A referee for the CHSH game sometimes swaps in better hardware: with probability the…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Nonlocality: EPR, Bell inequalities and the CHSH game

Problem

A referee for the CHSH game sometimes swaps in better hardware: with probability

q

the players share a Popescu-Rohrlich box,

P(a,b|x,y)=\tfrac12

whenever

a\oplus b=xy

and

0

otherwise; with probability

1-q

they fall back on the simplest classical strategy, always answering

a=b=0

. A single shared random bit, generated before the round and independent of

x

and

y

, decides which of the two the players use in that round. ((a)) Show that this mixed strategy cannot be used to signal: Bob's marginal outcome probability, averaged over the shared randomness, does not depend on Alice's input

x

. ((b)) Find the winning probability

p_{\rm win}(q)

of the mixed strategy. ((c)) Find the value of

q

at which

p_{\rm win}(q)

equals the quantum optimum

\cos^2(\pi/8)

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