(
n
An )
(i.i.d.) Quantum Hypothesis Testing

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
n
(The
I state
Anis))
isn ) : tr ( 
- Type I error: Thestate
n(A
An , I  An  .


- Type II error:
 n ( An ) : tr(
n
An )
(i.i.d.) Quantum Hypothesis Testing

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:
An , I  An 
 n ( An ) : tr(  ( I  An ))
n
 n ( An ) : tr(
n
An )
(i.i.d.) Quantum Hypothesis Testing

Several possible settings, depending on the constraints
imposed on the probabilities of error  n ( An ),  n ( An )

E.g. in symmetric hypothesis testing,
rn : min p n ( An )  (1  p) n ( An )
0 An  I
Quantum Chernoff bound
(Audenaert, Nussbaum, Szkola, Verstraete 07)
log rn
1 s s
lim
 log inf tr(   )
n 
s[ 0 ,1]
n
(i.i.d.) Quantum Hypothesis Testing

Several possible settings, depending on the constraints
imposed on the probabilities of error  n ( An ),  n ( An )

E.g. in symmetric hypothesis testing,
rn : min p n ( An )  (1  p) n ( An )
0 An  I
Quantum Chernoff bound
(Audenaert, Nussbaum, Szkola, Verstraete 07)
log rn
1 s s
lim
 log inf tr(   )
n 
s[ 0 ,1]
n
(i.i.d.) Quantum Hypothesis Testing

Several possible settings, depending on the constraints
imposed on the probabilities of error  n ( An ),  n ( An )

E.g. in symmetric hypothesis testing,
rn : min p n ( An )  (1  p) n ( An )
0 An  I
Quantum Chernoff bound
(Audenaert, Nussbaum, Szkola, Verstraete 07)
log rn
1 s s
lim
 log inf tr(   )
n 
s[ 0 ,1]
n
Quantum Stein’s Lemma

Asymmetric hypothesis testing
rn ( ) : min  n ( An ) :  n ( An )  
0 An  I

Quantum Stein’s Lemma
(Hiai and Petz 91; Ogawa and Nagaoka 00)
log rn ( )
  0, lim
 S (  ||  )
n 
n
 tr(  (log   log  ))
Quantum Stein’s Lemma

Asymmetric hypothesis testing
rn ( ) : min  n ( An ) :  n ( An )  
0 An  I

Quantum Stein’s Lemma
(Hiai and Petz 91; Ogawa and Nagaoka 00)
log rn ( )
  0, lim 
 S (  ||  )
n 
n
 tr(  (log   log  ))
Quantum Stein’s Lemma

Most general setting
- Null hypothesis (null):
 n   n  D( H
- Alternative hypothesis (alt.):
 n   n  D( H  n )
n
)
Quantum Stein’s Lemma

Known results
-

n

:     D( H )
(null)
versus

n
(alt.)
rn ( ) : min  n ( An ) :  n ( An )  
0 An  I
tr(
n
An )
sup tr( 
 
n
( I  An ))
Quantum Stein’s Lemma

Known results
-

n

:     D( H )
(null)
versus

n
(alt.)
rn ( ) : min  n ( An ) :  n ( An )  
0 An  I
tr(
n
An )
sup tr(
 
n
( I  An ))
Quantum Stein’s Lemma

Known results
-

n

:     D( H )
(null)
versus

n
(alt.)
log rn ( )
  0, lim 
 inf S ( ||  )
n 
 
n
(Hayashi 00; Bjelakovic et al 04)
Quantum Stein’s Lemma

Known results
-

n

:     D( H )
(null)
versus

n
(alt.)
log rn ( )
  0, lim 
 inf S ( ||  )
n 
 
n
(Hayashi 00; Bjelakovic et al 04)
- Ergodic null hypothesis versus i.i.d. alternative hypothesis
(Hiai and Petz 91)
Quantum Stein’s Lemma

Known results
-

n

:     D( H )
(null)
versus

n
(alt.)
log rn ( )
  0, lim 
 inf S ( ||  )
n 
 
n
(Hayashi 00; Bjelakovic et al 04)
- Ergodic null hypothesis versus i.i.d. alternative hypothesis
(Hiai and Petz 91)
- General sequence of states: Information spectrum
(Han and Verdu 94; Nagaoka and Hayashi 07)
Quantum Stein’s Lemma

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  n ( An )
(Shields 93)
Quantum Stein’s Lemma

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  n ( An )
(Shields 93)
This talk: A setting where the optimal rate can be
determined for varying correlated alternative hypothesis
A generalization of Quantum Stein’s Lemma

