Quantum thermodynamic uncertainty relations establish fundamental trade-offs between the precision achievable in quantum systems and associated thermodynamic quantities such as entropy production or dynamical activity. While foundational, empirical demonstrations have thus far been confined to specific cases, either assuming time-reversal symmetry or involving particular measurement types, leaving the verification of their universal validity unrealized. This work leverages a quantum computer to report the first empirical verification of a general quantum thermodynamic uncertainty relation, valid for arbitrary dynamics and observables. We theoretically derive the relation, revealing survival activity as the pivotal thermodynamic quantity governing the precision bound. The verification is demonstrated on IBM's cloud-based quantum processor, which is treated as a real thermodynamic system. To achieve accurate results despite substantial device errors, we introduce a generic protocol for measuring survival activity and employ circuit reduction techniques based on the relation's properties. This strategy allows us to empirically identify survival activity for the first time and confirm the derived relation. Furthermore, the quantum computer's versatility enables the implementation of optimal observables, leading to the saturation of the relation and demonstrating the sharpness of our bound on a physical device. The method's broad applicability is further illustrated by verifying the trade-off for quantum time correlators. Our findings establish quantum computers as effective platforms for investigating fundamental thermodynamic trade-off relations.
Fluctuation theorems are fundamental results in non-equilibrium thermodynamics. Considering the fluctuation theorem with respect to the entropy production and an observable, we derive a new thermodynamic uncertainty relation which also applies to non-cyclic and time-reversal non-symmetric protocols. Furthermore, we investigate the relation between the thermodynamic uncertainty relation and the correlation between the entropy and the observable.
Entropy production characterizes irreversibility. This viewpoint allows us to consider the thermodynamic uncertainty relation, which states that a higher precision can be achieved at the cost of higher entropy production, as a relation between precision and irreversibility. Considering the original and perturbed dynamics, we show that the precision of an arbitrary counting observable in continuous measurement of quantum Markov processes is bounded from below by Loschmidt echo between the two dynamics, representing the irreversibility of quantum dynamics. When considering particular perturbed d…
Understanding current fluctuations is of fundamental importance and paves the way for the development of practical applications. According to the thermodynamic and kinetic uncertainty relations, the precision of currents can be constrained solely by total entropy production or dynamical activity. In this study, we derive a tighter bound on the precision of currents in terms of both thermodynamic and kinetic quantities, demonstrating that these quantities jointly constrain current fluctuations. The thermodynamic and kinetic uncertainty relations become particular cases of our result in asymptot…
We derive a thermodynamic uncertainty relation for first passage processes in quantum Markov chains. We consider first passage processes that stop after a fixed number of jumps, which contrasts with typical quantum Markov chains which end at a fixed time. We obtain bounds for the observables of the first passage processes in quantum Markov chains by the Loschmidt echo, which quantifies the extent of irreversibility in quantum many-body systems. Considering a particular case, we show that the lower bound corresponds to the quantum Fisher information, which plays a fundamental role in uncertaint…
We use quantum estimation theory to derive a thermodynamic uncertainty relation in Markovian open quantum systems, which bounds the fluctuation of continuous measurements. The derived quantum thermodynamic uncertainty relation holds for arbitrary continuous measurements satisfying a scaling condition. We derive two relations; the first relation bounds the fluctuation by the dynamical activity and the second one does so by the entropy production. We apply our bounds to a two-level atom driven by a laser field and a three-level quantum thermal machine with jump and diffusion measurements. Our re…
Thermodynamic uncertainty relations (TURs) place strict bounds on the fluctuations of thermodynamic quantities in terms of the associated entropy production. In this work we identify the tightest (and saturable) matrix-valued TUR that can be derived from the exchange fluctuation theorems describing the statistics of heat and particle flow between multiple systems. Our result holds for both quantum and classical systems, undergoing general non-Markovian and non-stationary processes. Moreover, it provides bounds not only for the variances, but also for the correlations between thermodynamic quan…
We study two-mode bosonic engines undergoing an Otto cycle. The energy exchange between the two bosonic systems is provided by a tunable unitary bilinear interaction in the mode operators modeling frequency conversion, whereas the cyclic operation is guaranteed by relaxation to two baths at different temperature after each interacting stage. By means of a two-point-measurement approach we provide the joint probability of the stochastic work and heat. We derive exact expressions for work and heat fluctuations, identities showing the interdependence among average extracted work, fluctuations and…
In classical Markov jump processes, current fluctuations can only be reduced at the cost of increased dissipation. To explore how quantum effects influence this trade-off, we analyze the uncertainty of steady-state currents in Markovian open quantum systems. We first consider three instructive examples and then systematically minimize the product of uncertainty and entropy production for small open quantum systems. As our main result, we find that the thermodynamic cost of reducing fluctuations can be lowered below the classical bound by coherence. We conjecture that this cost can be made arbi…
It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase…
Thermodynamics serves as a universal means for studying physical systems from an energy perspective. In recent years, with the establishment of the field of stochastic and quantum thermodynamics, the ideas of thermodynamics have been generalized to small fluctuating systems. Independently developed in mathematics and statistics, the optimal transport theory concerns the means by which one can optimally transport a source distribution to a target distribution, deriving a useful metric between probability distributions, called the Wasserstein distance. Despite their seemingly unrelated nature, a…
