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1 Introduction↩︎

Thermodynamics describes the transport of energy and other quantities between systems that have equilibrated individually but not with each other. These quantities are conserved according to principles such as the first law of thermodynamics. The second law decrees which transport can and cannot occur spontaneously. Over the last three decades, researchers have revolutionized the study of transport within microscopic systems, where fluctuations dominate [1]–[7].

More-recent results have prompted questions about how such exchanges occur in quantum systems, which can have coherences and nonclassical correlations [8]–[15]. To infer about quantum currents, one must measure quantum systems. But measurement back-action can destroy coherences. Thermodynamic processes therefore depend on our measurements of them. This conundrum has inspired considerable research, but no resolution is universally agreed upon.

Quantum coherences result from the noncommutation of operators. In classical thermodynamics, a system’s energy and particle number are commuting quantities; they can be measured simultaneously. What if the globally conserved thermodynamic quantities (charges) fail to commute?

This recently posed question has upended intuitions and engendered a burgeoning subfield of quantum thermodynamics [16]–[49]; see [50] for a recent Perspective. For example, charges’ noncommutation hinders arguments for the thermal state’s form [20], [23] and alters the eigenstate thermalization hypothesis, which explains how quantum many-body systems thermalize internally [35]. Other results span resource theories [20]–[23], [39]–[41], [43]–[45], heat engines [38], metrology [51] and a trapped-ion experiment [27], [48]. In the linear-response regime, noncommuting charges reduce entropy production  [28].

We analyze entropy produced by arbitrarily far-from-equilibrium exchanges of noncommuting charges. Three formulae for entropy production, which equal each other when all charges commute, have been used widely [52]. We show that these formulae fail to equal each other when charges fail to commute. This incommensurability stems from measurement disturbance: Noncommuting charges’ currents cannot be measured simultaneously.

An example illustrates the key idea: Consider two classical thermodynamic observables, such as energy and particle number. Let classical systems \({\tt A}\) and \({\tt B}\) begin in thermal (grand canonical) ensembles: System \({\tt X} = {\tt A}, {\tt B}\) has energy \(E^{\tt X}\) and particle number \(N^{\tt X}\) with a probability \(\propto e^{- \beta^{\tt X} ( E^{\tt X} - \mu^{\tt X} N^{\tt X} ) }\). \(\beta^{\tt X}\) denotes an inverse temperature; and \(\mu^{\tt X}\), a chemical potential. An interaction can shuttle energy and particles between the systems. If conserved globally, the quantities are called charges. Any charge (e.g., particle) entering or leaving a system produces entropy. The entropy produced in a trial—the stochastic entropy production (SEP)—is a random variable. Its average over trials is non-negative, according to the second law of thermodynamics. The SEP obeys constraints called exchange fluctuation theorems—tightenings of the second law of thermodynamics [6]. Classical fluctuation theorems stem from a probabilistic model: System \({\tt A}\) begins with energy \(E^{{\tt A}}_i\) and with \(N^{{\tt A}}_i\) particles, \({\tt A}\) ends with energy \(E^{{\tt A}}_f\) and with \(N^{{\tt A}}_f\) particles, and \({\tt B}\) satisfies analogous conditions, with a joint probability \(p ( E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i \, ; \, E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f )\). The progression \(( E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i )\) \(\mapsto\) \(( E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f )\), we view as a two-step stochastic trajectory between microstates (Fig. 1).

In the commonest quantum analog, quantum systems \({\tt X}= {\tt A}, {\tt B}\) begin in reduced grand canonical states \(\propto e^{ - \beta^{\tt X} ( \hat{H}^{\tt X} - \mu^{\tt X} \hat{N}^{\tt X} ) }\). \(\hat{H}^{\tt X}\) denotes a Hamiltonian; and \(\hat{N}^{\tt X}\), a particle-number operator. In the two-point measurement scheme [4], [5], one strongly measures each system’s Hamiltonian and particle number. Then, a unitary couples the systems, conserving \(\hat{H}^{{\tt A}} + \hat{H}^{{\tt B}}\) and \(\hat{N}^{{\tt A}} + \hat{N}^{{\tt B}}\). Finally, one measures \(\hat{H}^{{\tt A}}\), \(\hat{H}^{{\tt B}}\), \(\hat{N}^{{\tt A}}\), and \(\hat{N}^{{\tt B}}\) again.³ A joint probability distribution governs the four measurement outcomes. Using the outcomes and distribution, one can similarly define SEP, prove fluctuation theorems, and define stochastic trajectories [6]. Each measurement may disturb the quantum system, however [12], [53], [54].

Noncommutation introduces a twist into this story. Let the quantum systems above exchange charges that fail to commute with each other. For example, consider qubits \({\tt X}={\tt A},{\tt B}\) exchanging spin components \(\hat{\sigma}_{x,y,z}^{\tt X}\). The corresponding thermal states are \(\propto \exp ( - \sum_{\alpha = x,y,z} \beta_\alpha^{\tt X} \hat{\sigma}_\alpha^{\tt X} )\), wherein \(\beta_\alpha^{\tt X}\) denotes a generalized inverse temperature [23], [27], [35], [48]. One cannot implement the two-point measurement scheme straightforwardly, as no system’s \(\hat{\sigma}_{x,y,z}^{\tt X}\) operators can be measured simultaneously. One can measure each qubit’s three spin components sequentially, couple the systems with a charge-conserving unitary, and measure each qubit’s \(\hat{\sigma}_{x,y,z}\) sequentially again. Yet these measurements wreak havoc worse than if the charges commute: They disturb not only the systems’ states, but also the subsequent, noncommuting measurements [55].

Weak measurements would disturb the systems less [56], [57] at the price of extracting less information [58]. As probabilities describe strong-measurement experiments, quasiprobabilities describe weak-measurement experiments (App. 7). Quasiprobabilities resemble probabilities—being normalized to 1, for example. They violate axioms of probability theory, however, as by becoming negative. Consider, then, replacing the above protocol’s strong measurements with weak measurements. We may loosely regard \({\tt A}{\tt B}\) as undergoing stochastic trajectories weighted by quasiprobabilities, rather than probabilities. Levy and Lostaglio applied quasiprobabilities in deriving a fluctuation theorem for energy exchanges [12]. Their fluctuation theorem contains the real part of a Kirkwood–Dirac quasiprobability (KDQ) [57], [59]–[62].

KDQs have recently proven useful across quantum thermodynamics [12], [56], [57], [63]–[66], information scrambling [63], [67]–[70], tomography [71]–[77], metrology [78]–[80], and foundations [81]–[86].

When employed as work or heat distributions, KDQs have multiple desirable properties not achieved by joint probability distributions [56], [57], [87], [88]

Negative and nonreal KDQs can reflect nonclassicality [89], [90] and measurement disturbance [73], [91]–[95]. In [12], negative real KDQs signal anomalous heat currents, which flow spontaneously from a colder to a hotter system. Generalizing from energy to potentially noncommuting charges, our results cover a more fully quantum setting. They also leverage the KDQ’s ability to become nonreal.

To accommodate noncommuting charges, we generalize the SEP. In conventional quantum thermodynamics, three common SEP formulae equal each other [52]. Entropy is cast as an extensive thermodynamic variable by a “charge formula,” as quantifying missing information by a “surprisal formula,” and as quantifying irreversibility by a “trajectory formula.” We generalize all three formulae using KDQs. If the charges commute, the generalizations reduce to the usual formulae. The generalizations satisfy four sanity checks, including by equaling each other when the charges commute. Yet we find deductively that noncommuting charges break the equivalence, generating three species of SEP.

Different SEPs, we find, highlight different ways in which charges’ noncommutation impacts transport:

1. Charge SEP: Charges’ noncommutation enables individual stochastic trajectories to violate charge conservation. These violations underlie commutator-dependent corrections to a fluctuation theorem.

2. Surprisal SEP: Initial coherences, relative to eigenbases of the charges, enable the average surprisal entropy production to become negative. Such negativity simulates a reversal of time’s arrow. Across thermodynamics, a ubiquitous initial state is a product of thermal states. Only if the charges fail to commute can such a state have the necessary coherences. Hence charges’ noncommutation enables a resource, similar to work, for effectively reversing time’s arrow in a common setup.

3. Trajectory SEP: The generalized trajectory SEP can become nonreal, thanks to the KDQ. Such nonreality signals contextuality—a strong form of nonclassicality [96]–[98]—in a noncommuting-current experiment.

This work opens the field of stochastic thermodynamics [99] to noncommuting charges. Furthermore, our work advances the widespread research program of critically comparing thermodynamic with information-theoretic entropies and leveraging information theory for thermodynamics [100]–[103].

