The Operational Meaning of

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29 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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The Operational Meaning of

Min
-

and Max
-
Entropy


Christian

Schaffner



CWI Amsterdam, NL


joint work with

Robert
König



Caltech, USA

Renato

Renner


ETH Zürich, Switzerland

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http://arxiv.org/abs/0807.1338

2

/19

Agenda


von Neumann Entropy


Min
-

and Max
-
Entropies


Operational Meaning


Conclusion


3

/19


quantum
setting:

finite
-
dimensional Hilbert spaces



classical
-
quantum
setting:





classical
setting:





Notation

4

/19

von Neumann Entropy


simple definition


“handy” calculus


operational:


useful in many
asymptotic

iid

settings:


data compression rate


channel capacities


randomness extraction rate


secret
-
key rate


….



one
-
shot
setting?










5

/19

Conditional Min
-

and Max
-
Entropy


conditional von Neumann entropy:





conditional min
-
entropy:






conditional max
-
entropy:




for pure





[Renner 05]



for pure





Goal of this talk:

Understanding these quantities!

operator inequality:



6

/19








for


product
state:








measure for the rank of
½
A


Warm
-
Up Calculations


for a
product
state




classically:




7

/19


“smooth” variants
can be defined


handy calculus (as for von Neumann entropy)


operational interpretation
in many
one
-
shot
scenarios:


Data Compression


Privacy Amplification

(with applications in cryptography)


Decoupling


State Merging




Smooth Min
-
/Max
-
Entropies

8

/19

Agenda


von Neumann Entropy


Min
-

and Max
-
Entropies


Operational Meaning


Conclusion


9

/19

Conditional Min
-

and Max
-
Entropy


conditional van Neumann entropy:





conditional min
-
entropy:






conditional max
-
entropy:




for pure





[Renner 05]



for pure





Goal of this talk:

Understanding these quantities!

10

/19

The Operational Meaning of Min
-
Entropy


for classical states:
guessing probability





for
cq
-
states:
guessing probability






for a
POVM {
M
x
}

11

/19

The Operational Meaning of Min
-
Entropy


for
cq
-
states:
guessing probability




for
qq
-
states:
achievable quantum correlation







F( , )
2

12

/19

Proof: Operational
Interpr

of Min
-
Entropy

Proof uses:


duality

of semi
-
definite programming


Choi
-
Jamiolkowski

isomorphism





for
qq
-
states:
achievable quantum correlation

F( , , )
2

13

/19

F( , )
2

The Operational Meaning of Max
-
Entropy


for
cq
-
states:
security of a key





for

½
X
B
14

/19

F( , )
2

The Operational Meaning of Max
-
Entropy


for
cq
-
states:
security of a key






for
qq
-
states:
decoupling accuracy




for

15

/19

Proof: Operational
Interpr

of Max
-
Entropy

for

follows using





monotonicity

of fidelity


unitary relation of purifications





F( , )
2

16

/19


connections

between operational quantities,

e.g. randomness extraction




additivity

of min
-
/max
-
entropies:








·

follows from definition




Implications of our Results

17

/19


subadditivity

of min
-
entropy:











Implications of our Results


implies
subadditivity

of von Neumann entropy



concrete applications
in the
noisy
-
quantum
-
storage model












18

/19

Summary

19

/19

Summary

½
X
B