Advantage of flexible catalysis for entanglement and quantum thermodynamics

Abstract: Understanding the fundamental limits of state convertibility is crucial for establishing the boundaries of quantum information processing and thermodynamic efficiency. While auxiliary systems, catalysts, can facilitate otherwise impossible transformations, standard catalysis rigidly requires the auxiliary system to return to its exact initial state. In this work, we investigate the power of flexible catalysis, where the catalyst evolves through a cycle of states, restoring its initial configuration only after a finite number of steps. Focusing on the regime of fixed, finite dimensions, we analyze the capabilities of flexible catalysis within the resource theories of entanglement and quantum thermodynamics. In the context of entanglement, we derive conditions limiting flexible catalysts and demonstrate that they offer a strict advantage in the success probability of stochastic local operations and classical communication. Conversely, in quantum thermodynamics, we prove that flexible catalysis strictly outperforms standard catalysis even in deterministic settings. We provide an example identifying state transformations that are impossible with any standard catalyst of fixed dimension and Hamiltonian but become achievable via a flexible cycle.

• The paper shows that flexible catalysis is constrained in deterministic entanglement transformations, with proven no-advantage cases for two-dimensional catalysts and selected three-dimensional settings, while a broader no-advantage claim remains conjectural.
• The paper demonstrates a stochastic LOCC advantage: a two-qubit flexible cycle raises success probability from about 0.7299 for the best standard catalyst to 0.7666, a gain of roughly 5%.
• The paper establishes a deterministic thermodynamic advantage using non-degenerate catalyst energy levels, enabling a transition forbidden to all two-dimensional standard catalysts under thermo-majorization.

Overview

The paper investigates flexible catalysis in two resource theories: bipartite pure-state entanglement under LOCC and quantum thermodynamics under thermal operations. In standard catalysis, an auxiliary system enables a transformation while returning to its exact initial state in a single step; flexible catalysis relaxes this by allowing the catalyst to traverse a cycle of states $\{\vec{c}_i\}_{i=1}^n$ with $\vec{c}_{n+1} = \vec{c}_1$, where each step satisfies $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$ (or the thermo-majorized analogue). Without dimension constraints, flexible catalysis is equivalent to standard catalysis via a direct-sum catalyst, so the question is whether a strict advantage exists in the fixed, finite-dimensional regime. The authors find a theory-dependent answer: in entanglement theory flexible catalysis is heavily constrained (with a conjectured no-go for deterministic transformations), yet yields a strict advantage in stochastic LOCC, while in quantum thermodynamics it strictly outperforms standard catalysis even deterministically.

Limitations of flexible catalysis in entanglement theory

The authors establish several no-advantage and degeneracy results for majorization-based transformations. The central analytical result states that if a sequence of $k$-dimensional flexible catalysts enables $\vec{x} \to \vec{y}$, then at least one member of the sequence is already a valid standard catalyst, whenever either (i) $k=2$, or (ii) $k=3$ with $x_1 = y_1$ and $x_d = y_d > 0$. The $k=2$ case follows from total ordering of qubit Schmidt vectors under majorization: any cycle contains a minimal element, and transitivity of majorization under tensor products promotes it to a standard catalyst. The $\vec{c}_{n+1} = \vec{c}_1$0 case follows from cyclic boundary inequalities that force all catalyst components to be equal.

These results rest on structural lemmas proved in the supplementary material. If $\vec{c}_{n+1} = \vec{c}_1$1, the flexible-catalysis condition implies component-wise bounds $\vec{c}_{n+1} = \vec{c}_1$2 and $\vec{c}_{n+1} = \vec{c}_1$3; multiplying around the cycle yields the necessary conditions $\vec{c}_{n+1} = \vec{c}_1$4 and $\vec{c}_{n+1} = \vec{c}_1$5 — identical to those for standard catalysis, hence not separating the two paradigms. Additional results include: all catalysts in a sequence must share the same number of non-zero components; the uniform vector cannot appear in any non-trivial flexible cycle; the maximally entangled state cannot belong to any flexible sequence; and for system dimension $\vec{c}_{n+1} = \vec{c}_1$6, no flexible catalysis is possible between incomparable states in either direction.

The authors also show that one of the two dimension-lower-bound criteria from standard catalysis theory (the adjacent-component ratio bound) fails for flexible catalysis, via an explicit counterexample in $\vec{c}_{n+1} = \vec{c}_1$7 where a valid flexible cycle contains a catalyst state with adjacent-component ratio $\vec{c}_{n+1} = \vec{c}_1$8 exceeding the standard-catalytic limit of $\vec{c}_{n+1} = \vec{c}_1$9. Flexible catalysts can therefore exhibit steeper probability drops than any standard catalyst for the same transformation — a genuine relaxation of structure, though one that does not translate into deterministic advantage in the cases analyzed.

