Entanglement-Resolved Fluctuation Theorems

• Entanglement-resolved fluctuation theorems are a framework that generalizes classical fluctuation relations to quantify fluctuating entanglement in quantum processes.
• The approach employs an entanglement battery and detailed probability distributions to capture work-like entanglement costs during pure and mixed state transformations.
• These theorems impact reversible entanglement manipulation, experimental quantum thermodynamics, and the certification of quantum resource contributions in operational protocols.

Entanglement-resolved fluctuation theorems extend the framework of nonequilibrium statistical mechanics and quantum thermodynamics by explicitly incorporating quantum entanglement as a fluctuating resource. These theorems generalize classical fluctuation relations (such as the Jarzynski equality and Crooks relation) to quantify the conversion, dissipation, and irreversibility of entanglement during quantum processes. This approach provides new operational insight into reversibility, resource interconversion, and the constraints imposed by the laws of thermodynamics in quantum and quantum information-theoretic settings.

1. Foundational Principles and Main Results

Fluctuation theorems characteristically relate the probabilities of observing entropy production or work fluctuations in forward and reverse processes, both in classical and quantum domains. Entanglement-resolved fluctuation theorems adapt this structure to bipartite quantum systems, resolving the statistical fluctuations in entanglement consumed or generated in a process. The paradigm was established for pure-state transformations under local operations and classical communication (LOCC), supplemented by an “entanglement battery” storing ebits, which can fluctuate coherently between protocol instances (Alhambra et al., 2017).

Given an initial pure state $|\psi\rangle_{AB}$ (Schmidt coefficients $p_i$) and a target pure state $|\phi\rangle_{AB}$ (Schmidt coefficients $q_j$), the protocol seeks to implement the transformation

$$|\psi\rangle_{AB} \otimes |n\rangle_{A'B'} \longrightarrow |\phi\rangle_{AB} \otimes \sum_w \sqrt{P(w)} |n+w\rangle_{A'B'}$$

where $|n\rangle_{A'B'}$ is the state of the auxiliary entanglement battery containing $n$ ebits, and $w$ parameterizes the change (“work” analog) in stored entanglement (“ebit-number eigenstates”).

The main assisted-stochasticity theorem provides necessary and sufficient conditions for the existence of such a transformation, in terms of conditional probability distributions $P(i, w | j)$, satisfying three constraints: normalization over outcomes, entanglement-weighted stochasticity, and conservation of the reduced-state spectrum (Alhambra et al., 2017).

2. Entanglement Jarzynski Equality and Crooks Relation

The entanglement Jarzynski equality is an exact constraint on the statistical moments of the entanglement fluctuation distribution $P(w)$:

$$\left\langle 2^{w - [ -\log q_j + \log p_i ] } \right\rangle = 1$$

For a maximally entangled target $|\phi\rangle$ (of local dimension $d'$), the equality reduces to $\sum_w 2^w P(w) = d/d'$, where $d$ is the Schmidt rank of $|\psi\rangle_{AB}$. This relation mirrors the nonequilibrium work relations in thermodynamics, constraining fluctuations at the microscopic level and recovering average second-law inequalities via Jensen’s inequality: $\langle w \rangle \leq \Delta S$, where $\Delta S = S(p) - S(q)$ is the von Neumann entropy difference (Alhambra et al., 2017).

The Crooks-type fluctuation theorem for entanglement transformations relates the forward and reverse distributions:

$\frac{P_F(w)}{P_R(-w)} = 2^{-w}$

This quantifies irreversibility in entanglement-processing protocols, with $w$ interpreted as the stochastic “entanglement cost.” The mean cost in the forward process is matched by entanglement extraction in the reverse, with the detailed fluctuation symmetry capturing single-shot resource interconversion (Alhambra et al., 2017, Kwon et al., 2018).

3. Second-Law and Third-Law Analogues; Reversibility

From the Jarzynski equality, one obtains a strong-converse (second-law-type) inequality for the fluctuation distribution:

$\Pr[w > W_0] \leq 2^{-W_0} (d/d')$

showing that attempts to extract more entanglement than allowed by the average entropic difference are exponentially suppressed.

A third-law analogue is derived by bounding the maximal attainable entanglement increase:

$2^{w_{\max}} \geq (q_{\min} d') / p_{\min}$

which expresses that finite battery-shifts cannot increase the Schmidt rank in the $p_{\min} \rightarrow 0$ limit—mirroring the impossibility of reaching zero entropy (a pure state) with finite thermodynamic resources. These fluctuation-theorem-based statements unify single-shot reversibility, the structure of resource monotones, and categorical thermodynamic laws in entanglement transformations (Alhambra et al., 2017).