The paper is organized as follows. Section 2 details the setup and reviews background material. Section 3 presents the three SEP formulae, the physical insights they imply, and their fluctuation theorems and averages. Section 4 numerically illustrates our main results with a two-qubit example. In Sec. 5, we sketch a trapped-ion experiment based on our results. Section 6 concludes with avenues for future work.

As a semantic disclaimer and to provide broader context: Studies of the second law have engendered debates about what should be called the entropy production. We provide an alternative perspective: We show that three SEP formulae, despite being equivalent in the commuting case, become inequivalent if charges cease to commute. Each of these formulae encapsulates a different aspect of entropy production. This perspective harmonizes with demonstrations that, for small systems, the conventional second law splits into multiple second “laws” [20], [23], [104]–[108].

2 Preliminaries↩︎

We specify the physical setup in Sec. 2.1. Section 2.2 reviews KDQs; Sec. 2.3, information-theoretic entropies; and Sec. 2.4, fluctuation theorems.

2.1 Setup↩︎

Consider two identical quantum systems, \({\tt A}\) and \({\tt B}\). (From now on, we omit hats from operators.) Each system corresponds to a copy of a Hilbert space \(\mathcal{H}\). The bipartite initial state \(\rho\) leads to the reduced states \(\require{physics} \rho^{{\tt A}} \mathrel{\vcenter{:}}= \Tr_{{\tt B}}(\rho)\) and \(\require{physics} \rho^{{\tt B}}\mathrel{\vcenter{:}}= \Tr_{{\tt A}}(\rho)\).

On \(\mathcal{H}\) are defined single-system observables \(Q_{\alpha = 1, 2, \ldots, c}\), assumed to be linearly independent [47]. For \({\tt X}= {\tt A}, {\tt B}\), we denote \({\tt X}\)’s copy of \(Q_\alpha\) by \(Q_\alpha^{\tt X}\). The dynamics will conserve the global observables \(Q_\alpha^\text{tot}\mathrel{\vcenter{:}}= Q_\alpha^{\tt A}\otimes \mathbb{1}^{\tt B}+ \mathbb{1}^{\tt A}\otimes Q_\alpha^{\tt B}\), so we call the \(Q_\alpha\)’s charges. In the commuting case, all the charges commute: \([Q_\alpha, Q_{\alpha'}] = 0\) \(\forall \alpha, \alpha'\). At least two charges do not in the noncommuting case: \([Q_\alpha, Q_{\alpha'}] \neq 0\) for some \(\alpha, \alpha'\). Each charge eigendecomposes as \(Q_\alpha = \sum_j \lambda_{\alpha,j} \, \Pi_{\alpha, j}\). We denote tensor-product projectors by \(\Pi_{\alpha, {k}} = \Pi^{{\tt A}}_{\alpha, k^{\tt A}} \otimes \Pi^{{\tt B}}_{\alpha,k^{\tt B}}\), invoking the composite index \({k}=(k^{{\tt A}},k^{{\tt B}})\). To simplify the formalism, we assume that the \(Q_\alpha\) are nondegenerate, as in [6], [12], [109], [110]. Appendix 8 concerns extensions to degenerate charges. The nondegeneracy renders every projector rank-one: \(\Pi_{\alpha, k} =\ketbra{\alpha_k}{\alpha_k}\). We call the \(Q_\alpha^\text{tot}\) eigenbasis \(\{\ket{\alpha_k}\}_k\) the \(\alpha^\text{th}\) product basis.

Having introduced the charges, we can expound upon the initial state. The reduced states \(\rho^{\tt X}\) are generalized Gibbs ensembles (GGEs) [111]–[115] \[\begin{align} \label{eq95GGE} \rho^{\tt X}_{\rm GGE} \mathrel{\vcenter{:}}= \frac{1}{Z^{\tt X}} \exp \Bigg( - \sum_{\alpha = 1}^c \beta^{\tt X}_\alpha Q_\alpha^{\tt X} \Bigg) . \end{align}\tag{1}\] \(\beta^{\tt X}_\alpha\) denotes a generalized inverse temperature. The partition function \(Z^{\tt X}\) normalizes the state. If the \(Q_\alpha^{\tt X}\)’s fail to commute, the GGE is often called the non-Abelian thermal state [23], [27], [35], [48]. Across most of this paper, \(\rho\) equals the tensor product \[\label{initstate} \rho^{{\tt A}}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}.\tag{2}\] Such a product of thermal states, being a simple nonequilibrium state, surfaces across thermodynamics [116]. (In Sections 3.2.2 and 3.3.3, we consider initial states \(\rho\) that deviate from this form but retain GGE reduced states.)

Our results stem from the following protocol: Prepare \({\tt A}{\tt B}\) in \(\rho\). Evolve \(\rho\) under a unitary \(U\) that conserves every global charge: \[\begin{align} \label{chargecons} [U, Q_\alpha^\text{tot}]=0 \; \forall \alpha . \end{align}\tag{3}\] \(U\) can bring the state arbitrarily far from equilibrium. The final state, \(\rho_\text{f} = U \rho U^\dagger\), induces the reduced states \(\require{physics} \rho_\text{f}^{\tt A}\mathrel{\vcenter{:}}= \Tr_{\tt B}( \rho_\text{f} )\) and \(\require{physics} \rho_\text{f}^{\tt B}\mathrel{\vcenter{:}}= \Tr_{\tt A}( \rho_\text{f} )\). The average amount of \(\alpha\)-type charge in \({\tt A}\) changes by \(\require{physics} \Delta \langle Q_\alpha \rangle \mathrel{\vcenter{:}}= \Tr( Q_\alpha^{\tt A}[\rho^{\tt A}_\text{f} - \rho^{\tt A}] )\); and the average amount in \({\tt B}\), by \(\require{physics} \Tr( Q_\alpha^{\tt B}[\rho^{\tt B}_\text{f} - \rho^{\tt B}] ) = - \Delta \langle Q_\alpha \rangle\). The systems’ generalized inverse temperatures differ by \(\Delta \beta_\alpha \mathrel{\vcenter{:}}= \beta_\alpha^{\tt A}- \beta_\alpha^{\tt B}\).

2.2 Kirkwood-Dirac quasiprobability↩︎

The KDQ’s relevance stems from the desire to reason about charges flowing between \({\tt A}\) and \({\tt B}\). One can ascribe to \({\tt A}\) some amount of \(\alpha\)-type charge only upon measuring \(Q_\alpha^{\tt A}\). Measuring \(Q_\alpha^{\tt A}\) strongly would disturb \({\tt A}\) and subsequent measurements of noncommuting \(Q_{\alpha'}^{\tt A}\)’s [55]. Therefore, we consider sequentially measuring charges weakly. We define the forward protocol by combining the preparation procedure and unitary with weak measurements (Sec. 3.3 introduces a reverse protocol):

1. Prepare \({\tt A}{\tt B}\) in \(\rho\).

2. Weakly measure the product basis of \(Q_1^{\tt A}\) and \(Q_1^{\tt B}\), then the product basis of of \(Q_2^{\tt A}\) and \(Q_2^{\tt B}\), and so on until \(Q_c^{\tt A}\) and \(Q_c^{\tt B}\). (One can implement these measurements using the detector-coupling technique in [67].)

3. Evolve \({\tt A}{\tt B}\) under \(U\).

4. Weakly measure the charges’ product bases in the reverse order, from \(c\) to 1. (This measurement ordering ensures that our SEP definitions satisfy sanity checks described in Sec. 3.)

Lacking strong measurements, this protocol differs qualitatively from two-point-measurement schemes. The forward protocol leads naturally to a KDQ, as shown in App. 7: \[\require{physics} \begin{align} \label{KDdef} & \Tr \left( U^\dagger \left[ \Pi_{1, f_1} \Pi_{2, f_2} \ldots \Pi_{c, f_c} \right] U \left[ \Pi_{c, i_c} \ldots \Pi_{2,i_2} \Pi_{1,i_1} \right] \rho \right) \nonumber \\ & \quad \eqqcolon \tilde{p}_{\rm F} (i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1) . \end{align}\tag{4}\] The list \((i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1)\) defines a stochastic trajectory, as in Sec. 1. (Recall the definition, in Sec. 2.1, of composite-system indices.) Loosely speaking, we might view the trajectory as occurring with a joint quasiprobability \(\tilde{p}_{\rm F} (i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1)\).

