Testing the equivalence to thermal states via extractable work under LOCC

Technical Summary: Testing the Equivalence to Thermal States via Extractable Work under LOCC

Problem Statement
A central problem in quantum thermodynamics is understanding how thermal behavior emerges in isolated quantum many-body pure states. It is well-established that typical pure states are indistinguishable from thermal states when restricted to strictly local operations (LO), as the extractable work from such states vanishes in the thermodynamic limit, mirroring the behavior of thermal states. This equivalence arises because local operations cannot access correlations between subsystems. Conversely, under arbitrary global operations, extensive work can be extracted from any pure state by fully exploiting multipartite correlations.

The critical open question addressed in this work is whether this equivalence to thermal states persists under Local Operations and Classical Communication (LOCC). LOCC represents an intermediate operational regime where classically accessible correlations can be exploited for work extraction, yet genuinely quantum correlations remain inaccessible. Determining if pure states remain "thermal" (yielding subextensive work) or become "non-thermal" (yielding extensive work) under LOCC is nontrivial, as it requires distinguishing between classical and quantum multipartite correlations.

Methodology
The authors employ a work-extraction framework in the infinite-temperature regime ($k_B T \gg \Delta E$), where extractable work is determined solely by the state's purity rather than energy changes. The framework defines three regimes:

1. Global Operations ($W_{\text{global}}$): $W_{\text{global}}(\rho) = N \ln d - S(\rho)$. For pure states, this is extensive ($\Theta(N)$).
2. Strictly Local Operations ($W_{\text{local}}$): $W_{\text{local}}(\rho) = N \ln d - \sum S(\rho_n)$. For states locally indistinguishable from thermal states, this vanishes ($o(1)$).
3. LOCC ($W_{\text{LOCC}}$): Defined as the supremum of extractable work over all finite-round LOCC protocols. The work is formulated as $W_{\text{LOCC}}(\rho) = N \ln d - \min_{\Lambda} H_{\Lambda, \rho}(Y^L)$, where the second term represents the work cost of erasing the classical memory of measurement outcomes.

To address the difficulty of optimizing over all LOCC protocols, the authors derive a tractable upper bound on $W_{\text{LOCC}}$ using geometric entanglement ($E_g$):
$$W_{\text{LOCC}}(|\psi\rangle) \leq N \ln d - E_g(|\psi\rangle)$$
where $E_g(|\psi\rangle) \equiv -\ln (\max_{|u\rangle \in \mathcal{P}_{SN}} |\langle u|\psi\rangle|^2)$ and $\mathcal{P}_{SN}$ is the set of product states. This inequality implies that if a state possesses asymptotically maximal multipartite entanglement ($E_g \approx N \ln d$), the extractable work under LOCC must be subextensive.

For states where this upper bound does not guarantee subextensivity, the authors construct explicit lower bounds by designing tailored work-extraction protocols (specifically, adaptive measurement strategies) to demonstrate extensive work extraction.

Key Contributions and Results
The paper establishes criteria for thermal equivalence under LOCC based on the scaling of extractable work ($W_{\text{LOCC}}$). A state is considered equivalent to a thermal state if $W_{\text{LOCC}} = o(N)$ and inequivalent if $W_{\text{LOCC}} = \Theta(N)$.

1. Highly Entangled States (Thermal Equivalence):
   The authors prove that states with asymptotically maximal geometric entanglement yield subextensive work under LOCC. This class includes:

   • Haar-random states: Proven to have maximal geometric entanglement with high probability, leading to $W_{\text{LOCC}} \leq 2 \ln N = o(N)$.
   • Approximate state $t$-designs: For ensembles with $t = \omega(\ln N)$, the geometric entanglement is also asymptotically maximal, resulting in $W_{\text{LOCC}} = o(N)$.
   • Random graph states: Graphs sampled uniformly from all $N$-vertex graphs exhibit maximal geometric entanglement, yielding $W_{\text{LOCC}} = o(N)$.
     These results confirm that despite being locally indistinguishable from thermal states, these highly entangled states remain thermally equivalent even when classical communication is allowed.

2. States with Limited Entanglement (Thermal Inequivalence):
   Conversely, the authors demonstrate that certain states with restricted multipartite entanglement structures allow for extensive work extraction under LOCC, despite being locally thermal:

   • Constant-degree graph states: For graph states where every vertex has a fixed degree $r = O(1)$ (e.g., hexagonal or square lattices), the authors construct a protocol yielding $W_{\text{LOCC}} \geq \frac{N \ln 2}{r+1} = \Theta(N)$.
   • Subset state ensemble: States defined as uniform superpositions over a subset $S$ of computational basis states with support size $K < 2^{cN}$ (where $c < 1$). A simple one-round protocol yields $W_{\text{LOCC}} \geq N \ln 2 - \ln K = \Theta(N)$.
   • Automaton circuits: The authors note that output states of automaton circuits (which do not generate coherence in the computational basis) fall into this category, exhibiting local thermalization but LOCC inequivalence.

Significance
The paper provides a refined operational notion of thermal equivalence that extends beyond the traditional local regime. By utilizing LOCC, the work distinguishes between pure states that are merely locally thermal and those that are globally thermal in the presence of classical communication.

The primary significance lies in linking the multipartite quantum correlation structure directly to thermodynamic behavior. The results show that:

• States with asymptotically maximal multipartite entanglement (like Haar-random states) are robustly thermal even under LOCC.
• States with limited multipartite entanglement (like constant-degree graph states or low-coherence subset states) can be distinguished from thermal states via LOCC, as they allow for extensive work extraction.

This work sharpens the understanding of thermalization in isolated quantum systems, offering an operational criterion to classify pure states based on their correlation structures. It highlights that the "thermal" nature of a state is not absolute but depends on the class of allowed operations, a distinction becoming increasingly relevant as experimental capabilities expand to include classical communication between subsystems.