# Different strategies for quantum thermometry in a squeezed thermal bath

###### Abstract

For the process of quantum thermometry with probes coupled to squeezed thermal baths via the nondemolition interaction, we address the dephasing dynamics of the quantum Fisher information computed analytically. This model for a single-qubit is exactly solvable, and the time-dependent reduced density matrix of the qubit can be obtained exactly. We also calculate the upper bound for the parameter estimation and investigate how the optimal estimation is affected by the initial conditions and decoherence, particularly the squeezing parameters. We then generalize the results for entangled probes and analyse the multi-qubit strategies for probing the temperature. Our results show that the squeezing can lead to decrease of uses of the channel for optimal thermometry. Comparing different scenarios for multi-qubit strategies, we find that increase of the qubits interacting with the channel does not necessarily vary the precision of estimating the temperature. Besides, if the probes are prepared initially in the W state, increase in the number of noiseless qubits attached to the interacting qubits, does not considerably affect the precision of thermometry.

ntangled probes, quantum metrology, dephasing model, squeezed thermal bath

## 1 Introduction

Acquiring information about the world is realized by observation and measurement, and the results of which are subject to error (1). The classical approach to decrease the statistical error is increase of the the resources for the measurement according to the central limit theorem; but this method is not always desirable or efficient (2). Quantum parameter estimation theory describes strategies allowing the estimation precision to surpass the limit of classical approaches (3, 4, 5, 6, 7, 8, 9, 10, 11, 12). When the quantum system is sampled N times, different strategies (13) allowing one to achieve the Heisenberg limit, can be designed such that the variance of the estimated parameter scales as . Initially entangled probes, however, may in principle offer a significant enhancement in precision of parameter estimation (13, 14, 15, 16, 17, 18). Those strategies have been realized experimentally in atomic spectroscopy (19, 20) in which the spin-squeezed states have been employed for improving frequency calibration precision (21, 22). Besides, the same quantum enhancement principle can be utilized in optical interferometry (23, 24) with exciting applications in the process of seeking the first direct detection of gravitational waves (25).

On the other hand, it has been also proven the entanglement is not always useful for parameter estimation (26) and there are highly entangled pure states that are not useful in the process of quantum estimation. Moreover, it has been discussed (3) while entanglement at the initial stage can be useful to enhance the precision of the estimation, entangled measurements are never necessary. Besides, there are some cases in which if the probes are initially prepared in an unentangled state, a better performance for the parameter estimation is attainable (27). Specifically, in Ref. (13) it has been illustrated that in the presence of Pauli x-y, depolarizing, and amplitude damping noise, unentangled probes perform better in the high-noise regime. Particularly, the view that entanglement is necessary for quantum-enhanced metrology have challenged in Refs. (28, 29) by demonstrating that the enhancement, obtained via entanglement, may be contingent on the final measurement and the way in which the unknown parameter is encoded into the probe quantum state. In addition, it has been illustrated that under certain conditions the entanglement may even lead to deleterious effects in quantum metrology (30). According to the above discussion, presenting a universal prescription, applicable for all quantum systems, about the relation between the QFI and entanglement, is not possible. These reasons motivate us to investigate more the role of the entanglement in different quantum scenarios for quantum metrology.

Recently, the quantum parameter estimation theory have attracted increasing attention in the field of quantum thermodynamics in which accurate estimation and control of the temperature are very significant (31, 32, 33, 34). In addition to the emergence of primary and secondary thermometers based on precisely machined microwave resonators (35, 36), recent studies have been focused on measuring temperature at even more smaller scales in which nanosize thermal baths are extremely sensitive to disturbances induced by the probes (37, 38, 39, 40, 41). Some interesting paradigms of nanoscale thermometry are quantum harmonic oscillators (42) nanomechanical resonators (43), and atomic condensates (44, 45). On the other hand, temperature plays an important role in realizing phase-matching condition in non-linear optical materials. Besides, thermal processes may result in large nonlinear optical effects originated from temperature dependence of the material refractive index (46). Accordingly, precise determination of the temperature is of great importance in all branches of modern science and technology (47, 48, 49). A scheme for enhancing the sensitivity of quantum thermometry is proposed in (50) where the sensing quantum system used to estimate the temperature of an external bath is dynamically coupled with an external ancilla (a meter) via a Hamiltonian term. Moreover, the dephasing dynamics of a single-qubit as an effective process in order to estimate the temperature of its environment is addressed in (51). Here, we generalize the results by using the entangled probes.

