Structure of the thermodynamic arrow of time in classical and quantum theories

In this work we analyse the structure of the thermodynamic arrow of time, defined by transformations that leave the thermal equilibrium state unchanged, in classical (incoherent) and quantum (coherent) regimes. We note that in the infinite-temperature limit the thermodynamic ordering of states in both regimes exhibits a lattice structure. This means that when energy does not matter and the only thermodynamic resource is given by information, the thermodynamic arrow of time has a very specific structure. Namely, for any two states at present there exists a unique state in the past consistent with them and with all possible joint pasts. Similarly, there also exists a unique state in the future consistent with those states and with all possible joint futures. We also show that the lattice structure in the classical regime is broken at finite temperatures, i.e., when energy is a relevant thermodynamic resource. Surprisingly, however, we prove that in the simplest quantum scenario of a two-dimensional system, this structure is preserved at finite temperatures. We provide the physical interpretation of these results by introducing and analysing the history erasure process, and point out that quantum coherence may be a necessary resource for the existence of an optimal erasure process.Comment: 14 pages, 10 figures. Published version. Expanded discussion and a new section on history erasure process adde

Tunneling transfer protocol in a quantum dot chain immune to inhomogeneity

We propose a quantum dot (QD) implementation of a quantum state transfer channel. The proposed channel consists of N vertically stacked QDs with the nearest neighbor tunnel coupling, placed in an axial electric field. We show that the system supports high-fidelity transfer of the state of a terminal dot both by free evolution and by adiabatic transfer. The protocol is to a large extent insensitive to inhomogeneity of the energy parameters of the dots and requires only a global electric field.Comment: 3 pages, 6 figure

Quantum-state transfer in spin chains via isolated resonance of terminal spins

We propose a quantum-state transfer protocol in a spin chain that requires only the control of the spins at the ends of the quantum wire. The protocol is to a large extent insensitive to inhomogeneity caused by local magnetic fields and perturbation of exchange couplings. Moreover, apart from the free evolution regime, it allows one to induce an adiabatic spin transfer, which provides the possibility of performing the transfer on demand. We also show that the amount of information leaking into the central part of the chain is small throughout the whole transfer process (which protects the information sent from being eavesdropped) and can be controlled by the magnitude of the external magnetic field.Comment: 7 pages, 5 figures. Published versio

Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.Comment: 31 pages, 10 figures. Final version, to be published in Quantu

Distinguishing classically indistinguishable states and channels

We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between $M$ quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of $M$ quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of $M$ perfectly distinguishable states (channels) that are classically indistinguishable.Comment: 22 pages, 8 figures. Published versio

Coherifying quantum channels

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels $\Phi^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamio{\l}kowski state $J_{\Phi}$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamio{\l}kowski state $J_\Phi^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process $\Phi^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_\Phi^{\mathcal{C}}$, and relate the latter to the unitarity of $\Phi^{\mathcal{C}}$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.Comment: 20 pages, 8 figures. Published versio

Resource engines

In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of thermodynamics that studies heat engines operating between two baths at different temperatures. Here, we investigate the performance of resource engines, which replace the access to two heat baths at different temperatures with two arbitrary constraints on state transformations. The idea is to imitate the action of a two--stroke heat engine, where the system is sent to two agents (Alice and Bob) in turns, and they can transform it using their constrained sets of free operations. We raise and address several questions, including whether or not a resource engine can generate a full set of quantum operations or all possible state transformations, and how many strokes are needed for that. We also explain how the resource engine picture provides a natural way to fuse two or more resource theories, and we discuss in detail the fusion of two resource theories of thermodynamics with two different temperatures, and two resource theories of coherence with respect to two different bases.Comment: 25 pages, 4 figures, comments welcom

Thermal recall: Memory-assisted Markovian thermal processes

We develop a resource-theoretic framework that allows one to bridge the gap between two approaches to quantum thermodynamics based on Markovian thermal processes (which model memoryless dynamics) and thermal operations (which model arbitrarily non-Markovian dynamics). Our approach is built on the notion of memory-assisted Markovian thermal processes, where memoryless thermodynamic processes are promoted to non-Markovianity by explicitly modelling ancillary memory systems initialised in thermal equilibrium states. Within this setting, we propose a family of protocols composed of sequences of elementary two-level thermalisations that approximate all transitions between energy-incoherent states accessible via thermal operations. We prove that, as the size of the memory increases, these approximations become arbitrarily good for all transitions in the infinite temperature limit, and for a subset of transitions in the finite temperature regime. Furthermore, we present solid numerical evidence for the convergence of our protocol to any transition at finite temperatures. We also explain how our framework can be used to quantify the role played by memory effects in thermodynamic protocols such as work extraction. Finally, our results show that elementary control over two energy levels at a given time is sufficient to generate all energy-incoherent transitions accessible via thermal operations if one allows for ancillary thermal systems.Comment: 20 pages, 14 figures. Substantially extended results with new sections and application. Published versio