\(\tilde{p}_{\rm F}\) can assume negative and nonreal values. One can infer \(\tilde{p}_{\rm F}\) experimentally by performing the forward protocol many times, performing strong-measurement experiments, and processing the outcome statistics [63]. Angle brackets \(\langle . \rangle\) denote averages with respect to \(\tilde{p}_{\rm F}\), unless we specify otherwise.

Two cases further elucidate \(\tilde{p}_{\rm F}\) and the forward protocol: the commuting case and weak-measurement limit. In the commuting case, if \(\rho\) is diagonal with respect to the charges’ shared eigenbasis, \(\tilde{p}_{\rm F}\) is a joint probability.

2.3 Information-theoretic entropies↩︎

We invoke four entropic quantities from information theory [117]. Let \(X = x_1, x_2, \ldots, x_n\) denote a discrete random variable; \(P = \{ p_1, p_2, \ldots, p_n \}\) and \(R = \{ r_1, r_2, \ldots, r_n \}\), probability distributions over \(X\); and \(\omega_1\) and \(\omega_2\), quantum states. Suppose that \(X\) evaluates to \(x_j\). The surprisal \(- \log (p_j)\) quantifies the information we learn. (This paper’s logarithms are base-\(e\).) Averaging the surprisal yields the Shannon entropy, \(S_{\rm Sh}(P) \mathrel{\vcenter{:}}= - \sum_j p_j \log(p_j)\). The quantum analog is the von Neumann entropy, \(\require{physics} S_{\rm vN}(\omega_1) \mathrel{\vcenter{:}}= - \Tr \boldsymbol{(} \omega_1 \log(\omega_1) \boldsymbol{)}\).

The quantum relative entropy quantifies the distance between states: \(\require{physics} D(\omega_1 || \omega_2) = \Tr \boldsymbol{(} \omega_1 [ \log(\omega_1) - \log(\omega_2)] \boldsymbol{)}\). \(D\) measures how effectively one can distinguish between \(\omega_1\) and \(\omega_2\), on average, in an asymmetric hypothesis test. \(D (\omega_1 || \omega_2) \geq 0\) vanishes if and only if \(\omega_1 = \omega_2\).

Analogously, the classical relative entropy distinguishes probability distributions: \(D(P || R) = \sum_j p_j \left[ \log(p_j) - \log(r_j) \right]\). We will substitute KDQ distributions for \(P\) and \(R\) in Sec. 3.3.1. The logarithms will be of complex numbers. We address branch-cut conventions and the complex logarithm’s multivalued nature there.

2.4 Exchange fluctuation theorems↩︎

An exchange fluctuation theorem governs two systems trading charges. (We will drop the exchange from the name.) Consider two quantum systems initialized and exchanging energy as in the introduction, though without the complication of particles. Each trial has a probability \(p(\sigma)\) of producing entropy \(\sigma\). Denote by \(\langle . \rangle_P\) averages with respect to the probability distribution. The fluctuation theorem \(\require{physics} \expval{ e^{-\sigma} }_P = 1\) implies the second-law-like inequality \(\require{physics} \expval{\sigma}_P \geq 0\) [6]. Unlike the second law, fluctuation theorems are equalities arbitrarily far from equilibrium. More-general (e.g., correlated) initial states engender corrections: \(\require{physics} \expval{ e^{-\sigma} }_P = 1 + \ldots\) [12], [110].

3 Three generalized SEP formulae↩︎

We now present and analyze the generalized SEP formulae: the charge SEP \(\sigma_\text{chrg}\) (Sec. 3.1), the surprisal SEP \(\sigma_\text{surp}\) (Sec. 3.2), and the trajectory SEP \(\sigma_\text{traj}\) (Sec. 3.3). To simplify notation, we suppress the indices that identify the trajectory along which an SEP is produced.

Each formula satisfies four sanity checks: (i) Each \(\langle \sigma \rangle\) has a clear physical interpretation. (ii) Each \(\sigma\) satisfies a fluctuation theorem (Sec. 2.4). Any corrections depend on charges’ commutators. (iii) If the charges commute, all three SEPs coincide. (iv) Suppose that no current flows: \([U,Q_\alpha^{\tt X}]=0\) \(\forall \alpha\). As expected physically, the average entropy production vanishes: \(\langle \sigma \rangle = 0\). Also, the fluctuation theorems lack corrections: \(\langle e^{-\sigma} \rangle = 1\).

3.1 Charge stochastic entropy production↩︎

First, we motivate the charge SEP’s definition (Sec. 3.1.1). \(\sigma_\text{chrg}\) portrays entropy as an extensive thermodynamic quantity (Sec. 3.1.2). Furthermore, \(\sigma_\text{chrg}\) satisfies a fluctuation theorem whose corrections depend on commutators of the \(Q_\alpha\)’s (Sec. 3.1.3). Corrections arise because noncommuting charges enable individual stochastic trajectories to violate a “microscopic,” or “detailed,” notion of charge conservation. [Nevertheless, charge conservation as defined in Eq. 3 is not violated.]

3.1.1 Charge SEP formula↩︎

The fundamental relation of statistical mechanics [118] motivates \(\sigma_\text{chrg}\)’s definition. In this paragraph, we reuse quantum notation (Sec. 2.1) to denote classical objects, for simplicity. The fundamental relation governs large, classical systems \({\tt X} = {{\tt A}}, {{\tt B}}\) that have extensive charges \({Q}^{{\tt X}}_\alpha\) and intensive parameters \({\beta}^{{\tt X}}_\alpha\). Let an infinitesimal interaction conserve each \({Q}^{{\tt A}}_\alpha + {Q}^{{\tt B}}_\alpha\). Since \(d {Q}_\alpha^{{\tt B}} = - d{Q}_\alpha^{{\tt A}}\), the total entropy changes by \[\begin{align} dS^{ {\tt A}{\tt B}} & = \sum_\alpha \left( \beta_\alpha^{{\tt A}} \, d{Q}_\alpha^{{\tt A}} + \beta_\alpha^{{\tt B}} \, d {Q}_\alpha^{{\tt B}} \right) \\ &= \sum_\alpha \Delta {\beta}_\alpha \, d {Q}_\alpha^{{\tt A}} \,. \label{ftr} \end{align}\tag{5}\] According to the second law of thermodynamics, \(dS^{ {\tt A}{\tt B}} \geq 0\) during spontaneous processes [119]. We posit that \(\require{physics} \expval{\sigma_\text{chrg}}\) should assume the form 5 . We reverse-engineer such a formula, using the charges’ eigenvalues: \[\sigma_\text{chrg} \mathrel{\vcenter{:}}= \sum_\alpha \left[ \beta^{\tt A}_\alpha \left( \lambda_{\alpha, f^{\tt A}_\alpha} - \lambda_{\alpha, i^{\tt A}_\alpha} \right) + \beta^{\tt B}_\alpha \left( \lambda_{\alpha, f^{\tt B}_\alpha} - \lambda_{\alpha, i^{\tt B}_\alpha} \right) \right] . \label{schrg}\tag{6}\]

3.1.2 Average of charge SEP↩︎

By design, the charge SEP averages to \[\require{physics} \label{avgc} \expval{\sigma_\text{chrg}} = \sum_\alpha \Delta \beta_\alpha \, \Delta \expval{Q_\alpha} .\tag{7}\] This average is non-negative, equaling a relative entropy [52]: \[\require{physics} \label{relentr} \expval{\sigma_\text{chrg}} = D(\rho_{\rm f}||\rho) \geq 0 \, .\tag{8}\] This inequality echoes the second-law statement written just below Eq. 5 .

3.1.3 Fluctuation theorem for charge SEP↩︎

\(\sigma_\text{chrg}\) obeys the fluctuation theorem \[\require{physics} \begin{align} \label{noncommft} & \expval{e^{-\sigma_\text{chrg}}} \\ \nonumber &= \Tr( U^\dagger \left[ e^{-\Delta \beta_1 Q_1^{{\tt A}}} \ldots e^{-\Delta \beta_c Q_c^{{\tt A}} } \right] U \left[ e^{\Delta \beta_c Q_c^{{\tt A}}} \ldots e^{\Delta \beta_1 Q_1^{{\tt A}}} \right] \rho ) \\ \nonumber & \quad \; + \left( \text{c-1 terms dependent on commutators} \right) . \end{align}\tag{9}\] We prove the theorem and present the correction’s form in App. 9.1. Below, we show that the right-hand side (RHS) evaluates to 1 in the commuting case. In the noncommuting case, corrections arise from two physical sources: (i) \(\rho^{\tt A}\) and \(\rho^{\tt B}\) are non-Abelian thermal states. (ii) Individual stochastic trajectories can violate charge conservation.