In this work, we propose a thermometer, consisting N qubits for probing the temperature of a squeezed thermal bath and calculate the bounds on the quantum thermometry in the presence of squeezing. It is supposed that qubits are directly coupled to the thermal bath of interest and qubits are not directly coupled to the bath but instead serve as an information storage which may be read out at the final time . Our scheme relies on the possibility of performing joint measurements on all of qubits. The model is well adopted to describe physical systems such as the molecular oscillation, exciton-phonon interaction, and photosynthesis process (52, 53, 54). Analytically, we investigate the effects of the initial state or squeezing and other environmental parameters on the estimation of the temperature. Besides, we extend our study to multiqubit estimation realized by initially entangled probes and address the role of entanglement in the process of thermometry on the squeezed thermal bath.

This paper is organized as follows: In Secion 2, we present a brief review of the QFI and obtaining the upper bound for the quantum parameter estimation. The model is introduced in Section 3. Different strategies for quantum thermometry are discussed completely in Section 4. Finally in Section 5, the main results are summarized.

## 2 The Preliminaries

### 2.1 (Quantum) Fisher information

The classical Fisher information is an important method of measuring the amount of information which an observable random variable carries about unknown parameter . Supposing that denotes the probability distribution with measurement outcomes , the classical Fisher information is defined as (55):

(1) |

characterizing the inverse variance of the asymptotic normality of a maximum-likelihood estimator. If observable is continuous, the summation should be replaced by an integral.

The quantum analog of the Fisher information can be formally generalized from Eq. (1) such that it is defined as (56):

(2) |

in which the symmetric logarithmic derivative (SLD) operator represents a Hermitian operator determined by

(3) |

where denotes the anti-commutator. Considering the density matrix spectral decomposition as , associated with and , we can rewrite the QFI as (57)

(4) |

where in the first and second summations we should exclude sums over all and , respectively.

According to quantum Cramér-Rao (QCR) theorem, a significant property of the QFI is that we can obtain the achievable lower bound of the mean-square error of the unbiased estimator for parameter T, i.e.,

(5) |

in which denotes the unbiased estimator, and represents the number of repeated experiments.

### 2.2 Upper bound for parameter estimation

Given an initial pure state , we know that it evolves according to the expression , where ’s represent -dependent Kraus operators (58). It has been derived that the upper bound to the QFI is given by (59):

(6) |

where

(7) |

(8) |

and .

## 3 The Model

At first, we introduce the dephasing model (60), composed of a two level system interacting with a boson bath . The interaction of system-bath (S-B) can be described by Hamiltonian , where and are the ground and excited states, respectively. Because ; the energy of the system is conserved, and hence the population of energy levels do not change with time.

Starting from , we focus on the case in which the initial state of the boson bath is a squeezed thermal state (61)

(9) |

in which T represents the temperature of the thermal bath prepared in the state , denotes the normalization constant, and represents the squeezing operator for the boson bath, where denotes the squeezing operator corresponding to mode (62):

(10) |

where and represent the squeezing strength and phase parameters, respectively.

In the interaction picture, the evolved reduced density matrix can be obtained as

(11) |

where is the population of excited level and denotes the decay factor. We focus on the Ohmic environment where its coupling spectral density is given by in which denotes the cutoff frequency and is a unitless number representing the coupling strength.

