论文标题
连续时间量子马尔可夫半群的热力学形式主义：详细的平衡状况，熵，压力和平衡量子过程
Thermodynamic formalism for continuous-time quantum Markov semigroups: the detailed balance condition, entropy, pressure and equilibrium quantum processes
论文作者
论文摘要
$ m_n（\ mathbb {c}）$表示$ n $复杂矩阵的$ n $集合。考虑连续时间量子量子份$ \ MATHCAL {p} _t = e^{t \，\ MATHCAL {l}} $，$ t \ geq 0 $，其中$ \ MATHCAL {l}：m_n（m_n（mathbb {c}）如果我们假设$ \ MATHCAL {l}（i）= 0 $，我们将调用$ e^{t \，\ Mathcal {l}} $，$ t \ geq 0 $ a Quantum Markov Semigroup。给定一个固定密度矩阵$ρ=ρ=ρ_{\ MATHCAL {l}} $，对于量子Markov Semigroup $ \ Mathcal {p} _t $，$ t \ geq 0 $，我们可以定义连续的时间固定的量子Markov进程，$ x_t $ tap $ t \ geq。操作员$ \ Mathcal {l} _0：m_n（\ Mathbb {c}）\ to m_n（\ Mathbb {C}）$，我们将提出$ m_n（\ Mathbb {c}）$ m_n（\ mathbb {c}）$的熵的自然概念。给定一个Hermitian操作员$ a：\ mathbb {c}^n \ to \ mathbb {c}^n $（扮演哈密顿量的角色），我们将研究$ a $ a $的压力的变性原理的版本。密度矩阵$ρ_A$最大化压力将称为平衡密度矩阵。从$ρ_a$，我们将得出一个新的无限发电机$ \ mathcal {l} _a $。最后，由semogroup $ \ MATHCAL {p} _t = e^{t \，\ Mathcal {l} _a} $，$ t \ geq 0 $定义的连续时间量子马尔可夫的过程将被称为连续的平均量式equilibribibium Quantibibim Markiltian $ a $ a $ a $ a $ a $ a $ a $ a $ a $它对应于哈密顿$ a $的作用的量子热力学平衡。
$M_n(\mathbb{C})$ denotes the set of $n$ by $n$ complex matrices. Consider continuous time quantum semigroups $\mathcal{P}_t= e^{t\, \mathcal{L}}$, $t \geq 0$, where $\mathcal{L}:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ is the infinitesimal generator. If we assume that $\mathcal{L}(I)=0$, we will call $e^{t\, \mathcal{L}}$, $t \geq 0$ a quantum Markov semigroup. Given a stationary density matrix $ρ= ρ_{\mathcal{L}}$, for the quantum Markov semigroup $\mathcal{P}_t$, $t \geq 0$, we can define a continuous time stationary quantum Markov process, denoted by $X_t$, $t \geq 0.$ Given an {\it a priori} Laplacian operator $\mathcal{L}_0:M_n(\mathbb{C}) \to M_n(\mathbb{C})$, we will present a natural concept of entropy for a class of density matrices on $M_n(\mathbb{C})$. Given an Hermitian operator $A:\mathbb{C}^n\to \mathbb{C}^n$ (which plays the role of an Hamiltonian), we will study a version of the variational principle of pressure for $A$. A density matrix $ρ_A$ maximizing pressure will be called an equilibrium density matrix. From $ρ_A$ we will derive a new infinitesimal generator $\mathcal{L}_A$. Finally, the continuous time quantum Markov process defined by the semigroup $\mathcal{P}_t= e^{t\, \mathcal{L}_A}$, $t \geq 0$, and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian $A$. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian $A$.
