Negative quasiprobability trajectories for Bell-diagonal states under local decoherence

Abstract: The fluctuation theorem (FT) relates microscopic trajectory distributions to macroscopic averages. Using quasiprobability trajectories, this framework has recently been extended to quantum information dynamics. Despite sharing the form of conventional thermodynamic FTs, information FTs involve quasiprobabilities whose negativity and statistical properties remain poorly understood. We analytically determine the properties of negative distributions based on the two-qubit Bell-diagonal states evolving under local dephasing, depolarizing, and amplitude-damping channels. We show that the occurrence of negative quasiprobabilities is not determined by the entanglement or Bell nonlocality of the initial state. Instead, a diagonal-interference decomposition of the transition quasiprobability provides a sufficient condition for negativity. More importantly, we prove that quasiprobability distributions containing negative weights can violate Horváth's necessary and sufficient criterion for the Jensen-Steffensen inequality to hold for every continuous convex function, even though the specific exponential Jensen-like relation enforced by the FT and the quantum data-processing inequality remains valid. This trajectory-level contrast with classical stochastic descriptions reveals a distinctive feature of quantum information dynamics that is invisible when only the corresponding average-level information inequalities are considered.

• The paper analytically characterizes Kirkwood–Dirac trajectory weights for two-qubit Bell-diagonal states under local dephasing, depolarizing, and amplitude-damping channels, finding no negativity for the first two and exactly two negative trajectories under generic amplitude damping.
• Amplitude damping produces sharp parameter-independent lower bounds of $(1-\sqrt{2})/4\approx-0.10355$ for transition quasiprobabilities and $(1-\sqrt{2})/8\approx-0.05178$ for reduced trajectory weights, with negativity determined by destructive interference rather than entanglement or Bell nonlocality.
• The results show that the integral fluctuation theorem and exponential Jensen relation can remain valid even when Horváth’s criterion for all convex functions fails, highlighting limits on extending average information inequalities to signed trajectory distributions.

The paper analyzes the negativity and statistical structure of Kirkwood–Dirac (KD) quasiprobability trajectories arising in the fluctuation theorem (FT) for quantum mutual information dissipation (2608.16358). Working with two-qubit Bell-diagonal states evolving under local dephasing, depolarizing, and amplitude-damping channels, the authors obtain fully analytic characterizations of when trajectory weights become negative, how negative they can become, and what this implies for convexity-based inequalities at the trajectory level.

Quasiprobability FT for mutual information dissipation

For a bipartite system $S=S_1S_2$ undergoing local channels $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$, the mutual information cannot increase, so $\Delta\mathcal{I}\geq 0$ by the data-processing inequality. The paper adopts a stochastic mutual information change $\Delta\iota[\gamma]$ built from eigenvalues of the initial and final global and local states, mirroring stochastic-entropy constructions. Because the spectral projectors of global and local states do not commute, a classical joint distribution is unavailable; instead, KD quasiprobabilities assign normalized weights to trajectories while reproducing correct marginals. The key identities are

$$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$

the latter being an integral FT whose form matches thermodynamic FTs but which concerns a purely informational quantity and rests on a signed quasiprobability distribution. Summing over environmental variables yields a Kraus operator-sum representation of the reduced trajectory weight, $\mathcal{Q}_S[\gamma]=p_k\,q[\gamma]$, where $$q[\gamma]=\mathrm{Tr}(\Pi_{k'}\Pi_{s'}\mathcal{E}_S(\Pi_s\Pi_k))$$ is a conditional transition quasiprobability.

Nonnegativity under dephasing and depolarizing noise

For local dephasing, only two transition values occur, $(1+\lambda^2)/4$ and $(1-\lambda^2)/4$, both nonnegative; for local depolarizing noise, four coefficients $\kappa_1,\dots,\kappa_4$ arise, all nonnegative on the complete-positivity interval $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$0. Hence all reduced trajectory weights remain nonnegative for every Bell-diagonal initial state under these two channels. This is already informative: incompatibility between the Bell projectors and fixed computational-basis projectors does not by itself force negativity.

Negativity under amplitude damping

Local amplitude damping breaks the Bell-diagonal structure: the final state is an $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$1 state with non-maximally-mixed marginals. For $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$2, exactly two trajectories can be negative whenever $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$3. The paper establishes two sharp results:

• Most negative transition quasiprobability: attained at $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$4 for $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$5 and at $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$6 otherwise, with the parameter-independent bound $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$7.
• Most negative reduced weight: the minimizing trajectory is $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$8 rather than the most negative transition trajectory, because the population factor $$\mathcal{E}_S=\mathcal{E}_{S_1}\otimes\mathcal{E}_{S_2}$$9 shifts the optimum; the bound is $\Delta\mathcal{I}\geq 0$0, exactly half the transition-level bound.

