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arXiv:2008.09319 · Physical Review A

Surpassing the thermal Cramér-Rao bound with collisional thermometry

Angeline Shu, Stella Seah, Valerio Scarani

Published Oct 28, 2020• 27 citations
arXivHTMLPDF

Abstract

In collisional thermometry, a system in contact with the thermal bath is probed by a stream of ancillas. Coherences and collective measurements were shown to improve the Fisher information in some parameter regimes, for a stream of independent and identically prepared (i.i.d.) ancillas in some specific states [Seah et al., Phys. Rev. Lett. 123, 180602 (2019)]. Here we refine the analysis of this metrological advantage by optimizing over the possible input ancilla states, also for block-i.i.d. states of block size $b=2$. For both an indirect measurement interaction and a coherent energy exchange channel, we show when the thermal Cram\'er-Rao bound can be beaten, and when a collective measurement of $N>1$ ancilla may return advantages over single-copy measurements.

DOI
10.1103/physreva.102.042417

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In collisional thermometry, a system in contact with the thermal bath is probed by a stream of ancillas. Coherences and collective measurements were shown to improve the Fisher information in some parameter regimes, for a stream of independent and identically prepared (i.i.d.) ancillas in some specific states [Seah et al., Phys. Rev. Lett. 123, 180602 (2019)]. Here we refine the analysis of this metrological advantage by optimizing over the possible input ancilla states, also for block-i.i.d. states of block size b=2b=2b=2. For both an indirect measurement interaction and a coherent energy exchange channel, we show when the thermal Cram'er-Rao bound can be beaten, and when a collective measurement of N>1N>1N>1 ancilla may return advantages over single-copy measurements.
We introduce a general framework for thermometry based on collisional models, where ancillas probe the temperature of the environment through an intermediary system. This allows for the generation of correlated ancillas even if they are initially independent. Using tools from parameter estimation theory, we show through a minimal qubit model that individual ancillas can already outperform the thermal Cramer-Rao bound. In addition, due to the steady-state nature of our model, when measured collectively the ancillas always exhibit superlinear scalings of the Fisher information. This means that even collective measurements on pairs of ancillas will already lead to an advantage. As we find in our qubit model, such a feature may be particularly valuable for weak system-ancilla interactions. Our approach sets forth the notion of metrology in a sequential interactions setting, and may inspire further advances in quantum thermometry.
Quantum thermometry exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. This highlights the need for constructing concrete estimation strategies. Of particular importance is collisional thermometry, where a series of ancillae are sent sequentially to probe the system's temperature. In this paper we put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cram\'er-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cram\'er-Rao bound associated with Van Trees and Sch\"utzenberger.
Measuring the temperature of a quantum system is an essential task in almost all aspects of quantum technologies. Theoretically, an optimal strategy for thermometry often requires measuring energy, which demands full accessibility over the entire system as well as a complex entangled measurement basis. In this paper, we take a different approach and show that single-qubit sequential measurements in the computational basis not only allow for precise thermometry of a many-body system, but may also achieve precision beyond the thermometry capacity of the probe at equilibrium, given by the Cram\'er-Rao bound. Thus, using consecutive single-qubit measurements of the probe out of equilibrium is, in most cases, very beneficial, as it achieves lower-temperature uncertainties and avoids demanding energy measurements when compared with probes at thermal equilibrium. To obtain such precision, the time between the two subsequent measurements should be smaller than the thermalization time so that the probe never thermalizes. Therefore, the nonequilibrium dynamics of the system continuously imprint information about temperature in the state of the probe. To demonstrate the generality of our findings, we consider thermometry in both spin chains and the Jaynes-Cummings model.
Abstract In probabilistic quantum metrology, one aims at finding measurements that concentrate the Fisher information on the resulting conditional quantum states, i.e. those states post-selected according to the outcome of the measurement. The quantum Cramér–Rao (QCR) bound itself cannot be surpassed this way, but it could be possible to improve the information-cost ratio (where cost refers to the experimental implementation), or even the total Fisher information. In this paper, we propose a post-selection protocol achieving this goal using single-photon subtraction and thermal states of radiation. In particular, we achieve a larger information-cost ratio for the estimation of temperature than the strategy attaining the QCR bound. Our scheme involves only classical states of radiation and this contrasts with (but does not contradict) recent results proving that for unitary quantum statistical models, post-selection strategies can outperform direct protocols only exploiting nonclassical states.
The metrological limits of thermometry operated in nonequilibrium dynamical regimes are analyzed. We consider a finite-dimensional quantum system, employed as a quantum thermometer, in contact with a thermal bath inducing Markovian thermalization dynamics. The quantum thermometer is initialized in a generic quantum state, possibly including quantum coherence with respect to the Hamiltonian basis. We prove that the precision of the thermometer, quantified by the Quantum Fisher Information, is enhanced by the quantum coherence in its initial state. We analytically show this in the specific case of qubit thermometers for which the maximization of the Quantum Fisher Information occurs at a finite time during the transient thermalization dynamics. Such a finite-time precision enhancement can be better than the precision that is achieved asymptotically.
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cram\'er-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.