Generalized Limits for Single-Parameter Quantum Estimation

Exceptions to the limitation of m include the Dicke model, the Lipkin-Meshkov-Glick model, and the Mølmer-Sørensen interaction.

“Exceptions to this include the Dicke model [40] where collective coherence leads to superradiance, the Lipkin-Meshkov-Glick model [41], where all particles interact with each other, and the Mølmer-Sørensen interaction [42], in which an ensemble of ions are effectively coupled by a spatially uniform electromagnetic field.”

For N-qubit dephasing, the bound on minimum evolution time is given by Eq. (18).

“where α ( t ), β ( t ), and δ ( t ) are optimization variables. We get then [47]”

Eq. (14) captures the fact that the evolved and initial states never become orthogonal for r under a critical value rcrit ≃ 2.6.

“Another feature of (14) is that it captures the fact that the evolved and initial states never become orthogonal for r under a critical value r crit ≃ 2 . 6 [47].”

For Markovian dephasing, the minimum of CQ(t) over all Hermitian 2x2 operators hE(t), written in terms of the constant variance of Z, is given by Eq. (13).

“In order to find the best possible bound on τ within our approach, we now minimize C Q ( t ) over the whole set of Hermitian 2 × 2 operators ˆ h E ( t ). The minimum, written in terms of the (constant) variance of ˆ Z , is [47]”

This transition to faster behavior can be corroborated via direct calculations on symmetric, separable states.

“This transition to faster behavior can be corroborated via direct calculations on symmetric, separable states [47].”

For q=1, the bound on the minimum evolution time scales as τ ∼ 1/N throughout, a prediction validated by exact calculations with GHZ states.

“For q = 1, the bound on the minimum evolution time scales as τ ∼ 1 /N throughout, a prediction validated by exact calculations with GHZ states, see Supplemental Material [47].”

The operator K_a := i(∂aU)U† = -iU∂aU† is an alternative operator to characterize the precision limit of unitary process.

“An alternative operator that can be used to characterize the precision limit of unitary process is [50,53,54]”