Integral quantum fluctuation theorems under measurement and feedback control

We derive integral quantum fluctuation theorems and quantum Jarzynski equalities for a feedback-controlled system and a memory which registers outcomes of the measurement. The obtained equalities involve the information content, which reflects the information exchange between the system and the memory, and take into account the back action of a general measurement contrary to the classical case. The generalized second law of thermodynamics under measurement and feedback control is reproduced from these equalities.

Figure 1

Schematic illustration of our setup. (a) The initial state of the system and memory is given by ${\rho }_{\mathrm{i}}^S\otimes {\rho }_0^M$, and its diagonal basis is given by $|{\psi }^S\left(x\right)\rangle \otimes |{\psi }^M\left(a\right)\rangle$. (b-1) A unitary transformation $U^{SM}$ is performed to entangle the system and memory. After this transformation, the state is given by $U^{SM}{\rho }_{\mathrm{i}}^S\otimes {\rho }_0^M{U}^{†SM}$. (b-2) A general measurement $M_{k,a}:=\sqrt{p_a^M}\langle {\varphi }_k^M|U^{SM}|{\varphi }_a^M\rangle$ on the system is performed through a projection measurement on the memory by using the basis $|{\varphi }_k^M\rangle$, and the measurement outcome $k$ is obtained. The postmeasurement state is given by $p_k^{-1}{\sum }_a{M}_{k,a}^S{\rho }_{\mathrm{i}}^S{M}_{k,a}^{†S}\otimes |{\varphi }_k^M\rangle \langle {\varphi }_k^M|$, and its diagonal basis is given by $|{\phi }_k^S\left(y\right)\rangle \otimes |{\varphi }_k^M\rangle$. The measurement process causes a quantum transition between states labeled by $(x,a)$ and $(y,k)$. (c) From the measurement outcome $k$, we acquire information about the system, which is characterized by the information gain $I$. It is utilized to perform a feedback control on the system described by a unitary transformation $U_k^S$, which depends on the measurement outcome $k$. The final state is given by ${\rho }_{\mathrm{f}}^{SM}=p_k^{-1}{\sum }_a{U}_k^S{M}_{k,a}^S{\rho }_{\mathrm{i}}^S{M}_{k,a}^{†S}{U}_k^{†S}\otimes |{\varphi }_k^M\rangle \langle {\varphi }_k^M|$. The entropy production or work can be evaluated from the matrix element $\langle {\varphi }_k^S\left(z\right)\left|{\rho }_{\mathrm{f}}^S\left(k\right)\right|{\varphi }_k^S\left(z\right)\rangle$, where ${\rho }_{\mathrm{f}}^S\left(k\right)=T{r}_M\left[{\rho }_{\mathrm{f}}^{SM}\left(k\right)\right]$, and $|{\varphi }_k^S\left(z\right)\rangle$ is the diagonal element of the reference state or the energy eigenstate of the final Hamiltonian. Thus, a feedback control provides a quantum transition between the states labeled by $(y,k)$ and $z$.