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The uncertainty principle in the presence of quantum memory

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Mario Berta

on 9 February 2015

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Transcript of The uncertainty principle in the presence of quantum memory

The uncertainty principle in the presence of quantum memory
An information theoretic view
Entropic uncertainty relations:
Let's add a quantum memory
Trade-off: uncertainty vs. entanglement:
Some extensions of:
Monogamy of entanglement:
Quantitative trade-offs between uncertainty and. entanglement:
The uncertainty principle in quantum mechanics
Position/Momentum relations:
03/20/14 Baton Rouge
Mario Berta, Caltech
An uncertainty game:
Renyi entropies (min- and max-entropy), general measurements (POVM's), larger sets of measurements, uncertainty equalities, etc.
Simple proofs based on monotonicity of relative entropy:
Uncertainty game with quantum Bob (quantum memory/observer):
[Heisenberg (27), Kennard (28)]
General observables:
Adversial uncertainty:
[Robertson (29)]
Measurement disturbance relations
Position/Momentum relations:
Some applications and connections
Quantum cryptography:
[Buscemi et al. (Phys. Rev. Lett. 13)]
Measurement disturbance relations:
Conjecture in [Boileau and Renes (Phys. Rev. Lett. 09)]
Witnessing entanglement:
[B. et al. (Nat. Phys. 09)]
[Deutsch (83), Kraus (87), Massen and Uffink (88)]
[Hirschmann (57), Beckner (75), Bialnicki-Birula and Mycielski (75)]
Position/Momentum relations:
(differential entropy)
Example: 1 qubit
The rules:
[B. et al. (Nat. Phys. 10)]
[Coles et al. (Phys. Rev. Lett. 12), Frank and Lieb (Comm. Math. Phys. 13)]
[Furrer et al, (Phys. Rev. Lett. 12/14), Frank and Lieb (Comm. Math. Phys. 13), B. et al. (arXiv.org 13)]
Operationally useful in quantum information theory and quantum cryptography
Connection to measurement disturbance relations
quantum key distillation
two-party crypgraphy
[B. et al. (Nat. Phys. 10), Prevedelel et al. (Nat. Phys. 10), Li et al. (Nat. Phys. 10)]
[Tomamichel and Renner (Phys. Rev. Lett 11), Tomamichel et al. (Nat. Comm. 13)]
[B. et al. (CRYPTO 12, IEEE Trans. 12), Dupuis et al. (CRYPTO 13)]
Review [Wehner and Winter (New J. of Phys. 09)]
Uncertainty game with classical Bob (observer dependent uncertainty):
Information theory meets physics and vice versa
Strenghtening of unconditional case:
Ex. qubits:
[Devetak and Winter (Proc. R. Soc. 05)]
[Hall (Phys. Rev. Lett. 95)]
[Boileau and Renes (Phys. Rev. Lett. 09)]
let to many new results in this direction: Coles and Piani (Phys. Rev. A 12), Werner et al. (Phys. Rev. Lett. 13), Coles and Furrer (arXiv.org 13), Renes and Scholz (arXiv.org 14)] etc.
(i) agree on X and Z
(ii) Bob prepares state and sends it to Alice
(iii) Alice measures X or Z
(iv) Alice tells Bob if she measured X or Z
(v) Bob tasks is to guess Alice's measurement outcomes
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