Robust Randomness Expansion

Upper and Lower Bounds

Matthew Coudron, Thomas Vidick, Henry Yuen

arXiv:1305.6626

The motivating question

Is it possible to test randomness?

The motivating question

Is it possible to test randomness?

The motivating question

Is it possible to test randomness?

1000101001111…..

The motivating question

Is it possible to test randomness?

1111111111111…..

The motivating question

Is it possible to test randomness?

1111111111111…..

No, not possible!

No-signaling offers a way…

No-signaling offers a way…

No-signaling constraint makes testing randomness possible!

CHSH gamex ϵ {0,1}

y ϵ {0,1}

a ϵ {0,1}

b ϵ {0,1}

CHSH condition: a+b = x Λ y

Classical win probability: 75%

Quantum win probability: ~85%

CHSH gamex ϵ {0,1}

y ϵ {0,1}

a ϵ {0,1}

b ϵ {0,1}

CHSH condition: a+b = x Λ y

Classical win probability: 75%

Quantum win probability: ~85%

Idea [EPR, Bell]: if the devices win the CHSH game

with > 75% success probability, then their outputs

must be randomized!

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

1 0

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

1 0

0 0

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

10 01

0 0

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

10 01

01 00

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

100 011

01 00

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

100 011

011 001

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

1001 0111

011 001

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

1001 0111

0110 0010

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

10010101010101010

0111010110101010

01101010101111000

0010111110101011

Won ~85% of rounds?

Certifying randomness via CHSHDevices play n rounds of the CHSH game [Colbeck].

10010101010101010

0111010110101010

01101010101111000

0010111110101011

Outputs have (W n) bits of certified min-entropy!

Certifying randomness via CHSH

10010101010101010

0111010110101010

01101010101111000

0010111110101011

Outputs have (W n) bits of certified min-entropy!

Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it!

1000101001

Short random seed

Long pseudorandom input

Certifying randomness via CHSH

10010101010101010

0111010110101010

01101010101111000

0010111110101011

Outputs have (W n) bits of certified min-entropy!

Protocols of [Colbeck ‘10][PAM+ ‘10][VV ’12][FGS13] not only certify randomness, but also expand it!

1000101001

Short random seed

Long pseudorandom input

State-of-the-art: Vazirani-Vidick protocol uses m bits of seed and produces 2O(m) certified

random bits! [VV12]

How do we measure randomness?

We use min-entropy. For a random variable X,

Hmin (X) := min log 1/Pr(X = x)

Why min-entropy? It characterizes the amount of uniformly random bits that one can extract from a random source X!

x

What are the possibilities? Limits?

• Doubly exponential expansion?

• …infinite expansion?

• Noise robustness?

Our results

• First upper bounds for non-adaptive randomness expansion

• Constructions of noise-robust protocols

The modelRandomness amplifier is an interactive protocol between a classical referee and 2 non-signaling devices.

• Randomness efficiency• Referee uses m random bits to sample inputs to devices

• Completeness• There exists an ideal strategy that passes the protocol

with probability > c

• Soundness• For all strategies S, if the devices using S, pass with

probability > s, then Hmin( device outputs ) > g(m)

c – completeness s – soundness g(m) - expansion

The modelRandomness amplifier is an interactive protocol between a classical referee and 2 non-signaling devices.

• Randomness efficiency• Referee uses m random bits to sample inputs to devices

• Completeness• There exists an ideal strategy that passes the protocol

with probability > c

• Soundness• For all strategies S, if the devices using S, pass with

probability > s, then Hmin( device outputs ) > g(m)

• Non-adaptive• Inputs to devices don’t depend on their outputs

c – completeness s – soundness g(m) - expansion

Upper bounds*

1. Noise-robust randomness amplifiers- g(m) < exp(exp(m))

2. Randomness amplifiers using XOR games and devices have non-signaling power

- g(m) < exp(m)

*IMpossibility results

XOR game: game win condition depends only on parity of players’ answers.

non-signaling strategies: strictly more powerful than quantum strategies.

How to prove upper bounds?

Exhibit a cheating strategy for the devices,

i.e. a strategy Scheat where

Pr ( Passing protocol with Scheat ) > sbut

Hmin ( device outputs ) < g(m)

An exp(exp(m)) upper bound

• Our main doubly-exp upper bound applies to non-adaptive, noise-robust randomness amplifiers

• A proof for a simplified setting:• Protocols based on perfect games (e.g.

Magic Square)• Referee check devices won every round

An exp(exp(m)) upper bound

Intuition: after exp(exp(m)) rounds, inputs to the devices will start repeating in predictable ways…

Independently of referee’s private randomness!

An exp(exp(m)) upper bound

Input Matrix

0000 0001 …. 1110 1111(1, 0) (0, 1) (1,0) (1,1)

(1,1) (0,1) (1,1) (1,1)

(0,0) (0,0) (0,0) (1,0)

…

(1,0) (0,1) (1,0) (1,1)

Referee’s random seed (2m columns)

Input to devices

in round i

After exp(exp(m)) rounds, rows must

start repeating

An exp(exp(m)) upper bound

Input Matrix

0000 0001 …. 1110 1111(1, 0) (0, 1) (1,0) (1,1)

(1,1) (0,1) (1,1) (1,1)

(0,0) (0,0) (0,0) (1,0)

…

(1,0) (0,1) (1,0) (1,1)

Referee’s random seed (2m columns)

Repeat answers

whenever rows

repeat!

An exp(exp(m)) upper bound

• Strategy Scheat

• Play “honestly” in round i when row i of Input Matrix is new

• If row i is a repeat of row j for some j < i, repeat answers from round j.

• Claim. Devices produce at most exp(exp(m)) bits of randomness, but pass protocol with probability 1.

Generalizing the upper bound

• What if the referee is more clever? • Checks for obvious answer repetitions• Uses a non-perfect game, like odd-cycle

game or CHSH*• Still have exp(exp(m)) upper bound!

• Requirement for noise robustness gives devices freedom to cheat!

* For quantum players

An exponential upper bound

• Cheating strategies that take advantage of the game structure

• XOR-game protocols• XOR game: f(x + y)• Devices can employ full non-signaling

strategies (i.e. super-quantum strategies)

• Referee checks devices won every round• g(m) < exp(m)

Open problems

• Better upper bounds?–More elaborate cheating strategies?– Show g(m) < exp(m) always?

• Better lower bounds?–Match the doubly exponential upper

bound?

• Adaptive protocols with infinite expansion?

Open problems

• Better upper bounds?–More elaborate cheating strategies?– Show g(m) < exp(m) always?

• Better lower bounds?–Match the doubly exponential upper

bound?

• Adaptive protocols with infinite expansion?

Thanks!

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