Optimal frequency measurements with maximally correlated states

Abstract

We show how maximally correlated states of N two-level particles can be used in spectroscopy to yield a frequency uncertainty equal to (NT) Ϫ1 , where T is the time of a single measurement. From the time-energy uncertainty relation we show that this is the best precision possible. We rephrase these results in the language of particle interferometry and obtain a state and detection operator which can be used to achieve a phase uncertainty exactly equal to the 1/N Heisenberg limit, where N is the number of particles used in the measurement. ͓S1050-2947͑96͒50712-2͔

Quantum Computation, Spectroscopy of Trapped Ions, and Schrödinger's Cat

“…An alternative method was suggested in Ref. [14] and in Refs. [5] and [15] methods to generate ψ max with a fixed number of steps (independent of L) are discussed.…”

“…[13] has the advantage that the appropriate states can be generated by acting on all the ions at once (thus not requiring focused laser beams), but has the disadvantage that these states are entangled with the motion, thereby requiring small motional decoherence. Reference [14] investigated the use of the generalized GHZ state, sometimes called the maximally entangled state, in spectroscopy. This state has the form…”

Frequentist and Bayesian Quantum Phase Estimation

“…Our results are illustrated with a simple test model [37,38]. We consider N qubits with basis states |0 and |1 , initially prepared in a (generalized) GHZ state |GHZ = (|0 ⊗N + |1 ⊗N )/ √ 2, with all particles being either in |1 or in |0 .…”

“…The phase-encoding is a rotation of each qubit in the Bloch sphere |0 → e −iθ/2 |0 and |1 → e +iθ/2 |1 , which transforms the |GHZ state into |GHZ(θ) = (e −iNθ/2 |0 ⊗N + e +iNθ/2 |1 ⊗N )/ √ 2. The phase is estimated by measuring the parity (−1) N 0 , where N 0 is the number of particles in the state |0 [37,[39][40][41]. The parity measurement has two possible results µ = ±1 that are conditioned by the "true value of the phase shift" θ 0 with probability p(±1|θ 0 ) = (1 ± cos (Nθ 0 ))/2.…”

Noisy metrology: a saturable lower bound on quantum Fisher information

“…It plays an important role at a very fundamental level, involving the measurement of fundamental constants of Nature like the Planck constant, the speed of light in vacuum and the gravitational constant. Furthermore, it has widespread practical implications ranging from determination of atomic transition frequency [1][2][3] to a phase shift in an interferometric measurement due to the presence of gravitational waves [4][5][6].…”