Reverse Relative Entropy of Entanglement

• Reverse relative entropy of entanglement is defined as the minimum of D(σ‖ρ) over separable states, providing a distinctive measure from the conventional relative entropy.
• It serves as the optimal asymptotic error exponent in entanglement distillation protocols under non-entangling operations, directly linking hypothesis testing with resource conversion.
• Its single-letter formulation eliminates the need for multi-copy regularization, thereby facilitating practical computations in quantum resource theories and device-independent cryptography.

Reverse relative entropy of entanglement is a quantum information measure that captures the distinguishability of a quantum state from the set of separable states, with the arguments in the quantum relative entropy reversed relative to the standard definition. Operationally, it has been identified as the optimal asymptotic error exponent for entanglement distillation protocols under non-entangling operations, and uniquely, it admits a single-letter expression that does not require regularization over many copies. Its mathematical definition and operational meaning position it as a central tool for benchmarking and understanding entanglement in the context of hypothesis testing, resource conversion, and quantum communication protocols.

1. Formal Definition

For a bipartite quantum state $\rho_{AB}$, the reverse relative entropy of entanglement is defined as

$$D(S_{A:B} \| \rho_{AB}) = \min_{\sigma_{AB} \in S_{A:B}} D(\sigma_{AB} \| \rho_{AB}) ,$$

where $S_{A:B}$ is the set of separable states on the bipartition $A:B$, and $$D(\sigma \| \rho) = \operatorname{Tr}[\sigma(\log \sigma - \log \rho)]$$ denotes the quantum relative entropy (Lami et al., 2024).

Unlike the standard relative entropy of entanglement, which minimizes $D(\rho_{AB}\|\sigma_{AB})$ over separable $\sigma_{AB}$, the reverse variant interchanges the roles of the arguments. The minimizer $\sigma_{AB}^\star$ (if it exists) is referred to as the "closest" separable state to $\rho_{AB}$ in the reverse sense.

2. Connection to Entanglement Distillation and Operational Meaning

The operational significance of the reverse relative entropy of entanglement was established by reframing entanglement distillation as a hypothesis testing task in the sense of composite quantum Sanov's theorem. Given $n$ copies of an unknown bipartite state, the task is to discriminate between the hypothesis $H_0$: "the state is $\rho_{AB}^{\otimes n}$" versus $H_1$: "the state is separable".

The key result is that the optimal asymptotic error exponent for this discrimination—i.e., in the regime where the type-II error (accepting $H_0$ when $H_1$ is true) is kept below a fixed threshold and the type-I error (rejecting $H_0$ when it is correct) decays exponentially fast—is exactly the reverse relative entropy of entanglement: $E_{d,\mathrm{err}} (\rho_{AB}) = \Sanov(\rho_{AB}\|S_{A:B}) = D(S_{A:B}\| \rho_{AB}) .$ This means that, for any protocol attempting to distill maximally entangled states from $\rho_{AB}^{\otimes n}$ using only non-entangling operations, the best achievable decay rate of the error probability is set by a single-copy optimization over separable $\sigma_{AB}$ (Lami et al., 2024).

3. Single-Letter Formulability and Practical Implications

The reverse relative entropy of entanglement is notable for being a "single-letter" quantity: it is evaluated directly on a single copy of the state without the need for a regularized limit over many copies. In contrast, many other fundamental measures, such as the standard relative entropy of entanglement or entanglement cost, require optimization or limits over $n\to\infty$ copies to achieve operational interpretations. Formally: $$D(S_{A:B}\| \rho_{AB}) = \min_{\sigma_{AB}\in S_{A:B}} D(\sigma_{AB} \| \rho_{AB}),$$ with no need to consider $n$-copy extensions or collective optimizations.

This property greatly facilitates computation and lends itself to direct benchmarking of quantum states in practical settings, such as device-independent quantum cryptography and finite-resource distillation protocols. The reverse relative entropy quantifies exactly how "inaccessible" the set of separable states is from $\rho_{AB}$ in the most discriminative hypothesis testing task.

4. Conceptual Comparison to Standard Relative Entropy of Entanglement

The standard relative entropy of entanglement,

$$E_R(\rho_{AB}) = \min_{\sigma_{AB}\in S_{A:B}} D(\rho_{AB} \| \sigma_{AB}),$$

measures how distinguishable the actual state $\rho_{AB}$ is from separable states. In contrast, the reverse measure

$$D(S_{A:B} \| \rho_{AB}) = \min_{\sigma_{AB}\in S_{A:B}} D(\sigma_{AB} \| \rho_{AB})$$

quantifies how distinguishable a separable state can be from $\rho_{AB}$. These two are generally not equal and have different continuity, convexity, and operational properties.

For pure states, the reverse relative entropy may diverge or vanish trivially, indicating the measure's sensitivity to the spectrum of the target state and the structure of the separable set (Takayanagi et al., 2018). In the perturbative regime (small perturbations about a reference state), the forward and reverse relative entropies agree to quadratic order, reflecting approximate symmetry in the distinguishability landscape.

5. Connections to Hypothesis Testing and Quantum Resource Theories

Theoretical advances in quantum hypothesis testing (generalized Sanov's theorem) inform the reverse relative entropy's operational status as an error exponent rate (Lami et al., 2024). More generally, this links resource-theoretic quantifiers—distinguishability measures to the set of "free" states—to the fundamental statistical limits of discrimination tasks in quantum information theory.

For the resource theory of entanglement, the reverse relative entropy sets a strict bound: no entanglement distillation protocol under non-entangling operations can achieve an error exponent exceeding $D(S_{A:B} \| \rho_{AB})$. This positions the reverse relative entropy as a sharp threshold for operational advantage in resource conversion protocols.

6. Implications for Restricted Measurements, Monogamy, and Multipartite Extensions

The structure underlying the reverse relative entropy appears in studies of restricted-measurement variants of relative entropy (e.g., LOCC or separable POVMs) (0904.2705, Berta et al., 2023, Li et al., 2012). In such contexts, dual (variational) formulations of the measured relative entropy permit analysis of operational measures (monogamy inequalities, squashed entanglement faithfulness) directly in terms of "reverse" optimizations—suprema over measurement outcome distributions or operator cones. These variational formulations provide streamlined matrix-analytic proofs for fundamental properties of multipartite entanglement and exhibit connections between the reverse relative entropy and constraints on distributed correlations.

7. Summary Table: Reverse Relative Entropy of Entanglement

Feature	Reverse Relative Entropy of Entanglement	Standard Relative Entropy of Entanglement
Formula	$$D(S_{A:B} \| \rho_{AB}) = \min_{\sigma_{AB} \in S_{A:B}} D(\sigma_{AB} \| \rho_{AB})$$	$$E_R(\rho_{AB}) = \min_{\sigma_{AB} \in S_{A:B}} D(\rho_{AB} \| \sigma_{AB})$$
Operational Meaning	Asymptotic error exponent for non-entangling distillation protocols	Entanglement cost, upper bound on distillable entanglement
Regularization Needed?	No (single-letter)	Yes (for many operational tasks)
Applicability	Error-exponent analysis, discrimination, single-copy quantification	Resource cost, many-copy conversion protocols
Continuity, Convexity, Additivity	May differ from forward variant; context-dependent	Well-studied; subject to known limitations and open problems

The reverse relative entropy of entanglement thus provides a uniquely direct operational quantification of quantum entanglement, with rigorous statistical meaning in hypothesis testing, explicit formulaic accessibility for single systems, and broad implications across entanglement theory, quantum resource theories, and quantum information processing (Lami et al., 2024).