Global Passivity in Microscopic Thermodynamics

The main thread that links classical thermodynamics and the thermodynamics of small quantum systems is the celebrated Clausius inequality form of the second law. However, its application to small quantum systems suffers from two cardinal problems. (i) The Clausius inequality does not hold when the system and environment are initially correlated—a commonly encountered scenario in microscopic setups. (ii) In some other cases, the Clausius inequality does not provide any useful information (e.g., in dephasing scenarios). We address these deficiencies by developing the notion of global passivity and employing it as a tool for deriving thermodynamic inequalities on observables. For initially uncorrelated thermal environments the global passivity framework recovers the Clausius inequality. More generally, global passivity provides an extension of the Clausius inequality that holds even in the presences of strong initial system-environment correlations. Crucially, the present framework provides additional thermodynamic bounds on expectation values. To illustrate the role of the additional bounds, we use them to detect unaccounted heat leaks and weak feedback operations (“Maxwell demons”) that the Clausius inequality cannot detect. In addition, it is shown that global passivity can put practical upper and lower bounds on the buildup of system-environment correlations for dephasing interactions. Our findings are highly relevant for experiments in various systems such as ion traps, superconducting circuits, atoms in optical cavities, and more.

Popular Summary

With the rise of quantum technologies, the laws of thermodynamics face new challenges. However, one formulation of the second law holds up even in microscopic situations. The Clausius inequality, which states that heat only moves from a hot body to a cooler one, seems to be a thread connecting classical and quantum thermodynamics. We introduce a thermodynamic framework that fixes some deficiencies in the Clausius inequality and leads to new predictions.

In small setups, the Clausius inequality suffers from two problems. First, it is not valid when the system and the environment are initially correlated. Second, even when the inequality is valid, it may yield useless predictions. To tackle these deficiencies, we introduce the global passivity framework, which uses the initial state of the setup to make predictions about observable quantities after application of some quantum evolution protocol. We then obtain new thermodynamic constraints on the possible measurement outcomes in nanoscopic setups.

To illustrate the role of the new constraints, we consider a “lazy Maxwell’s demon,” an entity that moves energy from a cold bath to a hot one. However, the demon often dozes off and lets energy flow back from hot to cold. When that happens, the demon cannot be detected using the Clausius inequality. This raises a fundamental question: Can some other thermodynamic principles be used to detect lazy demons? We show that the bounds derived from global passivity can detect demons that the Clausius inequality cannot.

Our formalism can predict what outcomes and processes can and cannot be achieved in experiments involving systems such as ion traps, superconducting circuits, and atoms in optical cavities.

Figure 1

(a) A typical scenario for our theory. A microbath with very small heat capacity (e.g., several spins) is initially prepared in a thermal state and then coupled to a system in a nonthermal state. In such scenarios the dynamics is highly non-Markovian and, in addition, the unitary transformation that generates the interactions may add or remove energy from the system-environment setup. (b) In the standard quantum thermodynamic setup all elements (microbaths and system) are initially uncorrelated and then a global unitary (a thermodynamic protocol) describes their interaction. (c) In this paper we consider any initial conditions including strong entanglement between the system and the microbaths. Moreover, we allow for a mixture of unitaries which include the possibilities of noise in the thermodynamic protocol.

Figure 2

(a) Passivity of a general Hermitian operator $\mathcal{A}$ with respect to (w.r.t.) initial density matrix ${\rho }_0$. ${\overrightarrow{p}}_i$ denote the eigenvalues of ${\rho }_0$. In passive distributions with respect to an observable $\mathcal{A}$, larger eigenvalues of $\mathcal{A}$ (measurement results ${\lambda }_i$) have lower probability to be observed (probability is given by the size of the circles). Panel (b) illustrates that starting from passive distribution, any permutation must increase the expectation value $⟨\mathcal{A}⟩$. This implies that $⟨\mathcal{A}⟩$ increases under a unitary operation ($\mathrm{\Delta }⟨\mathcal{A}⟩\ge 0$). (c) If $\mathcal{A}$ is passive with respect to ${\overrightarrow{p}}_i$, then, ${\mathcal{A}}^{\alpha }$ is passive with respect to ${\overrightarrow{p}}_i$ for any $\alpha >0$.

