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Quantum Thermometry

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Thermodynamics in the Quantum Regime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 195))

Abstract

We discuss the application of techniques of quantum estimation theory and quantum metrology to thermometry. The ultimate limit to the precision at which the temperature of a system at thermal equilibrium can be determined is related to the heat capacity when global measurements are performed on the system. We prove that if technical or practical limitations restrict our capabilities to local probing, the highest achievable accuracy to temperature estimation reduces to a sort of mesoscopic version of the heat capacity. Adopting a more practical perspective, we also discuss the relevance of qubit systems as optimal quantum thermometers, in order to retrieve the temperature, or to discriminate between two temperatures, characterizing a thermal reservoir. We show that quantum coherence and entanglement in a probe system can facilitate faster, or more accurate measurements of temperature. While not surprising given this has been demonstrated in phase estimation, temperature is not a conventional quantum observable, therefore these results extend the theory of parameter estimation to measurement of non-Hamiltonian quantities. Finally we point out the advantages brought by a less standard estimation technique based on sequential measurements, when applied to quantum thermometry.

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Notes

  1. 1.

    The maximum-likelihood principle selects the parameter values that make the data most probable. It stems from the definition of the likelihood function \(\mathcal{L}(\lambda )\) as the joint conditional probability of the observed data, that for the case of independent measurements reduces to the product of the probabilities of the single outcomes \(\theta _i\),

    $$\begin{aligned} \mathcal{L}(\lambda )=\mathcal{L}(\vec {\theta }|\lambda )=\prod _{i}^{N} p(\theta _i|\lambda ). \end{aligned}$$
    (21.3)

    The maximum-likelihood estimate is the value of \(\lambda \) that maximizes \(\mathcal{L}(\lambda )\) or equivalently its logarithm. This procedure selects the parameter values that make the data most probable. It results that the variance on the maximum-likelihood estimate of \(\lambda \), in the limit of large N, saturates the Cramér–Rao bound (21.1).

  2. 2.

    Given \(p_i\) and \(p'_i\) two probability density distributions, and \(\{\rho _i \}\) and \(\{\rho '_i\}\) two sets of density matrices, it results that \(\mathcal{F}\left( \sum _{i} p_i \rho _i, \sum _{i} p'_i \rho '_i \right) \ge \sum _{i}\sqrt{p_i p'_i}\mathcal{F} ( \rho _i, \rho '_i )\). This property is dubbed strong concavity property for the fidelity.

  3. 3.

    Although we note that the ‘phase operator’ of a harmonic oscillator is itself only properly defined in a limiting sense, much like the position operator of a free particle: one can write a limit of hermitian operators that localise the quantity, but they are accompanied by a divergence in the conjugate variable (number or position), which leads to unphysical energetic divergences. However, this is different from temperature, which cannot be defined as the limit of a sequence of hermitian operators.

  4. 4.

    It results that in this case the SLD commutes with the system Hamiltonian, implying that energy measurements are optimal. Moreover, thermal states represent a special case as they belong to the so-called exponential class [42].

  5. 5.

    Let \(\rho ^{\Gamma }\) be the state of a given system \(\Gamma \). It is always possible to introduce another system a (the so-called reference or ancillary system) and define a pure state \(|\rho ^{\Gamma }\rangle \) on \(\Gamma a\) such that \(\rho ^{\Gamma } = \mathrm {Tr}_{a}[|\rho ^{\Gamma }\rangle \langle \rho ^{\Gamma }| ]\). Furthermore, if \(|\rho ^{\Gamma }\rangle \) and \(|{\rho '}^{\Gamma }\rangle \) are two purifications of \(\rho ^{\Gamma }\) on \(\Gamma a\) there exists a unitary transformation U on a such that \(|{\rho '}^{\Gamma }\rangle = (\mathbb {I}_\Gamma \otimes U ) |\rho ^{\Gamma }\rangle \), being \(\mathbb {I}_\Gamma \) the identity operator on \(\Gamma \). The last property is known as “freedom in purifications” [49].

  6. 6.

