One-Shot Quantum Information
Theory I: Entropic Quantities
Nilanjana Datta
University of Cambridge,U.K.

In Quantum information theory, initially one evaluated:
optimal rates of info-processing tasks, e.g.,

data compression,

transmission of information through a channel, etc.
under the assumption of an “asymptotic, memoryless setting”
Assume:

information sources & channels are memoryless

They are available for asymptotically many uses

E.g. Transmission of classical information
classical
info
N
Noisy quantum
channel
Optimal rate (of classical information transmission):
classical capacity
C (N ) 
maximum number of bits transmitted per use of N
memoryless: there is no correlation in the noise acting
on successive inputs
N n
:
n
successive uses of the channel; independent
“asymptotic, memoryless setting”

To evaluate
C (N ) :
classical
info
x
En
 x( n )
input
N n
n
N
uses
n
encoding

One requires : prob. of error
C (N ) :
N  n (  x( n ) )
channel
output
decoding
POVM
(n)
e 0
p
Dn
x'
as
n
Optimal rate of reliable information transmission
Entropic Quantities
Optimal rates of information-processing tasks in the
“asymptotic, memoryless setting”

Compression of Information:
Memoryless quantum info. source
Data compression limit:

S ( )
  ,H 
[Schumacher]
von Neumann entropy
Info Transmission thro' a memoryless quantum channel
N
[Holevo, Schumacher, Westmoreland]
Classical capacity C (N )
--given in terms of the Holevo capacity ;
Quantum capacity
Q (N )
[Lloyd, Shor, Devetak]
--given in terms of the coherent information ;
These entropic quantities are all obtainable from a single
parent quantity;
Quantum relative entropy: For
 ,   0; Tr  1
D(  ||  ) : Tr   log    Tr   log  
supp   supp 
e.g. Data compression limit:
S (  ) : Tr   log     D(  || I )
(  I )
In real-world applications
“asymptotic memoryless setting” not necessarily valid

In practice: information sources & channels are used a
finite number of times;

there are unavoidable correlations between successive
uses (memory effects)
Hence it is important to evaluate optimal rates for a
finite number of uses (or even a single use)
of an arbitrary source or channel

Evaluation of corresponding optimal rates:
One-shot information theory
One-shot information theory
classical
info
x
E
x
N
single use
N
input
encoding
One-shot   error
classical capacity
C (N)
(1)
:
N(x )
channel
output
D
x'
decoding
POVM
max. number of bits that can be
transmitted on a single use of N
Prob. of
error:
pe  
for some
  0,
In the one-shot setting too…

Capacities, data compression limit etc. are
-- given in terms of entropic quantities
Min-/max-/0- entropies (R.Renner)

Obtainable from certain (generalized) relative entropies
Parent quantities for optimal ‘rates’ in the one-shot setting
D0 (  ||  )
Dmax (  ||  )
Max-relative entropy
0-relative Renyi entropy
D  (  ||  )
Dmin (  ||  )
Min-relative entropy
“super-parent”:
Quantum Renyi Divergence
(sandwiched Renyi relative entropy)
[Wilde et al; Muller-Lennert et al]
In the one-shot setting too…

Capacities, data compression limit etc. are
-- given in terms of entropic quantities
Min-/max- entropies (R.Renner)

Obtainable from certain (generalized) relative entropies
Dmax (  ||  )
 
D0 (  ||  )
D(  ||  )
  12
?
[ND, F.Leditzky]
D  (  ||  )
 1
Dmin (  ||  )
  relative Renyi entropy
Quantum Renyi Divergence
(sandwiched Renyi entropy)
[Wilde et al; Muller-Lennert et al]
D (  ||  )
if [ ,  ]  0
Outline
Semidefinite programming

Mathematical Tool:

Definitions of generalized relative entropies:

Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )

Properties & operational significances of them

Their “smoothed” versions



Their children:
the min-, max- and 0-entropies
 (  ||  )
The “super-parent” : Quantum Renyi Divergence D

Relationship between D  (  ||  ) & D0 (  ||  )
Notations & Definitions
A
HA
quantum system
Hilbert space
B (H ) : algebra of linear operators acting on H
P (H ) : set of positive operators……
D (H )  P (H ) :

set of density matrices (states)
 : B (HA )  B (HB )   : A  B 
Linear maps: If
* : B  A
*
Tr B  (  A )   Tr   (B )  A 
its adjoint map:
defined through

