For open quantum systems with static environments, Chen, Chen and Chen (Phys. Rev. A 96, 062114 (2017)) identified the system's information flux with the negative entropy production rate, equated accumulated entropy production with system-environment mutual information, and obtained channel-resolved fluxes proportional to the decay rates of a time-local master equation, revealing partial information retrieval in weakly non-Markovian regimes where trace-distance measures are blind. This problem asks for the extension of that framework beyond its static-environment, undriven regime: an instantaneous, thermodynamically consistent information-flux accounting that separates environment displacement from correlation build-up and incorporates irreversible work for driven systems, together with operational channel-resolved measurement protocols and non-Markovianity measures that detect weak (P-divisible but not CP-divisible) memory effects invisible to existing quantity-based and entropy-production-rate witnesses. Subsequent work provides major partial progress - strong-coupling thermodynamic frameworks, Araki-Lieb bounds showing mutual information cannot dominate entropy production for extended baths, and clarified conditions for negative entropy production rates - but the instantaneous accounting for driven, non-static quantum systems, the channel-resolved measurement, and the weak-non-Markovianity diagnostics remain open.

Background

In the theory of open quantum systems, the reduced state of a system S interacting with an environment E evolves under a dynamical map Λt\Lambda_t that is completely positive and trace preserving (CPTP: it maps valid density matrices to valid density matrices and does so even when applied to a subsystem entangled with an ancilla). Dynamics is conventionally called Markovian when it is memoryless; a dynamical criterion is CP-divisibility, meaning Λt\Lambda_t can be split as Λt=Λt,sΛs\Lambda_t = \Lambda_{t,s}\Lambda_s with CPTP intermediate maps for all t>st > s, and failure of this splitting signals non-Markovian memory. The measure of Breuer, Laine and Piilo (BLP) quantifies non-Markovianity by revivals of the trace distance (the distinguishability of two states) between pairs of initially distinct states, interpreting revivals as information flowing back from E to S. A finer hierarchy grades divisibility by k-positivity (positivity of the map when tensored with the identity on a k-dimensional system): dynamics may be P-divisible (all intermediate maps positive) without being CP-divisible - for a qubit this is called weakly non-Markovian - or may fail even P-divisibility, in which case it is called essentially (or strongly) non-Markovian.

On the thermodynamic side, for a system with Hamiltonian Hsys(t)H_{\mathrm{sys}}(t) in contact with a bath at temperature TT, the nonequilibrium information content of a state ρ\rho is measured by the relative entropy Ineq(ρ)=S(ρ∥ρeq(t))I^{\mathrm{neq}}(\rho) = S(\rho \| \rho^{\mathrm{eq}}(t)) to the instantaneous thermal equilibrium state, where S(ρ∥σ)=−Tr[ρ(ln⁡ρ−ln⁡σ)]S(\rho\|\sigma) = -\mathrm{Tr}[\rho(\ln\rho - \ln\sigma)] is the quantum Kullback-Leibler divergence. Its changes obey ΔIneq=−ΔSirr+ΔWirr/kBT\Delta I^{\mathrm{neq}} = -\Delta S^{\mathrm{irr}} + \Delta W^{\mathrm{irr}}/k_B T (in the manner of Deffner and Lutz), where ΔSirr≥0\Delta S^{\mathrm{irr}} \ge 0 is the irreversible entropy production expressing the second law and ΔWirr\Delta W^{\mathrm{irr}} is the irreversible work (work performed beyond the equilibrium free-energy change), which vanishes for time-independent Hamiltonians.