We investigate the thermodynamic uncertainty relation (TUR), \textit{i.e.} a trade-off between entropy production rate and relative power fluctuations, for nondegenerate three-level and degenerate four-level maser heat engines. In the nondegenerate case, we consider two slightly different configurations of three-level maser heat engine and contrast their degree of violation of standard TUR. In the high-temperature limit, standard TUR relation is always violated for both configurations. We also uncover an invariance of TUR under simultaneous rescaling of the matter-field coupling constant and s…
We derive a thermodynamic uncertainty relation for general open quantum dynamics, described by a joint unitary evolution on a composite system comprising a system and an environment. By measuring the environmental state after the system-environment interaction, we bound the counting observables in the environment by the survival activity, which reduces to the dynamical activity in classical Markov processes. Remarkably, the relation derived herein holds for general open quantum systems with any counting observable and any initial state. Therefore, our relation is satisfied for classical Markov…
To reveal the role of the quantumness in the Otto cycle and to discuss the validity of the thermodynamic uncertainty relation (TUR) in the cycle, we study the quantum Otto cycle and its classical counterpart. In particular, we calculate exactly the mean values and relative error of thermodynamic quantities. In the quasistatic limit, quantumness reduces the productivity and precision of the Otto cycle compared to that in the absence of quantumness, whereas in the finite-time mode, it can increase the cycle's productivity and precision. Interestingly, as the strength (heat conductance) between t…
We investigate the fine-grained uncertainty relations for qubit system by measurements corresponding respectively to two and three spin operators. Then we derive the general bound for a combination of two probabilities of projective measurements in mutually unbiased bases in $d$-dimensional Hilbert space. All of those uncertainty inequalities can be applied to construct different thermodynamic cycles such that the violation of those inequalities will lead to the violation of the second law of thermodynamics. This reveals the relationship between fine-grained uncertainty and the second law of t…
The thermodynamic uncertainty relation, originally derived for classical Markov-jump processes, provides a trade-off relation between precision and dissipation, deepening our understanding of the performance of quantum thermal machines. Here, we examine the interplay of quantum system coherences and heat current fluctuations on the validity of the thermodynamics uncertainty relation in the quantum regime. To achieve the current statistics, we perform a full counting statistics simulation of the Redfield quantum master equation. We focus on steady-state quantum absorption refrigerators where no…
The thermodynamic uncertainty relation (TUR) is expected to hold in nanoscale electronic conductors, when the electron transport process is quantum coherent and the transmission probability is constant (energy and voltage independent). We present measurements of the electron current and its noise in gold atomic-scale junctions and confirm the validity of the TUR for electron transport in realistic quantum coherent conductors. Furthermore, we show that it is beneficial to present the current and its noise as a TUR ratio in order to identify deviations from noninteracting-electron coherent dynam…
As a foundation of modern physics, uncertainty relations describe an ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertain relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position $x$ and momentum $p$ operators, Heisenberg's uncertainty principle points exactly to the area of the variances of $x$ and $p$. Moreover, for finite-dim…
The problem of time is a notable obstacle towards the recognition of quantum theory as the ultimate fundamental description of nature. Quantum theory may not be complete if founded upon classical notions. Louis de Broglie, seeming to be more or less convinced about the ontology of his proposed matter waves, tried to develop a theory of sub-quantum degrees of freedom relying on statistical thermodynamics. He realized a quantum particle as a fluctuating dense corpuscle formed via non-linear effects from a sub-quantum medium. A wave on the medium guides the vibrating corpuscle. He argued that an …
Historically, Kennard was the first to choose the standard deviation as a quantitative measure of uncertainty, and neither he nor Heisenberg explicitly explained why this choice should be appropriate from the experimental physical point of view. If a particle is prepared by a single slit of spatial width $Δx$, it has been shown that a finite standard deviation $σ_p<\infty$ can only be ensured if the wave-function is zero at the edge of $Δx$, otherwise it does not exist [8]. Under this circumstances the corresponding sharp inequality is $σ_p Δx\geq π\hbar$. This bound will be reconsidered from …
Quantitative measure of disorder or randomness based on the entropy production characterizes thermodynamical irreversibility, which is relevant to the conventional second law of thermodynamics. Here we report, in a quantum mechanical fashion, the first theoretical prediction and experimental exploration of an information-theoretical bound on the entropy production. Our theoretical model consists of a simplest two-level dissipative system driven by a purely classical field, and under the Markovian dissipation, we find that such an information-theoretical bound, not fully validating quantum rela…
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 backacti…
Thermodynamic Uncertainty Relations in Chaotic Andreev Billiards
Universidade Estadual Paulista（圣保罗州立大学）；Laboratório Nacional de Luz Síncrotron, Centro Nacional de Pesquisa em Energia e Materiais（巴西国家同步辐射光源实验室，国家能源与材料研究中心）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）
Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein-Hawking (black hole) entropy, which relates the entropy to the cross-sectional area of the black hole horizon. Using generalized uncertainty principle (GUP), corrections to the geometric entropy and thermodynamics of black hole will be introduced. The impact of GUP on the entropy near the horizon of three types of black holes; Schwarzschild, Garfinkle-Horowitz-Strominger and Reissner-Nordström is determined. It is found that the logarithmic divergence in the entropy-area relation turns to be …
Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are deriv…