Figure 3

The passivity-divergence relation offers an extension of the Clausius inequality (second law) that can handle initial correlation between environments or systems. (a) Hot and cold qubits are prepared with initial correlation in the form of coherence (superposition) between states $|01⟩$ and $|10⟩$ as shown by the form of the initial density. (b) In the absence of initial correlation ($C=0$), an energy-conserving interaction makes the hot spin colder and the cold spin hotter. (c) In the presence of initial correlation $C=-0.19$, the same interaction makes the cold spin colder and the hot one hotter. All parameters are taken from a recent NMR experiment [2]. The top graph shows that the CI “entropy production” (red) and the left-hand side (lhs) of the ICCI (blue) have the same values for this case and both are positive at all time. However, in the presence of initial correlation (bottom graph), the lhs of Eq. (2) attains negative values, and the CI fails. In contrast, the ICCI remains valid (positive) at all times.

Figure 4

(a) Heat leak detection in a three-qubit circuit with two cnot interactions. The qubits also undergo decay to the ground state at a rate $\gamma =1/1000$. This weak decay is not visible in the polarization dynamics shown in (b). For example, the polarization of qubit 1 (blue) looks perfectly flat. Panel (c) shows a validity check of the inequalities Eq. (30). Since the $\alpha =1$ curve that corresponds to the CI is always positive for the depicted time, it cannot be used to detect the heat leak. On the other hand, the $\alpha =5$ and $\alpha =6$ global passivity inequalities are violated in less than $3/1000$ of the decay time (which is $1/\gamma =1000$).

Figure 5

Bounds on the system-environment correlation dynamics derived from the global passivity inequalities Eq. (30). (a) The blue curve depicts $\mathrm{corr}[H_{\mathrm{env}},{\sigma }_x]$ between the energy of a three-spin microbath and the coherence $⟨{\sigma }_x⟩$ of a one-spin system (see upper right-hand box) as a function of time under a dephasing-type interaction. The red and green shaded area show lower and upper bound on this correlation as obtained from the $\alpha =2$ global passivity inequality. (b) The $\alpha =3$ passivity inequality leads to even tighter bound (red) on the correlation to higher energy moments of the environment $P(H_{\mathrm{env}}^2)={\beta }_x^2\beta {\tilde{H}}_{\mathrm{env}}+(\beta H_{\mathrm{env}}{)}^2$, where $\beta$ is the temperature of the environment and ${\beta }_x$ determines the magnitude of the initial coherence of the system.

Figure 6

(a) A schematic description of a process with a weak feedback (“lazy demon”). A hot and cold microbath are coupled, and evolve unitarily for a finite time. Then a feedback (see text) is applied to the setup with probability $\chi$ (feedback strength). (b) Lazy demon detection using global passivity inequalities Eq. (30). When the feedback strength exceeds ${\chi }_{\mathrm{crit}}$, the heat changes its sign, and flows from the (initially) cold qubits to the hot qubits. In this case the action of the demon can be detected by a violation of CI. For $\chi \le {\chi }_{\mathrm{crit}}$, the CI cannot detect the demon. The blue curve depicts the value of ${\chi }^{*}(\alpha )$ for which $\mathrm{\Delta }⟨B^{\alpha }⟩=0$. The global passivity inequalities Eq. (30) can detect the demon for $\chi \ge {\chi }^{*}(\alpha )$ (right green arrow). Since for $1\le \alpha \le {\alpha }_{\max }$, ${\chi }^{*}(\alpha )\le {\chi }_{\mathrm{crit}}$, we conclude that global passivity can be used to detect the demon that CI cannot. Interestingly, the best detection takes place at a noninteger value ${\alpha }_{\mathrm{opt}}\simeq 2.6$.