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Quantum Stein’s lemma for correlated states
and asymptotic entanglement transformations
Fernando G.S.L. Brandão and Martin B. Plenio
arXiv 0810.0026 (chapter 4)
arXiv 0902.XXXX
QIP 2009, Santa Fe
Multipartite entangled states
Non-entangled states
   p j  ...  
1
j
j
Can be created by LOCC
(Local Operations and
Classical Communication)
k
j
Multipartite entangled states

LOCC asymptotic
entanglement conversion
r is an achievable rate if

A
B
 
lim min || (  nr )   n ||1  0
n LOCC
LOCC optimal conversion rate

R(    )  inf r : achievable
  n  LOCC  n    n ( r o ( n ))
Asymptotically non-entangled states
 is asymptotically non-entangled if there is a state 
such that
R(   )  0
Is every entangled state asymptotically entangled?
• For distillable states:
R(  EPR )  0
Hence, they must be asymptotically entangled
• For bound entangled states,
(Horodecki, Horodecki, Horodecki 98)
Are they asymptotically entangled?
R(  EPR )  0
Asymptotically non-entangled states
 is asymptotically non-entangled if there is a state 
such that
R(   )  0
Is every entangled state asymptotically entangled?
• For distillable states:
R(  EPR )  0
Hence, they must be asymptotically entangled
• For bound entangled states,
(Horodecki, Horodecki, Horodecki 98)
Are they asymptotically entangled?
R(  EPR )  0
Asymptotically non-entangled states
• We can use entanglement measures to analyse the problem:
• Let r be an achievable rate:

 
lim min || (  nr )   n ||1  0
n LOCC
1
1
1
 nr
 nr
E (  )  E ((  ))  E ( n )  (|| (  nr )   n ||1 )
n
n
n
LOCC monotonicity
Asymptotic continuity
1
rE (  )  E ( ), E (  ) : lim E (  n )
n  n



R
R(    )  E  ( ) / E  (  )
• If
E  ( )  0
, then

is asymptotically entangled
Asymptotically non-entangled states
• We can use entanglement measures to analyse the problem:
• Let r be an achievable rate:

 
lim min || (  nr )   n ||1  0
n LOCC
1
1
1
 nr
 nr
E (  )  E ((  ))  E ( n )  (|| (  nr )   n ||1 )
n
n
n
LOCC monotonicity
Asymptotic continuity
1
rE (  )  E ( ), E (  ) : lim E (  n )
n  n



R
R(    )  E  ( ) / E  (  )
• If
E  ( )  0
, then

is asymptotically entangled
Asymptotically non-entangled states
• Every bipartite entangled state is
asymptotically entangled
(Yang, Horodecki, Horodecki, Synak-Radtke 05)
• Entanglement cost:
EF (   n )
R(EPR   )  EC (  )  lim
n 
n
Bennett, DiVincenzo, Smolin, Wootters 96, Hayden, Horodecki, Terhal 00
•
EC ( )  0
for every bipartite entangled states
Asymptotically non-entangled states
• This talk: Every multipartite entangled state is asymptotically entangled
• The multipartite case is not implied by the bipartite:
there are entangled states which are separable across
any bipartition
ex: State derived from the Shift Unextendible-Product-Basis
(Bennett, DiVincenzo, Mor, Shor, Smolin, Terhal 98)
Asymptotically non-entangled states
• Regularized relative entropy of entanglement:
(Vedral and Plenio 98, Vollbrecht and Werner 00)
ER (  )  min S (  ||  )  min tr(  (log   log  ))
 S
n
E
(

)

R
ER (  )  lim
n 
n
 S
R(   )  ER ( ) / ER ( )
ER (  )  0 for every entangled state

E
We show that by linking R to a certain quantum
• Rest of the talk:
•
hypothesis testing problem
Same result has been found by Marco Piani, with
different techniques
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 
Error probabilities Alternative
Null
- Type I error: hypothesis
hypothesis

 n ( An ) : tr(  ( I  An ))
n
- Type II error:
 n ( An ) : tr(
n
An )
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
state is
- Type I The
error:
 

An , I  An 
The state is

 n ( An ) : tr(  ( I  An ))
n
- Type II error:
 n ( An ) : tr(
n
An )
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

 
An , I  An 
- Type I error:
 n ( An ) : tr(  ( I  An ))
- Type II error:
 n ( An ) : tr(
n
n
An )
Several different instances depending on the constraints imposed in the
error probabilities: Chernoff distance, Hoeffding bound, Stein’s Lemma,
etc...
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
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
- 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
- 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
- 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
- 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

Consider the following two hypothesis
n
n
 n  S ( H1 n
H

...

H
2
k )
n
- Null hypothesis: For every n   we have 
- Alternative hypothesis: For every
state
n
n   n  D( H  n ) ,
1. Each
n
where
4. If
5. If
n n satisfies
is closed and convex
2. Each contains the maximally mixed state
3. If
we have an unknown
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
- (Direct Part)
n n
  0
satisfying properties 1-5 and
there is a
  D(H )
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
- (Strong Converse)
satisfying properties 1-5 and   D (H )
   0, An , I  An n
, if
 n  n n s.t. tr( Ann )  2
lim tr( An  )  0
n
n
 n ( E (  )  )
A generalization of Quantum Stein’s Lemma

theorem: Given
n n
- (Strong Converse)
satisfying properties 1-5 and   D (H )
   0, An , I  An n
, if
 n  n n s.t. tr( Ann )  2
 n ( E (  )  )
lim tr( An  )  0
n
n
Proof: Exponential de Finetti theorem (Renner 05) + duality convex
optimization + quantum Stein’s lemma; see arXiv 0810.0026
Corollary: strict positiveness of ER∞
For an entangled state

we construct a sequence of POVMs s.t.
lim tr( An  )  1
n
n
 n  Sn n ,
tr( Ann )  2
 n
,  0
Corollary: strict positiveness of ER∞
How we construct the An’s : we measure each copy with a local informationally
complete POVM M to obtain an empirical estimate  n
of the state. If
||  n   ||1   : min ||    ||1 / 2
 S
we accept, otherwise we reject
M
Corollary: strict positiveness of ER∞
• By Chernoff-Hoeffding’s bound, it’s clear that for some
tr ( Ann )  2
 n
n   (d )
,
n
for
 0
n of the form
, with
 supported on separable states
Corollary: strict positiveness of ER∞
• In general, by the exponential de Finneti theorem,
(Renner 05)
tr1,..., n ( S n (n ))   (d )"
 (1 ) n
"
almost power states
•
trA "
 (1 ) n
n
• We show that
"  2
( n)
,
for
||    ||1  
 (1 ) n
( n)

(
d

)
tr
(
A
"

"
)

2
n

||   ||1 
which implies the result
Corollary: strict positiveness of ER∞
• Let’s show that
  ( d )tr( A "
n
 (1 ) n
")  2
( n)
||  ||1 
• We measure an info-complete
POVM on all copies of
tr1,..., n ( S n (n ))
M
expect the first

• The estimated state is
close
from the post-selected state with
probability 1  2   ( n )
• As we only used LOCC, the post-selected state must be separable and
hence
2
far apart from

Thank you!
•x