Quantum Conditional Entropies

• Quantum conditional entropies are measures quantifying the uncertainty of one quantum system given another, allowing negative values to capture nonclassical correlations.
• They are defined via von Neumann entropy and generalized through f-divergences, Rényi, and Tsallis frameworks, providing robust operational insights.
• These entropies underpin critical applications in quantum cryptography, resource theories, and device-independent protocols, enhancing secure communication and data analysis.

Quantum conditional entropies quantify the amount of uncertainty about one quantum system given access to another, capturing quantum correlations (including, but not limited to, entanglement) and forming the backbone of quantum information theory, cryptography, and statistical mechanics. Unlike classical conditional entropy, which is always nonnegative, quantum conditional entropy can be negative, reflecting fundamentally nonclassical features in bipartite and multipartite quantum systems. This article reviews the principal definitions, properties, generalizations, and operational significance of quantum conditional entropies, emphasizing rigorous results and key methodologies from both finite and infinite-dimensional settings.

1. Foundational Definitions

The standard quantum conditional entropy of a bipartite state $\rho_{AB}$ on $A \otimes B$ is

$S(A|B)_{\rho} = S(\rho_{AB}) - S(\rho_B)$

where $S(\cdot)$ denotes the von Neumann entropy and $\rho_B = \mathrm{Tr}_A \rho_{AB}$.

In infinite-dimensional Hilbert spaces, direct subtraction can yield indeterminate forms due to potentially infinite entropies. To resolve this, the conditional entropy is defined via quantum relative entropy (Kuznetsova, 2010): $$H(C|A) = S(\rho_C) - S(\rho_{AC} \| \rho_A \otimes \rho_C)$$ where $S(\cdot \| \cdot)$ is the quantum relative entropy. This definition ensures that $H(C|A)$ takes values in $[-\infty, +\infty)$, accommodates divergent marginal entropies, and recovers the subtractive formula when all marginal entropies are finite.

Conditional entropies have been generalized using quantum $f$-divergences and via optimization formulations, leading to a family of conditional entropies parameterized by divergences, entropy functions, or measurement models (Rastegin, 2013, Rubboli et al., 29 Oct 2024).

2. Key Properties: Monotonicity, Concavity, and Subadditivity

Quantum conditional entropies—across von Neumann, Rényi, or Tsallis families—are known to satisfy several core properties essential for their utility in information theory (Kuznetsova, 2010, Rubboli et al., 29 Oct 2024):

• Monotonicity under Conditioning: For any tripartite state, conditioning on additional quantum systems does not increase the conditional entropy:

$S(ABC|D) \leq S(AB|D)$

More generally, conditional entropy is monotonic under partial traces and compatible quantum operations, a reflection of the data-processing inequality for quantum relative entropy.

• Concavity in State: Quantum conditional entropy is concave in the underlying state:

$$S(A|B)_{\lambda \rho^{(1)} + (1-\lambda) \rho^{(2)}} \geq \lambda S(A|B)_{\rho^{(1)}} + (1-\lambda) S(A|B)_{\rho^{(2)}}$$

The proof in infinite dimensions relies on finite-rank truncations and dominated convergence arguments.

• Subadditivity: For composite systems,

$S(AB|CD) \leq S(A|C) + S(B|D)$

This is established using approximation techniques and leveraging properties of the relative entropy.

• Chain Rules and Duality: Modern treatments organize conditional entropies within families (e.g., Petz, sandwiched Rényi, $\alpha$-$z$ Rényi) which obey chain rules, additivity under tensor products, and duality relations (Rubboli et al., 29 Oct 2024).

These principles are foundational for developing robust information-theoretic and cryptographic applications, ensuring that conditional entropies behave predictably under composition and coarse-graining of quantum systems.

3. Generalizations: f-divergences, Rényi, and Tsallis Conditional Entropies

Beyond the von Neumann entropy, several alternative conditional entropy families have been introduced:

• Conditional Entropies from $f$-divergences: The general construction is

$$H_f(\rho_{AB}|B) = -\inf_{\sigma_B,\,\operatorname{Tr}(\sigma_B) \leq 1} D_f(\rho_{AB} \| I_A \otimes \sigma_B)$$

where $D_f$ is a quantum $f$-divergence, and subnormalized $\sigma_B$ makes the definition invariant under Hilbert space extensions (Rastegin, 2013). The function $f$ must be operator convex, with $f(1) = 0$ and $f(0)=0$.

• Tsallis and Rényi Conditional Entropies: For $\alpha > 0, \alpha\neq 1$,

$$S_\alpha(\rho) = \frac{1}{1-\alpha} \log \operatorname{Tr}(\rho^\alpha)$$

The conditional version is $$S_\alpha(A|B) = S_\alpha(\rho_{AB}) - S_\alpha(\rho_B)$$. For the Tsallis case,

$$H_\alpha(\rho_{AB}|B) = -\inf_{\sigma_B, \operatorname{Tr}(\sigma_B) = 1} D_\alpha(\rho_{AB}\|I_A \otimes \sigma_B)$$

with $D_\alpha$ the Tsallis $\alpha$-divergence (Rastegin, 2013). These quantities inherit chain rules and (when $\alpha$ is in an appropriate range) monotonicity properties.

• Variational Formulation for $\alpha$-$z$ Rényi (Rubboli et al., 29 Oct 2024):

$$\operatorname{Tr}\left[(A^{p/2} B^{q} A^{p/2})^{1/(p+q)}\right] = \sup_{X > 0} \ldots$$

For $\alpha > 1$ (and $z \geq 1$), this variational approach underpins key results about monotonicity and data-processing for generalized conditional entropies.