In the commuting case, the first term in Eq. 9 equals 1. The reason is, in \(\rho = \rho_{\rm GGE}^{\tt A}\otimes \rho_{\rm GGE}^{\tt B}\), each \(\rho_{\rm GGE}^{\tt X} \propto \exp \left( - \sum_\alpha \beta^{\tt X}_\alpha Q_\alpha^{\tt X} \right) \propto \prod_\alpha \exp( -\beta^{\tt X}_\alpha Q_\alpha^{\tt X} )\). The \({\tt A}\) exponentials cancel the \(e^{\beta_\alpha^{\tt A}Q_\alpha^{\tt A}}\) factors in Eq. 9 . No such cancellation occurs in the noncommuting case, since \(\rho_{\rm GGE}^{\tt X} \not\propto \prod_\alpha \exp( -\beta^{\tt X}_\alpha Q_\alpha^{\tt X} )\). The first term in Eq. 9 can therefore deviate from 1, quantifying the noncommutation of the charges in \(\rho_{\rm GGE}^{\tt X}\).

The second term vanishes in the commuting case, since all commutators of charges vanish. This second term can deviate from 0 in the noncommuting case, due to nonconserving trajectories, which we introduce now. Define a trajectory \((i^{\tt A}_1,i^{\tt B}_1,i^{\tt A}_2, i^{\tt B}_2,\ldots, i^{\tt B}_c)\) \(\mapsto\) \((f^{\tt A}_c,f^{\tt B}_c,f^{\tt A}_{c-1},f^{\tt B}_{c-1},\ldots,f^{\tt B}_1\)) as conserving if the corresponding charge eigenvalues satisfy \[\label{dcc} \lambda_{\alpha,i^{\tt A}_\alpha}+\lambda_{\alpha,i^{\tt B}_\alpha} =\lambda_{\alpha,f_\alpha^{\tt A}} +\lambda_{\alpha,f_\alpha^{\tt B}} \quad \forall \alpha .\tag{10}\] Any trajectory that violates this condition, we call nonconserving. Loosely speaking, under 10 , \({\tt A}{\tt B}\) has the same amount of \(\alpha\)-type charge at the trajectory’s start and end.

In the commuting case, \(\tilde{p}_{\rm F} = 0\) when evaluated on nonconserving trajectories. We call this vanishing detailed charge conservation. Earlier work has relied on detailed charge conservation [12], [56]. In the noncommuting case, \(\tilde{p}_{\rm F}\) need not vanish when evaluated on nonconserving trajectories (App. 9.2). The mathematical reason is, different charges’ eigenprojectors in \(\tilde{p}_{\rm F}\) [Eq. 4 ] can fail to commute. A physical interpretation is that measuring any \(Q_\alpha^{\tt X}\) disturbs any later measurements of noncommuting \(Q_{\alpha'}^{\tt X}\)’s. Nonconserving trajectories resemble classically forbidden trajectories in the path-integral formulation of quantum mechanics.

We now show that nonconserving trajectories underlie the final term in Eq. 9 . Every stochastic function \(g(i_1, i_2, \ldots,i_c \, ; \, f_c, f_{c-1}, \ldots, f_1)\) averages to \(\require{physics} \expval{g} = \expval{g}_\text{cons} + \expval{g}_\text{noncons}\). Each term equals an average over just the conserving or nonconserving trajectories. For example, the average 7 decomposes as \(\require{physics} \expval{\sigma_\text{chrg}} = \expval{\sigma_\text{chrg}}_\text{cons} + \expval{\sigma_\text{chrg}}_\text{noncons}\). The second term on the RHS of Eq. 9 takes the form \(\require{physics} \expval{g}_\text{noncons}\) (App. 9.2.3). In the commuting case, all trajectories are conserving, so the second term equals zero—as expected, since the term depends on commutators \([Q_\alpha^{\tt A},Q_{\alpha'}^{\tt A}]\). Therefore, the fluctuation theorem’s final term (second correction) stems from violations of detailed charge conservation. The average 7 contains contributions from conserving and nonconserving trajectories.

We have identified two corrections to the fluctuation theorem 9 . On the equation’s RHS, the first term can deviate from 1, and the second term can deviate from 0. The first deviation originates in noncommutation of the charges in \(\rho^{\tt X}_{\rm GGE}\). The second deviation stems from nonconserving trajectories. An example illustrates the distinction between noncommuting charges’ influences on initial conditions and dynamical trajectories. Let \(c = 3\), and let \([Q_1,Q_2] \neq 0\), while \([Q_1,Q_3],[Q_2,Q_3]=0\). Let the noncommuting charges correspond to uniform \(\beta\)’s: \(\beta^{\tt A}_1=\beta^{\tt B}_1 \, , \quad \text{and} \quad \beta^{\tt A}_2=\beta^{\tt B}_2 \, .\) Thus, there exists no temperature gradient that directly drives noncommuting-charge currents. Accordingly, the fluctuation theorem’s first term equals 1. Yet the second term—an average over nonconserving trajectories—is nonzero.

3.2 Surprisal stochastic entropy production↩︎

\(\sigma_\text{surp}\) casts entropy as missing information (Sec. 3.2.1). The average \(\langle \sigma_\text{surp}\rangle\) can be expressed in terms of relative entropies (Sec. 3.2.2), as one might expect from precedent [52]. Yet initial coherences, relative to charges’ product bases, can render \(\langle \sigma_\text{surp}\rangle\) negative. If \(\rho\) is a product \(\rho^{\tt A}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}\)—as across much of thermodynamics—\(\rho\) has such coherences only if the \(Q_\alpha\)’s fail to commute. Furthermore, positive \(\langle \sigma_\text{surp}\rangle\) values accompany time’s arrow. Hence charges’ noncommutation enables a resource, similar to work, for effecting a seeming reversal of time’s arrow. Charges’ noncommutation also engenders a correction to the \(\sigma_\text{surp}\) fluctuation theorem (Sec. 3.2.3).

3.2.1 Surprisal SEP formula↩︎

Information theory motivates the surprisal SEP formula. Averaging the surprisal \(-\log(p_j)\) yields the Shannon entropy (Sec. 2.3), so the surprisal is a stochastic (associated-with-one-trial) entropic quantity. A difference of two surprisals forms our \(\sigma_\text{surp}\) formula. The probabilities follow from preparing \(\rho\) and projectively measuring the \(\alpha^ {\rm th}\) product basis, for any \(\alpha\). Outcome \(i_\alpha \mathrel{\vcenter{:}}= (i_\alpha^{\tt A}, i_\alpha^{\tt B})\) obtains with a probability \(\require{physics} p_\alpha (i_\alpha^{\tt A}, i_\alpha^{\tt B}) \mathrel{\vcenter{:}}= \Tr( \Pi_{\alpha, i_\alpha} \rho )\); and outcome \(f_\alpha \mathrel{\vcenter{:}}= (f_\alpha^{\tt A}, f_\alpha^{\tt B})\), with a probability \(\require{physics} p_\alpha (f_\alpha^{\tt A}, f_\alpha^{\tt B}) \mathrel{\vcenter{:}}= \Tr( \Pi_{\alpha, f_\alpha} \rho )\). Appendix 10.1 shows how these probabilities generalize those in the conventional surprisal-SEP formula [52]. The surprisal SEP quantifies the information gained if we expect to observe \(i_\alpha\) but we obtain \(f_\alpha\): \[\begin{align} \label{ssurp} \sigma_\text{surp} \mathrel{\vcenter{:}}= \log \left( \frac{p_\alpha(i^{\tt A}_\alpha, i^{\tt B}_\alpha) }{p_\alpha(f^{\tt A}_\alpha,f^{\tt B}_\alpha) } \right) . \end{align}\tag{11}\] All results below hold for arbitrary \(\alpha\). Nevertheless, App. 12 introduces a variation on Eq. 11 —an alternative definition that contains an average over all \(\alpha\). Appendix 10.2 confirms that \(\sigma_\text{surp}\) reduces to \(\sigma_\text{chrg}\) if the \(Q_\alpha\)’s commute.

3.2.2 Average of surprisal SEP↩︎

\(\langle \sigma_\text{surp}\rangle\) demonstrates that charges’ noncommutation can enable a seeming reversal of time’s arrow. Time’s arrow manifests in, e.g., spontaneous flows of heat from hot to cold bodies. This arrow accompanies positive average entropy production. Hence negative \(\langle \sigma \rangle\)’s simulate reversals of time’s arrow. These simulations cost resources, such as the work traditionally used to pump heat from colder to hotter bodies. Quantum phenomena, such as entanglement, serve as such resources, too [8], [120]. We identify another such resource: initial coherences relative to charges’ product bases, present in the common initial state 1 only if charges fail to commute.