4. Extension to Mixed States, General Quantum Channels, and Resource Hierarchies

The basic fluctuation-theorem structure extends from pure states and LOCC to arbitrary quantum channels and mixed states via the Petz recovery map, with entropy production generalized to complex quantities incorporating coherence and entanglement. In such extended settings, the forward and reverse statistics are explicitly related by:

$$\frac{P_F(\sigma)}{P_R(-\sigma^*)} = \exp[\sigma_R - 2i\theta \sigma_I]$$

where $\sigma$ is the complex entropy production per microtrajectory; the real part encodes resource dissipation, and the imaginary part signals transfer of quantum coherence and broken symmetry (Kwon et al., 2018). When restricted to quantities purely describing entanglement jumps $\Delta E_S$, the detailed fluctuation theorem simplifies to

$$\frac{P_F(\Delta E_S)}{P_R(-\Delta E_S)} = e^{- \Delta E_S}$$

with an associated integral theorem $\langle e^{-\Delta E_S}\rangle = 1$, and monotonicity $\langle \Delta E_S\rangle \leq 0$ under LOCC (Kwon et al., 2018).

Resource-resolved relations have been generalized for arbitrary quantum resource theories within endpoint measurement (EPM) schemes. Using decompositions such as the appended correlation operator and the Best Separable Approximation (BSA), genuine entanglement contributions to entropy production and fluctuation relations can be isolated and operationalized using Kullback-Leibler-based distances, which quantify the thermodynamic relevance of entanglement in arbitrary protocols (Mondkar et al., 17 Dec 2025).

5. Experimental and Operational Realizations

The two-point measurement (TPM) framework for entropy production—implemented in photonic interferometer experiments with entangled photon pairs—has demonstrated the entanglement-resolved fluctuation theorems directly. Here, one photon enables a non-destructive measurement of the initial state via entanglement, while the other undergoes a quantum channel (e.g., finite-temperature amplitude damping), allowing for full statistics of entropy production to be reconstructed from coincidence counts. Such experimental setups validate both the integral and detailed fluctuation theorems for entropy production in systems where quantum back-action is circumvented by entanglement (Aguilar et al., 2021).

In autonomous system+device models, inclusion of initial entanglement and coherence between the system and an actuator (“device”) leads to explicit modifications of Jarzynski and Crooks relations for work. The corrections are analytically tractable for Gaussian states and quantified by the deviation $\langle e^{-\beta W} \rangle_{\rho_e} - 1$, directly linking entanglement to enhanced or reduced average work and demonstrating operational significance in quantum thermodynamics (Silva et al., 2022).

6. Resource Fluctuation Distances and Hierarchies

Fluctuation distances, defined as minimal Kullback-Leibler divergences between work or entropy production distributions generated by resourceful initial states and those generated by the closest resource-free (product or separable) states,

$$D_E(\rho_{AB};\Phi,H) = \min_{\sigma \in \mathrm{SEP}} D_{KL}(P_F(\cdot|\rho_{AB}) \Vert P_F(\cdot | \sigma))$$

serve as quantitative measures of the operational impact of entanglement in arbitrary processes. Vanishing $D_E$ implies that entanglement has no thermodynamic signature for the protocol; large $D_E$ signals potentially exploitable entanglement for work extraction or enhanced nonequilibrium performance. These distances satisfy natural data-processing inequalities and are bounded by standard entanglement monotones (e.g., relative entropy of entanglement) (Mondkar et al., 17 Dec 2025).

7. Applications in Reversible Entanglement Manipulation and Quantum Thermodynamics

Entanglement-resolved fluctuation theorems enable fully reversible entanglement concentration and dilution protocols at the single-shot level via the entanglement battery model. The fluctuation framework subsumes previous results in asymptotic reversibility, catalytic recovery, and optimal dilution, under a unified single-copy paradigm. Furthermore, protocols such as partial entanglement recovery and streaming transformation approach the theoretical lower limits set by the fluctuation theorems, with no dimension bounds or specific catalyst requirements (Alhambra et al., 2017).

Operational implications include the certification of quantum resource contributions in thermodynamic work statistics, the establishment of single-shot resource monotones, and the systematic tightening of second-law inequalities by accounting for genuine quantum correlations. The fluctuation theorem perspective is thus foundational for the resource-theoretic treatment of quantum information, quantum thermodynamics, and the statistical mechanics of complex quantum systems (Alhambra et al., 2017, Mondkar et al., 17 Dec 2025, Kwon et al., 2018, Silva et al., 2022).