PPT Entanglement with Correlated Catalysis: Monotones and Irreversibility

Abstract: Quantum catalysts can overcome otherwise impossible quantum state transformations without being consumed, and allowing them to become correlated with the output makes this assistance substantially more powerful. This raises a fundamental question for entanglement theory: which limitations on state manipulation remain when such correlated catalysts are freely available? We answer this question in the positive-partial-transpose (PPT) resource theory, which allows a substantially broader class of operations than local operations and classical communication (LOCC). We identify general conditions under which regularized relative-entropy measures become strongly superadditive, and use them to construct monotones that constrain correlated catalytic PPT transformations without any knowledge of the catalyst. In particular, we prove that the regularized PPT relative entropy is fully additive and strongly superadditive, resolving an open problem in entanglement theory. Most importantly, these constraints show that even arbitrary correlated catalysts cannot restore asymptotic reversibility: for an explicit state, the optimal entanglement distillation rate remains strictly smaller than the entanglement cost. Thus, substantial catalytic assistance does not remove some of the fundamental limitations of mixed-state entanglement manipulation.

• The paper establishes a general composite-hypothesis-testing criterion that makes regularized relative-entropy measures strongly superadditive and valid monotones under arbitrary correlated catalysis.
• The paper applies the criterion to every finite level of the PPT_k hierarchy, producing the first systematic family of additive, strongly superadditive monotones for correlated-catalytic PPT and LOCC transformations.
• The paper proves that PPT entanglement remains irreversible under correlated catalysis, with the explicit state ρ_v satisfying E_d,PPT^cc = log₂(1 + 1/√2) ≈ 0.643 < E_c,PPT^cc = 1.

Overview

This paper by Ao, Philip, and Streltsov addresses a structural question at the intersection of entanglement theory and catalytic resource manipulation: which resource monotones constrain state transformations when arbitrary correlated catalysts are freely available, and do such catalysts restore asymptotic reversibility? The setting is the resource theory of positive-partial-transpose (PPT) entanglement, where free states are PPT states and free operations are completely PPT-preserving maps — a strictly larger class than LOCC, so that impossibility results here automatically apply to LOCC as well. The paper delivers three main contributions: a general criterion for strong superadditivity of regularized relative-entropy measures; an application of this criterion to the $PPT_k$ hierarchy of comparison sets, yielding correlated-catalytic monotones; and a proof that $D_{PPT}^{\infty}$ is fully additive and strongly superadditive, resolving an open problem. These results are then combined to show that PPT entanglement manipulation remains irreversible even under arbitrary correlated catalysis.

Background: correlated catalysis and monotone requirements

A catalyst is an auxiliary state $\tau_{CD}$ that enables otherwise impossible transformations while being returned intact. In correlated catalysis, the catalyst must be recovered exactly but may end up correlated with the transformed system, enlarging the set of achievable transformations. A key difficulty is operational: the enabling catalyst need not be known in advance, so one seeks monotones depending only on the initial and final system states.

Rubboli and Tomamichel established that, under mild continuity assumptions, additivity on tensor products plus strong superadditivity on arbitrary correlated states suffice for a monotone to remain valid under correlated catalysis: before the transformation, additivity lets the independent catalyst's contribution be evaluated separately; after the transformation, strong superadditivity lower-bounds the joint resource despite correlations, and the catalyst's contribution cancels since it is recovered unchanged.

Monotones satisfying all required properties simultaneously are scarce. Under LOCC only two examples were known — the squashed entanglement [cond-mat/0306443] and the conditional entanglement of mutual information (0805.1441) — and whether these two coincide remains open. No such monotone was previously known for the PPT resource theory. Rather than constructing examples ad hoc, the paper identifies general structural conditions guaranteeing strong superadditivity for regularized relative-entropy distances to suitable comparison sets.