Consider the following two hypothesis
- Null hypothesis: For every
n
- Alternative hypothesis: For every
state
n
n   n  D( H  n ) ,
1. Each
n
we have
where
4. If
5. If
we have an unknown
n n satisfies:
is closed and convex
2. Each contains the maximally mixed state
3. If
 n
I n / dim( H ) n
  n1 , then trn1 ( )  n
   n and    m , then     n m
   n , then S n ( )  n
A generalization of Quantum Stein’s Lemma

Consider the following two hypothesis
- Null hypothesis: For every
n
- Alternative hypothesis: For every
state
n
n   n  D( H  n ) ,
1. Each
n
we have
where
4. If
5. If
we have an unknown
n n satisfies
is closed and convex
2. Each contains the maximally mixed state
3. If
 n
I n / dim( H ) n
  n1 , then trj ( )  n j  1,..., n  1
   n and    m , then     n m
   n , then S n ( )  n
A generalization of Quantum Stein’s Lemma

Consider the following two hypothesis
- Null hypothesis: For every
n
we have
 n
n   we have an unknown
n   n  D( H  n ) , where
satisfies
S n (*)n 
n P
 * P
- Alternative hypothesis: For every
state
1. Each
n
 SYM ( n )
is closed and convex
2. Each contains the maximally mixed state
3. If
4. If
5. If
I n / dim( H ) n
  n1 , then trj ( )  n j  1,..., n  1
   n and    m , then     n m
   n , then S n ( )  n
A generalization of Quantum Stein’s Lemma

theorem: Given  n n satisfying properties 1-5 and
  D(H ) ,
- (Direct Part)
  0
there is a An , I  An n s.t.
lim tr( An  n )  1
n
 n  n n ,
tr( Ann )  2
 n ( E (  )  )
n
S
(

||  )

E (  )  lim min
n    n
n
A generalization of Quantum Stein’s Lemma

theorem: Given  n n satisfying properties 1-5 and
  D(H ) ,
- (Strong Converse) 
  0, An , I  An n
 n  n n s.t. tr( Ann )  2
lim tr( An  n )  0
n
s.t.
 n ( E (  )  )
A motivation: Entanglement theory

We say   D( H  H1  ...  H k ) is separable if
   p j 1j  ...   kj
j
If it cannot be written in this form, it is entangled
n
, S (H
n
)

The sets of separable sates over H
satisfy properties 1-5

The rate function of the theorem is a well-known
entanglement measure, the regularized relative
(Vedral and Plenio 98)
entropy of entanglement
Regularized relative entropy of entanglement

E (  )  lim minn

R


  D( H  H1  ...  H k )
Given an entangled state
n   S ( H
S (
)
n
n
||  )
(Vedral and Plenio 98)
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

Cor: For every entangled state
  D( H  H1  ...  H k )
ER (  )  0
Regularized relative entropy of entanglement

Rate of conversion of two states by local operations and
classical communication:


kn


k n
n
R(    )  inf lim sup : lim min || (  )   ||1  0
k n   n 
n n LOCC


The corollary implies that if
 is entangled,
R(    )  0

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
Some elements of the proofs

Asymptotic continuity: Let En (  ) : min S (  ||  ), E (  )

  n
E (  )  E ( )  f (||    ||1 )n, for  ,   D( H n )
(Horodecki and Synak-Radtke 05; Christandl 06)

Non-lockability: Let
   pj j
j
p E
j
j
n
(  j )  h ( p j )  E n (  )   p j E  n (  j )
j
(Horodecki3 and Oppenheim 05)
Some elements of the proofs

Lemma: Let
  D(H ) and Y ,   0 s.t.   Y  
Then  ' D( H ) s.t. F (  ' ,  )  1  tr (  )
and

' Y
Lemma: Let
(Datta and Renner 08)
 ,   D( H ) ,   0
lim tr( 
n
n
2
( S (  || )  ) n
 )  0
n
(Ogawa and Nagaoka 00)
Some elements of the proofs

