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Entanglement sampling and applications. Process. Omar Fawzi (ETH Zürich ) Joint work with Frédéric Dupuis (Aarhus University) and Stephanie Wehner (CQT, Singapore ) arXiv:1305.1316. Uncertainty relation game. Eve. Alice. Choose n- qubit state. Choose random. EVE. ….

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entanglement sampling and applications

Entanglement sampling and applications

Process

Omar Fawzi (ETH Zürich)

Joint work with FrédéricDupuis (Aarhus University)

and Stephanie Wehner (CQT, Singapore)

arXiv:1305.1316

uncertainty relation game
Uncertainty relation game

Eve

Alice

Choose n-qubit state

Choose random

EVE

Choose n-qubit state

X1

Xn

X2

Xn-1

Choose random

Guess X

Maximum ?

X1

Xn

X2

Xn-1

Guess X

Maximum Pguess?

uncertainty relation game1
Uncertainty relation game
  • Can Eve do better with different ?
  • No [Damgard, Fehr, Salvail, Shaffner, Renner, 2008]

Measure in

X

Guess X

Between 0and n

Notation:

uncertainty relations with quantum eve
Uncertainty relations with quantum Eve

Eve has a quantum memory

A

Measure in

E

X

Guess X

using E and

Maximum ?

[Berta, Christandl, Colbeck, Renes, Renner, 2010]

uncertainty relations with quantum eve1
Uncertainty relations with quantum Eve

Measure in

A

X

E

Measure in

X

uncertainty relations with quantum eve2
Uncertainty relations with quantum Eve

E.g., if storage of Eve is bounded?

Uncertainty relation + chain rule 

using maximal entanglement

Converse

Is maximal entanglement necessary for large Pguess?

Main result: YES

At least n/2 qubits of memory necessary

the uncertainty relation
The uncertainty relation

E=X

between –n and n

Max entangled

Max entangled

  • Measure for closeness to maximal entanglement
  • Log of guessing prob.

between 0 and n

the uncertainty relation1
The uncertainty relation

Max entanglement

general statement
General statement

X

C

More generally:

Gives bounds on Q Rand Access Codes

Meas in Θ

M

A

A

Example:

E

E

application to two party cryptography
Applicationtotwo-party cryptography

??

??

“I’m Alice!”

password

Stored password

Equal?

Malicious ATM: tries to learn passwords

Yes/No

Malicious user: tries to learn other customers passwords

application to secure two party computation
Application to secure two-party computation
  • Unconditional security impossible

[Mayers 1996; Lo, Chau, 1996]

  • Physical assumption:

bounded/noisy quantumstorage

[Damgard, Fehr, Salvail, Schaffner 2005; Wehner, Schaffner, Terhal 2008]

    • Security if

Using new uncertainty relation

    • Security if

n: number of communicated qubits

proof of uncertainty relation
Proof of uncertainty relation

Step 1:

X

Conditional state

Meas in Θ

A

E

proof of uncertainty relation1
Proof of uncertainty relation

Step 2: Write by expanding in Pauli basis

proof of uncertainty relation2
Proof of uncertainty relation

Relate

and

Observation 1:

Not good enough

proof of uncertainty relation3
Proof of uncertainty relation

Relate

and

Observation 1:

Observation 2:

Combine 1 and 2  done!

conclusion
Conclusion
  • Summary
    • Uncertainty relation with quantum adversary

for BB84 measurements

    • Generic tool to lower bound output entropy using input entropy
  • Open question
    • Combine with other methods to improve?

?