Time-reversal symmetry and its violation (T-violation) are fundamental across diverse physics domains, from particle physics to fluctuation theorems and reciprocity. While time-reversal symmetry for isolated systems is simply determined by the Hamiltonian, quantifying the magnitude of T-violation, especially in noisy quantum processes, has been a subject of ongoing debate. Here, we propose an operationally meaningful framework to quantify the intrinsic T-violation of quantum channels. We integrate operationally defined time-reversal transformations into dynamical resource theory to formulate T-violation as a resource. T-violation decomposes into two distinct components: kinematic T-violation originating from antiunitary motion-reversal, and nonunitality driven purely by thermodynamics. This decomposition resolves the longstanding confusion between broken time-reversal symmetry and non-invertibility. Finally, we analyze the resulting resource theory of kinematic T-violation, showing that it is neither quality-like nor quantity-like, and proving the existence of a universal golden unit even within classical channels.
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24 pages, 5 figures, 1 table. Comments are welcome
Time-reversal symmetry and its violation (T-violation) are fundamental across diverse physics domains, from particle physics to fluctuation theorems and reciprocity. While time-reversal symmetry for isolated systems is simply determined by the Hamiltonian, quantifying the magnitude of T-violation, especially in noisy quantum processes, has been a subject of ongoing debate. Here, we propose an operationally meaningful framework to quantify the intrinsic T-violation of quantum channels. We integrate operationally defined time-reversal transformations into dynamical resource theory to formulate T-violation as a resource. T-violation decomposes into two distinct components: kinematic T-violation originating from antiunitary motion-reversal, and nonunitality driven purely by thermodynamics. This decomposition resolves the longstanding confusion between broken time-reversal symmetry and non-invertibility. Finally, we analyze the resulting resource theory of kinematic T-violation, showing that it is neither quality-like nor quantity-like, and proving the existence of a universal golden unit even within classical channels.
Deriving an arrow of time from time-reversal symmetric microscopic dynamics is a fundamental open problem in many areas of physics, ranging from cosmology, to particle physics, to thermodynamics and statistical mechanics. Here we focus on the derivation of the arrow of time in open quantum systems and study precisely how time-reversal symmetry is broken. This derivation involves the Markov approximation applied to a system interacting with an infinite heat bath. We find that the Markov approximation does not imply a violation of time-reversal symmetry. Our results show instead that the time-reversal symmetry is maintained in the derived equations of motion. This imposes a time-symmetric formulation of quantum Brownian motion, Lindblad and Pauli master equations, which hence describe thermalisation that may occur into two opposing time directions. As a consequence, we argue that these dynamics are better described by a time-symmetric definition of Markovianity. Our results may reflect on the formulations of the arrow of time in thermodynamics, cosmology, and quantum mechanics.
The Bayesian inverse of a quantum channel E is a channel F in the reverse direction of E that yields time-symmetric correlations for sequential measurements performed on open quantum systems. Such an operational form of time-reversal symmetry for open quantum systems is quite remarkable, as the dynamics of open quantum systems are inherently irreversible due to system-environment interactions. Similar to the Petz map, a Bayesian inverse F is defined with respect to a fiducial reference state ρ for the channel E. However, Bayesian inverses do not always exist, and it is often a non-trivial task to determine the set of states ρ for which a Bayesian inverse of E exists. In this work, we solve the general problem of quantum Bayesian inversion for unital channels acting on a single qubit. Our analysis is streamlined by demonstrating that finding a Bayesian inverse for a unital qubit channel may be reduced to finding a Bayesian inverse of a Pauli channel, which is simply a mixture of unitary channels associated with the Pauli matrices. As such, we provide a complete description of when operational time-reversal symmetry is attainable for sequential measurements of a single qubit in the presence of unital noise.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. For mixed states, however, conventional diagnostics of weak symmetry breaking can miss a stronger form of symmetry breaking associated with the possibility of exchanging conserved charges with an environment. We develop a resource theory of strong symmetry breaking by identifying the appropriate free states and free operations, and, for channels on a fixed system, by showing that the latter are precisely the operations realizable without exchanging the conserved charge with the environment. We systematically construct measures of strong symmetry breaking, including strong entanglement asymmetry and covariance matrices of symmetry generators, for a broad class of symmetry groups. We further completely characterize i.i.d. convertibility between arbitrary states under any compact Lie group strong symmetry. This identifies the combinations of resource-theoretically valid measures that quantify strong symmetry breaking for arbitrary states, including all mixed states. In particular, for U(1) symmetry, if the states have positive variances of the conserved quantity and equal strong-symmetry periods, the conversion rate is completely determined by the variance ratio whenever the input is pure or the output is weak symmetric. Thus, for these state conversions, the variance of the conserved quantity plays an operational role analogous to that of entanglement entropy in entanglement theory or quantum Fisher information in the resource theory of weak asymmetry. We further show how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open-system dynamics. We illustrate the framework with examples from quantum field theory, strong-to-weak spontaneous symmetry breaking, and a strong-symmetry analogue of Mpemba-type dynamics.