Almost power states:
V (H
n
,


nr
 
) : P 
[ , n , r ]
nr
 Sym( H
n

  r :   SYM (n),  r  H r
)  span(V ( H
n
,
nr

))
Exponential de Finetti theorem: For any permutation-
nk


D
(
H
) there exists a measure
symmetric state n  k

 over H  H E and states  n  

tr1,..., k (  n  k )    (d  )trE  n  n

,n,r
s.t.
 n dim( H ) 2
2

k ( r 1)
nk
1
(Renner 05)
Some elements of the proofs

Almost power states:
V (H
n
,


nr
 
) : P 
[ , n , r ]
nr
 Sym( H
n

  r :   SYM (n),  r  H r
)  span(V ( H
n
,
nr

))
Exponential de Finetti theorem: For any permutation-
nk


D
(
H
) there exists a measure
symmetric state n  k

 over H  H E and states  n  

tr1,..., k (  n  k )    (d  )trE  n  n

[ , n , r ]
s.t.
 n dim( H ) 2
2

k ( r 1)
nk
1
(Renner 05)
Elements of the proof

(Proof sketch) We can write the statement of the theorem as


1
,
y

E

 ( )
n
yn
lim n (  ,2 )  

n 
0
,
y

E

 ( )
n (  n ,2 yn )  max tr( A n ) : tr( A)  2 yn   n
0 A I
The dual formulation of the convex optimization above reads
n (  n ,2 yn )  min tr(  n  2bn  )   2n (b y )
 n ,b
It is then clear that it suffices to prove


1
,
y

E

 ( )
n
yn
lim min tr (   2  )  

n   n
0, y  E (  )
Elements of the proof

(Proof sketch) We can write the statement of the theorem as


1
,
y

E

 ( )
n
yn
lim n (  ,2 )  

n 
0
,
y

E

 ( )
n (  n ,2 yn )  max tr( A n ) : tr( A)  2 yn   n
0 A I
The dual formulation of the convex optimization above reads
n (  n ,2 yn )  min tr(  n  2bn  )   2n (b y )
 n ,b
It is then clear that it suffices to prove


1
,
y

E

 ( )
n
yn
lim min tr (   2  )   

n   n
0, y  E (  )
Elements of the proof

(Proof sketch) We first show that for every
lim min tr( 
n
2
n n
( E (  )  ) n
)  0

n
Take n sufficiently large such that E (  )  E n (  ) / n   / 2
Let n   n be such that E n (  )  S (  ||  )
By the strong converse of quantum Stein’s Lemma
lim tr( 
 nm
n 
As n
m
2
( E (  )  / 2 ) n
n m )   0
 nm we find
lim inf min tr( 
n
n
n
2
( E (  )  ) n
)  0
Elements of the proof

(Proof sketch) We first show that for every
lim min tr( 
n
n n
2
( E (  )  ) n
)  0

n
Take n sufficiently large such that E (  )  E n (  ) / n   / 2
Let n   n be such that E (  n )  S (  || n )
n
By the strong converse of quantum Stein’s Lemma
lim tr( 
 nm
n 
As n
m
2
( E (  )  / 2 ) nm
n m )   0
 nm we find
lim inf min tr( 
n
n
n
2
( E (  )  ) n
)  0
Elements of the proof

(Proof sketch) We first show that for every
lim min tr( 
n
n n
2
( E (  )  ) n
)  0

n
Take n sufficiently large such that E (  )  E n (  ) / n   / 2
Let n   n be such that E (  n )  S (  || n )
n
By the strong converse of quantum Stein’s Lemma
lim tr( 
 nm
n 
As n
m
2
( E (  )  / 2 ) nm
n m )   0
 nm we find
lim inf min tr( 
n
n
n
2
( E (  )  ) n
)  0
Elements of the proof

(Proof sketch) We now show
lim min tr( 
n n
n
2
( E (  )  ) n
)    0
Let n   n be an optimal sequence in the eq. above
( E (  )  ) n
( E (  )  ) n
n  (   2
n ) 
We can write   2
Assuming conversely that the limit is zero, we find
n
n  2
( E (  )  ) n
n
n ,  n    n 1  0
Elements of the proof

(Proof sketch) We now show
lim min tr( 
n n
n
2
( E (  )  ) n
)    0
Let n   n be an optimal sequence in the eq. above
( E (  )  ) n
( E (  )  ) n
n  (   2
n ) 
We can write   2
Assuming conversely that the limit is zero, we find
n
n  2
( E (  )  ) n
n
n ,  n    n 1  0
Elements of the proof

(Proof sketch) We now show
lim min tr( 
n n
n
2
( E (  )  ) n
)    0
Let n   n be an optimal sequence in the eq. above
( E (  )  ) n
( E (  )  ) n
n  (   2
n ) 
We can write   2
Assuming conversely that the limit is zero, we find
n
n  2
( E (  )  ) n
n
n ,  n    n 1  0
Elements of the proof

