The contact temperature of arbitrary quantum states

Technical Summary: The Contact Temperature of Arbitrary Quantum States

Problem Statement
The paper addresses the challenge of assigning a temperature to an arbitrary, non-equilibrium quantum state $\rho$ of a finite-dimensional system. While equilibrium states are well-defined by the Gibbs density matrix $\gamma_\beta = e^{-\beta H}/\text{Tr}(e^{-\beta H})$, extending the concept of temperature to general states remains an open problem. Existing approaches include the "effective temperature" $\beta^*$, defined as the Gibbs temperature yielding the same average energy, or the minimizer of the relative entropy. However, these definitions are often abstract or lack an operational physical interpretation. The authors aim to define a "contact temperature" $\beta_c$ through an operational scheme: coupling the system to a universal thermometer and determining the thermometer's temperature at which the net heat flow vanishes.

Methodology
The authors propose a specific model of a "universal thermometer" (System B) coupled to the system of interest (System A).

1. System A: A $d$-level quantum system with a non-degenerate Hamiltonian $H_A$ and energies $\varepsilon_1 < \dots < \varepsilon_d$.
2. System B (Thermometer): An infinite-dimensional quantum system with a Hamiltonian $H_B$ having equally spaced eigenvalues $\eta_k = k/N$ (where $N$ is large). The thermometer is prepared in a Gibbs state $\tau_\beta = e^{-\beta H_B}/\text{Tr}(e^{-\beta H_B})$ at inverse temperature $\beta$.
3. Interaction: The systems interact via a unitary operator $U$ acting on the composite Hilbert space, satisfying the energy conservation condition $[U, H_A + H_B] = 0$.
4. Genericity Assumption: To ensure a "good" thermal contact, the authors impose two conditions:
   • Resonance: The energy differences (Bohr frequencies) of System A must be multiples of the thermometer's level spacing $1/N$.
   • Equiprobability: The transition probabilities between any two eigenstates of the uncoupled Hamiltonian $H_A + H_B$ with the same total energy are equal. This is mathematically modeled by assuming $U$ acts as a random unitary (Haar measure) within each energy eigenspace.

The central quantity is the heat flow $Q_A(\beta)$, defined as the change in the expected energy of System A after the interaction:
$$Q_A(\beta) = \text{Tr}_{AB}(U(\rho \otimes \tau_\beta)U^* H_A) - \text{Tr}_A(\rho H_A)$$
The contact temperature $\beta_c$ is defined as the unique value of $\beta$ for which $Q_A(\beta) = 0$.

Key Results and Contributions

1. Explicit Expression for Heat Flow:
   The authors derive an explicit formula for the heat flow $Q_A(\beta)$ for $\beta > 0$. The flow depends only on the energy spectrum of $H_A$ and the populations $p_j$ of the state $\rho$ in the energy eigenbasis, independent of the specific details of the thermometer's degeneracy or the gap size $N$ (in the limit $N \to \infty$).
   $$Q_A(\beta) = \sum_{1 \le j < k \le d} \frac{e^{-\beta(\varepsilon_k - \varepsilon_j)}}{k(k-1)} p_j \sum_{l=1}^{k-1} (\varepsilon_k - \varepsilon_l) - \sum_{k=2}^d \frac{p_k}{k} \sum_{l=1}^{k-1} (\varepsilon_k - \varepsilon_l)$$

2. Existence and Uniqueness of Contact Temperature:

   • Case $Q_A(0) \ge 0$: If the heat flow at infinite temperature is non-negative, there exists a unique non-negative root $\beta_c \ge 0$. The function $\beta \mapsto Q_A(\beta)$ is strictly decreasing and continuous.
   • Case $Q_A(0) < 0$: For states where the system transfers heat even to an infinitely hot reservoir (violating the equilibrium second law intuition), the standard formula for $\beta > 0$ yields no root. The authors address this by considering a finite-level thermometer (with $K$ levels) defined for all $\beta \in \mathbb{R}$, and then taking the limit $K \to \infty$. This yields an extended heat flow function $\bar{Q}_A(\beta)$ defined for all $\beta \in \mathbb{R}$.
   • Result: For any state $\rho$ (excluding the ground state and the most excited state), there exists a unique real root $\beta_c \in \mathbb{R}$ such that $\bar{Q}_A(\beta_c) = 0$.
     • If $\rho$ is the ground state, $\beta_c = +\infty$.
     • If $\rho$ is the most excited state, $\beta_c = -\infty$.
     • For all other states, $\beta_c$ is finite. The sign of $\beta_c$ matches the sign of $Q_A(0)$.

3. Structural Analysis via Stochastic Matrices:
   The paper analyzes the interaction through the lens of reduced thermal quantum channels $\Phi_\beta$. The heat flow is related to a stochastic matrix $G(\beta)$ describing the transition probabilities between energy levels.

   • The matrix $G(\beta)$ is shown to be strictly positive and satisfy detailed balance.
   • The spectrum of the channel $\Phi_\beta$ is analyzed: it has a simple eigenvalue 1 (corresponding to the Gibbs state) and all other eigenvalues have modulus strictly less than 1.
   • This spectral gap ensures that repeated interactions with a fresh thermometer at temperature $\beta$ drive the system exponentially fast to the Gibbs state $\gamma_\beta$. Consequently, the contact temperature of the system's state converges to $\beta$ under repeated thermalization.

4. Comparison with Effective Temperature:
   The authors demonstrate that the contact temperature $\beta_c$ is generally distinct from the effective temperature $\beta^*$ (defined via energy matching or relative entropy minimization). In specific examples (e.g., a 3-level system), $\beta_c$ and $\beta^*$ can differ significantly, and their relationship depends on the specific populations of the state.

Significance and Claims
The paper claims to provide a universal and operational definition of temperature for arbitrary finite-dimensional quantum states.

• Universality: The thermometer model yields a unique contact temperature regardless of the specific physical nature of the system, provided the system is finite-dimensional.
• Operational Definition: Unlike abstract definitions based on entropy maximization, $\beta_c$ is defined by a physical process: the vanishing of heat flow upon coupling to a specific type of thermal reservoir.
• Handling Non-Equilibrium: The framework successfully assigns a temperature (potentially negative) to states that are "far from equilibrium" and would otherwise violate standard thermodynamic intuitions (e.g., transferring heat to an infinitely hot bath).
• Mathematical Rigor: The work establishes the existence and uniqueness of this temperature through rigorous analysis of the heat flow function and the spectral properties of the associated quantum channels.

The authors do not propose experimental implementations or specific applications to quantum machines beyond noting the connection to "virtual temperatures" in thermal machines. The work remains a theoretical construction defining a new thermodynamic quantity for quantum states.