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Generalized Relative Entropies, Entanglement Monotones and a family of quantum protocols
Nilanjana DattaUniversity of Cambridge,U.K.
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Quantum Relative Entropy
A fundamental quantity in Quantum Mechanics & Quantum
Information Theory is the Quantum Relative Entropy
of w.r.t.
It acts as a parent quantity for the von Neumann entropy:
: Tr ( log ) Tr ( log )( || )S
, 0, Tr 1
well-defined if supp supp
: Tr ( log )( ) ( || )S S I ( )I
2log log
0: (state / density matrix)
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It also acts as a parent quantity for other entropies:
e.g. for a bipartite state :
Conditional entropy
Mutual information
: ( ) ( ) ( )( : ) ( || )AA A BB B A BS S SI A B S
AB
: ( )( | ) ( | )( ) |AB A BAB BS A B S IS S
A B
Tr B A AB
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Some Properties of
( || ) 0 if & only i0 f S
( || )S
Joint convexity:
||( ) ( || )kk
k kpS S
“distance”
Invariance under
joint unitaries
* *( || ) ( || )S U U U U S
, states
1
n
i ii
p
For two mixtures of states1
n
i ii
p
&
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Monotonicity of Quantum Relative Entropy under a completely positive trace-preserving (CPTP) map
( ( ) || ( )) ( || )S S
ABC B AB BC( ) ( ) ( ) ( )S S S S
:
Many properties of other entropies can be proved using (1)
……….(1)
e.g. Strong subadditivity of the von Neumann entropy
Lieb & Ruskai ‘73
CB A
v.powerful!
( | ) ( | )S A BC S A B ABC
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Outline
Define 2 generalized relative entropy quantities
Discuss their properties and operational significance
Define 2 entanglement monotones
Discuss their operational significance
Consider a family tree of quantum protocols
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Two new relative entropies
Definition 1 : The max- relative entropy of a state & a positive operator is
max ( || ) : log min :S
( ) 0
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Definition 2: The min- relative entropy of a state & a positive operator is
where denotes the projector onto the support of
min ( || ) : log Tr ( )S
(supp )
supp supp
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Remark: The min- relative entropy
is the quantum relative Renyi entropy of order 0 :
where
min S ( || )
min ( || ) : log Tr S
11( || ) : log Tr ( )1
S
0( || ) S 0
lim ( || )S
quantum relative Renyi entropy of order ( 1)
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max min( || ) ( || )S S
max ( || ) : log min :S log 0
0 , 0( ) 0
0 0Tr [ ( )] 0
0log log [ ) 0Tr( ]
0log log [Tr( )]
min ( || )S max ( || )S
1
Proof:
0Tr [ ] Tr [ ]
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Operational significance of
Tr(( ) )I
He does a measurement to infer which state it is
POVM
Tr( )
min ( || )S
State Discrimination: Bob receives a state
or
( )I [ ] [ ]&
Possible errors
Type I
actual stateinference
Type II
Error
probabilities
Type I
Type II
hypothesis testing
0 I
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Tr(( ) )I
Suppose
Tr( )
min ( || )S
: log Tr
Prob(Type I error) Prob(Type II error)
0 Tr( )
(POVM element)
Bob never infers the state
to be when it is
min ( || )2 S
BUT
Hence
= Prob(Type II error |Type I error = 0)
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Compare with the operational significance of ( || )S
arises in asymptotic hypothesis testing
Suppose Bob is given many identical copies of the state
He receives
( )nn
n
For n large enough,
Prob(Type II error |Type I error
( || ) 2 Sn
)for any fixed
0 1
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Hence,min ( || ) & ( || )S S
have similar interpretations in terms of Prob(Type II error)
min ( || ) :S ( || ) :S
a single copy of the state
Prob(Type I error)
copies of the state
Prob(Type I error)
0n
0n
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Like we have for
Also
Most interestingly
* *( ( ) || ( )) ( || )S S for any CPTP map
for *( || ) 0S * max, min
( || )S
min max( || ) ( || ) ( || )S S S
* ** *( || ) ( || )S S U U U U
for any unitary
operator U
, states
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The min-relative entropy is jointly convex in its arguments.