To prove this result, we denote by \(\Phi_\alpha\) the channel that dephases states \(\omega\) with respect to the \(\alpha^\text{th}\) product basis: \(\Phi_\alpha(\omega) \mathrel{\vcenter{:}}= \sum_k \Pi_{\alpha,k} \, \omega\, \Pi_{\alpha,k}\). \(\sigma_\text{surp}\) averages to \[\require{physics} \label{avgs}\expval{\sigma_\text{surp}} = D \boldsymbol{(} \rho_{\rm f} || \Phi_\alpha \left( \rho \right) \boldsymbol{)} - D \boldsymbol{(} \rho || \Phi_\alpha(\rho) \boldsymbol{)}\tag{12}\] (App. 10.3). The relative entropy is non-negative (Sec. 2.3). Therefore, initial coherences relative to charges’ product bases can reduce \(\require{physics} \expval{\sigma_\text{surp}}\). Such coherences can even render \(\require{physics} \expval{\sigma_\text{surp}}\) negative. Since \(\require{physics} \expval{\sigma_\text{chrg}} \geq 0\) [Eq. 8 ], \(\sigma_\text{surp}\) is sensitive to the resource of coherence, while \(\sigma_\text{chrg}\) is not.

This result progresses beyond three existing results. First, coherences engender a correction to a heat-exchange fluctuation theorem [12]. Those coherences are relative to an eigenbasis of the only charge in [12], energy. If the dynamics conserve only one charge, or only commuting charges, then the thermal product \(\rho^{\tt A}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}\) [Eq. 1 ] lacks the necessary coherences. \(\rho^{\tt A}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}\) has those coherences only if charges fail to commute. Across thermodynamics, \(\rho^{\tt A}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}\) is ubiquitous as an initial state. Hence noncommuting charges underlie a resource for effectively reversing time’s arrow in a common thermodynamic setup.

Second, [28] shows that noncommuting charges can reduce entropy production in the linear-response regime. Our dynamics \(U\) can be arbitrarily far from equilibrium. Third, just as we attribute \(\require{physics} \expval{\sigma_\text{surp}} < 0\) to initial coherences, so do [8], [110], [120], [121] attribute negative average entropy production to initial correlations. Our framework recapitulates such a correlation result, incidentally: If \(\rho\) encodes correlations but retains GGE marginals, \(\require{physics} \expval{\sigma_\text{chrg}}\) [Eq. 7 ] shares the form of 12 , to within one alteration. The decorrelated state \(\rho^{\tt A}\otimes \rho^{\tt B}\) replaces the dephased state \(\Phi_\alpha (\rho)\): \(\require{physics} \expval{\sigma_\text{chrg}} = D( \rho_{\rm f} || \rho^{\tt A}\otimes \rho^{\tt B}) - D(\rho || \rho^{\tt A}\otimes \rho^{\tt B})\). Initial correlations can therefore render \(\require{physics} \expval{\sigma_\text{chrg}}\) negative. Just as initial correlations can render a \(\langle \sigma \rangle\) negative, \(\sigma_\text{surp}\) reveals coherences attributable to noncommuting charges can.

3.2.3 Fluctuation theorem for surprisal SEP↩︎

To formulate the fluctuation theorem, we define the coherent difference \(\Delta \rho_\alpha \mathrel{\vcenter{:}}= \Phi_\alpha(\rho)^{-1} -\rho^{-1}\). It quantifies the coherences of \(\rho\) with respect to the \(\alpha^ {\rm th}\) product basis. If and only if \(\rho\) is diagonal with respect to this basis, \(\Delta \rho_\alpha = 0\). The surprisal SEP obeys the fluctuation theorem \[\require{physics} \begin{align} \label{surpft} \expval{e^{-\sigma_\text{surp}}} = 1+\Tr \boldsymbol{(} U^\dagger \Phi_\alpha(\rho) U \, \Delta \rho_\alpha \, \rho \boldsymbol{)} \end{align}\tag{13}\] (App. 10.4). The second term—the correction—arises from \(\rho\)’s coherences relative to the charges’ product eigenbases. If \(\rho =\rho^{\tt A}_{\rm GGE}\otimes \rho^{\tt B}_{\rm GGE}\), as throughout much of thermodynamics, then \(\rho\) can have such coherences only if charges fail to commute. Our correction resembles that in [12] but arises from distinct physics: noncommuting charges, rather than initial correlations.

3.3 Trajectory stochastic entropy production↩︎

\(\sigma_\text{traj}\) evokes how entropy accompanies irreversibility (Sec. 3.3.1). \(\sigma_\text{traj}\) can assume nonreal values, signaling contextuality in a noncommuting-current experiment (Sec. 3.3.2). Despite the unusualness of nonreal entropy production, \(\sigma_\text{traj}\) satisfies two sanity checks: \(\require{physics} \expval{\sigma_\text{traj}}\) has a sensible physical interpretation (Sec. 3.3.3), and \(\sigma_\text{traj}\) obeys a correction-free fluctuation theorem (Sec. 3.3.4). Complex-valued entropy production appeared also in Ref. [11].

3.3.1 Trajectory SEP formula↩︎

\(\sigma_\text{traj}\) generalizes the conventional trajectory SEP formula, which we review now. Recall the classical experiment in Sec. 1: Classical systems \({\tt A}\) and \({\tt B}\) begin in grand canonical ensembles, then exchange energy and particles. In each trial, \({\tt A}{\tt B}\) undergoes the trajectory \(( E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i )\) \(\mapsto\) \(( E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f )\) with some joint probability. Let us add a subscript F to that probability’s notation: \(p_{\rm F} ( E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i \, ; \, E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f )\). Imagine preparing the grand canonical ensembles, then implementing the time-reversed dynamics. One observes the reverse trajectory, \(( E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f )\) \(\mapsto\) \(( E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i )\), with a probability \(p_{\rm R}( E^{{\tt A}}_f, N^{{\tt A}}_f, E^{{\tt B}}_f, N^{{\tt B}}_f \, ; \, E^{{\tt A}}_i, N^{{\tt A}}_i, E^{{\tt B}}_i, N^{{\tt B}}_i )\). The probabilities’ log-ratio forms the conventional trajectory SEP formula [1]–[3], [6]: \(\log(p_{\rm F} / p_{\rm R})\). In an illustrative forward trajectory, heat and particles flow from a hotter, higher-chemical-potential \({\tt A}\) to a colder, lower-chemical-potential \({\tt B}\). This forward trajectory is likelier than its reverse: \(p_{\rm F} > p_{\rm R}\). Hence \(\log(p_{\rm F} / p_{\rm R}) > 0\), as expected from the second law of thermodynamics.

Let us extend this formula to quasiprobabilities. Section 2.2 established a forward protocol suitable for potentially noncommuting \(Q_\alpha\)’s. That section attributed to the forward trajectory \((i_1, i_2, \ldots, i_c)\) \(\mapsto\) \((f_c, f_{c-1}, \ldots, f_1)\) the KDQ \(\tilde{p}_{\rm F} (i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1)\) [Eq. 4 ]. The reverse protocol features \(U^\dagger\), rather than \(U\), in step [step95U], with a reversed list of measurement outcomes. The reverse trajectory \((f_1, f_2, \ldots, f_c)\) \(\mapsto\) \((i_c, i_{c-1}, \ldots, i_1)\) corresponds to the quasiprobability \(\require{physics} \tilde{p}_{\rm R} (f_1, f_2, \ldots, f_c \, ; \, i_c, i_{c-1}, \ldots, i_1) \mathrel{\vcenter{:}}= \Tr( [ \Pi_{1,f_1} \ldots \Pi_{c, f_c} ] U [ \Pi_{c, i_c} \ldots \Pi_{1, i_1} ] U^\dagger \rho)\). This definition captures the notion of time reversal, we argue in App. 11.1, while enabling \(\sigma_\text{traj}\) to agree with \(\sigma_\text{chrg}\) and \(\sigma_\text{surp}\) in the commuting case (App. 11.2). Both quasiprobabilities feature in the trajectory SEP: \[\require{physics} \begin{align} \tag{14} & \sigma_\text{traj}\mathrel{\vcenter{:}}= \log \left( \frac{ \tilde{p}_\text{F} (i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1) }{ \tilde{p}_\text{R} (f_1, f_2, \ldots, f_c \, ; \, i_c, i_{c-1}, \ldots, i_1) } \right) \\ & = \log \left( \frac{\Tr \left( U^\dagger \left[ \Pi_{1,f_1} \ldots \Pi_{c, f_c} \right] U \left[ \Pi_{c,i_c} \ldots \Pi_{1, i_1} \right] \rho \right)}{ \Tr \left( \left[ \Pi_{1,f_1} \ldots \Pi_{c, f_c} \right] U \left[ \Pi_{c, i_c} \ldots \Pi_{1, i_1} \right] U^\dagger \rho \right)} \right) \nonumber \\ &=\log \left( \frac{ \bra{i_1} \rho U^\dagger \ket{f_1}}{ \bra{i_1} U^\dagger \rho \ket{f_1} } \right) . \tag{15} \end{align}\] The final equality follows from the nondegeneracy of the local charges \(Q_\alpha^{\tt X}\): The projectors \(\Pi_{\alpha, k} = \ketbra{\alpha_k}{\alpha_k}\), so factors cancel between the numerator and denominator. The \(\bra{i_1}\)’s and \(\ket{f_1}\)’s distinguish \(Q_1\) from the other charges. However, 15 holds for every possible labeling of the \(Q_\alpha\)’s (holds under the labeling of any charge as 1) and for every ordering of the projectors in 4 . Moreover, App. 12 introduces a variation on Eq. 14 —a definition that contains an average over all measurement orderings, removing the dependence on the ordering.