Quantum operations (quantum channels) : linear CPTP map
Mathematical Tool
Semi-definite programming (SDP)



A well-established form of convex optimization
The objective function is linear in an input
constrained to a semi-definite cone
Efficient algorithms have been devised for its
solution
Mathematical Tool
(formulation:Watrous)
(2) Semi-definite programming (SDP)
(, A, B); A, B  P (H ),
 : P (HA )  P (HB ) positivity-preserving map

Primal problem

Dual problem
minimize
Tr( AX )
maximize
subject to
 ( X )  B;
X  0;
subject to
Optimal solutions:

Tr( BY )
* (Y )  A;
Y  0;


IF Slater’s duality
condition holds.
Outline


Mathematical Toolkit:

Semidefinite programming
Definitions of generalized relative entropies:
Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )
Definitions of generalized relative entropies
  D (H ),   P (H ); supp   supp  ;

Max-relative entropy [ND]
Dmax (  ||  ) : inf  :   2  


 log  max (

1/2

1/2
1/2


2 I
1/2
)
Min-relative entropy [Dupuis et al]
Dmin (  ||  ) : 2 log ||   ||1
 2 log F (  ,  ) fidelity
Definitions of generalized relative entropies
  D (H ),   P (H ); supp   supp  ;

0-relative Renyi entropy

D0 (  ||  ) :  log Tr (   )
where


contd.

denotes the projector onto supp
 -relative Renyi entropy

(  1)
1
 1
D (  ||  ) :
log Tr (  )
 1
lim D (  ||  ) =D0 (  ||  )
 0
Properties of generalized relative entropies
Positivity:

If
 ,   D (H ),
D* (  ||  )  0
for
*  max, 0, min
just as
D(  ||  )
Data-processing inequality:

D* ( (  ) ||  ( ))  D* (  ||  )
for any CPTP map
Invariance under joint unitaries:

D* (U U || U  U )  D* (  ||  )
†

†

for any unitary
operator
U
Interestingly,
D0 (  ||  )  Dmin (  ||  )  D(  ||  )  Dmax (  ||  )
Operational interpretation of the max-relative entropy
Multiple state discrimination problem:
its state
& told
a quantum
system
Bob
with prob.
1
2
1
m
1
m
m
He does measurements to infer the state: POVM

E1 ,.., Em  :

0  Ei  I ;
m
E
i 1
i
I
His optimal average success probability:
p
*
succ
: max
E1 ,.., Em
m
Tr  E  

m
1
i 1
i
i
Theorem 3 [M.Mosonyi & ND]:
The optimal average success probability in this multiple
state discrimination problem is given by:
p
*
succ

1
m
min max

1i  m
Dmax ( i || )
2

Let
p
*
succ
i 
Sketch of proof:
m
basis in
:
i 1
: max
E1 ,.., Em
C
m

Let
1 m
 :  i i  i
m i 1
 P (Cm  H )
m
Tr  E  

m
1
 max
m
Ei i1:POVM
i 1
i
i
m
Tr[ ( i i  Ei )]
i 1
m
Y
  i i  Ei  P (Cm  H );
i 1
TrCm Y

max
m
Y P ( C H );
Tr m Y  IH
C
Tr  Y 
  Ei  IH ;
i

Let
p
*
succ
i 
Sketch of proof:
m
basis in
:
i 1
: max
E1 ,.., Em
C
m

Let
1 m
 :  i i  i
m i 1
 P (Cm  H )
m
Tr  E  

m
1
 max
m
Ei i1:POVM
i 1
i
i
m
Tr[ ( i i  Ei )]
i 1
m
Y
  i i  Ei  P (Cm  H );
i 1
TrCm Y

max
m
Y P ( C H );
Tr m Y  IH
C
Tr  Y 
  Ei  IH ;
i

Let
p
*
succ
i 
Sketch of proof:
m
basis in
:
i 1
: max
E1 ,.., Em
C
m

Let
1 m
 :  i i  i
m i 1
 P (Cm  H )
m
Tr  E  

m
1
 max
m
Ei i1:POVM
i 1
i
i
m
Tr[ ( i i  Ei )]
i 1
m
Y
  i i  Ei  P (Cm  H );
i 1
TrCm Y

max
m
Y P ( C H );
Tr m Y  IH
C
Tr  Y 
  Ei  IH ;
i

SDP

duality condition holds
[Koenig, Renner, Schaffner]