The max-relative entropy is quasiconvex:
For two mixtures of states &
max max1( || ) max ( || )i ii n
S S
1
n
i ii
p
1
n
i ii
p
Also act as parent quantities for other entropies………..
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mmin ax( ) : ( || )SH I
log || ||
mmax in( ) : ( || )SH I log rank( )
( ) ( || )S S I
Min- and Max- entropies
Just as:
max min( ) ( )H H
[Renner]
von Neumann entropy
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mam xin : (( | || )) AB A BIH A SB
minmin : (: || )( ) AB A BI A B S
just as:
just as:
( | ) ( || )AB A BS A B S I
( : ) ( || )AB A BI A B S
For a bipartite state
etc.
etc.
:AB
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Min- and Max- Relative Entropies satisfy the:
(1) Strong Subadditivity Property
min min( | ) ( | )H A BC H A B
CB
A
(2) Subadditivity Property
max AB max A max B( ) ( ) ( )H H H AB
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(Q) What are the operational significances of the min- and max- relative entropies in
Quantum Information Theory?
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A class of important problems
the evaluation of:
data compression, transmission of information through a channel entanglement manipulation etc.
Initially evaluated in the
under the following assumptions:
information sources & channels were memoryless they were used an infinite number of times (asymptotic limit)
Optimal rates -- entropic quantities
obtainable from the relative entropyparent quantity
“asymptotic, memoryless setting”
optimal rates of info-processing tasks
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optimal rate of data compression:
: the minimum number of qubits needed to
store (compress) info emitted per use of a
quantum info source : reliably
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, Hsignals (pure states)
with probabilities
Then source characterized by:
Hilbert space
density matrix
1
k
i i ii
p
1 2, ...., k
1 2, ,..., kp p p H
Quantum Data Compression
Quantum Info source signals
,i ip
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To evaluate data compression limit : Consider a sequence
If the quantum info source is memoryless
,n n n H
;n nH H n
n ( ) B He.g. A memoryless quantum info source emitting qubits
Consider successive uses of the source ; qubits emitted
Stored in qubits
rate of data compression = nmn
n nnm ; nm n (data compression)
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: Tr ( ( log )) S
,n n n H
Optimal rate of data compression
under the requirement that ( ) 0nerror n
p
lim :n
nR mn
von Neumann entropyof the source
WHAT IF : :nstate of a quantum spin system
( interacting spins)nn
n
( )R S
Asymptotic, Memoryless Setting
( || ) S I (parent)
not memoryless !
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Quantum channelinformation
Transmission of information through a quantum channel
Optical fibre : through which polarized photons are transmitted
mobile particles which carry the info
A quantum spin chain – governed by a suitable Hamiltonian
info carriers (spin-1/2 particles) not mobile
instead the dynamical properties of the spin chain are exploited to transmit info
Examples:
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( )2i i N iJ
with
2J1J 1NJ A B
1 2 NN-1
1 11
12
Nx x y y
i i i i ii
JH
[Christandl, ND, Ekert, Landahl]
Perfect transfer of state through a quantum spin chain
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Alice Bob
Quantum channelinformation
noisy!distorted
information
( )n n
( )n n
Let : successive uses of a quantum channel
no correlation in the noise affecting states transmitted through successive usesof the channel:
memoryless if:
the max. amount of info that can bereliably transmitted per use of the channel
Transmission of information through a quantum channel
( ) 0ne n
p
Optimal rate/capacity :
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Alice Bobinfo
Input states:
-- product states
-- entangled statesor
Measurements:
individual
collective
Additional resource:
e.g. shared entanglement
( )...1 2
nn
n
( )nuses of a
noisy channel
Information: --- classical or quantum
E D
encoding
input output
decoding
( )n ( )n
These capacities evaluated in : asymptotic, memoryless setting
Parent quantity = quantum relative entropy
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In real-world applications “asymptotic memoryless setting”
In practice: info. sources & channels are used a finite
number of times;
there are unavoidable correlations between successive
uses (memory effects)
e.g. “Spin chain model for correlated quantum channels”
Rossini et al, New J.Phys. 2008
not necessarily valid
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Hence it is important to evaluate optimal rates for
finite number of uses (or even a single use)
of an arbitrary source, channel or entanglement resource
Corresponding optimal rates:
optimal one-shot rates
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(Q) How can memory effects (effects of correlated noise) arise in a single use (of a source or channel) ?
