Entanglement sampling and applications Omar Fawzi (ETH Zürich) Joint work with Frédéric Dupuis (Aarhus University) and Stephanie Wehner (CQT, Singapore) arXiv:1305.1316 Process
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Entanglement sampling and applications
Omar Fawzi (ETH Zürich)
Joint work with Frédéric Dupuis (Aarhus University) and Stephanie Wehner (CQT, Singapore)
arXiv:1305.1316
Process
Uncertainty relation game
Choose n-qubit state
Choose random
Guess X
…
X1 X2 Xn-1 Xn
Eve Alice
Maximum ?
Uncertainty relation game
• Can Eve do better with different ?• No [Damgard, Fehr, Salvail, Shaffner, Renner, 2008]
Measure in
XGuess X
Notation:
Between 0 and n
Uncertainty relations with quantum Eve
Eve has a quantum memory
Measure in
X
A
E
Guess Xusing E and
Maximum ?
[Berta, Christandl, Colbeck, Renes, Renner, 2010]
Uncertainty relations with quantum Eve
Measure in
X
A
E
Measure in
X
Uncertainty relations with quantum Eve
E.g., if storage of Eve is bounded?Uncertainty relation + chain rule
Converse Is maximal entanglement necessary for large Pguess?
At least n/2 qubits of memory necessary
using maximal entanglement
Main result: YES
The uncertainty relation
• Measure for closeness to maximal entanglement
• Log of guessing prob. E=X
Max entangled
between –n and n
between 0 and n
Max entangled
The uncertainty relation
Max entanglement
General statementMeas in Θ
E
A X
M
E
A C
More generally:
Example:
Gives bounds on Q Rand Access Codes
Application to two-party cryptography
Equal?
password Stored password
Yes/No
“I’m Alice!”
Malicious ATM: tries to learn passwords
Malicious user: tries to learn other customers passwords
????
Application to secure two-party computation
• Unconditional security impossible [Mayers 1996; Lo, Chau, 1996]
• Physical assumption:
bounded/noisy quantum storage[Damgard, Fehr, Salvail, Schaffner 2005; Wehner, Schaffner, Terhal 2008]
o Security if
Using new uncertainty relationo Security if
n: number of communicated
qubits
Proof of uncertainty relation
Step 1:
Conditional state
Meas in Θ
E
A X
Proof of uncertainty relation
Step 2: Write by expanding in Pauli basis
Proof of uncertainty relation
Relate
and
Observation 1:
Not good enough
Proof of uncertainty relation
Relate
and
Observation 1:
Observation 2:
Combine 1 and 2 done!
Conclusion• Summary
o Uncertainty relation with quantum adversary for BB84 measurementso Generic tool to lower bound output entropy
using input entropy
• Open questiono Combine with
other methods to improve?
?
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