A general criterion for strong superadditivity

The central technical tool (Lemma 1 of the paper) considers a family $\mathcal F(A:B)$ of compact convex sets of subnormalized positive operators, one per bipartite system, satisfying four conditions: (i) each contains a full-rank state; (ii) the $n$-copy sets are permutation invariant; (iii) tensor-product closure, $$\mathcal F(A:B)\otimes\mathcal F(A':B')\subseteq\mathcal F(AA':BB')$$; and (iv) closure of positive polars under tensor products, where $$\mathcal F^\circ=\{X\geq0:\operatorname{Tr}(X\tau)\leq1\ \forall\tau\in\mathcal F\}$$.

Under these hypotheses, the regularized relative entropy $$D_{\mathcal F}^{\infty}(\rho)=\lim_{n}\frac1n\min_{\tau_n\in\mathcal F(A^n:B^n)}D(\rho^{\otimes n}\Vert\tau_n)$$ is strongly superadditive:

$$D_{\mathcal F}^{\infty}(\omega_{AA'BB'})\geq D_{\mathcal F}^{\infty}(\omega_{AB})+D_{\mathcal F}^{\infty}(\omega_{A'B'}).$$

The proof is a composite hypothesis-testing argument. Conditions (i)–(iv), together with permutation invariance, verify the assumptions needed to apply the generalized quantum Stein lemma — whose original proof contained a gap identified by Berta et al. (Dehghani et al., 2023) and which has since been re-established rigorously (Yimer et al., 14 Feb 2025) — characterizing $D_{\mathcal F}^{\infty}$ via optimal error exponents against the composite alternative $D_{PPT}^{\infty}$0. For arbitrary (possibly correlated) $D_{PPT}^{\infty}$1, product tests built from near-optimal tests on the two marginals achieve vanishing type-I error by a union bound, while condition (iv) implies submultiplicativity of support functions, so the type-II exponent adds across marginals. Notably, the argument never requires $D_{PPT}^{\infty}$2 to factorize, which is precisely what yields strong superadditivity rather than mere additivity.

This criterion applies beyond entanglement: the supplemental material shows it also establishes strong superadditivity and correlated-catalytic monotonicity of the regularized thauma in the odd-dimensional magic resource theory, results not previously observed.

Correlated-catalytic monotones from the $D_{PPT}^{\infty}$3 hierarchy

The authors apply the criterion to the hierarchy $D_{PPT}^{\infty}$4 introduced by Wang, Jing, and Zhu (Banerjee et al., 2024), where $D_{PPT}^{\infty}$5 is the Rains set and higher levels are defined recursively through nested partial-transpose constraints, with

$D_{PPT}^{\infty}$6

For every finite $D_{PPT}^{\infty}$7, the sets $D_{PPT}^{\infty}$8 satisfy conditions (i)–(iv): tensor closure and polar-tensor closure were established by Fang, Fawzi, and Fawzi (Fang et al., 2024) and Beigi, Rubboli, and Tomamichel [2506.xxxx]; the remaining properties follow directly from the definition. Consequently:

Theorem 1: For every finite $D_{PPT}^{\infty}$9, the regularized $\tau_{CD}$0 relative entropy $\tau_{CD}$1 is monotone under correlated catalytic transformations implemented by completely PPT-preserving maps.

Tensor-product additivity follows from Beigi et al.'s single-copy criterion; asymptotic continuity (hence lower semicontinuity) follows from Winter's uniform continuity bounds [quant-ph/0511085]. Since every LOCC map is completely PPT-preserving, these monotones also constrain correlated-catalytic LOCC transformations. This supplies the first systematic family of additive, strongly superadditive monotones for PPT entanglement theory.

Full additivity of the regularized PPT relative entropy

The regularized PPT relative entropy $\tau_{CD}$2 was known to be weakly additive [quant-ph/0205087], but full additivity (distinct states) and strong superadditivity remained open. The obstacle is that the positive polar of the PPT set itself is not closed under tensor products, so the criterion cannot be applied directly.

