Trading coherence and entropy by a quantum Maxwell demon

The second law of thermodynamics states that the entropy of a closed system is nondecreasing. Discussing the second law in the quantum world poses different challenges and provides different opportunities, involving fundamental quantum-information-theoretic questions and interesting quantum-engineered devices. In quantum mechanics, systems with an evolution described by a so-called unital quantum channel evolve with a nondecreasing entropy. Here, we seek the opposite, a system described by a nonunital and, furthermore, energy-conserving channel that describes a system whose entropy decreases with time. We propose a setup involving a mesoscopic four-lead scatterer augmented by a microenvironment in the form of a spin that realizes this goal. Within this nonunital and energy-conserving quantum channel, the microenvironment acts with two noncommuting operations on the system in an autonomous way. We find that the process corresponds to a partial exchange or swap between the system and environment quantum states, with the system's entropy decreasing if the environment's state is more pure. This entropy-decreasing process is naturally expressed through the action of a quantum Maxwell demon and we propose a quantum-thermodynamic engine with four qubits that extracts work from a single heat reservoir when provided with a reservoir of pure qubits. The special feature of this engine, which derives from the energy conservation in the nonunital quantum channel, is its separation into two cycles, a working cycle and an entropy cycle, allowing us to run this engine with no local waste heat.

Figure 1

Purification of an energetically isolated system with the help of a quantum demon. The system qubit on the left undergoes a transition (arrow) to a mixed state (full circles), e.g., by decoherence or through thermal excitation. The demon makes use of a pure demon qubit (right, open circles) and swaps its state with the system qubit, thereby transferring entropy or coherence but no energy.

Figure 2

Nonunital quantum channel realizing maximal purification by two consecutive orthogonal rotations ${\widehat{u}}_1={\sigma }_x$ and ${\widehat{u}}_3={\sigma }_z$. The three-dimensional (3D) sketch illustrates the scattering of an electron incident from leads 1 and 2 and scattered into the leads 3 and 4; $s_{ij}$ denote the scattering amplitudes. The microenvironment is defined through the spin $\mathbf{S}$ which interacts with the magnetic fields $\mathbf{B}$ generated by the electron traveling in the close-by leads 1 (generating the field ${\mathbf{B}}_1$) and 3 (generating the field ${\mathbf{B}}_3$); we ignore the interaction with currents in the distant leads 2 and 4. The geometry of the leads and the spin position is chosen such that the magnetic fields of the currents in leads 1 and 3 are orthogonal. The initial (or operational) spin states $|\uparrow \rangle$ and $|\downarrow \rangle$ are chosen parallel to $z$ (the field generated by a current in lead 3) such that the passage of the electron in lead 1 flips the spin and generates the unitary ${\widehat{u}}_1$. The passage of the electron through lead 3 induces a rotation around $z$ and adds the phases as required for the unitary ${\widehat{u}}_3$.

Figure 3

An electron in an incoming superposition state $a|0,+\rangle +b|1,-\rangle$ undergoes scattering in a quantum point contact (a QPC, here implemented in a quantum Hall bar device) described by the scattering matrix $\widehat{s}$ with symmetric scattering coefficients $s_{ij},i,j=\pm$. The states $|0,\pm \rangle$ interact with a double dot residing in one of the operational states $|\uparrow \rangle$ or $|\downarrow \rangle$ which are superpositions of the physical states $|{\uparrow }_z\rangle$ and $|{\downarrow }_z\rangle$. The interaction between the double dot and the scattering states $|0,\pm \rangle$ generates the conditional (on $|\Uparrow \rangle$) $\pi$-phase shift $\widehat{u}$. Combining this operation with the scattering of the electron at the QPC and a $\pi /2$ rotation of the double dot, see text, swaps the electron and double-dot states.

Figure 4

Schematic version of a double Mach-Zehnder interferometer with quantum demon. The incoming wave packet is decohered (arrow) in the first loop and restored to a pure state $|\varphi \rangle$ by the action of the demon (double dot). The second loop tests the coherence of the wave function by shifting the relative phase in the two arms with the help of a tunable flux $\mathrm{\Phi }$. The appearance of Aharonov-Bohm oscillations in the outgoing arms testifies for the successful action of the demon.

