Fluctuation Theorems for a Quantum Channel

We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary nonequilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry, and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.

Popular Summary

In everyday experience, thermodynamics is deterministic: Heat flows from hot to cold, and a dropped glass remains shattered. However, in the quantum realm, where objects can exist in many states simultaneously, these rules might be bent or even broken. To better understand how well thermodynamics applies to quantum systems, we take a look at fluctuation theorems, which underlie the second law of thermodynamics and establish relationships between transition probabilities of processes that move forward and backward in time. We show that fluctuation theorems can be generalized for an arbitrary quantum process, which allows us to establish a powerful framework to understand quantum information theory and thermodynamics.

To construct fluctuation theorems for a quantum channel, we need to understand how the reverse process can be defined for a quantum channel and what the fluctuating quantities are. Our fluctuation theorem relates the entropy production throughout the forward and backward quantum processes, while generalized notions of entropy—modified to include coherence, as defined by 20th-century polymath von Neumann—and information exchange in the quantum regime play a central role. Conventional thermodynamic fluctuation theorems are special cases of our results, and heat can be understood as a particular type of information exchange in the form of energy.

The second law of thermodynamics can be expressed in terms of the quantum relative entropy, providing an important physical insight connecting the arrow of time and recoverability of the quantum channel. In return, the nonincreasing theorems of quantum resources such as entanglement and coherence can be understood under this unified framework.

Figure 1

Schematic for the standard TPM scheme for the forward $A\stackrel{\mathcal{N}}{\to }B$ and backward $\tilde{A}\stackrel{\mathcal{R}}{\leftarrow }B$ processes. The distributions $P_{\to }(x,y^{\prime })$ and $P_{\leftarrow }(x,y^{\prime })$ are defined in the TPM between two different times 0 and $\tau$.

Figure 2

Detailed balance of pure states for the JC Hamiltonian described in Eq. (5) with the parameters $\beta =1$, $\hslash {\omega }_0=1$, and $g=0.1$. Panels (a)–(c) represent the results without noise, and panels (d)–(f) represent the results with noise ${\mathcal{L}}_{\text{noise}}$ given by Eq. (10) with $\mathrm{\Gamma }=0.1$. Panels (a) and (d) refer to the forward (blue solid lines) and backward (red dashed lines) transition probabilities of $|g⟩\to |e⟩$ versus the evolution time $\tau$. Panels (b) and (e) refer to the state transition $|g⟩+|e⟩\to |g⟩-|e⟩$. Panales (c) and (f) show the ratio $T(|\psi ⟩\to |{\varphi }^{\prime }⟩)/\tilde{T}(|\tilde{\psi }⟩\leftarrow |\widetilde{{\varphi }^{\prime }}⟩)$ between the forward and backward transition probabilities from $|\psi ⟩=\sqrt{r}|g⟩+\sqrt{1-r}|e⟩$ to $|{\varphi }^{\prime }⟩=\sqrt{1-r}|g⟩-\sqrt{r}|e⟩$ versus energy difference $\mathrm{\Delta }E=E_{{\varphi }^{\prime }}-E_{\psi }$ (blue solid lines). Dashed lines refer to the classical thermodynamic detailed balance $e^{-\beta \mathrm{\Delta }E}$. The green and red dots are obtained for the transition with $\mathrm{\Delta }E=0$ for the coherent and incoherent initial fields, respectively. The additional factor $\mathrm{\Upsilon }\approx 1.27$ can be observed for the case with coherence. This ratio depends only on the values of $\beta$ and ${\omega }_0$ but not on $g$, $\mathrm{\Gamma }$, or $\tau$.

Figure 3

Quantum information exchange ($\delta q$) is equal to the entropy difference ($\delta s$) for the reference state $\gamma \stackrel{\mathcal{N}}{\to }\mathcal{N}(\gamma )$ (top). When the recovery map ${\mathcal{R}}_{\gamma }$ does not fully recover the quantum state $\rho$, there is entropy production, $\sigma =\delta s-\delta q\ne 0$ (bottom).

Figure 4

(Top) Distribution of the forward $P_{\to }(\sigma )$ (blue) and backward $P_{\leftarrow }(-\sigma )$ (red) entropy productions in the JC Hamiltonian. The same parameters for $H_{\mathrm{JC}}$ are chosen as in Fig. 2 with $\tau =18.66$. The initial atomic state is given by $\rho =(1/2)|\psi ⟩⟨\psi |+\mathbb{1}/4$, with $|\psi ⟩=(|g⟩+|e⟩)/\sqrt{2}$. When the field is initially in the incoherent thermal state ${\gamma }_f$, $P_{\to }(\sigma )$ and $P_{\leftarrow }(-\sigma )$ are always positive (inset). When the initial field is in the coherent Gibbs state $|{\gamma }_f⟩$, $P_{\to }(\sigma )$ and $P_{\leftarrow }(-\sigma )$ can be negative (main). (Bottom) For both coherent (green) and incoherent (blue) cases, the entropy production satisfies the fluctuation theorem $\log [P_{\to }(\sigma )/P_{\leftarrow }(-\sigma )]=\sigma$ in Eq. (16).

Figure 5

Rotated Petz recovery maps (top). The reference state $\gamma$ is fully recovered by the Petz recovery map, i.e., $({\mathcal{R}}_{\gamma }^{\theta }\circ \mathcal{N})(\gamma )=\gamma$. For a covariant quantum channel $\mathcal{N}$ satisfying ${\mathcal{J}}_{\mathcal{N}(\gamma )}^{-i\theta }\circ \mathcal{N}\circ {\mathcal{J}}_{\gamma }^{i\theta }=\mathcal{N}$, all the rotated Petz recovery maps are the same, i.e., ${\mathcal{R}}_{\gamma }^{\theta }={\mathcal{R}}_{\gamma }$ (bottom).

Figure 6

Quantum information exchange for the two-level atom with the fixed reference state ${\gamma }_a=e^{-\beta H_a}/Z_a$. All possible transitions are described by the arrows, and some transformations are selected to show the values of quantum information exchange. This can be generalized to a higher-dimensional system.

Figure 7

(a) Real and (b) imaginary parts of $P_{\to }({\sigma }_R+i{\sigma }_I)$ for the JC model interacting with the coherent Gibbs state $|{\gamma }_f⟩$. The initial state and the parameters are the same as in Fig. 4. The red points indicate the negativities. (d) The transition probability between the rotated eigenstates $T(e^{-i\beta H_a\theta /2}|{\psi }_{\mu }⟩\to e^{-i\beta H_a\theta /2}|{\varphi }_{{\nu }^{\prime }}^{\prime }⟩)$ versus $\theta$ and (e) its Fourier transform with normalization. The peaks indicate the points where the imaginary entropy productions occur (i.e., ${\sigma }_I=0,\pm \beta \hslash {\omega }_0/2,\pm \beta \hslash {\omega }_0$). (f) When the initial field is in the Gibbs state, the transition probability does not depend on the rotation, and (c) no imaginary entropy production occurs.

Figure 8

Free-energy loss (blue) versus asymmetry loss (brown) in the JC Hamiltonian with thermal noise. The initial state and the parameters are the same as in Fig. 4, and $P_{\to }({\sigma }_R)$ for each case is given by the summation over ${\sigma }_I$. Both the free-energy and coherence loss satisfy the integral QFT: $⟨e^{\beta \delta F}⟩=e^{\beta \mathrm{\Delta }{F}_{\mathrm{eq}}}=1$ and $⟨e^{\delta C_R}⟩=1$.