The Operational Meaning of

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29 Νοε 2013 (πριν από 4 χρόνια και 7 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

of min
-
/max
-
entropies:

·

follows from definition

Implications of our Results

17

/19

of min
-
entropy:

Implications of our Results

implies

of von Neumann entropy

concrete applications
in the
noisy
-
quantum
-
storage model

18

/19

Summary

19

/19

Summary

½
X
B