Regularized Relative Entropy of Resource

• Regularized relative entropy of resource is defined as the asymptotic per-copy distinguishability between a quantum state and free states, capturing conversion rates and resource costs.
• It operationally characterizes optimal hypothesis testing error exponents and state transformation rates across various resource theories such as entanglement and coherence.
• Efficient numerical methods leveraging convexity and symmetry reductions enable accurate approximations for both static states and quantum channels in these theories.

The regularized relative entropy of resource is a fundamental asymptotic quantity in quantum resource theories, unifying diverse operational tasks such as state conversion, hypothesis testing, and resource cost rates. It arises by regularizing the quantum relative entropy between a state and the set of resource-free states, capturing the optimal per-copy distinguishability and conversion rates in the many-copy limit. This measure is central to entanglement, coherence, asymmetry, thermodynamic, and computational resource theories, and admits rigorous operational meanings in both static and dynamical (channel) settings.

1. Formal Definition and Core Properties

Given a closed convex set of “free” states $\mathcal F$ (such as separable states in entanglement theory), the single-copy relative entropy of resource for a state $\rho$ is

$$D_R(\rho) := \inf_{\sigma\in\mathcal F} D(\rho\|\sigma),$$

where $$D(\rho\|\sigma) = \operatorname{Tr}[\rho(\log\rho - \log\sigma)]$$ is the Umegaki quantum relative entropy, defined for $$\operatorname{supp}(\rho)\subseteq \operatorname{supp}(\sigma)$$ and $+\infty$ otherwise (Beigi et al., 8 Jul 2025). The regularized version is

$$D_R^\infty(\rho) := \lim_{n\to\infty} \frac{1}{n} D_R(\rho^{\otimes n}) = \lim_{n\to\infty} \frac{1}{n} \inf_{\sigma^{(n)}\in\mathcal F_n} D\left(\rho^{\otimes n} \| \sigma^{(n)}\right),$$

where $\mathcal F_n$ is the free set on $n$-copy systems, typically closed under tensor products and convex combinations (Anshu et al., 2017, Li et al., 2012).

Key properties:

• Monotonicity: $D_R(\rho)$ and $D_R^\infty(\rho)$ are non-increasing under free operations (completely positive trace-preserving maps that map free states to free states) (Anshu et al., 2017).
• Faithfulness: $D_R^\infty(\rho)>0$ if and only if $\rho$ is not free; zero if and only if $\rho\in\mathcal F$ (0904.2705).
• Convexity: Both $D_R$ and $D_R^\infty$ are convex as functions of the state.
• Additivity: $D_R^\infty(\rho)$ is in general subadditive: $D_R(\rho\otimes\tau) \leq D_R(\rho) + D_R(\tau)$. Additivity holds ($D_R^\infty(\rho)=D_R(\rho)$) under specific optimizer conditions (Beigi et al., 8 Jul 2025).

2. Resource-Theoretic and Operational Interpretation

The regularized relative entropy of resource is operationally the unique monotone quantifying the asymptotic cost/rate of state transformations in a wide class of quantum resource theories (Anshu et al., 2017, Matsumoto, 2010). In the context of catalytic resource theories—which allow the use of free ancillary systems as catalysts and free unitaries—it quantifies the minimal per-copy rate of “free noise” or randomness required to erase or create the resource content of $\rho$ (Anshu et al., 2017). Explicitly, the minimum randomness rate $R_{\mathrm{optimal}}$ required to erase the resource in $n$ copies of $\rho$ is

$R_{\mathrm{optimal}} = D_R^\infty(\rho).$

Moreover, $D_R^\infty(\rho)$ governs the optimal hypothesis testing error exponents for discrimination between $\rho^{\otimes n}$ and all $\sigma^{(n)}\in\mathcal F_n$, and characterizes asymptotic interconvertibility of resources (Li et al., 2012, Beigi et al., 8 Jul 2025).

3. Additivity, Approximation, and Efficient Computation

Additivity of $D_R$ under tensor products is a critical criterion for determining whether single-shot or regularized quantities are operationally relevant: $$D_R^\infty(\rho) = D_R(\rho) \quad \text{if and only if} \quad D_R(\rho\otimes\tau) = D_R(\rho) + D_R(\tau) \ \forall \tau.$$ (Beigi et al., 8 Jul 2025) gives necessary and sufficient conditions for this additivity in terms of the optimizer $$\sigma^* = \arg\min_{\sigma\in\mathcal F} D(\rho\|\sigma)$$: a “twirled overlap” criterion that requires

$$\sup_{t\in\mathbb{R}}\,\sup_{\tau\in\mathcal F}\; \operatorname{Tr}\!\left[\tau\,\sigma^{*-\frac{1}{2}(1+it)}\,\rho\,\sigma^{*-\frac{1}{2}(1-it)}\right] \leq 1.$$

Failure of this condition requires full regularization to compute asymptotic rates.

Efficient numerical approximation of regularized relative entropy is enabled under structural assumptions: convexity, compactness, permutation invariance, tensor-product closure, and polar-set closure (Fang et al., 21 Feb 2025). For sets of states or channels satisfying these, quantum relative entropy programs (QREPs)—generalized semidefinite or conic programming formulations—can approximate $D_R^\infty$ within any prescribed additive error in polynomial time, leveraging symmetry-reduction and Schur–Weyl dualities.

