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Communications in Theoretical Physics

Optimal probe states for phase estimation with a fixed mean particle number

Jin-Feng Qin, Bo Liu

Published Jan 21, 2025• 0 citations
Publisher

Abstract

Abstract Quantum phase estimation reveals the power of quantum resources to beat the standard quantum limit and has been widely used in many fields. To improve the precision of phase estimation, we discuss the optimal probe states for phase estimation with a fixed mean particle number. By searching for the maximum quantum Fisher information, we optimize the probe states, which are superior to the path-entangled Fock states. Comparing the mean particle number ( n ¯ ) with the dimension of the probe states in Fock space (N + 1), when n ¯ ≤ N , our optimal probe states can provide a better performance than the n ¯ 00 n ¯ states. When n ¯ > N , our optimal probe states can also remain optimal if the dimension of the probe states is large enough.

DOI
10.1088/1572-9494/adac3c

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Abstract Quantum phase estimation reveals the power of quantum resources to beat the standard quantum limit and has been widely used in many fields. To improve the precision of phase estimation, we discuss the optimal probe states for phase estimation with a fixed mean particle number. By searching for the maximum quantum Fisher information, we optimize the probe states, which are superior to the path-entangled Fock states. Comparing the mean particle number ( n ¯ ) with the dimension of the probe states in Fock space (N + 1), when n ¯ ≤ N , our optimal probe states can provide a better performance than the n ¯ 00 n ¯ states. When n ¯ > N , our optimal probe states can also remain optimal if the dimension of the probe states is large enough.
Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the NOON state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.
We address the problem of continuous-variable quantum phase estimation in the presence of linear disturbance at the Hamiltonian level by means of Gaussian probe states. In particular we discuss both unitary and random disturbance by considering the parameter which characterizes the unwanted linear term present in the Hamiltonian as fixed (unitary disturbance) or random with a given probability distribution (random disturbance). We derive the optimal input Gaussian states at fixed energy, maximizing the quantum Fisher information over the squeezing angle and the squeezing energy fraction, and we discuss the scaling of the quantum Fisher information in terms of the output number of photons, nout{n}_{\mathsf{out}}nout​. We observe that, in the case of unitary disturbance, the optimal state is a squeezed vacuum state and the quadratic scaling is conserved. As regards the random disturbance, we observe that the optimal squeezing fraction may not be equal to one and, for any nonzero value of the noise parameter, the quantum Fisher information scales linearly with the average number of photons. Finally, we discuss the performance of homodyne measurement by comparing the achievable precision with the ultimate limit imposed by the quantum Cram\'er--Rao bound.
Abstract Phase diffusion invariably accompanies all phase estimation strategies, quantum or classical. A precise estimation of the former can often provide valuable understanding of the physics of the phase generating phenomena itself. We theoretically examine the performance of fixed-particle number probe states in the simultaneous estimation of phase and collective phase diffusion. We derive analytical quantum limits associated with the simultaneous local estimation of phase and phase diffusion within the quantum Cramér-Rao bound framework in the regimes of large and small phase diffusive noise. The former is for a general fixed-particle number state and the latter for Holland-Burnett states, for which we show quantum-enhanced estimation of phase as well as phase diffusion. We next investigate the simultaneous attainability of these quantum limits using projective measurements acting on a single copy of the state in terms of a trade-off relation. In particular, we are interested how this trade-off varies as a function of the dimension of the state. We derive an analytical bound for this trade-off in the large phase diffusion regime for a particular form of the measurement, and show that the maximum of 2 , set by the quantum Cramér-Rao bound, is attainable. Further, we show numerical evidence that as diffusion approaches zero, the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime and suggest that the trade-off for estimating one parameter with quantum-limited precision leads to a complete lack of precision for the other parameter as the diffusion strength approaches zero. Finally, we provide numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett probe states.
Using the quantum Fisher information (QFI), we present an improved phase estimation scheme employing power-law potential (PLP) systems in comparison to NOON states under perfect and lossy conditions for a fixed mean photon number. The study is devoted to the phase estimation of the quantum states resulting from tightly binding, harmonic and loosely binding potentials for modest particle numbers. Finally, we discuss the saturation of the optimal phase uncertainty of this interferometric setting with parity detection.
Phase diffusion invariably accompanies all phase estimation strategies – quantum or classical. A precise esti- mation of the former can often provide valuable understanding of the physics of the phase generating phenom- ena itself. We theoretically examine the performance of fixed-particle number probe states in the simultaneous estimation of phase and collective phase diffusion. We derive analytical quantum limits associated with the si- multaneous local estimation of phase and phase diffusion within the quantum Crame ́r-Rao bound framework in the regimes of large and small phase diffusive noise. The former is for a general fixed-particle number state and the latter for Holland Burnett states, for which we show quantum-enhanced estimation of phase as well as phase diffusion. We next investigate the simultaneous attainability of these quantum limits using projective measure- ments acting on a single copy of the state in terms of a trade-off relation. In particular, we are interested how this trade-off varies as a function of the dimension of the state. We derive an analytical bound for this trade-off in the large phase diffusion regime for a particular form of the measurement, and show that the maximum of 2, set by the quantum Crame ́r-Rao bound, is attainable. Further, we show numerical evidence that as diffusion approaches zero, the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime and suggest that the trade-off for estimating one parameter with quantum-limited precision leads to a complete lack of precision for the other parameter as the diffusion strength approaches zero. Finally, we provide numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett probe states.
The quantum Fisher information of a quantum state with respect to a certain parameter quantifies the sensitivity of the quantum state to changes in that parameter. Maximizing the quantum Fisher information is essential for achieving the optimal estimation precision of quantum sensors. A typical quantum sensor involves a qubit(e.g. a spin-1/2) probe undergoing an unknown rotation, here the unknown rotation angle is the parameter to be estimated. A well known limitation is that if the rotation axis is unknown, the maximal quantum Fisher information is impossible to attain. This limitation has been lifted recently by leveraging entanglement between the probe qubit and an ancilla qubit. Namely, through measurement of the ancilla after the axis is revealed, one can prepare the probe that is optimal for any unknown rotation axis. This proposal, however, works only for a spin-1/2. Considering large spin probes can achieve a larger quantum Fisher information, offering enhanced metrological advantage, we here utilize the entanglement between a large spin probe and an ancilla to achieve optimal quantum Fisher information for estimating the rotation angle, without prior knowledge of the rotation axis. Different from the previous spin-1/2 case, achieving the optimal precision with large spins generally requires post-selection, resulting in a success probability dependent on the dimension of the Hilbert space. Furthermore, we extend the encoding state from the maximally entangled case to general entangled states, showing that optimal metrology can still be achieved with a certain success probability.