Quantum conditional entropies from convex trace functionals

1. Problem Statement

Quantum conditional entropies are fundamental measures of uncertainty in quantum information theory, quantifying the uncertainty of a system $A$ given access to a correlated side information system $B$. While the von Neumann entropy is standard, non-asymptotic scenarios (e.g., quantum cryptography, finite blocklength coding) require more refined measures, typically based on quantum Rényi divergences.

Existing literature primarily focuses on two canonical constructions of conditional Rényi entropies derived from the $\alpha$-$z$ relative entropy:

1. Petz-type (Arrow Down, $\downarrow$): Defined by fixing the side information state to the marginal $\rho_B$.
2. Sandwiched-type (Arrow Up, $\uparrow$): Defined by optimizing (infimum or supremum) over all possible side information states $\sigma_B$.

These two families arise from distinct parameter regimes and constructions. A significant gap exists in unifying these families into a single framework that:

• Interpolates between the Petz and Sandwiched types.
• Satisfies fundamental operational properties (Data-Processing Inequality, Additivity, Duality, Chain Rules) across a broad parameter space.
• Reveals the underlying geometric structure of the trace functionals involved.

2. Methodology

The authors introduce a three-parameter family of quantum conditional entropies, denoted $H^\lambda_{\alpha,z}(A|B)_\rho$, defined via a novel optimization over a trivariate trace functional.

Key Methodological Tools:

• Trivariate Trace Functionals: The core of the work involves analyzing functionals of the form $\Phi_{p,q,r,s}(A, B, C) = \text{Tr}((A^{p/2} K B^{q/2} C^r B^{q/2} K^\dagger A^{p/2})^s)$. The new entropy is expressed as an optimization of such a functional where the optimization is performed locally on one subsystem ($C$).
• Complex Interpolation Theory: Used extensively to prove convexity/concavity properties of the trace functionals and to establish chain rules via Hadamard's three-line theorem.
• Variational Characterizations: The authors utilize variational forms of Schatten norms and trace functionals (extending results by Frank, Lieb, Zhang, and Beigi) to convert optimization problems into fixed-point equations.
• Spectral Pinching and Universal States: Asymptotic techniques involving permutation-invariant "universal states" ($\omega_{B^n}$) are employed to handle limits and prove monotonicity properties.
• Fixed-Point Equations: The optimizers for the conditional entropy are characterized by specific fixed-point equations, which are crucial for proving additivity.

3. Key Contributions

A. Definition of a Unified Family

The paper defines a new conditional entropy $H^\lambda_{\alpha,z}(A|B)_\rho$ with parameters $(\alpha, z, \lambda)$.

• Interpolation: The parameter $\lambda$ acts as an interpolation/extrapolation parameter.
  • $\lambda = 0$ recovers the "Arrow Down" ($\downarrow$) family (Petz-type).
  • $\lambda = 1$ recovers the "Arrow Up" ($\uparrow$) family (Sandwiched-type).
  • Intermediate values of $\lambda$ provide a continuous bridge between these distinct regimes.
• Generality: This framework encompasses all previously studied conditional Rényi entropies (Petz, Sandwiched, Min, Max, von Neumann) as special cases or limits.

B. Fundamental Mathematical Properties

The authors establish that this family satisfies the core axioms required for operational significance within a specific Data-Processing Inequality (DPI) region $\mathcal{D}$:

1. Data-Processing Inequality (DPI):

   • The entropy is monotonic under a broad class of channels: specifically, bipartite channels where the action on $A$ is a convex combination of unitaries (or sub-unital maps) and the action on $B$ is an arbitrary quantum instrument.
   • Novelty: Unlike standard proofs that rely on the DPI of the underlying divergence, the authors prove this by first establishing the convexity/concavity of the function $\rho \mapsto \exp((1-\alpha)H^\lambda_{\alpha,z})$ and then lifting this to monotonicity. This is necessary because the underlying trivariate functional itself is not monotone under quantum channels for the relevant parameters.

2. Additivity:

   • The entropy is additive under tensor products: $H^\lambda_{\alpha,z}(AA'|BB')_{\rho \otimes \sigma} = H^\lambda_{\alpha,z}(A|B)_\rho + H^\lambda_{\alpha,z}(A'|B')_\sigma$.
   • This is proven by characterizing the optimizer $\sigma_B$ via a fixed-point equation and showing that the tensor product of optimizers is the optimizer for the tensor product state.

3. Entropic Duality:

   • The paper proves a complete duality relation for pure tripartite states $\rho_{ABC}$:
     $$H^\lambda_{\alpha,z}(A|B)_\rho + H^{\hat{\lambda}}_{\hat{\alpha},\hat{z}}(A|C)_\rho = 0$$
   • Significance: This resolves an open question regarding the duality of the "Arrow Up" and "Arrow Down" entropies. The authors show that the duals of the standard Petz and Sandwiched entropies do not belong to the canonical forms (1) (i.e., $\hat{\lambda} \notin \{0, 1\}$). Instead, they belong to the new family with non-trivial $\lambda$, completing the picture of conditional entropies.

4. Chain Rules:

   • General chain rules are derived relating $H(AB|C)$, $H(A|BC)$, and $H(B|C)$.
   • These rules generalize known inequalities for von Neumann entropy and provide new inequalities for Rényi entropies, including hybrid rules mixing Petz and Sandwiched types.

5. Monotonicity in Parameters:

   • The entropy is shown to be monotonic with respect to $\alpha$, $z$, and $\lambda$ within the DPI region.

4. Key Results

• The DPI Region ($\mathcal{D}$): The authors explicitly define the region in the $(\alpha, z, \lambda)$ parameter space where the DPI holds. This region is the union of two sets, $\mathcal{D}_1$ (for $0 < \alpha < 1$) and $\mathcal{D}_2$ (for $\alpha > 1$), defined by linear inequalities involving the parameters.
• Completeness under Duality: The family is "complete" under duality, meaning the dual of any member in the family is also a member of the family (with transformed parameters). This contrasts with previous families where the duals fell outside the defined class.
• Operational Applications: The paper notes that these entropies have already been applied to characterize strong converse exponents for randomness extraction in both commutative and general quantum settings.
• Classical Limit: In the classical (commutative) limit, the new entropy recovers a weighted quasi-norm of the conditional distribution, interpolating between the definitions by Hayashi/Skoric and Arimoto.

5. Significance

• Unification: The work provides a unified geometric framework for quantum conditional entropies, revealing that the previously distinct Petz and Sandwiched families are merely boundary cases of a broader, continuous structure.
• Resolution of Open Problems: It answers the open question of whether the duals of standard conditional entropies admit canonical forms, demonstrating that they require the generalized $\lambda$ parameter.
• Methodological Advancement: The paper advances the toolkit of quantum information theory by successfully applying complex interpolation and multivariate trace inequalities to a setting where standard divergence-based proofs fail.
• Operational Relevance: By establishing the DPI and chain rules, the authors ensure these quantities are suitable for use in quantum Shannon theory, cryptography, and resource theories, particularly in non-asymptotic regimes where fine-grained entropy measures are critical.

In summary, Rubboli, Goodarzi, and Tomamichel have constructed a comprehensive family of quantum conditional entropies that unifies existing theories, satisfies all necessary operational axioms, and reveals deep structural connections through duality and convexity properties.