Chen, Chen and Chen proposed a thermodynamic definition of the information flux of an open system, valid under a static-environment hypothesis: the environment's reduced state ρenv(t)\rho_{\mathrm{env}}(t) is time independent during the interaction and its Hamiltonian constant, while correlations with the system may still build up. In this regime they defined the total information flux as the negative entropy production rate, F=−∂tSirr\mathcal{F} = -\partial_t S^{\mathrm{irr}}; showed that the accumulated entropy production equals the system-environment mutual information Imut(t)I^{\mathrm{mut}}(t) (the shared information Ssys+Senv−StotS_{\mathrm{sys}} + S_{\mathrm{env}} - S_{\mathrm{tot}}), as a special case of the general decomposition ΔSirr=Imut(t)+Ienvneq(t)−Ienvneq(0)\Delta S^{\mathrm{irr}} = I^{\mathrm{mut}}(t) + I^{\mathrm{neq}}_{\mathrm{env}}(t) - I^{\mathrm{neq}}_{\mathrm{env}}(0) derived in the manner of Esposito, Lindenberg and Van den Broeck; and, within the time-local (Lindblad-form) master equation description, obtained channel-resolved fluxes Fi,j\mathcal{F}_{i,j} each proportional to the instantaneous decay rate γi,j(t)\gamma_{i,j}(t) of its dissipation channel. In a CNOT-gate model they showed that partial information retrieval through a single channel can occur in the weakly non-Markovian (P-divisible but not CP-divisible, 'PD1') regime while the total flux remains outgoing, so net quantity-based measures such as BLP return zero and cannot detect it.

The open problem is that this framework is restricted to static environments and undriven systems, and its operational use (measurement, diagnostics) was not demonstrated. Later work has shown how delicate the extension is: the system-environment mutual information is bounded by the Araki-Lieb inequality (which limits the difference of the subsystem entropies by the entropy of the total state) and need not dominate the entropy production; the link between negative entropy production rates and non-Markovianity holds only under specific conditions and fails at the rate level for driven quantum systems at strong coupling; and heat (energy) backflow and information backflow are distinct phenomena. A thermodynamically consistent, channel-resolved, operationally accessible information-flux characterization of non-Markovianity beyond the static-environment regime is still missing.

Problem Statement

Extend the thermodynamic information-flux characterization of non-Markovian open-system dynamics beyond the static-environment, undriven regime of Chen, Chen and Chen (Phys. Rev. A 96, 062114 (2017)), and make it operational. For an open quantum system, generally with a time-dependent Hamiltonian, interacting with an environment whose reduced state changes appreciably during the interaction (a non-static, 'vigorous' environment), the problem asks for:

(1) An instantaneous information flux that is thermodynamically consistent: it must reproduce, in the static-environment and time-independent-Hamiltonian limit, the definition F=−∂tSirr\mathcal{F} = -\partial_t S^{\mathrm{irr}} and the identity ΔSirr=Imut(t)\Delta S^{\mathrm{irr}} = I^{\mathrm{mut}}(t); it must respect the first and second laws and the exact integrated decomposition ΔSirr=Imut(t)+Ienvneq(t)−Ienvneq(0)\Delta S^{\mathrm{irr}} = I^{\mathrm{mut}}(t) + I^{\mathrm{neq}}_{\mathrm{env}}(t) - I^{\mathrm{neq}}_{\mathrm{env}}(0); it must separate and quantify the contributions from environment-state displacement versus system-environment correlation build-up (including regimes where the entropy production is comparable to the Araki-Lieb bound 2ln⁡N2\ln N on the mutual information, with NN the system's Hilbert-space dimension, where no general rule is known); and it must incorporate, rather than neglect, the irreversible-work contribution ΔWirr/kBT\Delta W^{\mathrm{irr}}/k_B T for driven systems.

(2) Operational measurement protocols that resolve the channel-resolved fluxes Fi,j\mathcal{F}_{i,j} of the time-local description, in particular observing channel-resolved partial information retrieval in weakly non-Markovian (P-divisible but not CP-divisible) regimes.

(3) Operational measures of non-Markovianity constructed from channel-wise thermodynamic fluxes that detect and quantify weak non-Markovianity, which net quantity-based measures (BLP-type) and negative-entropy-production-rate witnesses provably cannot see.

These three components are jointly mandatory and serve one overarching objective: a thermodynamically grounded, operationally accessible, channel-resolved information-flux framework for non-Markovian dynamics valid for general environments and drivings; a partial deliverable that covers only one or two of them does not resolve the task.

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