An -ion GHZ state is held in a trap where each ion feels two kinds of phase noise at…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Physical realizations and quantum technologies

Quantum Information and Technology, Ch Noise as a quantum operation

Problem

N

-ion GHZ state

(|0\cdots0\rangle+|1\cdots1\rangle)/\sqrt2

is held in a trap where each ion feels two kinds of phase noise at once: an independent random phase of variance

\sigma_i^2=0.01

on every ion, and a single common phase, shared by all ions because they sit in the same stray field, of variance

\sigma_c^2=0.001

. Treat the two noise sources as statistically independent, so the coherence between the two branches of the GHZ state is the product of the independent-noise factor and the common-noise factor derived in the text. ((a)) Write the coherence factor

C(N)

as a function of

N

\sigma_i^2

and

\sigma_c^2

. ((b)) Find the crossover number of ions

N^*

at which the independent-noise contribution to the exponent of

C(N)

equals the common-noise contribution, in terms of

\sigma_i^2

and

\sigma_c^2

, and evaluate it for the stated values. ((c)) Evaluate

C(N)

N=N^*

. ((d)) State, in one sentence, which noise source dominates the loss of coherence for

N\gg N^*

ions, and which for

N\ll N^*

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