A notable subtlety is that optimization over the initial state and the weak-damping limit do not commute: for every fixed $\Delta\mathcal{I}\geq 0$1 the negative weights vanish as $\Delta\mathcal{I}\geq 0$2, yet the optimized minimum approaches $\Delta\mathcal{I}\geq 0$3 because the optimizing state simultaneously approaches the degenerate point $\Delta\mathcal{I}\geq 0$4.

Crucially, negativity is not tied to entanglement or Bell nonlocality. The maximally entangled, CHSH-violating state $\Delta\mathcal{I}\geq 0$5 has $\Delta\mathcal{I}\geq 0$6 and produces no negative trajectories, whereas the separable state $\Delta\mathcal{I}\geq 0$7 has $\Delta\mathcal{I}\geq 0$8 and does.

Interference dominance condition

Decomposing the transition quasiprobability into diagonal and off-diagonal parts relative to the final local basis, $\Delta\mathcal{I}\geq 0$9, the paper proves a sufficient condition ("interference dominance"): if both contributions are real, $\Delta\iota[\gamma]$0, and $\Delta\iota[\gamma]$1, then the weight is negative. Dephasing and depolarizing satisfy $\Delta\iota[\gamma]$2 throughout their parameter ranges, explaining their nonnegativity; amplitude damping violates it precisely on the four negative trajectories. A pre-rotated amplitude-damping channel provides a second explicit example with $\Delta\iota[\gamma]$3. The authors are careful to note that this criterion is sufficient rather than universal: for general channels $\Delta\iota[\gamma]$4 need not be real or nonnegative, so other mechanisms of negativity may exist.

Violation of Horváth's criterion

Although the exponential Jensen-like relation $\Delta\iota[\gamma]$5 holds—guaranteed jointly by the integral FT and the data-processing inequality—the paper asks whether the signed grouped distribution satisfies Horváth's necessary and sufficient conditions for the Jensen–Steffensen inequality to hold for every continuous convex function. The answer depends on degeneracy structure:

• For the initial Bell state $\Delta\iota[\gamma]$6, the unique negative trajectory shares its support value of $\Delta\iota[\gamma]$7 with a positive partner whose combined grouped weight is positive; Horváth's criterion is satisfied. Microscopic negativity alone therefore does not imply violation.
• For the family producing the most negative reduced weight ($\Delta\iota[\gamma]$8, $\Delta\iota[\gamma]$9), the smallest support value $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$0 is unique and carries negative weight, so Horváth's endpoint test fails.
• Beyond this extremal family, a structured slice $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$1, $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$2 exhibits violation whenever $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$3; for $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$4 the violation covers the entire physical region, and for $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$5 it occupies a computable subregion bounded by the curve $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$6.

This result sharpens a claim in the underlying multipartite FT work: the FT plus data processing guarantee the exponential relation for $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$7 only, and do not extend to arbitrary convex functions. Consequently, once Horváth's criterion fails, Jensen-type inequalities for other information measures must be verified independently.

Limitations and open questions

The analysis is confined to two-qubit Bell-diagonal states and three unital-or-relaxing local channels with product environments; the interference-dominance condition assumes real-valued diagonal and interference contributions, so complex-valued KD distributions fall outside its scope. The paper leaves open whether the decomposition and the bounds $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$8 and $$\langle\Delta\iota\rangle_{\mathcal{Q}_S}=\Delta\mathcal{I}, \qquad \langle e^{-\Delta\iota}\rangle_{\mathcal{Q}_S}=1,$$9 generalize to multipartite states and correlated, non-Markovian, or collective dynamics; whether Rényi data-processing inequalities admit a trajectory-level counterpart tied to generalized Jensen–Steffensen conditions; and whether some generalized quantum Jensen's inequality can recover the data-processing inequality from the KD FT even when Horváth's criterion is violated.

Conclusion

The paper shows that trajectory-level negativity in informational FTs is governed not by entanglement or nonlocality but by destructive interference between local-basis components of the final global eigenstate, quantified through exact parameter-independent lower bounds. Its central structural finding is that classical and quantum information inequalities can coexist at the average level while resting on different stochastic foundations: the exponential relation survives via the logarithmic information variable and the integral FT, even when the signed trajectory distribution violates the conditions required for convexity arguments valid for all convex functions.