( )m
Hence, one-shot capacity encompasses the capacity of a channel for a finite number of its uses!
scenario of practical interest!
uses of a channel m
(A) e.g. for a channel: We could have :
with memory(finite)
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min max( || ), ( || )S S Min- & Max relative entropies:
act as parent quantities for one-shot rates of protocols
acts as a parent quantity for asymptotic rates of protocols
Quantum relative entropy: ( || )S
just as
e.g. Quantum Data Compression
asymptotic rate:
one-shot rate:
( )S ( || )S I
max ( )H min ( || )S I [Koenig & Renner]
max ( )H
more precisely
, H memoryless source
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(max
)max( ) : min ( )
BH H
1: : || |) |(B
max ( )H
0 < 1 Optimal rate of one-shot data compressionfor a maximum probability of error
smoothed max- entropy
max ( ) ?H
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[Wang & Renner] : one-shot classical capacityof a quantum channel
Further Examples min ( || )S
Parent quantity for the following:
[ND & Buscemi] : one-shot entanglement cost under LOCC
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[Buscemi & ND] : one-shot quantum capacity of a quantum channel
max ( || )S
Parent quantity for the following:
[Buscemi & ND] : one-shot entanglement distillation
[ND & Hsieh] : one-shot entanglement-assisted classical capacity of a quantum channel
etc.
(today!)
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One-shot results yield the known results of the
asymptotic, memoryless case, on taking:
n and then 0
In fact, one-shot results can be looked upon as the
fundamental building blocks of Quantum Info. Theory
One-shot results also take into account effects of correlation (or memory) in sources, channels etc.
Why are one-shot results important?
Hence the one-shot analysis is more general !
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Entanglement monotones
Min- & Max relative
entropies
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= a measure of how entangled a state is ;
( )E
SD
i.e., the amount of entanglement in the state
( )E
Entanglement monotones
:“minimum distance” of from the set of separable states.S
AB Let
set of all states
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One of the most important and fundamental entanglement measures for a bipartite state
AB
( ) : min ( || )RE S
SQuantum Relative Entropy
Relative Entropy of Entanglement
DS
( )RE “distance”
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max max( ) : min ( || )E S
S
( ) min ( || )RE S
S
Entanglement Monotones
relative entropy of entanglement
We can define two quantities:
Max-relative entropy of entanglement
min min( ) : min ( || )E S
SMin-relative entropy of
entanglement
these can be proved to be entanglement monotones!
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Properties of
*( ) 0E if is separable
LOCC* *( ( )) ( )E E
is not changed by a local change of basis*( )E
monotonicity
max min( ) , ( )E E
* max, min
(local operations & classical communication)
etc.