Defining where and assuming that and , we can find that the decay factor is given by (63)

(12) |

where, as described before, represents the squeezing strength, characterizes the S-B coupling strength, and denotes the phase difference between the squeezing phase relative to the phase of the coupling strength. Moreover, the time-dependent coefficients are of the form

(13) |

where

and where

(14) |

(15) |

in which and .

## 4 Strategies for quantum thermometry

### 4.1 Single-qubit strategy

Our model describes a dephasing channel such that the system can be used for probing temperature of the thermal bath. In the first scenario in which the single-qubit is used for thermometry, the corresponding QFI is obtained as follows:

(16) |

where the partial derivative is given by

(17) |

in which

(18) |

where .

In the high-temperature limit that and , we find that

(19) |

Preparing the qubit probe in a pure state , the QFI reduces to the following expression:

(20) |

saturating the upper bound obtained from Eq. (6) for . Throughout this paper, we set and normalize the QFI for clearer illustration.

As illustrated in Figs. 1-3, there is no trace of memory effects and it is seen that the evolution of the open system is completely Markovian. Besides, investigating the QFI as a function of time, we find that at first it increases, reaching a maximum. According to theory of quantum metrology, an increase in QFI means that the precision of quantum estimation is improved. This originates from the fact that interaction of the qubit with the bath encodes the information about the temperature into the quantum state of the probe, hence the QFI increases. However, because of the decoherence effects, the encoded information flows from the system to the environment, and hence its destructive influence appears and the QFI falls, thus, the quantum thermometry becomes more inaccurate.

We first investigate the behaviour of the QFI dynamics with respect to cutoff frequency and S-B coupling strength . As shown in Figs. 1 and 1, the variation of theses parameters can not change the maximum value of the QFI. Therefore, the optimum precision obtained in the process of thermometry does not vary. However, the figure shows when the cutoff frequency or the coupling increases, the maximum point of the QFI is shifted to the left, and hence the QFI reaches its maximum value sooner. Although increase of the coupling between the probe and the bath or increasing the cutoff frequency decreases the interaction time for obtaining the optimal estimation precision, the QFI decreases more quickly, and hence the time period that we can extract the information about the temperature is shortened.

Squeezing effects of increasing squeezing strength are shown in Fig. 2. We can achieve sooner the optimal precision of thermometry probing more squeezed fields such that the optimal value of the QFI also remains invariant. Interestingly, Fig. 2 illustrates that in more squeezed fields the optimal estimation is attainable with more weak coupling between the probe and the bath. This may leads to important results in improving the control of decoherence in the process of quantum communication.

Now we investigate how may affect the dynamics of the QFI. As shown in Fig. 3, when the relative phase varies from to rad, the interaction time between the probe and the bath can be reduced and hence the optimal value of estimation, not affected by variation of the relative phase, is obtained sooner. However, varying the relative phase from to , we see that the maximum point, at which the QFI is maximized, can be shifted to the right. Therefore, the QFI reaches its maximum value at a later time-point. Nevertheless, Fig. 3 exhibits a positive and interesting consequence of increasing the relative phase from to . We see that larger relative phases lead to retardation of the QFI loss during the time evolution and therefore enhance the estimation of the parameter at a later time.

### 4.2 Multi-qubit strategy

Two usual strategies in which probes are submitted to independent processes are shown in Fig. 4 for estimating the temperature. In the parallel and ancilla-assisted strategies shown in Figs. 4 and 4, the upper bound for the QFI associated with the temperature is obtained as follows:

(21) |

where in the parallel strategy, we put . Clearly, use of the noiseless ancillas can not improve the upper bound for the QFI.

