Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy

两级开放量子系统中的完全正性、正性与长时间渐近行为

Complete positivity, positivity and long-time asymptotic behavior in a two-level open quantum system

本文研究两级开放量子系统中完全正性、正性与非马尔可夫性的条件，建立了弛豫率条件以确保动力学映射的完全正性和正性，并分析了系统趋于平衡态的长时间渐近行为。

Positivity and complete positivity of differentiable quantum processes

We study quantum processes, as one parameter families of differentiable completely positive and trace preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-divisibility only depend on the dissipation matrix in the master equation form…

Dynamical matrix for arbitrary quadratic fermionic bath Hamiltonians and non-Markovian dynamics of one and two qubits in an Ising model environment

We obtain the analytical expression for the Kraus decomposition of the quantum map of an environment modeled by an arbitrary quadratic fermionic Hamiltonian acting on one or two qubits, and derive simple functions to check the non-positivity of the intermediate map. These functions correspond to two different sufficient criteria for non-Markovianity. In the particular case of an environment represented by the Ising Hamiltonian, we discuss the two sources of non-Markovianity in the model, one due to the finite size of the lattice, and another due to the kind of interactions.

开放量子系统的非平衡动力学中熵产的局部方法

Local approach to entropy production in the nonequilibrium dynamics of open quantum systems

本文研究了开放量子系统非平衡动力学中熵产的局部方法，证明了马尔可夫动力学与熵产率正性之间的关系，并展示了非马尔可夫动力学的熵产率正性反例。

Non-Markovianity and negative entropy production rates

Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovian dynamics. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovian dynamics and when it does not. This is established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a system in cont…

Monotonicity of the Quantum Relative Entropy Under Positive Maps

We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Be…

Positivity of the assignment map implies complete positivity of the reduced dynamics

Consider the set $\mathcal{S}=\lbraceρ_{SE}\rbrace$ of possible initial states of the system-environment. The map which assigns to each $ρ_{S}\in \mathrm{Tr}_{E}\mathcal{S}$ a $ρ_{SE}\in \mathcal{S}$ is called the assignment map. The assignment map is Hermitian, in general. In this paper, we restrict ourselves to the case that the assignment map is, in addition, positive and show that this implies that the so-called reference state is a Markov state. Markovianity of the reference state leads to existence of another assignment map which is completely positive. So, the reduced dynamics of the sy…

量子热力学的表述及其在开放系统中的应用

Formulations of Quantum Thermodynamics and Applications in Open Systems

本文提出基于ergotropy的热力学表述，通过热量与von Neumann熵的直接联系，更准确地测量非马尔可夫性，并保持非平衡温度非负。

量子相对熵的第二定律型不等式：导论

Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction

本文以量子相对熵的正定性和单调性为核心，系统介绍其与热力学第二定律相关的不等式，并推广到量子涨落定理、Hatano-Sasa不等式、数据处理不等式及带反馈控制的广义第二定律。

Five approaches to exact open-system dynamics: Complete positivity, divisibility and time-dependent observables

To extend the classical concept of Markovianity to an open quantum system, different notions of the divisibility of its dynamics have been introduced. Here we analyze this issue by five complementary approaches: equations of motion, real-time diagrammatics, Kraus-operator sums, as well as time-local (TCL) and nonlocal (Nakajima-Zwanzig) quantum master equations. As a case study featuring several types of divisible dynamics, we examine in detail an exactly solvable noninteracting fermionic resonant level coupled arbitrarily strongly to a fermionic bath at arbitrary temperature in the wideband l…

Monotonicity with Volume of Entropy and of Mean Entropy for Translationally Invariant Systems as Consequences of Strong Subadditivity

We consider some questions concerning the monotonicity properties of entropy and mean entropy of states on translationally invariant systems (classical lattice, quantum lattice and quantum continuous). By taking the property of strong subadditivity, which for quantum systems was proven rather late in the historical development, as one of four primary axioms (the other three being simply positivity, subadditivity and translational invariance) we are able to obtain results, some new, some proved in a new way, which appear to complement in an interesting way results proved around thirty years ago…

相对熵与挠率耦合

Relative Entropy and Torsion Coupling

本文在爱因斯坦-嘉当引力微扰框架下，通过要求全息CFT中相对熵的正性，约束了轴流-挠率耦合、费米子质量及状态方程。

海森堡图像中不可逆量子动力学的可除性

Divisibility of Non-Invertible Quantum Dynamics in the Heisenberg Picture

本文研究海森堡图像中不可逆量子动力学的可除性，证明CP可除性等价于算子范数单调性，刻画广义逆的完全正性障碍，并给出基于单位完全正投影的充分构造，且对量子比特动力学保证完全正广义逆传播子。

Phase covariant qubit dynamics and divisibility

Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates $γ_z(t)$, $γ_-(t)$, and $γ_+(t)$, respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase …