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arXiv:2305.15480 · PRX Quantum

Non-Abelian Transport Distinguishes Three Usually Equivalent Notions of Entropy Production

Twesh Upadhyaya, William F. Braasch, Gabriel Teixeira Landi, Nicole Yunger Halpern

Published Sep 23, 2024• 23 citations
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Abstract

We extend entropy production to a deeply quantum regime involving noncommuting conserved quantities. Consider a unitary transporting conserved quantities (“charges”) between two systems initialized in thermal states. Three common formulas model the entropy produced. They respectively cast entropy as an extensive thermodynamic variable, as an information-theoretic uncertainty measure, and as a quantifier of irreversibility. Often, the charges are assumed to commute with each other (e.g., energy and particle number). Yet quantum charges can fail to commute. Noncommutation invites generalizations, which we posit and justify, of the three formulas. The noncommutation of charges, we find, breaks the equivalence of the formulas. Furthermore, different formulas quantify different physical effects of the noncommutation of charges on entropy production. For instance, entropy production can signal contextuality—true nonclassicality—by becoming nonreal. This work opens up stochastic thermodynamics to noncommuting—and so particularly quantum—charges. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.030355

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We extend entropy production to a deeply quantum regime involving noncommuting conserved quantities. Consider a unitary transporting conserved quantities (“charges”) between two systems initialized in thermal states. Three common formulas model the entropy produced. They respectively cast entropy as an extensive thermodynamic variable, as an information-theoretic uncertainty measure, and as a quantifier of irreversibility. Often, the charges are assumed to commute with each other (e.g., energy and particle number). Yet quantum charges can fail to commute. Noncommutation invites generalizations, which we posit and justify, of the three formulas. The noncommutation of charges, we find, breaks the equivalence of the formulas. Furthermore, different formulas quantify different physical effects of the noncommutation of charges on entropy production. For instance, entropy production can signal contextuality—true nonclassicality—by becoming nonreal. This work opens up stochastic thermodynamics to noncommuting—and so particularly quantum—charges. Published by the American Physical Society 2024
Abstract Noncommuting conserved quantities have recently launched a subfield of quantum thermodynamics. In conventional thermodynamics, a system of interest and an environment exchange quantities—energy, particles, electric charge, etc.—that are globally conserved and are represented by Hermitian operators. These operators were implicitly assumed to commute with each other, until a few years ago. Freeing the operators to fail to commute has enabled many theoretical discoveries—about reference frames, entropy production, resource-theory models, etc. Little work has bridged these results from abstract theory to experimental reality. This paper provides a methodology for building this bridge systematically: we present a prescription for constructing Hamiltonians that conserve noncommuting quantities globally while transporting the quantities locally. The Hamiltonians can couple arbitrarily many subsystems together and can be integrable or nonintegrable. Our Hamiltonians may be realized physically with superconducting qudits, with ultracold atoms, and with trapped ions.
Modern quantum experiments provide examples of transport with noncommuting quantities, offering a tool to understand the interplay between thermal and quantum effects.Here we set forth a theory for non-Abelian transport in the linear response regime.Our key insight is to use generalized Gibbs ensembles with noncommuting charges as the basic building blocks and strict charge-preserving unitaries in a collisional setup.The linear response framework is then built using a collisional model between two reservoirs.We show that the transport coefficients obey Onsager reciprocity.Moreover, we find that quantum coherence, associated with the noncommutativity, acts so as to reduce the net entropy production, when compared to the case of commuting transport.This therefore provides a clear connection between quantum coherent transport and dissipation.As an example, we study heat and squeezing fluxes in bosonic systems, characterizing a set of thermosqueezing coefficients with potential applications in metrology and heat-to-work conversion in the quantum regime.
The pillars of quantum theory include entanglement and operators' failure to commute. The Page curve quantifies the bipartite entanglement of a many-body system in a random pure state. This entanglement is known to decrease if one constrains extensive observables that commute with each other (Abelian ``charges''). Non-Abelian charges, which fail to commute with each other, are of current interest in quantum thermodynamics. For example, noncommuting charges were shown to reduce entropy-production rates and may enhance finite-size deviations from eigenstate thermalization. Bridging quantum thermodynamics to many-body physics, we quantify the effects of charges' noncommutation---of a symmetry's non-Abelian nature---on Page curves. First, we construct two models that are closely analogous but differ in whether their charges commute. We show analytically and numerically that the noncommuting-charge case has more entanglement. Hence charges' noncommutation can promote entanglement.
Noncommutativity of observables is a defining feature of quantum physics. It plays a fundamental role in the formulation of the uncertainty principle for complementary variables and strongly affects the thermodynamic behavior of systems with noncommuting, that is, non-Abelian, conserved quantities. We here derive extensions of the first and second laws of thermodynamics, including nonequilibrium fluctuation relations, for driven quantum systems. We identify an additional non-Abelian contribution to the energy and entropy balances, without which these relations would be violated. This contribution can be harnessed to enhance both work extraction and nonequilibrium currents compared to what is achieved in commuting thermodynamics. These results demonstrate that noncommutativity is a potentially valuable thermodynamic resource.
Thermodynamic uncertainty relations (TURs) bound the precision of currents by entropy production, but quantum transport of noncommuting (non-Abelian) charges challenges standard formulations because different charge components cannot be monitored within a single classical frame. We derive a process-level matrix TUR starting from the operational entropy production Σ=D(ρSE′∥ρS′ ⁣⊗ ⁣ρE)Σ= D(ρ'_{SE}\|ρ'_S\!\otimes\!ρ_E)Σ=D(ρSE′​∥ρS′​⊗ρE​). Isolating the experimentally accessible bath divergence Dbath=D(ρE′∥ρE)D_{\mathrm{bath}}=D(ρ'_E\|ρ_E)Dbath​=D(ρE′​∥ρE​), we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors ΔqΔqΔq: Dbath≥B(Δq,V,V′)D_{\mathrm{bath}} \ge B(Δq,V,V')Dbath​≥B(Δq,V,V′), where the bound depends only on the transported-charge signal ΔqΔqΔq and the pre/post collision covariance matrices VVV and V′V'V′. In the small-fluctuation regime Dbath≥12 ΔqTV−1Δq+O(∥Δq∥4)D_{\mathrm{bath}}\geq\frac12\,Δq^{\mathsf T}V^{-1}Δq+O(\|Δq\|^4)Dbath​≥21​ΔqTV−1Δq+O(∥Δq∥4), while beyond linear response it remains accurate. Numerical strong-coupling qubit collisions illustrate the bound and demonstrate near-saturation across broad parameter ranges using only local measurements on the bath probe.
We derive detailed and intergral fluctuation relations as well as a thermodynamic uncertainty relation constraining the exchange statistics of an arbitrary number of noncommuting conserved quantities among two quantum systems in transport setups arbitrary far from equilibrium. These universal relations, valid without the need of any efficacy parameter, extend the well-known heat exchange fluctuation theorems for energy and particle transport to the case of non-Abelian quantum transport, where the noncommutativity of the charges allows going beyond standard thermodynamic behavior. In particular, we show that this can lead to enhanced precision in the current fluctuations, and it allows for the inversion of all currents against their affinity biases.