Primal problem

Dual problem
minimize
Tr( AX )
maximize
subject to
 ( X )  B;
X  0;
subject to
minimize
subject to
Tr X
I Cm  X  
X  0;
Tr( AX )  Tr( IH X )  Tr X
Tr( BY )
* (Y )  A;
Y  0;

max
m
Y P ( C H );
Tr m Y  IH
Tr  Y 
C
*

B   ; A  IH ; =TrCm
* =TrCm : P (Cm  H )  P (H )
  : P (H )  P (Cm  H );
 ( X )  I Cm  X
Sketch of proof contd:

*
psucc
 min Tr X : X  0,   I Cm

1 m
 X ;  :  i i  i
m i 1
 min Tr X : X  0, i  mX i  1, 2,.., m
  I C
m



Tr
X

 X  mX ; TrX  mTrX  TrX 

m 
X

1
1  min Tr mX : X  0, i  X i  1, 2,.., m
 mi i  i   i i  X

m
m
i 1
i 1


X


; X   
  =TrX ;  

TrX


1  i  X i

minm  :   0, i   i  1, 2,.., m
m  D (H )
inf  : i     2 D (  || ) 

1
m
min max

1iM
2

Dmax ( i || )
max
i


Operational interpretation of D0 (  ||  ) :  log Tr (   )
 Quantum binary
Bob receives a state
hypothesis testing:

(null hypothesis)

or (alternative
hypothesis)
He does a measurement to infer which state it is

POVM

A [ ]
Possible errors
Type I
Type II

Error
probabilities
( I  A) [ ]
&
inference

actual state



  Tr(( I  A)  )
Type I
  Tr( A )
Type II


Suppose
(POVM element)
A  
Prob(Type I error)
Prob(Type II error)
  Tr( A )
 Tr(  )
  Tr(( I  A)  )
0
Bob never infers the state
to be
BUT

when it is

D0 (  ||  ) :  log Tr  
Hence

 0
2
 D0 (  || )
= Prob(Type II error |Type I error = zero)

(POVM element)
Suppose
A  
Prob(Type I error)
Prob(Type II error)
  Tr( A )
 Tr(  )
  Tr(( I  A)  )
0
Bob never infers the state
to be

BUT
In fact,
when it is

D0 (  ||  ) :  log Tr  
min Prob(Type II error |Type I error = zero)

*
 0
2
 D0 (  || )
Smoothed relative entropies

What if Bob has a single copy of the state but one allows
*|
 min Tr (A )
non-zero but small value of the Prob(Type I error)
 0
i.e., let  0A Ifor some   0.
Tr (A ) 1
?
D0 (  ||  )   log  * | 0   log Tr   
 * |   log
min
Tr (?A )

*
|
0  A I  
Tr (A ) 1
  Tr(( I  A)  );   0  Tr (A )  Tr ( )  1

For
 
choose
A
such that
Tr (A )  1  

D0 (  ||  )   log  * |   max
Hypothesis testing relative entropy
[Wang & Renner]
0  A I
Tr (A ) 1
  log(Tr (A )) 
 DH (  ||  )

D
Compare operational significances of H (  ||  ) & D (  ||  )
D(  ||  )

arises in asymptotic binary hypothesis testing
Suppose Bob is given many ( n) identical copies of the state


He receives

*( n )

| ( n ) :
  [0,1) :
n
n
Bob’s POVM
 An , ( I  An )
Minimum type II error when
type I error  
 1

*( n )
lim  log  | ( n )   D (  ||  )
n 
 n

[Quantum Stein’s Lemma]
Operational interpretations in binary hypothesis testing
DH (  ||  )

D(  ||  )
One-shot setting;
Asymptotic memoryless setting;
Single copy of the state:
Multiple copies of the state:
  log  * | 
 1

*( n )
 lim  log  | ( n ) 
n 
 n

  [0,1) :
(Bob receives identical copies of the state:
 n
or

n
)
Smooth max-relative entropy
  0.
Dmax (  ||  )
Dmax (  ||  ) : min

B ( )

B (  ) :    0, Tr  1: F (  ,  )  1   

2
fidelity


Dmax
(  || )
&
2
 DH (  || )
can both be formulated as SDPs
Outline


Mathematical Toolkit:

Semidefinite programming
Definitions of generalized relative entropies:
Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )


Properties & operational significances of them
Their children:
the min-, max- and 0-entropies
Dmax (  ||  ), D0 (  ||  ) & Dmin (  ||  )
as parent quantities for other entropies
Just as:
von Neumann
entropy
S (  )   D(  || I )
(  I )
H 0 (  ) :  D0 (  || I )
H min (  ) :  Dmax (  || I )
  log max   
 log rank( )
Hmax (): Dmin ( || I )