    The same conclusions can be driven by noticing an interesting connection between the LQTS, related to the reconstruction of T, and the more studied case of phase estimation mentioned in the former section. Indeed by comparing relations (21.7)–(21.8) with (21.19)–(21.20) we have

    $$\begin{aligned} \mathfrak {S}_\mathcal{A}[\rho _{\beta }] + \mathcal{Q}_{a}(\lambda )= \mathcal{Q}_\mathcal{S S'}(\lambda ) \end{aligned}$$
    (21.23)

    where \(\mathcal{Q}_{a}(\lambda )\) and \(\mathcal{Q}_\mathcal{S \mathcal S'}(\lambda )\) are the QFI associated to the estimation of \(\lambda \) encoded on \(|\rho _\beta \rangle \) via the unitary superoperator \(\mathcal{E}_{\lambda /2}\) such that \(\mathcal{E}_{\lambda /2}(|\rho _\beta \rangle \langle \rho _\beta | )=e^{-i H' \lambda /2}|\rho _\beta \rangle \langle \rho _\beta | e^{i H' \lambda /2}= |\rho _\beta ^{(\lambda )}\rangle \langle \rho _\beta ^{(\lambda )}|\), assuming to have access at the measurement stage to subsystem \(a=\mathcal{B A' B'}\) and \(\mathcal{S \mathcal S'} = \mathcal{A} a\), respectively:

    $$\begin{aligned} \mathcal{Q}_{a}(\lambda )=\sum _{i<j} \frac{(e_i - e_{j})^2}{e_i + e_{j}}|\langle e_i | H' | e_j\rangle |^2, \quad \mathcal{Q}_\mathcal{S S'}(\lambda )\,{=}\,\langle {\rho _\beta }| H'^2 | \rho _\beta \rangle - \langle {\rho _\beta }| H' |\rho _\beta \rangle ^2 =\langle {\rho _\beta }| H^2 | \rho _\beta \rangle - \langle {\rho _\beta }| H |\rho _\beta \rangle ^2. \end{aligned}$$
    (21.24)

    In other words the accuracies corresponding to the temperature estimation on \(\mathcal{A}\) and the phase estimation on its complementary counterpart a, which are both positive quantities, are forced to sum up to the energy variance of the global system, thus establishing a sort of complementarity relation. Indeed, as already mentioned, the LQTS \(\mathfrak {S}_\mathcal{A}[\rho _{\beta }]\) and similarly \(\mathcal{Q}_{a}(\lambda )\) are increasing functions of the dimension of \(\mathcal{A}\) and a, respectively.

  7. 7.

    By definition the elements \(\Pi _\theta \) of a POVM are positive operators. This implies that for each \(\Pi _\theta \) there exists an other positive operator \(M_\theta \), determined up to a unitary transformation, such that \(\Pi _{\theta }=M_\theta ^\dagger M_\theta \) and \(\int d\theta M_\theta ^\dagger M_\theta = \mathbb {I}\). Therefore the probability of measuring \(\theta \) on a state \(\rho \) is given by \(p(\theta |\rho )=\mathrm {Tr}[\Pi _\theta \rho ]= \mathrm {Tr}[ M_\theta \rho M_\theta ^\dagger ]\), and the normalized state of the system after the measurement reads \(\rho _\theta = M_\theta \rho M_\theta ^\dagger /p(\theta |\rho )\). Finally, to each operator \(M_\theta \) we can associate a superoperator \(\mathcal{M}_\theta \) such that \(\mathcal{M}_\theta (\rho )=M_\theta \rho M_\theta ^\dagger \) and \(\int d\theta \mathcal{M}_\theta = \mathcal{I}\), with \(\mathcal{I}\) being the identity superoperator.

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Acknowledgements

ADP acknowledges financial support from the University of Florence in the framework of the University Strategic Project Program 2015 (project BRS00215).

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Florence, Via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy

    Antonella De Pasquale

  2. INFN Sezione di Firenze, via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy

    Antonella De Pasquale

  3. NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126, Pisa, Italy

    Antonella De Pasquale

  4. ARC Centre for Engineered Quantum System, Department of Physics, University of Queensland, Brisbane, QLD, 4072, Australia

    Thomas M. Stace

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Editors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Felix Binder

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Luis A. Correa

  3. ICFO: Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, Castelldefels, Spain

    Christian Gogolin

  4. CEMPS, Physics and Astronomy, University of Exeter, Exeter, UK

    Janet Anders

  5. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Gerardo Adesso

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De Pasquale, A., Stace, T.M. (2018). Quantum Thermometry. In: Binder, F., Correa, L., Gogolin, C., Anders, J., Adesso, G. (eds) Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-99046-0_21

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