The key insight is to work with the intersection $\tau_{CD}$3. Writing $\tau_{CD}$4 for the minimal trace of the recursive partial-transpose domination sequence, the supplemental material shows that membership in $\tau_{CD}$5 is governed by the zero-error exact PPT entanglement cost $\tau_{CD}$6 via Lami, Mele, and Regula's result (Álvarez-Caudevilla et al., 2024):

$\tau_{CD}$7

An "exact-cost domination" lemma then gives, for all states $\tau_{CD}$8,

$\tau_{CD}$9

proved by preparing $\mathcal F(A:B)$0 from maximally entangled states under completely PPT-preserving maps and using operator monotonicity of the logarithm. This yields the identity

$\mathcal F(A:B)$1

Since $\mathcal F(A:B)$2 inherits the required tensor and polar-tensor conditions — the latter via a compactness argument showing support functions converge along the decreasing sequence of $\mathcal F(A:B)$3 — the criterion transfers full additivity and strong superadditivity to $\mathcal F(A:B)$4 itself. Theorem 2 therefore resolves the open problem: $\mathcal F(A:B)$5 is additive on tensor products, strongly superadditive on arbitrary correlated states, and monotone under correlated catalysis. As a corollary, nontrivial convex combinations of the squashed entanglement with each $\mathcal F(A:B)$6 and $\mathcal F(A:B)$7 yield faithful, additive, strongly superadditive LOCC entanglement monotones.

Irreversibility persists under correlated catalysis

Whether asymptotic entanglement manipulation is reversible depends sensitively on the allowed operations: Brandão and Plenio obtained reversibility under asymptotically non-entangling operations (0705.3868, 0810.4171), whereas Lami and Regula showed irreversibility under non-entangling and dually non-entangling operations (Lami et al., 2021, Lami et al., 2023), and Regula and Lami showed probabilistic protocols restore reversibility (Regula et al., 2023). Lami, Regula, and Streltsov proved irreversibility under correlated-catalytic LOCC (Wu et al., 2023), but their argument relies on PPT bound entangled states being undistillable yet costly — reasoning that fails in the PPT resource theory itself, where PPT states are free with zero cost.

Theorem 3 closes this gap. For the explicit rank-two state

$\mathcal F(A:B)$8

introduced by Wang and Duan (Koberinski et al., 2017), the correlated-catalytic distillation rate and entanglement cost satisfy

$\mathcal F(A:B)$9

The proof combines both preceding theorems with the general catalytic rate bounds of Lami–Regula–Streltsov: $n$0 upper-bounds distillation (via Theorem 1 at $n$1) and $n$2 lower-bounds formation (via Theorem 2); Wang and Duan's exact values show both bounds coincide with the uncatalyzed rates, and trivial-catalyst inequalities pin down the catalytic rates exactly. The implication is direct: no amount of correlated catalytic assistance — with catalysts of arbitrary size and correlation structure — closes the gap between distillation and formation for this state. Substantial catalytic freedom does not remove the fundamental irreversibility of mixed-state entanglement manipulation under completely PPT-preserving operations.

Limitations and open questions

Several points remain open. First, it is unknown whether the family $n$3 is complete for correlated catalytic transformations under completely PPT-preserving maps, i.e., whether these monotones fully characterize achievable transformations. Second, Lemma 1 provides a mechanism rather than a classification: identifying further comparison sets satisfying conditions (i)–(iv) — potentially yielding new correlated-catalytic monotones in other resource theories — is left unaddressed. Third, the role of probabilistic transformations in the PPT setting is unresolved: probabilistic protocols restore reversibility under asymptotically resource-non-generating operations, but whether they do so when operations must be exactly completely PPT-preserving is open, as is the correct formulation combining probabilistic success with certain catalyst recovery (e.g., requiring the catalyst to be returned in every branch). Finally, the irreversibility result is established for a specific state; whether the gap is generic across PPT bound entangled states is not determined here.

Conclusion

The paper contributes a general criterion under which regularized relative-entropy measures become strongly superadditive, applies it to produce the first family of correlated-catalytic monotones for PPT entanglement, proves full additivity and strong superadditivity of $n$4 via the exact identification $n$5, and demonstrates with an explicit numerical gap ($n$6) that arbitrary correlated catalysts cannot restore asymptotic reversibility of PPT entanglement manipulation. The results sharpen the picture of how much operational freedom is compatible with irreversible entanglement manipulation and provide reusable machinery for constructing catalytically robust monotones in other quantum resource theories.