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The Operational Meaning of Min- and Max-Entropy
Christian Schaffner – CWI Amsterdam, NL
joint work with Robert König – Caltech, USARenato Renner – ETH Zürich, Switzerland
http://arxiv.org/abs/0807.1338
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Agenda
•von Neumann Entropy
•Min- and Max-Entropies
•Operational Meaning
•Conclusion
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• quantum setting:finite-dimensional Hilbert spaces
• classical-quantum setting:
• classical setting:
Notation
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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?
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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:
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• for
product state:
• measure for the rank of ½A
Warm-Up Calculations• for a product state
• classically:
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• “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
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Agenda
von Neumann Entropy
Min- and Max-Entropies
•Operational Meaning
•Conclusion
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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!
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The Operational Meaning of Min-Entropy
• for classical states: guessing probability
• for cq-states: guessing probability
for a POVM {Mx}
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The Operational Meaning of Min-Entropy
• for cq-states: guessing probability
• for qq-states: achievable quantum correlation
F( , )2
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Proof: Operational Interpr of Min-Entropy
Proof uses:
• duality of semi-definite programming
• Choi-Jamiolkowski isomorphism
• for qq-states: achievable quantum correlation
F( , , )2
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F( , )2
The Operational Meaning of Max-Entropy
• for cq-states: security of a key
for
½X B
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The Operational Meaning of Max-Entropy
• for cq-states: security of a key
• for qq-states: decoupling accuracy
for
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Proof: Operational Interpr of Max-Entropyfor
follows using
•
• monotonicity of fidelity
• unitary relation of purifications
F( , )2
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• connections between operational quantities, e.g. randomness extraction
• additivity of min-/max-entropies:
· follows from definition
Implications of our Results
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• subadditivity of min-entropy:
Implications of our Results
• implies subadditivity of von Neumann entropy
• concrete applications in the noisy-quantum-storage model
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Summary
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Summary
½X B