Initial-state dependence of thermodynamic dissipation for any quantum process

Exact results about the nonequilibrium thermodynamics of open quantum systems at arbitrary timescales are obtained by considering all possible variations of initial conditions of a system. First we obtain a quantum-information theoretic equality for entropy production, valid for an arbitrary initial joint state of system and environment. For any finite-time process with a fixed initial environment, we then show that the system's loss of distinction—relative to the minimally dissipative state—exactly quantifies its thermodynamic dissipation. The quantum component of this dissipation is the change in coherence relative to the minimally dissipative state. Implications for quantum state preparation and local control are explored. For nonunitary processes—like the preparation of any particular quantum state—we find that mismatched expectations lead to divergent dissipation as the actual initial state becomes orthogonal to the anticipated one.

Figure 1

Whether preparing a Bell state, erasing quantum memory, or extracting work, no control protocol can be simultaneously thermodynamically optimal for all initial quantum states on which it operates. Inset: A quantum system ${\rho }_t$ of two qubits is driven by a time-dependent Hamiltonian $H_{x_t}$ and time-dependent interaction $H_{x_t}^{\text{int}}$ with thermal reservoirs. Main: Suppose the control protocol $x_{0:\tau }$ resets the two qubits in finite time, and achieves minimal dissipation when operating on the noisy Bell state: ${\sigma }_0=\frac{1-\alpha }{2}\left(|01\rangle +|10\rangle \right)\left(\langle 01|+\langle 10|\right)+\frac{\alpha }{4}I$, with some particular $\alpha \in [0,1]$. If the reset protocol is designed to be optimal for erasing the ${\mathrm{\Psi }}^{+}$ Bell state ($\alpha =0$), then the same protocol approaches infinite dissipation as ${\rho }_0$ approaches any of the ${\mathrm{\Phi }}^{+}, {\mathrm{\Phi }}^{-}$, or ${\mathrm{\Psi }}^{-}$ Bell states orthogonal to it. For $0<\alpha \ll 1$, the dissipation ${\mathit{\Sigma }}_{{\rho }_0}$ scales as $ln(1/\alpha )$ near these three Bell states. If $\alpha =1$—i.e., the reset protocol is optimized for erasing completely randomized classical bits—then any initial quantum state can be erased with no more than $k_{\text{B}}ln2$ of dissipation beyond ${\mathit{\Sigma }}_{{\sigma }_0}$ (where ${\mathit{\Sigma }}_{{\sigma }_0}$ can be engineered to be arbitrarily small). This dissipation is distinct and additive to the Landauer cost of erasure. Dissipation is plotted over a triangular domain, which is only a subset of possible input states. This triangular domain is the convex hull of input states $I/4, {\mathrm{\Phi }}^{+}$, and ${\mathrm{\Psi }}^{+}$.