3.3.2 Nonreal trajectory SEP witnesses nonclassicality↩︎

Nonreal \(\sigma_\text{traj}\) values signal nonclassicality in a noncommuting-current experiment. To prove this result, we review weak values (conditioned expectation values) and contextuality (provable nonclassicality). We then express \(\sigma_\text{traj}\) in terms of weak values. Finally, we prove that nonreal \(\sigma_\text{traj}\) values herald contextuality in an instance of the forward or reverse protocol.

Figure 2 motivates the weak value’s form [58], [126]. Consider preparing a quantum system in a state \(\omega\) at a time \(t=0\). The system evolves under a unitary \(V\). An observable \(\mathcal{Y}\) is then measured, yielding an outcome \(y\). Denote by \(\mathcal{X}\) an observable that neither commutes with \(\omega\) nor shares the \(\mathcal{Y}\) eigenstate \(\ket{y}\). Which value is retrodictively most reasonably attributable to \(\mathcal{X}\) immediately after the state preparation? Arguably the weak value [122]–[125] \(\require{physics} \Tr(\Pi'_y \mathcal{X} \omega)/\Tr(\Pi_y' \omega)\), wherein \(\Pi'_y \mathrel{\vcenter{:}}= V^\dagger | y\rangle\langle y| V\). Weak values can be anomalous, lying outside the spectrum of \(\mathcal{X}\). Anomalous weak values actuate metrological advantages [78], [127]–[131] and signal contextuality [89], [90], [132].

Contextuality is a strong form of nonclassicality [98], [133]. One can model quantum systems as being in unknown microstates akin to classical statistical-mechanical microstates. One might expect to model operationally indistinguishable procedures identically. No such model reproduces all quantum-theory predictions, however. This impossibility is quantum theory’s contextuality, which underlies quantum-computational speedups, for example [134]. Anomalous weak values signal contextuality in the process prepare \(\omega\), measure \(\mathcal{X}\) weakly, evolve with \(V\), measure \(\mathcal{Y}\) strongly, and postselect on \(y\) [89], [90], [132].

Having introduced the relevant background, we prove that \(\sigma_\text{traj}\) depends on weak values and signals contextuality. Define the evolved projectors \(\Pi_{1, i_1}' \mathrel{\vcenter{:}}= U \Pi_{1, i_1} U^\dagger\) and \(\Pi_{1, f_1}'' \mathrel{\vcenter{:}}= U^\dagger \Pi_{1, f_1} U\). Define also the weak values \[\require{physics} \begin{align} & {}_{f_1}\langle \Pi_{1,i_1} \rangle_{\rho} \mathrel{\vcenter{:}}= \Tr(\Pi_{1, f_1}'' \Pi_{1, i_1} \rho) / \Tr(\Pi_{1, f_1}'' \rho) \quad \text{and} \\ & {}_{i_1}\langle \Pi_{1, f_1} \rangle_{\rho} \mathrel{\vcenter{:}}= \Tr(\Pi_{1, i_1}' \Pi_{1,f_1} \rho)/\Tr(\Pi_{1,i_1}' \rho). \end{align}\] Each is a complex number with a phase \(\phi\): \({}_{f_1}\langle \Pi_{1,i_1} \rangle_{\rho} = | {}_{f_1}\langle \Pi_{1,i_1} \rangle_{\rho} | e^{i \phi_{\rm F} }\), and \({}_{i_1}\langle \Pi_{1, f_1} \rangle_{\rho} = | {}_{i_1}\langle \Pi_{1, f_1} \rangle_{\rho} | e^{-i \phi_{\rm R} }\). We suppress the phases’ indices for conciseness. Equation 15 becomes \[\require{physics} \begin{align} \label{eq:straj95cartesian} \sigma_\text{traj} &=\log \left( \big\lvert {}_{f_1}\langle \Pi_{1,i_1} \rangle_{\rho} \big\rvert / \big\lvert {}_{i_1}\langle \Pi_{1, f_1} \rangle_{\rho} \big\rvert \right) + i(\phi_\text{F} - \phi_\text{R}) \nonumber \\ & \quad \; + \log \boldsymbol{(} \Tr ( \Pi_{1, f_1}'' \rho ) / \Tr (\Pi_{1,i_1}' \rho ) \boldsymbol{)} . \end{align}\tag{16}\] The final log is of mere probabilities. The phase difference is defined modulo \(2 \pi\). For convenience, we assume the complex logarithm’s branch cut lies along the negative real axis.⁴ Appendix 11.3 explains how to choose the complex logarithm’s value if \(\rho\) is pure and describes subtleties concerning mixed \(\rho\)’s.

If \({\rm Im}(\sigma_\text{traj}) \neq 0\), we call \(\sigma_\text{traj}\) anomalous, and at least one weak value is anomalous. Hence at least one of two protocols is contextual:

1. Forward compressed protocol: Prepare \(\rho\). Measure \(\Pi_{1, i_1}\) weakly.⁵ Evolve under \(U\). Measure the \(Q_1\) product basis strongly. Postselect on \(f_1\).

2. Reverse compressed protocol: Prepare \(\rho\). Measure \(\Pi_{1, f_1}\) weakly. Evolve under \(U^\dagger\). Measure the \(Q_1\) product basis strongly. Postselect on \(i_1\).

The forward compressed protocol is a simplification of the forward protocol in Sec. 2.2: Only measurements pertaining to charge \(Q_1\) are performed. Analogous statements concern the reverse compressed protocol. Hence \(\sigma_\text{traj}\) joins a sparse set of thermodynamic quantities known to signal contextuality [12], [135].

\(\sigma_\text{traj}\)’s signaling of contextuality exhibits irreversibility—fittingly for an entropic phenomenon—algebraically and geometrically. First, for \(\sigma_\text{traj}\) to signal contextuality—to become nonreal—nonreality of a weak value does not suffice. Rather, the weak values’ phases must fail to cancel: \(\phi_{\rm F} \neq \phi_{\rm R}\).⁶ This failure mirrors how, conventionally, entropy is produced only when \(p_{\rm F} \neq p_{\rm R}\).

Second, suppose that \(\rho\) is pure. \(\phi_{\rm F} - \phi_{\rm R}\) equals the geometric phase imprinted on a state manipulated as follows: \(\rho\) is prepared, the forward compressed protocol is performed, the state is reset to \(\rho\), the reverse compressed protocol is performed, and the state is reset to \(\rho\) [136], [137]. All measurements here are performed in the strong limit. The state acquires the phase \(e^{i \phi_{\rm F} }\) during the forward step and acquires \(e^{-i \phi_{\rm R} }\) during the reverse. Only if the forward and reverse steps fail to cancel does the geometric phase \(\neq 1\)—does \(\sigma_\text{traj}\) signal contextuality. Hence \(\sigma_\text{traj}\) heralds contextuality in the presence of irreversibility, appropriately for a thermodynamic quantity.

