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arXiv:1902.01551 · Physical Review A

Quantum metrology with generalized cat states

Mamiko Tatsuta, Yuichiro Matsuzaki, Akira Shimizu

Published Sep 13, 2019• 22 citations
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Abstract

We show a general relationship between a superposition of macroscopically distinct states and sensitivity in quantum metrology. Generalized cat states are defined by using an index which extracts the coherence between macroscopically distinct states, and a wide variety of states, including a classical mixture of an exponentially large number of states, has been identified as the generalized cat state with this criterion. We find that if we use the generalized cat states for magnetic field sensing without noise, the sensitivity achieves the Heisenberg scaling. More importantly, we even show that sensitivity of generalized cat states achieves the ultimate scaling sensitivity beyond the standard quantum limit under the effect of dephasing. As an example, we investigate the sensitivity of a generalized cat state that is attainable through a single global manipulation on a thermal equilibrium state and find an improvement of a few orders of magnitude from the previous sensors. Clarifying a wide class that includes such a peculiar state as metrologically useful, our results significantly broaden the potential of quantum metrology.

DOI
10.1103/physreva.100.032318

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We show a general relationship between a superposition of macroscopically distinct states and sensitivity in quantum metrology. Generalized cat states are defined by using an index which extracts the coherence between macroscopically distinct states, and a wide variety of states, including a classical mixture of an exponentially large number of states, has been identified as the generalized cat state with this criterion. We find that if we use the generalized cat states for magnetic field sensing without noise, the sensitivity achieves the Heisenberg scaling. More importantly, we even show that sensitivity of generalized cat states achieves the ultimate scaling sensitivity beyond the standard quantum limit under the effect of dephasing. As an example, we investigate the sensitivity of a generalized cat state that is attainable through a single global manipulation on a thermal equilibrium state and find an improvement of a few orders of magnitude from the previous sensors. Clarifying a wide class that includes such a peculiar state as metrologically useful, our results significantly broaden the potential of quantum metrology.
We show a general relationship between a superposition of macroscopically distinct states and sensitivity in quantum metrology. Generalized cat states are defined by using an index which extracts the coherence between macroscopically distinct states, and a wide variety of states, including a classical mixture of an exponentially large number of states, has been identified as the generalized cat state with this criterion. We find that, if we use the generalized cat states for magnetic field sensing without noise, we achieve the Heisenberg limited sensitivity. Moreover, we even show that sensitivity of generalized cat states achieves the ultimate scaling sensitivity beyond the standard quantum limit under the effect of dephasing. As an example, we investigate the sensitivity of a generalized cat state that is attainable through a single global manipulation on a thermal equilibrium state and find an improvement of a few orders of magnitude from the previous sensors. Clarifying a wide class that includes such a peculiar state as metrologically useful, our results significantly broaden the potential of quantum metrology.
Sensitivity in metrology without entanglement is limited by the standard quantum limit (SQL). Recent studies have found that the Heisenberg-limited scaling, the ultimate sensitivity in quantum metrology, can be achieved by generalized cat states, which are characterized by an index that indicates coherence among macroscopically distinct states and are associated with additive observables. Although generalized cat states include diverse states, encompassing classical mixtures of exponentially large numbers of states, the preparation of large generalized cat states has not been demonstrated yet. Here we characterize SQL-beating catlike states using the index qqq indicating macroscopic coherence and prove that any state with q>1.5q>1.5q>1.5 has a potential to surpass the SQL when used as a sensor. We propose a protocol to generate them through repetitive measurements on a quantum spin system of NNN spins, which we call a spin ensemble. Starting from a thermal equilibrium state of the spin ensemble, we demonstrate that we can increase the coherence among the spin ensemble via repetitive weak measurements of its total magnetization, which is indirectly measured through an ancillary qubit collectively coupled to the ensemble. Notably, our method for creating the SQL-beating catlike states requires no dynamical control over the spin ensemble. As a potential experimental realization, we discuss a hybrid system composed of a superconducting flux qubit and donor spins in silicon. Our results pave the way for the realization of entanglement-enhanced quantum metrology in state-of-the-art technology.
Quantum sensors have the potential to outperform their classical counterparts. For classical sensing, the uncertainty of the estimation of the target fields decreases 1/T1/\sqrt{T}1/T​ with measurement time TTT. On the other hand, by using quantum resources, one can decrease the uncertainty with time to 1/T1/T1/T - the Heisenberg limit. However as quantum states are susceptible to dephasing, it has been not clear whether one can achieve the Heisenberg limit for a measurement time scaling longer than the coherence time of those states. Here, we propose a scheme that estimates the amplitude of globally applied fields with Heisenberg limited sensitivity in principle for an arbitrary time scale under the effect of dephasing. We use one way quantum computing based teleportation between qubits to prevent the correlation between the quantum state and its local environment from increasing, and have shown that such a teleportation protocol can suppress the local dephasing while the information from the target fields keeps growing. Our methods has the potential to realize a quantum sensor with a sensitivity far beyond that of any classical sensors.
The diverging responses to parameter variations of systems at quantum critical points motivate schemes of quantum critical metrology (QCM), which feature sub-Heisenberg scaling of the sensitivity with the system size (e.g., the number of particles). This sensitivity enhancement by quantum criticality is rooted in the formation of Schr\"odinger cat states, or macroscopic superposition states at the quantum critical points. The cat states, however, are fragile to decoherence caused by coupling to local environments, since the local decoherence of any particle would cause the collapse of the whole cat state. Therefore, it is unclear whether the sub-Heisenberg scaling of QCM is robust against the local decoherence. Here we study the effects of local decoherence on QCM, using a one-dimensional transverse-field Ising model as a representative example. We find that the standard quantum limit is recovered by single-particle decoherence. Using renormalization group analysis, we demonstrate that the noise effects on QCM is general and applicable to many universality classes of quantum phase transitions whose low-energy excitations are described by a ${\ensuremath{\phi}}^{4}$ effective field theory. Since in general the many-body entanglement of the ground states at critical points, which is the basis of QCM, is fragile to quantum measurement by local environments, we conjecture that the recovery of the standard quantum limit by local decoherence is universal for QCM using phase transitions induced by the formation of long-range order. This work demonstrates the importance of protecting macroscopic quantum coherence for quantum sensing based on critical behaviors.
Quantum sensors have the potential to outperform their classical counterparts. For classical sensing, the uncertainty of the estimation of the target fields scales inversely with the square root of the measurement time T. On the other hand, by using quantum resources, we can reduce this scaling of the uncertainty with time to 1/T. However, as quantum states are susceptible to dephasing, it has not been clear whether we can achieve sensitivities with a scaling of 1/T for a measurement time longer than the coherence time. Here, we propose a scheme that estimates the amplitude of globally applied fields with the uncertainty of 1/T for an arbitrary time scale under the effect of dephasing. We use one-way quantum-computing-based teleportation between qubits to prevent any increase in the correlation between the quantum state and its local environment from building up and have shown that such a teleportation protocol can suppress the local dephasing while the information from the target fields keeps growing. Our method has the potential to realize a quantum sensor with a sensitivity far beyond that of any classical sensor.