4. Asymptotic Continuity and Faithfulness

$D_R^\infty$ is asymptotically continuous: for any $\rho, \rho'$, if $\|\rho-\rho'\|_1\leq\epsilon$, then

$$|D_R^\infty(\rho) - D_R^\infty(\rho')| \leq c\, \epsilon \log\frac{k}{\epsilon}$$

with $c$ universal and $k$ the Hilbert space dimension (Li et al., 2012). This ensures stability under small perturbations of the state. $D_R^\infty$ is also strictly positive on every non-free (e.g., entangled) state, even in multipartite scenarios (0904.2705).

Restricted-measurement variants—where only a subset of quantum operations (e.g., LOCC, SEP) are permitted in the divergence definition—yield lower bounds to $D_R^\infty$ that remain faithful and nontrivial, providing strictly positive lower bounds for the resource content in all non-free states.

5. Role in Quantum Channel Resource Theories

The notion of regularized relative entropy generalizes to quantum channels, with significant subtleties (Yuan, 2018, Gour et al., 2019). For a channel $\mathcal N$ and a convex, closed set of free channels $\mathcal F$, one defines several one-shot and asymptotic divergences. The primary version is

$$D_F^{(\infty)}(\mathcal N) := \lim_{n\to\infty} \frac{1}{n} \min_{\mathcal M \in \mathcal F(A^{\otimes n} \to B^{\otimes n})} D\left(\mathcal N^{\otimes n} \| \mathcal M \right),$$

where optimization runs over inputs and channels.

The regularized channel relative entropy satisfies strong monotonicity under free superchannels, asymptotic continuity, and an asymptotic equipartition property (AEP), paralleling the static (state) case (Gour et al., 2019). It is operationally the exponent in Stein’s lemma for channel discrimination and quantifies the asymptotic dilution/distillation rates for dynamical resources such as entanglement and coherence of channels.

Multiple distinct channel divergence measures emerge in the channel context; for several of these, product-state regularization and “liberal smoothing” techniques guarantee existence, continuity, and AEP (Gour et al., 2019).

6. Extensions, Variants, and Limitations

In probabilistic or stochastic protocols, especially those with vanishing asymptotic success probability, the regularized relative entropy of resource can fail to capture the true asymptotic transformability, necessitating alternative monotones such as the regularized Hilbert projective (or “projective relative”) entropy (Regula et al., 2022). In deterministic transformations, however, $D_R^\infty$ remains the definitive rate-limiting monotone.

Axiomatic studies establish that, under the basic requirements of monotonicity, normalization on classical distributions, weak additivity, and lower asymptotic continuity, the regularized relative entropy is the essentially unique asymptotic monotone (up to scale) (Matsumoto, 2010).

Restricted-measurement and resource-theory-specific variants (e.g., relative entropy of entanglement, relative entropy of coherence) correspond precisely to the regularized quantity with respect to the appropriate free set (Li et al., 2012, 0904.2705). For certain state families (such as maximally entangled or pure bipartite states), analytic evaluation is possible and restricted-measurement versions collapse to the Shannon entropy of entanglement (Li et al., 2012).

7. Examples and Applications

Resource Theory	Free Set $\mathcal F$	Operational Task Characterized by $D_R^\infty$
Entanglement	Separable states	Rate of entanglement erasure or distillation under non-entangling or catalytic LOCC operations (Anshu et al., 2017, Li et al., 2012)
Coherence	Diagonal states in a fixed basis	Cost of erasing coherence, coherence distillation rate
Magic-State	Stabilizer polytope	Optimal magic distillation rates (Beigi et al., 8 Jul 2025)
Channel entanglement	Separable/PPT channels	Asymptotic discrimination, channel entanglement cost (Yuan, 2018, Gour et al., 2019)
General resource theory	Arbitrary convex, compact $\mathcal F$	General resource erasure, catalysis, cost, and asymptotic conversion

Numerical methods based on QREPs can approximate $D_R^\infty$ efficiently for sets satisfying the requisite structure, and exhibit fast convergence in canonical resource theories. For instance, in the entanglement theory, using the Rains set as a relaxation of PPT or SEP can yield practical bounds for entanglement cost and distillation (Fang et al., 21 Feb 2025).

• "Additivity of quantum relative entropies as a single-copy criterion" (Beigi et al., 8 Jul 2025)
• "Efficient approximation of regularized relative entropies and applications" (Fang et al., 21 Feb 2025)
• "Quantifying resource in catalytic resource theory" (Anshu et al., 2017)
• "Relative entropy and squashed entanglement" (Li et al., 2012)
• "Relative Entropy of Entanglement and Restricted Measurements" (0904.2705)
• "Reverse Test and Characterization of Quantum Relative Entropy" (Matsumoto, 2010)
• "Relative entropies of quantum channels with applications in resource theory" (Yuan, 2018)
• "How to quantify a dynamical quantum resource" (Gour et al., 2019)
• "Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols" (Regula et al., 2022)