3.3.3 Average of trajectory SEP↩︎

Formally, averaging \(\sigma_\text{traj}\) [Eq. 14 ] with respect to \(\tilde{p}_{\rm F}\) yields a quasiprobabilistic relative entropy of Sec. 2.3:⁷ \[\require{physics} \begin{align} \expval{\sigma_\text{traj}} = D \boldsymbol{(} & \tilde{p}_\text{F} (i_1, i_2, \ldots, i_c \, ; \, f_c, f_{c-1}, \ldots, f_1) \\ & \nonumber || \, \tilde{p}_\text{R} (f_1, f_2, \ldots, f_c \, ; \, i_c, i_{c-1}, \ldots, i_1) \boldsymbol{)} . \end{align}\] The average can be negative and even nonreal. Yet \(\require{physics} \expval{\sigma_\text{traj}}\) has a particularly crisp physical interpretation when \(\rho\) is a pure state that retains GGE marginals 1 .⁸

A pure \(\rho\) is not as restricted as it may seem: Thermodynamics often features pure global states whose reduced states are thermal [8], [101], [120], [138], [139]. The average becomes \[\require{physics} \begin{align} \label{avgt} \expval{\sigma_\text{traj}} & = \tfrac{1}{2} \big[ D\boldsymbol{(}\rho || \Phi_1( U^\dagger \rho U)\boldsymbol{)} + D\boldsymbol{(}\rho|| U^\dagger \Phi_1(\rho) U\boldsymbol{)} \nonumber \\ & \qquad \; \: - D\boldsymbol{(}\rho || \Phi_1(\rho)\boldsymbol{)} - D\boldsymbol{(}\rho_{\rm f} || \Phi_1(\rho_{\rm f})\boldsymbol{)} \big] \nonumber \\ & \quad \, + i \langle \phi_\text{F} - \phi_\text{R}\rangle \end{align}\tag{17}\] (App. 11.3). Each \(\phi\) implicitly depends on indices \(i_1\) and \(f_1\). \(\require{physics} \expval{\sigma_\text{traj}}\)’s real and imaginary components have physical significances that we elucidate now.

In App. 11.3, we prove that \(\langle \phi_\text{F} - \phi_\text{R}\rangle\) is real.⁹ Hence, if \(\require{physics} \Im(\expval{\sigma_\text{traj}}) \neq 0\), at least one \(\phi_{\rm F}\) [associated with one tuple \((i_1, f_1)\)] or one \(\phi_{\rm R}\) is nonzero. Therefore, at least one weak value—at least one \({}_{f_1}\langle \Pi_{1, i_1}\rangle_\rho\) or \({}_{i_1}\langle \Pi_{1, f_1}\rangle_\rho\)—is anomalous. At least one instance of the forward or reverse compressed protocol is therefore contextual. In conclusion, \(\require{physics} \expval{\sigma_\text{traj}}\), beyond \(\sigma_\text{traj}\), witnesses contextuality.

\(\Re(\langle \sigma_\text{traj}\rangle)\) has two familiar properties. First, \(\Re(\langle \sigma_\text{traj}\rangle) \geq 0\), suggestively of the second law of thermodynamics. Second, \(\Re(\langle \sigma_\text{traj}\rangle)\) depends on relative-entropy differences, similarly to Eq. 8 .

Pairs of relative entropies have recognizable physical significances. Each negated \(D\) is a relative entropy of coherence, comparing a state (\(\rho\) or \(\rho_{\rm f}\)) to its dephased counterpart [140]. Hence states’ coherences relative to \(\{ \Pi_{1, k} \}\) reduce \(\require{physics} \expval{\sigma_\text{traj}}\). This reduction resembles the reduction of \(\require{physics} \expval{\sigma_\text{surp}}\) by initial coherence (Sec. 3.2.2). Not only initial coherences, though, but also final coherences reduce \(\require{physics} \expval{\sigma_\text{traj}}\).

Equation 17 ‘s first two relative entropies imprint noncommutation on \(\require{physics} \expval{\sigma_\text{traj}}\). In each such \(D\), the compared-to state (the final argument) results from a dephasing and a time-reversed evolution. The operations’ ordering differs between the \(D\)’s. The \(1/2\) in Eq. 17 averages over the orderings. Hence \(\require{physics} \expval{\sigma_\text{traj}}\) translates into sensible physics.

3.3.4 Fluctuation theorem for trajectory SEP↩︎

Passing another sanity check, \(\sigma_\text{traj}\) satisfies the correction-free fluctuation theorem \[\require{physics} \begin{align} \label{trajft} \expval{e^{-\sigma_\text{traj}}} = 1. \end{align}\tag{18}\] The proof follows from the KDQ’s normalization.

4 Two-qubit example↩︎

This section numerically illustrates the SEP definitions’ key properties. First, we introduce the system simulated. Afterward, we analyze the calculated SEPs using generic parameter choices.

In our example system, \({\tt A}\) and \({\tt B}\) are qubits. The charges are the Pauli operators \(Q_1=\sigma_z\), \(Q_2=\sigma_y\), and \(Q_3=\sigma_x\). By the Schur–Weyl duality, the most general charge-conserving unitary is a linear combination of the permutations of two objects—a linear combination of the identity and SWAP operators [47], [117]. The SWAP operator acts on states \(\ket{\psi}_{\tt A}\) and \(\ket{\phi}_{\tt B}\) as \(\text{SWAP} \ket{\psi}_{\tt A}\otimes \ket{\phi}_{\tt B}= \ket{\phi}_{\tt A}\otimes \ket{\psi}_{\tt B}\). We parameterize the unitary with an angle \(\theta\): \[\label{eq:qubit95unitary} U_\theta = \cos (\theta) \, \mathbb{1}+ i \sin(\theta) \;\text{SWAP} \, .\tag{19}\]

4.1 Charge SEP↩︎

Section 3.1.3 introduced nonconserving trajectories and the following results. On the RHS of the fluctuation theorem 9 are two terms. The first, notated as \(\require{physics} \expval{e^{-\kappa}}\) (App. 9.1), encodes the noncommutation of the charges in the initial state [Eqs. 1 and 2 ]. The fluctuation theorem’s final term, notated as \(\require{physics} \expval{e^{-\sigma_\text{chrg}}-e^{-\kappa}}_\text{noncons}\) (App. 9.1), is an average over nonconserving trajectories.

We support these claims by showing how the two terms change as \(\theta\) varies from \(0\) to \(\frac{\pi}{2}\) in Eq. 19 . Figure 3 shows the terms’ real and imaginary parts, as well as the average entropy production \(\require{physics} \expval{\sigma_\text{chrg}}\). The generalized inverse temperatures’ values (listed in Fig. 3’s caption) ensure that a strong thermodynamic force pushes the \(\sigma_x\) and \(\sigma_y\) charges from \(\tt B\) to \(\tt A\), whereas a weak force pushes \(\sigma_z\) oppositely.

The unitary parameter \(\theta\) increases along the \(x\)-axis. As per a sanity check in Sec. 3, when \(\theta=0\), the fluctuation theorem’s terms sum to one. The first term, \(\require{physics} \expval{e^{-\kappa}}\), equals one; and the second term, \(\require{physics} \expval{e^{-\sigma_\text{chrg}}-e^{-\kappa}}_\text{noncons}\), vanishes. As \(\theta\) increases, charges flow more, as evidenced by the increasing \(\require{physics} \expval{\sigma_\text{chrg}}\). \(\require{physics} \expval{e^{-\kappa}}\) changes little. This near-constancy reflects the origination of \(\require{physics} \expval{e^{-\kappa}}\) in the noncommutation of the charges in the initial state, which remains constant as \(\theta\) changes. In contrast, as \(\theta\) grows, \(\require{physics} \expval{e^{-\sigma_\text{chrg}}-e^{-\kappa}}_\text{noncons}\) increases in magnitude—due to both its real and imaginary parts. This growth reflects nonconserving trajectories’ growing contribution to the fluctuation theorem’s RHS.

4.2 Surprisal SEP↩︎

The initial state \(\rho\) can have coherences relative to each charge’s product basis. As discussed in Sec. 3.2.2, these coherences can reduce \(\langle \sigma_\text{surp}\rangle\), even rendering it negative. \(\sigma_\text{surp}\) is defined in terms of an arbitrary charge index \(\alpha\), which we choose to be 1: \(Q_1 = \sigma_z\).

Figure 4 shows \(\require{physics} \expval{\sigma_\text{surp}}\) as a function of \(\beta_x^{\tt A}\) and \(\beta_z^{\tt A}\). If \(\beta_z^{\tt A} \gg \beta_x^{\tt A}\), \(\rho^{{\tt A}}\) is nearly diagonal relative to the \(\sigma_z\) eigenbasis. Hence \(\require{physics} \expval{\sigma_\text{surp}}\) is positive, as evidenced in the plot’s top left-hand corner. In the opposite regime (\(\beta_x^{\tt A} \gg \beta_z^{\tt A}\)), \(\rho^{{\tt A}}\) has large coherences relative to the \(\sigma_z\) eigenbasis. These coherences drive \(\require{physics} \expval{\sigma_\text{surp}}\) below zero, as evidenced in the bottom right-hand corner.

