An intuitive scheme to assign a temperature to an arbitrary state of a quantum system is to investigate the heat flow resulting from the coupling to a thermometer. We introduce a simple model of a universal thermometer with the following property. When it is prepared in a Gibbs equilibrium state at inverse temperature $β\in\mathbb R$ and brought into thermal contact with a system in any state, the heat flow between the system and thermometer vanishes for a unique value of $β$. We call this value the contact temperature $β_{\rm op}\in\mathbb R$ of the system state. The thermometer is universal in that it yields a unique contact temperature for arbitrary states of finite dimensional quantum systems.
An intuitive scheme to assign a temperature to an arbitrary state of a quantum system is to investigate the heat flow resulting from the coupling to a thermometer. We introduce a simple model of a universal thermometer with the following property. When it is prepared in a Gibbs equilibrium state at inverse temperature β∈R and brought into thermal contact with a system in any state, the heat flow between the system and thermometer vanishes for a unique value of β. We call this value the contact temperature βop∈R of the system state. The thermometer is universal in that it yields a unique contact temperature for arbitrary states of finite dimensional quantum systems.
A system, or a set of systems, is said to be in thermal equilibrium when no net flow of heat energy occurs between its constituent parts when they are connected by a path permeable to heat (a diathermal wall).1 This condition is the endpoint of a spontaneous process: if two bodies at different temperatures are brought into thermal contact, energy will flow from the higher-temperature object to the lower-temperature object until this flow ceases and the equilibrium state is achieved.3 Within a single system, thermal equilibrium is characterized by a temperature that is both spatially uniform (the same at all points) and temporally constant (unchanging over time).4 This uniformity of macroscopic properties—such as temperature and pressure—is what defines the state of the system.1 It is important to distinguish this from the broader state of thermodynamic equilibrium. Thermal equilibrium (uniform temperature) is a necessary component, but full thermodynamic equilibrium also requires mechanical equilibrium (uniform pressure) and diffusive equilibrium (uniform chemical potential).6 Systems can achieve thermal equilibrium without achieving full thermodynamic equilibrium.6 The Zeroth Law of Thermodynamics: The "Discovery" of Temperature The concept of temperature, while intuitive, requires a formal physical principle to exist as a measurable, fundamental property. This principle is the Zeroth Law of Thermodynamics. Although it was formulated and named after the First and Second Laws, its foundational role was recognized as a logical prerequisite for the entire structure of thermodynamics.8 The law states: If system A is in thermal equilibrium with system C, and system B is in thermal equilibrium with system C, then systems A and B are in thermal equilibrium with each other.1 In this construction, system C plays the role of a thermometer.13 The profound implication of this law is mathematical: it establishes thermal equilibrium as a transitive relation 9, or more formally, an equivalence relation.14 In mathematics, any equivalence relation partitioning a set of objects implies the existence of a common, scalar property shared by all members of a given equivalence class. The Zeroth Law thus proves the existence of a scalar function of state—a property we call temperature—that is common to all systems in mutual thermal equilibrium.9 The law is, therefore, the fundamental basis for defining temperature and building consistent, reproducible temperature scales.1 This axiomatic foundation also provides the logical origin for one of temperature's most critical characteristics: it is an intensive property. An intensive property is one that does not depend on the size or amount of the system.15 If a large system 'A' is in internal thermal equilibrium 6, any two parts of it, subsystems 'A1' and 'A2', must also be in thermal equilibrium with each other. By the Zeroth Law, a thermometer 'C' that is in thermal equilibrium with the whole system 'A' must also be in thermal equilibrium with 'A1' and 'A2' individually.13 The thermometer, therefore, gives the same reading for all three. This demonstrates that the temperature of a system in thermal equilibrium is the same as the temperature of any part of it.16
A classical thermometer typically works by exchanging energy with the system being measured until it comes to equilibrium, at which point the readout is related to the final energy state of the thermometer. A recent paper noted that with a quantum thermometer consisting of a single spin/qubit, temperature discrimination is better achieved at finite times rather than once equilibration is essentially complete. Furthermore, preparing a qubit thermometer in a state with quantum coherence instead of an incoherent one improves its sensitivity to temperature differences. Implementing a recent proposal for efficiently emulating an arbitrary quantum channel, we use the quantum polarisation state of individual photons as models of "single-qubit thermometers" which evolve for a certain time in contact with a thermal bath. We investigate the optimal thermometer states for temperature discrimination, and the optimal interaction times, confirming that there is a broad regime where quantum coherence provides a significant improvement. We also discuss the more practical question of thermometers composed of a finite number of spins/qubits (greater than one), and characterize the performance of an adaptive protocol for making optimal use of all the qubits.
