Entanglement sampling and applications Omar Fawzi (ETH Zürich) Joint work with Frédéric Dupuis (Aarhus University) and Stephanie Wehner (CQT, Singapore)

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?

?