In standard physics, time reversal (T) is a mathematical abstraction: an antiunitary operator that formally reverses the direction of evolution. But what does it mean physically? How do you "reverse time" when you can't reverse scrambled eggs, un-mix gases, or un-bake a cake? This paper provides the answer that physics has been missing for a century. Using nothing but the language of cardboard tubes, colored corks, and paper sheets—objects you could find in a child's workshop—we solve what the most sophisticated quantum field theories cannot: what pure time reversal actually means when every transformation must be physically realizable, not just mathematically definable. The discovery is both radical and simple: T isn't about going backward. It's about stopping. ḱ → 0⁺: the geometric frame rate drops to zero. Billions of intermediate quantum states—the "movie" of a process—vanish into geometric silence. What remains when the frames stop generating? Only the bookends of reality: the beginning and the end. The proton and its anti-proton. The initial configuration and its final counterpart. No rewinding, no backtracking—just the collapse of history into its boundary conditions. This is more than a reinterpretation; it's an ontological revolution. Time isn't a dimension you travel along—it's a process you can halt. The arrow isn't reversed; it's erased. This work completes the "A Tale of Deep Symmetry" trilogy, providing the missing geometric interpretation of T that preserves the sacred CPT theorem while respecting a rigorous mechanical ontology. For physicists tired of mathematical abstractions that obscure rather than illuminate... For philosophers of science wrestling with the nature of time and reality... For AI researchers watching machines ask better questions than humans... For anyone who believes that the deepest truths about our universe can—and should—emerge from models simple enough to build at your kitchen table. The tubes and corks were never just toys. They were the Rosetta Stone for time itself. **Keywords:** Time reversal paradox, operational physics, geometric ontology, AI-driven discovery, CPT symmetry, foundational crisis, simplicity in science
Advancing quantum sensing tools for investigating atomic and nanoscale systems is crucial for the progress of quantum technologies. While many protocols use quantum probes to extract information from stationary or weakly coupled environments, challenges intensify at the atomic scale and the nanoscale, where the environment is inherently out of equilibrium and/or strongly coupled with the sensor. In this work, we demonstrate that time-reversal symmetry in the control dynamics of a quantum sensor is broken when the qubit sensor interacts with environments that are out of equilibrium (with nonstationary fluctuations), contain quantum non-Gaussian correlations, or exhibit intrinsic time-reversal symmetry breaking. Leveraging this phenomenon, we introduce a quantum sensing paradigm based on time-asymmetric dynamical control sequences, enabling the probing of the distance of an environment from equilibrium, its time-reversal symmetry-breaking behavior, or its quantum non-Gaussian nature. Additionally, we propose sensing strategies to distinguish between these different sources of symmetry breaking. We validate our approach through proof-of-principle experimental quantum simulations using solid-state nuclear magnetic resonance, where we drive the environment of a qubit sensor out of equilibrium and use our protocol to quantify the nonstationary characteristics of the generated states. Our findings highlight the potential of this framework for quantifying nonstationary behavior, designing tailored pump-probe experiments (including measurements of quantum information scrambling), and detecting quantum states that exhibit time-reversal symmetry breaking, time-crystal structures, or quantum non-Gaussian fluctuations. Overall, this work constitutes a step forward in designing quantum devices for atomic scale and nanoscale sensing, broadening their applicability to complex and dynamic quantum environments.