 2 log Tr 

[Renner]
Other min- & max- entropies
For a bipartite state  AB :
Conditional entropy
A
B
S ( A | B)  S (  AB )  S (  B )  max  D(  AB || I A   B )
B
Conditional min-entropy
H min ( A | B)  : max  Dmax (  AB || I A   B )
B
Max-conditional entropy
H max ( A | B)  : max  Dmin (  AB || I A   B )
B
0-conditional entropy
H 0 ( A | B)  : max  D0 (  AB || I A   B )
B


They have interesting mathematical properties:
e.g. Duality relation: [Koenig, Renner, Schaffner]:
For any purification
 ABC
of a bipartite state
 AB :
H max ( A | B)    H min ( A | C ) 
(just as for the
von Neumann entropy):
H ( A | B)    H ( A | C ) 
-- and -- interesting operational interpretations:
Operational interpretations


Conditional min-entropy
maximum achievable singlet fraction

Conditional max-entropy

[Koenig, Renner, Schaffner]
decoupling accuracy

Conditional 0-entropy

one-shot entanglement cost under LOCC
[F.Buscemi, ND]
Operational interpretations contd.

Conditional min-entropy
 AB
1

d
d

i 1
 Max. achievable singlet fraction
iA iB  HA  HB :
 AB   AB  AB
2
 H min ( A| B ) 
max. entangled
state
[Koenig, Renner, Schaffner]
 d max F  ((id A   B )  AB ),  AB 
 B :CPTP
Given the bipartite state
with the singlet state
quantum operations
fidelity
 AB ,
it is the maximum overlap
 AB , that can be achieved by local
 B on the subsystem B.
2
Operational interpretations contd.

Conditional max-entropy
Distance of
 AB ,

Decoupling accuracy
from a product state
 A  B
no correlations;
I
A 
dA
2
completely mixed state on
HA
decoupled
[Koenig, Renner, Schaffner]
H max ( A| B ) 
 d A max F   AB , A   B 
B
2
fidelity
From the cryptographic point of view:
How random
A
appears from the point of view of an
adversary who has access to
B.
Operational interpretations contd.

Conditional 0-entropy
 one-shot entanglement cost
One-shot Entanglement Dilution
Bell states
 Bell 
m
 : LOCC
Alice
 AB
Bob
One-shot entanglement cost
E (  AB ) : min m
(1)
C
= minimum number of Bell states needed to
prepare a single copy of  AB via LOCC
Operational interpretations contd.

Theorem [F.Buscemi & ND]: One-shot perfect entanglement
cost of a bipartite state
 AB
under LOCC:
E (  AB )  min H 0 ( A | R)  E
(1)
C
E
Pure-state ensembles:

E  pi , 
and
i
AB

i
conditional 0-entropy
i
i


p


AB
; AB  i AB
i
E
i
i
 RAB
  pi iR iR   AB
 AB
i
E
E
 RA
 TrB  RAB
,
classical-quantum state
Outline


Mathematical Toolkit:

Semidefinite programming
Definitions of generalized relative entropies:
Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )
Properties & operational significances of them
and their children: the min-, max- and 0-entropies



Their “smoothed” versions
 (  ||  )
The “super-parent” : Quantum Renyi Divergence D

D0 (  ||  )
Dmax (  ||  )
Max-relative entropy
O-relative Renyi entropy
D  (  ||  )
“super-parent”:
Quantum Renyi Divergence
(sandwiched Renyi entropy)
[Wilde et al; Muller-Lennert et al]
Dmin (  ||  )
Min-relative entropy
“super-parent”:
Quantum Renyi Divergence
(sandwiched Renyi entropy)
[Wilde et al; Muller-Lennert et al]
D0 (  ||  )
Dmax (  ||  )
 
D(  ||  )
D  (  ||  )
 1
Dmin (  ||  )
  12
D (  ||  )
if
[ ,  ]  0
Quantum Renyi Divergence
  D (H ),   P (H ); supp   supp  ;
For
  (0,1)  (1, ) :


1

 

D (  ||  ) :
log Tr  (  )  ;
 1
where  

Note: If
Tr  (  )