By organizing these generalizations within a broader mathematical framework, unified proofs of key properties and new generalized chain rules have been obtained, encompassing Petz-type and sandwiched Rényi conditional entropies.

4. Device-Independent and Semidefinite-Relaxed Conditional Entropies

Recent progress on device-independent quantum cryptography has leveraged conditional entropies adapted to optimizations over families of quantum states or behaviors constrained by observed data (Brown et al., 2020). The introduction of "iterated mean Rényi divergences" with parameters

$\alpha_k = 1 + \frac{1}{2^k-1}$

allows one to rewrite the associated conditional entropies as maximizations over noncommutative Hermitian polynomials subject to polynomial constraints independent of Hilbert space dimension. This independence is exploited in the Navascués–Pironio–Acín (NPA) hierarchy, relaxing the device-independent optimization problem to a sequence of semidefinite programs (SDPs). Each hierarchy level gives a tighter device-independent lower bound on, for instance, $H^{(\downarrow)}(A|B)$, efficiently computable via moment matrices associated with the operator-valued measurement structure.

In summary, new conditional entropy families can be numerically approximated device-independently via the NPA hierarchy and SDP relaxations, a critical advancement for quantum cryptography protocols where dimension independence and robust optimization are required (Brown et al., 2020).

5. Continuity, Stability, and Infinite-Dimensional Extensions

Continuity bounds for quantum conditional entropy are fundamental for robustness in information processing tasks. The tightest uniform continuity bounds, applicable when marginal states coincide, refine the Audenaert–Fannes (AF) inequality via an inequality based on the Jordan–Hahn decomposition (Audenaert et al., 27 Aug 2024): $$S(\rho_1) - S(\rho_2) \leq S(\rho_+) - S(\rho_-) + h(\epsilon)$$ where $\rho_1 - \rho_2 = \Delta_+ - \Delta_-$ (Jordan–Hahn), $S(\rho_+)$ and $S(\rho_-)$ are entropies of the positive/negative parts, and $h(\epsilon)$ is the binary entropy with $\epsilon = \tfrac{1}{2}\|\rho_1 - \rho_2\|_1$. Applied to conditional entropies for bipartite states $(\rho_{1})_{AB}$, $(\rho_{2})_{AB}$ with identical marginals on $B$, this leads to

$$|S(A|B)_{\rho_1} - S(A|B)_{\rho_2}| \leq \log(d_A^2 - 1) + h(\epsilon)$$

where $d_A$ is the dimension of $A$. These bounds extend (in appropriately modified form) to infinite dimensions.

The fundamental properties, continuity, and stability of conditional entropy in infinite-dimensional settings are critical for quantum optics, continuous-variable QKD, and other applications (Kuznetsova, 2010, Winter, 2015, Audenaert et al., 27 Aug 2024).

6. Operational Significance and Applications

Quantum conditional entropy is operationally significant in:

• Resource Theories: Negative conditional entropy is a quantifier of resources for superdense coding, quantum state merging, and entanglement distillation (Vempati et al., 2021, Rubboli et al., 29 Oct 2024).
• Channel/Process Analysis: Definitions have been lifted from states to quantum channels and processes via optimization over input states, resulting in conditional entropies for channels connected to causal influence and information flow (Das et al., 2 Oct 2024). The conditional entropy of a bipartite channel reveals when it is signaling and satisfies strong subadditivity for quantum processes.
• Cryptography and Device-Independent Protocols: Variationally and SDP-defined conditional entropies now directly enable finite-key security proofs, device-independent randomness generation, and robust protocols in the presence of imperfect detectors (Brown et al., 2020, Liu et al., 2022).
• Entropic Inequalities: New entropy power inequalities in both infinite and finite dimensions provide fundamental lower bounds, with saturation by Gaussian extremal states, and have direct implications for channel capacities and uncertainty relations (Palma et al., 2017, Jeong et al., 2017, Palma et al., 2018, Palma, 2018).

Conditional entropy appears as a central term in entropic uncertainty relations, as well as in operational and thermodynamic settings (e.g., relating to the amount of extractable work in recursive quantum information processing (Song, 17 Oct 2024)).

7. Chain Rules, Duality, and Unified Frameworks

Modern theory organizes quantum conditional entropies (including Petz, $\alpha$-$z$ Rényi, min/max, and more) into a single family characterized by:

• Unified Chain Rules: Generalized to hold across broad classes, with new insights into equality and monotonicity relations.
• Duality Relations: Many conditional entropies obey dualities under system partitioning, crucial for multipartite settings.
• Parameter Monotonicity: Monotonicity in the entropy parameters (e.g., Rényi $\alpha$) has been established via variational formulas (Rubboli et al., 29 Oct 2024).

This unified picture enables systematic analysis, generalizes operational interpretations, and streamlines technical proofs for existing and new results in quantum information theory.

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The paper and application of quantum conditional entropies—across definitions, properties, operational meaning, continuity, and computational aspects—remain foundational for advances in quantum information processing, cryptography, resource theory, and the analysis of quantum dynamical systems, especially in high-dimensional and device-independent scenarios. Recent developments demonstrate the increasing reach and power of variational methods, SDP relaxations, and unified axiomatic frameworks for organizing and extending the landscape of quantum conditional entropy.