Notably, the authors concede that their strongest deterministic claim is a conjecture, not a theorem: based on analytical results and an exhaustive numerical search that found no counterexample, they conjecture that any transformation achievable via $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$0-dimensional flexible catalysts is also achievable via a $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$1-dimensional standard catalyst. Proving or disproving this remains open.

Strict advantage in stochastic LOCC

Moving to probabilistic transformations, the paper defines the per-step success probability of a flexible cycle as the maximum geometric mean $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$2 over valid cyclic sequences, and proves a strict separation from the optimal standard catalyst of the same dimension. For a $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$3 system with Schmidt vectors $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$4 and $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$5, the base (uncatalyzed) success probability is $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$6. With a two-qubit flexible catalyst pair ($$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$7, $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$8), the optimal standard catalyst on the diagonal $$\vec{x} \otimes \vec{c}_i \prec \vec{y} \otimes \vec{c}_{i+1}$$9 achieves $k$0, whereas the global maximum lies strictly off-diagonal at $k$1, giving $k$2 — a relative gain of roughly 5%. Since $k$3 trivially (standard catalysis is the diagonal restriction), the substantive content is the demonstrated strictness of the inequality. The implication is that even two-qubit auxiliary systems, operated cyclically, extend the reachable SLOCC conversion rates beyond any statically constrained catalyst of the same size.

Strict deterministic advantage in quantum thermodynamics

For energy-incoherent states, deterministic transformations under closed thermal operations, Gibbs-preserving operations, and thermo-majorization are equivalent, so the analysis reduces to relative majorization with the Gibbs vector $k$4 as reference. A key structural observation is that, unlike in ordinary majorization, fixing the catalyst's dimension does not fix its reference state: the Hamiltonian spectrum must also be specified, and this degree of freedom is where flexible catalysis gains traction.

The main thermodynamic result is an explicit construction with system energy levels $k$5 at $k$6, transforming $k$7 (which violates thermo-majorization, $k$8). With a two-level catalyst of non-degenerate spectrum $k$9, the feasible set of catalyst pairs $\vec{x} \to \vec{y}$0 satisfying $\vec{x} \to \vec{y}$1 and $\vec{x} \to \vec{y}$2 is non-empty but lies entirely off the diagonal $\vec{x} \to \vec{y}$3 — for instance $\vec{x} \to \vec{y}$4, $\vec{x} \to \vec{y}$5. No two-dimensional standard catalyst enables the transition. This is a deterministic, exact separation, stronger than the probabilistic entanglement advantage.

The construction also yields a sharp structural insight: setting the catalyst spectrum to be degenerate ($\vec{x} \to \vec{y}$6) makes the feasible region vanish. Thus, whereas a trivial (infinite-dimensional, maximally mixed) catalyst suffices in the unbounded regime, non-degenerate catalyst energy levels are indispensable for finite-dimensional catalytic assistance here. The authors note that the thermodynamic advantage is tied to the non-uniformity of the Gibbs state — thermo-majorization reduces to majorization only when the reference is maximally mixed — and suggest a connection to improved work extraction, without quantifying it.

Limitations and open questions

Several caveats bound the results. The deterministic entanglement no-advantage claim is conjectural; only special cases ($\vec{x} \to \vec{y}$7; $\vec{x} \to \vec{y}$8 with matched extreme Schmidt coefficients; $\vec{x} \to \vec{y}$9 incomparable states) are proven, and the numerical search, while exhaustive over the cases considered, does not constitute a proof. The stochastic advantage is demonstrated for a single $k=2$0, $k=2$1, $k=2$2 instance, and the general scaling of the gain with system and catalyst dimension is unknown. The thermodynamic separation is likewise a specific example rather than a characterization of when flexible catalysis helps. The analysis is restricted to energy-incoherent states and classical probability vectors; whether the advantages persist for coherent states under fully quantum thermal operations is left open, as are transformations of general mixed states under LOCC and the quantification of the flexible–standard gap as a function of catalyst dimension and Hamiltonian complexity.

Conclusion

The paper maps the power of cyclic, finite-dimensional catalysis and finds a sharp resource-theory-dependent dichotomy. In entanglement theory, flexible catalysis is provably degenerate in several regimes and conjectured to offer no deterministic advantage at fixed dimension, but it strictly improves stochastic conversion success probabilities. In quantum thermodynamics, it enables deterministic transformations impossible for any standard catalyst of fixed dimension and Hamiltonian, with non-degenerate catalyst spectra playing an essential role. The work thereby identifies catalyst-state cycles as a physically motivated relaxation of catalytic constraints whose value depends on the reference-state structure of the underlying resource theory.