Figure 5

(a) Quantum circuit describing the demon action as a partial swap or pswap and its extension to a full swap operation. The dotted lines on the cnot operations indicate that no energy is exchanged between the system and demon qubits. (b) Minimal pswap and swap operations using two and three cnot operations with exchanged controller [39].

Figure 6

Quantum thermodynamic engine with two energetically separated cycles. The working cycle (left) starts with two working qubits (wits) in the pure (empty dots) ground state $|g\rangle$. The contact with the thermal reservoir excites them into a mixed (solid dots) state. After the demon's swap, the wits have taken over the pure states from the demon qubits (dits), $|\uparrow \rangle \to |e\rangle$ and $|\downarrow \rangle \to |g\rangle$. The energy ${\mathrm{\Delta }}_{\mathrm{w}}$ of the excited wit can be used for work extraction. Different from the pswap in Fig. 1, the present $\tilde{p}\text{swap}$ has an energy context involving a classical field (wavy line). The entropy cycle (right) starts with two pure dits in states $|\uparrow \rangle$ and $|\downarrow \rangle$ which are “consumed” by the wits and then reprepared for the next cycle.

Figure 7

Optimal values ${\epsilon }_{\eta }$ (solid line) and ${\epsilon }_{\mathrm{W}}$ (dashed lines) for the dit impurity $\epsilon$ vs occupation $p_e$ of excited states. The three dashed lines correspond to different values ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}=1.0$ (upper curve), ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}=2ln2$ below which the ideal engine with $\epsilon =0$ cannot produce positive work, and ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}=2$ (bottom curve). At point A, the efficiency ${\eta }_{2-\mathrm{cy}}^{\epsilon }$ approaches unity but the generated work $W_{2-\mathrm{cy}}^{\epsilon }$ goes to zero. At the points point B, ${\mathrm{B}}^{\prime }$ and ${\mathrm{B}}^{\prime \prime }$ both ${\eta }_{2-\mathrm{cy}}^{\epsilon }$ and $W_{2-\mathrm{cy}}^{\epsilon }$ approach zero at fixed values of ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}$ along the lines ${\epsilon }_{\eta }\left(p_e\right)$ and ${\epsilon }_{\mathrm{W}}(p_e,{\beta }_{\mathrm{d}})$ [note that ${\epsilon }_{\eta }$ does not depend on ${\beta }_{\mathrm{d}}$ and $\eta$ approaches zero at different points along ${\epsilon }_{\eta }\left(p_e\right)$ for different values of ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}]$. At point C, the work per cycle reaches a value $W_{2-\mathrm{cy}}^{\epsilon }\approx 0.43{\mathrm{\Delta }}_{\mathrm{w}}$ with an efficiency ${\eta }_{2-\mathrm{cy}}^{\epsilon }=0.57$. The points A, B, and C reappear in Fig. 8 below.

Figure 8

Engine efficiencies ${\eta }_{2-\mathrm{cy}}^{\epsilon }$ and work per cycle $W_{2-\mathrm{cy}}^{\epsilon }$ (inset) at a fixed demon temperature ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}=2$ as a function of the dimensionless working temperature $\beta {\mathrm{\Delta }}_{\mathrm{w}}$. Shown are the performances for the idealized machine with $\epsilon =0$ [dash-dotted line; this line decreases with decreasing ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}$ and disappears from the plot at ${\beta }_{\mathrm{d}}{\mathrm{\Delta }}_{\mathrm{w}}\le 2ln\left(2\right)\approx 1.39]$ and the efficiency- and power-optimized machines at $\epsilon ={\epsilon }_{\eta }$ (solid line) and $\epsilon ={\epsilon }_{\mathrm{W}}$ (dashed line). The thick solid upper line describes the Carnot efficiency ${\eta }_{\mathrm{C}}=1-T_{\mathrm{d}}/T$. Maximal efficiency but vanishing power is attained in A. On approaching B, both efficiency and power vanish. In C the power-optimized machine is characterized by both finite efficiency and power.