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min max( ) ( ) ( )RE E E
min max ( || ) ( || ) ( || )S S S
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What are the operational significances of
min max( ) & ( )?E E
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have interesting operational significances in
entanglement manipulation
What is entanglement manipulation ?
max ( )E min ( )E and
= Transformation of entanglement from one form to
another by local operations & classical communication
(LOCC) :
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Entanglement Distillation
nm
nAB
: LOCC
the maximum number of Bell states that
can be extracted from the state AB
Alice Bob
nlimsupn
mn
, nmnABF 1
n
= “distillable entanglement”
Bell states
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Entanglement Dilution
nm
nAB
: LOCC
the minimum number of Bell states needed to create the state
Alice Bob
n
nliminf
mn
AB= “entanglement cost”
Bell states : resource for creating a desired target state
Bell states
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have interesting operational significances in
entanglement manipulation
max ( )E min ( )E and
(separability-preserving maps)
n “one-shot” scenario
LOCC SEPP maps
( 1)nwhen
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Separability Preserving (SEPP) Maps
SEPP ( )AB
The largest class of CPTP maps which when acting
on a separable state yields a separable state
A SEPP map cannot create or increase entanglement
like a LOCC map !
If separable then separableAB
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Separability Preserving (SEPP)
Every LOCC operation is separability preserving
BUT the converse is not true
E.g. Consider the map :
SWAP is not a local operation
swap A B B AAB i i i i i i
i ip p
swapAB
LOCCSEPP
SWAP operation
separable state
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One-Shot Entanglement Distillation
m
AB
: SEPP
i.e., what is the maximum value of ?
“one-shot distillable entanglement of ”
What is the maximum number of Bell states that can be extracted from a single copy of using SEPP maps?AB
m
AB
Alice Bob
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AB“one-shot distillable entanglement of ”
min ABE
Result :
Min-relative entropy of entanglement
ND &F.Brandao
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One-Shot Entanglement Distillation
( ( ), )ABmF
AB
: SEPP
Then the maximum value of
0
:m
Alice Bobm
for some given1
min ( )ABE One-shot distillable entanglement
error
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min min( )
( ) : max ( )AB
AB ABB
E E
1: : || |) |(B
max max( ): min ( )( )
BE E
where
smoothed min-relative entropy of entanglement
similarly
smoothed max-relative entropy of entanglement
(operational significance in entanglement dilution
under SEPP maps)
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Summary Introduced 2 new relative entropies
(1) Min-relative entropy & (2) Max-relative entropy
min ( || )S max ( || )S ( || )S
Parent quantities for optimal one-shot rates for
(i) data compression for a quantum info source
(ii) transmission of (a) classical info & (b) quantum info
through a quantum channel
(iii) entanglement manipulation
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Entanglement monotones
min ABE
max :ABE
Min-relative entropy of entanglement
Max-relative entropy of entanglement
Operational interpretations:
min :ABE
max ABE
One-shot distillable entanglement of
One-shot entanglement cost of
AB
ABunder SEPP
under SEPP
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Family Tree of Quantum Protocols
FQSW
mother father
entanglement-assisted
classical communication
quantum communicationnoisy superdense
coding
noisy teleportation
entanglementdistillation
asymptotic, memorylessscenario S( || )
parent
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Apex of the Family Tree of Quantum Protocols
FQSW
mother father
entanglement-assisted
classical communication
quantum communicationnoisy superdense
coding
noisy teleportation
entanglementdistillation
One-shot FQSW
maxS ( || ) parent
ND & Hsieh
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With thanks to my collaborators:
Fernando Brandao (Rio, Brazil)
Francesco Buscemi (Nagoya, Japan)
Min-Hsiu Hsieh (Cambridge, UK)
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References: ND & M.H.Hsieh (2011): Apex of the family tree of quantum protocols:
resource inequalities and optimal rates
ND & M.H.Hsieh (2011): One-shot entanglement assisted classical communication
ND: Min- and Max- Relative Entropies; IEEE Trans. Inf. Thy,vol. 55, p.2807, 2009.
ND: Max-Relative Entropy alias Log Robustness; Intl. J.Info. Thy, vol, 7, p.475, 2009.
F. Brandao & ND: One-shot entanglement manipulation under non-entangling maps ( IEEE Trans. Inf. Thy, 2010)
F. Buscemi & ND: Entanglement cost in practical scenarios (PRL, 2010)