4.3 Trajectory SEP↩︎

As shown in Sec. 3.3.2, a nonreal \(\sigma_\text{traj}\) implies contextuality in a noncommuting-charge experiment. \(\sigma_\text{traj}\) carries indices \(i_1\) and \(f_1\), by the definition 14 . We numerically calculate the \(\sigma_\text{traj}\) evaluated on the trajectory defined by \(\ket{i_1} = \ket{0}^{\tt A}\otimes \ket{1}^{\tt B}\) and \(\ket{f_1} = \ket{0}^{\tt A}\otimes \ket{0}^{\tt B}\).

Figure 5 shows the imaginary part of \({\sigma_\text{traj}} (i_1,f_1)\) plotted against \(\beta_x^{\tt A}\) and \(\beta_y^{\tt A}\). \(\sigma_{\rm traj}\) typically has a nonzero imaginary component (of a magnitude similar to the real part’s), signaling contextuality. The imaginary component’s magnitude grows particularly large when \(\beta_y^{\tt A} \approx 0\). Furthermore, the imaginary component exhibits stability, changing smoothly with the swept parameters.

5 Experimental sketch↩︎

Our results can be tested experimentally. Several pieces of evidence indicate such an experiment’s feasibility. First, a trapped-ion experiment recently initiated the experimental testing of noncommuting-charge thermodynamics [48]. Second, other platforms have been argued to support such tests [27], [47]. Examples include superconducting qubits, neutral atoms, and possibly nuclear-magnetic-resonance systems. Third, sequential weak measurements have been realized with trapped ions [141], superconducting qubits [142], and photonics [143]–[147]. Fourth, we now sketch a trapped-ion experiment inspired by Sec. 4. We outline the setup, preparation procedure, evolution, measurement, and data processing. For concreteness, we tailor the proposal to the platform reported on in [148].

The platform consists of \({}^{171}\text{Yb}^+\) ions in a linear Paul trap. Each ion encodes a qubit in two hyperfine ground states (two energy levels that result from splitting a ground space with a hyperfine interaction). Our proposal calls for ten qubits: Two form the system of interest, two ancillas enable state preparation, and six ancillas enable weak measurements. The system of interest consists of a qubit \({\tt A}\) and a qubit \({\tt B}\). These qubits will exchange charges \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\). However, tracking just two noncommuting charges suffices for an initial experiment. We choose \(\sigma_x\) and \(\sigma_z\).

\({\tt A}\) and \({\tt B}\) must be prepared in a tensor product of GGEs [Eq. 2 ]. One procedure involves two ancilla qubits: Prepare \({\tt A}\) and an ancilla in a thermofield-double state (a purification of a GGE), using the trapped-ion protocol in [148]. Discard the ancilla qubit. Repeat these two steps with qubit \({\tt B}\) and another ancilla. Generalized inverse temperatures \(\beta^{\tt X}_\alpha\) parameterize the initial state. One chooses the parameters’ values as in Sec. 4, to observe the three results listed two paragraphs below.

After the state preparation, one weakly measures \(\sigma_x^{\tt A}\) and \(\sigma_x^{\tt B}\), then \(\sigma_z^{\tt A}\) and \(\sigma_z^{\tt B}\). One can implement a weak measurement using a qubit ancilla, using the circuit shown in Fig. 1(a) of [149]. Hence the initial weak measurements require four ancillas, total.

After these weak measurements, \({\tt A}\) and \({\tt B}\) evolve under a charge-preserving unitary \(U_\theta\) [Eq. 19 ]. The trapped-ion platform under consideration offers a gate set formed from arbitrary single-qubit rotations and \(XX\) gates [148], [150]. The gate set’s universality implies that \(U_\theta\) can be implemented, if the ions retain coherence for long enough. The two-qubit gate requires the most time—between 1 and 100s of \(\mu\)s [150]. However, coherence times range from 100s of ms to 100s of seconds. The time scales are significantly separated, although gate errors will further restrict circuit depth [151].

After the evolution, one weakly measures \(\sigma_x^{\tt A}\) and \(\sigma_x^{\tt B}\). Our abstract protocol (Sec. 2.2) ends with weak measurements of \(\sigma_z^{\tt A}\) and \(\sigma_z^{\tt B}\). An experimentalist can replace these weak measurements with strong measurements, without hindering the reconstruction of \(\tilde{p}_{\rm F}\) from the experimental data [63]. Furthermore, the replacement spares us the need for two extra ancillas.

One repeats the foregoing protocol many times. From the measurement outcomes, we infer the probability distribution over the possible measurement outcomes. One reconstructs the KDQ \(\tilde{p}_{\rm F}\) via the procedure in [63]. Analogously, one infers \(\tilde{p}_{\rm R}\) from another batch of trials, guided by the reverse protocol of Sec. 3.3.1. Since our theoretical results are deductive, the proposed experiment essentially checks quantum theory’s accuracy. However, observing three phenomena would highlight quantum features exhibited by the SEPs when charges fail to commute: (i) Observe a violation of detailed charge conservation. (ii) Observe a negative \(\require{physics} \expval{\sigma_\text{surp}}\). (iii) Observe an imaginary component of \(\sigma_\text{traj}\).

6 Outlook↩︎

Noncommuting charges challenge common expectations about entropy production. Three common SEP formulae, though equal when charges commute, separate when charges do not. The formulae also offer different physical insights into how noncommuting charges impact entropy production. First, the noncommutation enables stochastic trajectories to violate charge conservation individually. We introduce these nonconserving trajectories, possible only if charges fail to commute, as quantum phenomena in stochastic thermodynamics. The violations of detailed charge conservation underlie commutator-dependent corrections to a fluctuation theorem. Second, initial coherences relative to charges’ eigenbases can render \(\require{physics} \expval{\sigma_\text{surp}}\) negative. A common (two-thermal-reservoir) setup can entail such coherences only if the charges fail to commute. Hence, charges’ noncommutation source a resource that can, in a sense, effectively reverse time’s arrow. Third, nonreality of \(\sigma_\text{traj}\) signals contextuality—provable nonclassicality—in a noncommuting-current experiment. Such thermodynamic signatures of contextuality—a stringent criterion for nonclassicality—are rare. These results hold arbitrarily far from equilibrium. In addition to proving these results deductively, we illustrated them numerically and sketched a trapped-ion test.

Our work introduces noncommuting charges into stochastic thermodynamics [99]. The field’s results now merit reevaluation, in case charges’ noncommutation alters them. For example, thermodynamic uncertainty relations bound a current’s relative variance with entropy production [152]–[155]. Lowering entropy production, noncommuting charges may increase the relative variance, increasing currents’ unpredictability. Our work therefore motivates the derivation of thermodynamic uncertainty relations that highlight exchanges of noncommuting charges, using KDQs. Such relations may follow as extensions of [154].

As another avenue opened by our work within stochastic thermodynamics, nonconserving trajectories provide a new tool for unearthing genuinely quantum effects. Consider any charge exchange modeled with a KDQ. The KDQ decomposes into a conserving and nonconserving part. The former is the only piece that survives when the process has a classical (probabilistic) description. The average of every stochastic physical variable (analogous to entropy production), with respect to the KDQ, decomposes likewise. Hence the contribution of charge noncommutation to the average can be delineated clearly. For example, this technique could be applied to decompose the averaged stochastic work performed by a monitored quantum engine operating between noncommuting-charge reservoirs.

Quantum engines offer another possible application [156]: The more entropy an engine produces, the lesser the engine’s efficiency. We showed that charges’ noncommutation can lower entropy production even on average: Coherences relative to charges’ eigenbases serve as a resource for reducing average entropy production (Sec. 3.2). Hence engines might leverage such coherences. Such an engine could exchange not only heat, but also noncommuting charges, with reservoirs. One could extend the engine, proposed in [157], that leverages a spin reservoir in place of a heat reservoir.

To put the charges on even more-equal footing, one could average over them in the KDQ’s and SEPs’ definitions (App. 12). Finally, calculating a closed form for the average trajectory SEP, for mixed global states, remains an open problem.

Experimental opportunities complement the theoretical ones. In Sec. 5, we sketched a trapped-ion experiment for testing our results. Such a test would highlight the quantum features manifested by SEPs when charges fail to commute.

References↩︎