(Proof sketch) We now show
lim min tr( 
n
n n
2
( E (  )  ) n
)    0
Let n   n be an optimal sequence in the eq. above
( E (  )  ) n
( E (  )  ) n
n  (   2
n ) 
We can write   2
Assuming conversely that the limit is zero, we find
n
n  2
( E (  )  ) n
n
n ,  n    n 1  0
Then
E (  )  lim


n
En (  n )
n
 E (  )  
Elements of the proof

(Proof sketch) We now show
lim min tr( 
n
n n
2
( E (  )  ) n
)    0
Let n   n be an optimal sequence in the eq. above
( E (  )  ) n
( E (  )  ) n
n  (   2
n ) 
We can write   2
Assuming conversely that the limit is zero, we find
n
n  2
( E (  )  ) n
n
n ,  n    n 1  0
Then
E (  )  lim


n
En (  n )
n
 E (  )  
Elements of the proof

(Proof sketch) Finally we now show
lim min tr( 
n n
n
2
( E (  )  ) n
)    0
with   1 . Suppose conversely that   1 .From

n
2
( E (  )  ) n
n  ( 
n
2
( E (  )  ) n
n ) 
we can write
n  2
( E (  )  ) n
n , F (  n ,   n )  
Note that we can take  n , n
to be permutation-symmetric
Elements of the proof

(Proof sketch) Define n : tr1,..., n ( n ), n : tr1,..., n (n )
We have
 n  2
( E (  )  ) n
n , F ( n ,  (1 ) n )  
We can write
  (d  )




n
( E (  )  ) n
2


 n  trE  (1 ) n  (1 ) n ,
n  X n , X n 1  n

 (1 ) n  
10d 2
[ ,(1 ) n ,11d 2 1 log(n )]
Elements of the proof

(Proof sketch) Define n : tr1,..., n ( n ), n : tr1,..., n (n )
We have
n  2
( E (  )  ) n
n , F ( n ,  n )  
We can write
  (d  )




n
( E (  )  ) n
2


 n  trE  (1 ) n  (1 ) n ,
n  X n , X n 1  n

 (1 ) n  
10d 2
[ ,(1 ) n ,11d 2 1 log(n )]
Elements of the proof
(Proof sketch) Because F ( n ,   (1 ) n )  

B
  (d 
n1 / 8
)  (1)
( )
Therefore we can write
  ' (d 
B
n1 / 8

) n  O(1)2
( E (  )  ) n
n  O(1) X n , X n 1  n
10d 2
( )
and
B
  ' ' (d 
n4

) n  n
( ')
with  ' Bn 1 / 8 (  )
8d 2
2
( E (  )  ) n
n  n
8d 2
Xn, Xn 1  n
10d 2
Elements of the proof
(Proof sketch) Because F ( n ,   (1 ) n )  

B
  (d 
n1 / 8
)  (1)
( )
Therefore we can write
  ' (d 
B
n1 / 8

) n  O(1)2
( E (  )  ) n
n  O(1) X n , X n 1  n
10d 2
( )
and
B
  ' ' (d 
n4

) n  n
( ')
with  ' Bn 1 / 8 (  )
8d 2
2
( E (  )  ) n
n  n
8d 2
Xn, Xn 1  n
10d 2
Elements of the proof
(Proof sketch) Because F ( n ,  n )  

B
  (d 
n1 / 8
)  (1)
( )
Therefore we can write
  ' (d 
B
n1 / 8

) n  O(1)2
( E (  )  ) n
n  O(1) X n , X n 1  n
10d 2
( )
and
B
  ' ' (d 
n4

) n  n
( ')
with  ' Bn 1 / 8 (  )
8d 2
2
( E (  )  ) n
n  n
8d 2
Xn, Xn 1  n
10d 2
Elements of the proof

(Proof sketch) Therefore
'
n  n
with
'
8d 2
2
( E (  )  ) n
n  X n ' , X n 1  n 1
s.t. trE (  '  ' )   '
Finally
'
(1   ) E (  )  lim sup
n 
E n ( n )
n
 E (  )  
Elements of the proof
(Proof sketch) Because F ( n ,  n )  

B
  (d 
n1 / 8
)  (1)
( )
Therefore we can write
  ' (d 
B
n1 / 8

) n  O(1)2
( E (  )  ) n
n  O(1) X n , X n 1  n
10d 2
( )
and
B
  ' ' (d 
n4

) n  n
( ')
with  ' Bn 1 / 8 (  )
8d 2
2
( E (  )  ) n
n  n
8d 2
Xn, Xn 1  n
10d 2