1
2
[ ,  ]  0
 
  Tr  ( 
D  (  ||  ) 
2

)
 2   1  
  Tr  

1
log Tr (   1 )
 1


2 
  Tr  

 D (  ||  )
-relative Renyi entropy
 1 
Quantum Renyi Divergence
  D (H ),   P (H ); supp   supp  ;
For
  (0,1)  (1, ) :


1

 

D (  ||  ) :
log Tr  (  )  ;
 1
where  
1
2
 2   1  
[ ,  ]  0
Note:
If
Non-commutative generalization of
D (  ||  )

 
2 
 2 
Tr  (  )   Tr  (  )   Tr      Tr     1 

D  (  ||  ) 
1
log Tr (   1 )
 1

 D (  ||  )
-relative Renyi entropy
Two properties of Quantum Renyi Divergence
(1) Data-processing inequality;
For
[Frank & Lieb;
1
   ,
2
 ( (  ) ||  ( ))  D (  ||  )
D


Beigi 2013]
 : CPTP map
holds also for Dmin (  ||  ), D(  ||  ), Dmax (  ||  )

1
2
 1
 
Two properties of Quantum Renyi Divergence
(1) Monotonicity under CPTP maps
[Frank & Lieb;
Beigi 2013]
:
For
D  ( (  ) ||  ( ))  D  (  ||  )
1
   ,
2
holds also for Dmin (  ||  ), D(  ||  ), Dmax (  ||  )
1
   1,
(2) Joint convexity: For
2
[Frank & Lieb]
D  ( p i ||  p  i )   pi D  ( i ||  i )
i


i
i
i
i
Dmin (  ||  ) , D(  ||  )
Dmax (  ||  ) is quasi-convex: [ND]
holds also for
Note:
Dmax ( pi i ||  pi i )  max Dmax ( i ||  i )
i
i
1i  n

What about
D0 ( ||  ) ?
Outline

Mathematical Toolkit:

Semidefinite programming
Definitions of generalized relative entropies:

Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )


Properties & operational significances of them

Their children:
the min-, max- and 0-entropies

Their “smoothed” versions

 (  ||  )
The “super-parent” : Quantum Renyi Divergence D

Relationship between D  (  ||  ) & D0 (  ||  ) & its implication
Theorem: For  ,

[ND, F.Leditzky]
If
 0, Tr   1;
if
supp   supp  ;
 (  ||  ) =D (  ||  )  0.......(1)
then lim D

0
 0
supp   supp  , then (1) does not necessarily hold
D  (  ||  ) : Quantum Renyi
Divergence
Dmax (  ||  )
D0 (  ||  ) :
D0 (  ||  )
0-relative
Renyi entropy
Dmin (  ||  )
D  (  ||  )
D(  ||  )
D (  ||  )
if
[ ,  ]  0

Theorem: For  ,
[ND, F.Leditzky]
If

 0, Tr   1;
if
supp   supp  ;
 (  ||  ) =D (  ||  )  0.......(1)
then lim D

0
 0
supp   supp  , then (1) does not necessarily hold
Proof (key steps):
supp   supp  : D  (  ||  )  D (  ||  )   0.
(1) If

lim
D
D (  ||  )  D0 (  ||  ).....(a )
  0  (  ||  )  lim
0
(Araki-Lieb-Thirring inequality)
(2) If
supp   supp  : lim D  (  ||  )  D0 (  ||  )...(b)
(a ) & (b)  (1)
 0
if
supp   supp  ;
(a variant of the Pinching lemma)

Proof of the fact:
If
supp   supp  , then
lim D  (  ||  ) =D0 (  ||  )
 0
does not necessarily hold
A simple counterexample:
1 0
1 c
 
;   

0 0
 c 1
 ,   0,
c  (0,1).
  ,    0.
D0 ( ||  )  0
lim D  (  ||  ) = - log(1  c)  0;
 0
Summary
Generalized relative entropies:

Dmax (  ||  ), D0 (  ||  ), Dmin (  ||  )
Properties & some operational significances

Dmax (  ||  ) :in multiple state discrimination,
D0 (  ||  ) : in binary hypothesis testing
Their “smoothed” versions;





Their children:
D0 (  ||  )  DH (  ||  )


the min-, max- and 0-entropies
Operational significances of conditional entropies
 (  ||  )
The “super-parent” : Quantum Renyi Divergence D

 (  ||  ) & D (  ||  )
Relationship between D

0
Thank you!

Thanks also to:
F.Buscemi, F.Brandao, M-H.Hsieh, F.Leditzky, M.Mosonyi,
R.Renner, T.Rudolph,