Temperature uncertainty of a quantum system in canonical ensemble is inversely determined by its energy fluctuation, which is known as the temperature-energy uncertainty relation. No such uncertainty relation was discovered for a non-equilibrium open quantum system. In this article, we derive a universal temperature uncertainty relation for general non-equilibrium processes. We find that it is the fluctuation of heat, which is defined as the change in bath energy, determines the temperature uncertainty in non-equilibrium case. Specifically, the heat is divided into trajectory heat and backaction heat, which are associated with the system's trajectory of evolution and the backaction of partial measurement on system, respectively. Based on this decomposition, we reveal that both correlations between system and bath's process function and state function are the resources for enhancing temperature precision. Our findings are conductive to design ultrahigh sensitive quantum thermometer.
Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.
We provide a detailed description of the quantum interferometric thermometer, which is a device that estimates the temperature of a sample from the measurements of the optical phase. We rigorously analyze the operation of such a device by studying the interaction of the optical probe system prepared in a single-mode Gaussian state with a heated sample modeled as a dissipative thermal reservoir. We find that this approach to thermometry is capable of measuring the temperature of a sample in the nanokelvin regime. Furthermore, we compare the fundamental precision of quantum interferometric thermometers with the theoretical precision offered by the classical idealized pyrometers, which infer the temperature from a measurement of the total thermal radiation emitted by the sample. We find that the interferometric thermometer provides a superior performance in temperature sensing even when compared with this idealized pyrometer. We predict that interferometric thermometers will prove useful for ultraprecise temperature sensing and stabilization of quantum optical experiments based on the nonlinear crystals and atomic vapors.
Quantum thermometry aims to measure temperature in nanoscale quantum systems, paralleling classical thermometry. However, temperature is not a quantum observable, and most theoretical studies have therefore concentrated on analyzing fundamental precision limits set by the quantum Fisher information through the quantum Cramer-Rao bound. In contrast, whether a direct temperature readout can be achieved in quantum thermometry remains largely unexplored, particularly under the nonequilibrium conditions prevalent in real-world applications. To address this, we develop a direct temperature readout scheme based on a thermodynamic inference strategy. The scheme integrates two conceptual developments: (i) By applying the maximum entropy principle with the thermometer's mean energy as a constraint, we assign a reference temperature to the nonequilibrium thermometer. We demonstrate that this reference temperature outperforms a commonly used effective temperature defined through equilibrium analogy. (ii) We obtain positive semi-definite error functions that lower-bound the deviation of the reference temperature from the true temperature-in analogy to the quantum Cramer-Rao bound for the mean squared error-and vanish upon thermalization with the sample. Combining the reference temperature with these error functions, we introduce a notion of corrected dynamical temperature which furnishes a postprocessed temperature readout under nonequilibrium conditions. This corrected dynamical temperature can be evaluated adaptively without prior knowledge of the actual temperature. We validate the corrected dynamical temperature in a qubit-based thermometer under a range of nonequilibrium initial states, confirming its capability to estimate the true temperature. Importantly, we find that increasing quantum coherence can enhance the precision of this readout.