1. Sampling of min-entropy relative to quantum knowledge Robert König in collaboration with Renato Renner
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2. random access codes ? ? decoding probability for (random) subset time
Ambainis, Nayak, Ta-Shma, Vazirani 99/Nayak 99
Ben-Aroya, Regev, de Wolf 07:
measurement
adaptive
decoding probability
for
(random) subset ?
time
m-qubit state
storage ?
n coin tosses

3. Sampling min-entropy (pseudo) random subset time
vs random access codes
Claim: preservation of entropy-rate
vs decoding probability
(pseudo)
random subset
time
arbitrary quantum state
given : bound on entropy
vs bounded number of qubits
correlation
random variables,
large alphabet
vs coin flips

4. Min-entropy and secret keys
for classical-quantum states
equal to
extractable key length
(also equal to guessing entropy:
[K,Schaffner,Renner08])
Privacy amplification generates approximately [BBR88,BBCM95,Renner05]
bits of secure key from partially secret raw key X, against adversary holding Q (optimal)

5. temporarily available
Key expansion in the bounded storage model Sample-then-extract [Maurer92,….,Vadhan03]
previously only analysed
for classical adversary
known to work against quantum adversary
Source of randomness
temporarily available
resources
insecure channel
bits of key
bits of key
substring (sampled with S)
substring (sampled with S)
Privacy amplification
Quantum
storage
qubits

6. Implication for the bounded storage model
Sample-then-extract-approach for building locally computable extractors (Vadhan03) works against quantum adversaries!
Ingredients:
large source of randomness ( bits)
short initial shared key ( bits)
aim: generate bits of secure key
Validity against quantum adversaries cannot be established using classical extractor properties only [Gavinsky, Kempe, Kerenidis, Raz & de Wolf’06]
key
Sample: choose random subset
Extract:
``standard’’
hashing
Claim:
seed for “sampler”
seed for extractor

7. Main result: Sampling of min-entropy
Sample: choose random subset
(classical)
(randomly chosen)
subset of
rephrased: if
then
large alphabet size c needed! “blockwise sampling”
for any
e.g., for
Main result:
for any state
where

8. Why sampling (Shannon) entropy works
There is a simple proof for sampling of Shannon entropy.
Only uses
Subadditivity
Chain-rule
repeated application of chain-rule splits joint
entropy into sum of contributions
random subset hits “good” parts with
high probability
large
small

9. Why sampling (Shannon) entropy works
There is a simple proof for sampling of Shannon entropy.
Only uses
Subadditivity
Chain-rule
repeated application of chain-rule splits joint
entropy into sum of contributions
random subset hits “good” parts with
high probability
large
small
subadditivity helps to remove dependence
on variables not in subset
chain-rule shows that is large

10. (Min)-entropy(-)rules
subadditivity:
chain-rule (recombination):
Not true in general!
chain-rule (splitting):
Renato’s talk:
recursive application of
this rule impossible
Need three rules for entropy-sampling argument to work. Two of these hold trivially. The third rule has to be replaced for min-entropy.

11. Entropy-splitting and recombining
small
large
recombining
splitting
large entropy:
distance to original state:
Is a probability distribution
General strategy for showing
lower bound on smooth min-entropy
Separate splitting and smooth entropy
1. construct orthogonal decomposition
2. choose high-entropy subset
3. show that is large
Additional properties if split states constructed
using eigendecomposition of conditional operator

12. (Min)-entropy(-)rules
subadditivity:
chain-rule (recombination):
Not true in general!
chain-rule (splitting):
original
state
split states
(Approximate) chain-rule for appropriately chosen (discrete) splitting!

13. Recursive splitting and recombining
small
large
for a given subset choose high-entropy components

14. Conclusions/Application to BSM
sampling preserves smooth min-entropy rate - application
to the BSM: sample-then-hash approach achieves significant key expansion
to general key extraction/qkd schemes: building block (aka “condenser”) for constructing randomness-efficient quantum extractors
memory
bits of shared key
bits of shared key
against an adversary with qubits

15. THE END
Thank you for your attention!