Entangled Qubits Overcome Limits to Precision Measurement of Unknown Rotations

Scientists investigate methods for precisely estimating parameters in quantum systems, a challenge central to fields like quantum metrology and sensing. Huining Zhang, X. X. Yi, and colleagues at Northeast Normal University demonstrate a new approach to parameter estimation that overcomes limitations inherent in traditional techniques. Their research focuses on optimising the Fisher information, a measure of estimation precision, for large spin systems undergoing rotations with unknown axes. This work is significant because it extends a previously successful entanglement-based method, originally limited to spin-1/2 particles, to larger spins, potentially offering enhanced metrological advantages. The team reveals that achieving optimal precision with large spins necessitates post-selection, but importantly, they also show that optimal metrology remains possible even with general entangled states and a predictable success probability.

Entanglement and ancilla qubits enable optimal parameter estimation with unknown rotation axes

Scientists have developed a new method for achieving optimal precision in estimating physical parameters, overcoming a longstanding limitation in quantum metrology. This work introduces a technique leveraging entanglement between a large spin probe and an ancilla qubit to maximize the quantum Fisher information, even when the rotation axis of the system is unknown.
Traditionally, attaining peak estimation precision required complete prior knowledge of the system’s Hamiltonian, a constraint that has now been lifted through this innovative approach. The research demonstrates that by entangling a probe with an ancilla and performing a measurement on the ancilla after the axis is revealed, the probe can be optimally prepared for any unknown rotation axis.

This breakthrough extends previous work limited to spin-1/2 particles, successfully applying the principle to larger spin probes which inherently offer enhanced metrological advantages. Achieving optimal precision with these larger spins necessitates a post-selection process, resulting in a success probability linked to the dimension of the system’s Hilbert space.

Researchers have not only validated the method using maximally entangled states but have also broadened the scope to encompass general entangled states, confirming that optimal metrology remains attainable with a corresponding success probability. The study details how this entanglement-based protocol enables precise parameter estimation without prior knowledge of the generator Hamiltonian, a crucial advancement for quantum sensors.

Unlike conventional methods, this approach circumvents the need for complete upfront knowledge, offering a significant advantage in practical applications. The team derived the success probability for achieving optimal estimation when employing a maximally entangled state, analysing its relationship to the spin dimension.

Furthermore, the investigation extends beyond maximally entangled states, demonstrating that optimal estimation is still possible with arbitrary entangled states, albeit with a success probability dependent on both the entangled state’s characteristics and the Hamiltonian. This work establishes a robust framework for designing metrological protocols that utilise entanglement to overcome limitations imposed by incomplete information, paving the way for more sensitive and versatile quantum sensors. The findings are particularly relevant for applications requiring precise measurements in fields such as magnetic field sensing, atomic clocks, and gravitational wave detection.

Optimal estimation of spin rotations via probe-ancilla entanglement and post-selection

Researchers began by focusing on the quantum Fisher information, a key metric for assessing the precision of parameter estimation. The study investigates estimating an unknown rotation angle, β, applied to a large spin probe, building upon prior work limited to spin-1/2 systems. Central to their methodology is the entanglement of the spin probe with an ancilla qubit, a technique designed to overcome limitations arising from an unknown rotation axis.

Unlike previous approaches, this work does not require prior knowledge of the rotation axis, enabling optimal estimation precision regardless of its direction. The experimental design utilizes post-selection to achieve optimal precision with large spins, acknowledging that this introduces a success probability dependent on the Hilbert space dimension.

Initially, the team employed maximally entangled states for probe-ancilla entanglement, deriving the success probability required to attain optimal estimation and analysing its relationship to spin dimension. They then extended this to general entangled states, demonstrating that optimal metrology remains achievable, albeit with a success probability influenced by both the entangled state’s form and the Hamiltonian.

To quantify the sensitivity of the system, the researchers calculated the Fisher information, adhering to the Cramér-Rao bound which establishes a lower limit on the variance of the estimated parameter. The quantum Fisher information was then determined, representing the maximum attainable Fisher information achievable through quantum measurements.

The optimal probe encoding state was identified as an equally weighted superposition of eigenvectors corresponding to the maximum and minimum eigenvalues of the Hamiltonian, ensuring maximization of the quantum Fisher information. This approach allowed for precise estimation of β, even when the Hamiltonian is completely unknown a priori, through manipulation of probe-ancilla entanglement and probabilistic post-selection.

Optimal Fisher information via entanglement in high-dimensional spin systems

The research details a protocol achieving optimal Fisher information for estimating rotation angles without prior knowledge of the rotation axis, utilizing entanglement between a large spin probe and an ancilla. Specifically, the maximal Fisher information reaches 4s², where ‘s’ represents the spin quantum number, demonstrating a quadratic increase with spin size and a significant improvement over the limits achievable with spin-1/2 probes.

This enhancement stems from the quadratic dependence of the maximal quantum Fisher information on the spin quantum number, offering superior precision in parameter estimation. The study extends previous work limited to spin-1/2 systems by successfully implementing this protocol with large spins, generally requiring post-selection with a success probability dependent on the dimension of the Hilbert space.

Researchers utilized entangled states between the probe and ancilla, initialized as described by the equation |Ψ⟩AB = m X i,j=1 χij|i −1⟩A ⊗|j −1⟩B, where {χij} represents a set of coefficients defining the entanglement. The entangled state undergoes a Schmidt decomposition, expressed as |Ψ⟩AB = m X k=1 ξk|uk⟩A ⊗|vk⟩B, with ‘r’ denoting the Schmidt rank and ξk representing non-zero Schmidt coefficients satisfying P k ξ2 k = 1.

The optimal probe states are expressed as |n±⟩= 1 √ 2(|E1⟩± |Em⟩), maximizing the quantum Fisher information at a value of Fmax = (E1 −Em)2 = 4s². These states, along with the other eigenstates of the Hamiltonian, form a complete orthogonal basis. The basis vectors for the probe are related by the transformation matrix S = {Ski}, enabling the expression of the entangled state in terms of these optimal probe states: |Ψ⟩AB = m X i=1 ci|φi⟩A ⊗|ψi⟩B. The coefficients ci, defined as ci = q B⟨ ψi| ψi⟩B, are non-negative and determine the contribution of each basis state to the overall entangled state.

Optimal parameter estimation via entanglement without axis knowledge

Scientists have demonstrated a method for achieving optimal precision in estimating a parameter, specifically, a rotation angle, without needing prior knowledge of the rotation axis. This advancement utilises entanglement between a large spin probe and an ancilla qubit, extending previous work limited to spin-1/2 particles.

The research establishes that maximum quantum Fisher information can be attained even when the rotation axis is unknown, a significant challenge in precision measurement. The study details a scheme where the entanglement between the probe and ancilla enables optimal estimation, though it generally requires post-selection of measurement outcomes.

This post-selection process yields a success probability dependent on the spin dimension, differing from the guaranteed success observed in some spin-1/2 scenarios. Furthermore, the investigation extends beyond maximally entangled states, revealing that optimal estimation remains possible with general entangled states, albeit with a success probability influenced by the specific entangled state and the rotation generator.

The authors acknowledge that the post-selection process, while contributing to agnostic parameter estimation theory, is not always beneficial. They also note that the success probability, despite leveraging quantum resources like entanglement, is ultimately bounded. Future research could focus on mitigating the limitations of post-selection or exploring methods to enhance the success probability for larger spin systems. The demonstrated approach offers a pathway towards improved precision in parameter estimation, potentially benefiting optical setups and advancing quantum technology.