For our dephasing model, it is simple to obtain the operator-sum representation where the time-dependent Kraus operators are given by

(22) |

In the first scenario, we adopt the ancilla-assisted strategy in the sense that the probes are realized with two qubits such that one of which is noiseless ( see Fig. 4 with , and ). Because this is the model for two independent environments, the Kraus operators are just tensor products of Kraus operators of each of the qubits, . Preparing initially the qubits in the Bell state , we find that the output state of the channel that should be measured for estimating the temperature is given by

(23) |

Therefore, the corresponding QFI is obtained as

(24) |

saturating the upper bound. Using a three-qubit probe () with , and , initially prepared in the GHZ state , we again the same result for

(25) |

On the other hand, if the probes are initially in the W state the QFI is given by

(26) |

Therefore, and it cannot saturate the upper bound.

Natural generalizations of the GHZ and W states to N-qubit systems are

(27) |

(28) |

Provided that we start with state and qubits are affected by the channel, the QFI corresponding to the temperature estimation is obtained as

(29) |

As seen in Figs. 5 and 6, we find that in the entangled strategy with initial GHZ state, increase in , the number of uses of the channel, does not necessarily enhance the QFI and it may even lead to decrease of the precision of the temperature estimation. We address how the optimal number of interacting qubits with which the QFI is maximized, is affected by other parameters.

For initial GHZ state, we find that less squeezed fields needs more qubits for achieving optimal estimation of the temperature. Hence the squeezing leads to decrease of uses of the channel for optimal thermometry (see Fig. 5). Moreover, as shown in Fig. 5, when the relative phase increases from to rad, the number of uses of the channel can be reduced and the optimal value of estimation, not affected by variation of the relative phase, is obtained with less qubits interacting with the bath. On the other hand, increasing the relative phase from to , we observe that the QFI, associated with estimating the temperature, reaches its maximum value by using more qubits affected by the channel.

Figure 6 illustrates when the S-B coupling is strengthened or the interaction time between the qubits and the bath is increased the optimal for which the QFI is maximized decreases. Therefore, if the interacting qubits are weakly coupled to the bath or the interaction time decreases, the cost of quantum thermometry may increase.

On the other hand, if we start with state and one of the qubits is affected by the channel (n=1), the QFI is given by:

(30) |

where . It is obvious that the upper bound can be saturated if and only if is even. Moreover, we find when is odd and one of the qubits is affected by the channel, the single-probe strategy may lead to more precise estimation than the ancilla-assisted one started with a W state:

(31) |

Therefore, the entangled strategy is not necessarily more efficient than the single-qubit one for quantum thermometry. Besides, comparing Eqs. (29) and (30), we see when ,

(32) |

For , a compact expression for is not generally accessible except for some special cases presented here briefly:

(33) |

(34) |

(35) |

(36) |

It is seen that increase of the qubits interacting with the channel does not necessarily vary the precision of estimating the temperature. Moreover, our numerical calculation shows when the probes are prepared initially in the W state, increase in the number of noiseless qubits attached to the interacting qubits, does not considerably affect the precision of thermometry.

## 5 Summary and conclusions

To summarize, we investigated the quantum thermometry by using a qubit subjected to the dephasing dynamics via interacting with a squeezed thermal bath. we analysed the behaviour of the QFI and studied how the initial control and the environmental conditions can help us to achieve the optimal estimation. We obtained the general expression for the QFI associated with any parameter in the presence of the dephasing channel and investigated the corresponding upper bound for the parameter estimation. We illustrated that the optimum precision obtained in the process of thermometry is an invariant quantity. Moreover, it was shown that squeezing parameters lead to interesting and non-trivial effects on the quantum thermometry. Generalizing the results for entangled probes and analysing the multi-qubit strategies for estimating the temperature, we found that that in the entangled strategy with initial GHZ state, increase in the number of uses of the channel, does not necessarily enhance the QFI and it may even lead to decrease of the precision of the temperature estimation. Moreover, the squeezing may lead to decrease of the thermometry costs. In addition, when the probes are initially prepared in the W state, increase in the number of noiseless qubits attached to the sensors directly interacting with the baths, does not considerably affect the precision of temperature estimation.

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