Petz-Tsallis Relative Entropy

• Petz-Tsallis relative entropy is a parametric quantum divergence defined via deformed logarithms and power means, generalizing the Umegaki relative entropy.
• It exhibits key structural properties such as nonnegativity, data-processing inequality, and joint convexity, which are crucial for quantum state certification and resource quantification.
• Algorithmic estimation methods and robust operator inequalities enable its application in quantum information processing, nonextensive statistical mechanics, and matrix analysis.

The Petz-Tsallis relative entropy is a parametric family of quantum divergences generalizing the Umegaki (quantum Kullback-Leibler) relative entropy. It arises as a key object in the study of quantum state distinguishability, quantum information processing, and nonextensive statistical mechanics. Defined via a deformed logarithm and power means, it encompasses a rich structure of monotonicity, variational characterizations, operator inequalities, and algorithmic properties, and serves as the basis for coherence, discord, and correlation quantification.

1. Definition and Structural Framework

For density matrices $\rho, \sigma$ on a finite-dimensional Hilbert space $\mathcal{H}$ and parameter $0<\alpha<1$, the Petz-Tsallis $\alpha$-relative entropy is defined as

$$D_\alpha(\rho \| \sigma) = \frac{1 - \operatorname{Tr}(\rho^\alpha \sigma^{1-\alpha})}{1 - \alpha}.$$

This expression recovers the Umegaki quantum relative entropy in the limit $\alpha\to 1^-$: $$\lim_{\alpha \to 1^-} D_\alpha(\rho \| \sigma) = \operatorname{Tr}[\rho(\log\rho - \log\sigma)].$$ As an $f$-divergence in the sense of Petz, it is generated by the operator-convex function $f_\alpha(z) = (z^\alpha - z)/(\alpha - 1)$, making $$D_\alpha(\rho \| \sigma) = S_{f_\alpha}(\rho\|\sigma)$$ a Petz quasi-entropy (Bao et al., 1 Oct 2025, Rastegin, 2011).

An equivalent operator-theoretic formulation applies more generally, including to accretive operators, using the weighted geometric mean $A\sharp_t B$: $$T_t(A\|B) = \frac{A\sharp_t B - A}{t},\quad t\in(0,1)$$ yielding the standard $A^{1-t}B^t$ form for positive commuting $A,B$ (Raïssouli et al., 2017).

2. Core Properties and Operator Inequalities

The following properties are central:

• Nonnegativity and Equality Condition: $D_\alpha(\rho \| \sigma) \geq 0$, with equality iff $\rho = \sigma$ (Bao et al., 1 Oct 2025, Vershynina, 2019).
• Data-Processing Inequality: For any completely positive trace-preserving (CPTP) map $\mathcal{E}$,

$$D_\alpha(\rho \| \sigma) \geq D_\alpha(\mathcal{E}(\rho) \| \mathcal{E}(\sigma))$$

for $0<\alpha\leq 2$ (Bao et al., 1 Oct 2025, Rastegin, 2011, Vershynina, 2019).

• Joint Convexity: $(\rho, \sigma) \mapsto D_\alpha(\rho \| \sigma)$ is jointly convex (Vershynina, 2019, Raïssouli et al., 2017).
• Pinsker-Type Bounds: For $T=\tfrac12\|\rho - \sigma\|_1$,

$$2\alpha d_H^2(\rho, \sigma) + O(\alpha) d_{\mathrm{tr}}^4(\rho, \sigma) \leq D_\alpha(\rho \| \sigma) \leq d_{\mathrm{tr}}(\rho, \sigma)^{1-\alpha}$$

and, for classical distributions,

$$D_\alpha(P\|Q) \geq C_\alpha g(T),\quad C_\alpha = \frac{2\alpha}{1-\alpha} \text{ or } 2$$

with $g(t) = 1 - \sqrt{1-t^2}$ (Bao et al., 1 Oct 2025, Rastegin, 2011).

• Fannes-Type Continuity Bounds: For $\alpha>1$ in the commutative case, $D_\alpha(P\|Q)$ possesses upper continuity bounds in terms of the minimal probability in $Q$ (Rastegin, 2011).
• Golden-Thompson Inequality: Deformed q-exponential traces satisfy a generalized Golden–Thompson inequality for $q \in [0,1)$:

$$\operatorname{Tr} \exp_q(A+B) \le \operatorname{Tr} \exp_q(A)^{2-q}\left[A(q-1)+\exp_q B\right]$$

(Shi et al., 2019).

Operator inequalities, e.g., sandwich two-sided tangent inequalities, bound $T_v(A|B)$ by expressions involving operator means and logarithmic terms (Furuichi et al., 2020).

3. Variational Representations

The Petz-Tsallis relative entropy admits variational characterizations: $$D_p(X \| A) = \max_{L+\ln_q A>-\frac{1}{q-1}} \left\{\operatorname{Tr} X + \operatorname{Tr} X^{2-q} L - \operatorname{Tr} \exp_q(L+ \ln_q A)\right\}$$ with deformed logarithms and exponentials defined as

$$\ln_q(x) = \frac{x^{q-1} - 1}{q - 1}, \quad \exp_q(u) = [1 + (q-1)u]^{1/(q-1)}$$

extending the standard Gibbs variational principle (Shi et al., 2019). These variational forms extend to constrained, e.g., trace-one, settings relevant for quantum-state optimization.

4. Algorithmic Estimation and Complexity

Efficient estimation of $D_\alpha(\rho \| \sigma)$ is crucial in quantum information. For rank $r$ states and constant $\alpha \in (0,1)$:

• Sample Complexity (unknown circuits):

$$\widetilde{O}\left(\frac{r^{3.5}}{\epsilon^{10}}\right)$$

quantum samples are sufficient to estimate $D_\alpha$ to additive error $\epsilon$ (Bao et al., 1 Oct 2025).

• Query Complexity (known purification circuits):

$$\widetilde{O}\left(\frac{r^{1.5}}{\epsilon^4}\right)$$

with further scaling improvement for specific $\alpha$, including $$O~(r^{1+ \alpha}/\epsilon^{1/\alpha + 1/(1-\alpha)})$$ for $\alpha < 1/2$.

The estimation leverages techniques such as block-encodings, QSVT, the Hadamard test, quantum amplitude estimation, and the "quantum multi-samplizer" method.

Complexity-theoretic completeness: Decision problems based on $D_\alpha$ are $\mathsf{QSZK}$-complete in symmetric-parameter regimes and $\mathsf{BQP}$-complete in low-rank settings. For instance, the TsallisQSD$_\alpha(a, b)$ and HellingerQSD$(a, b)$ problems exhibit these complexity-theoretic distinctions (Bao et al., 1 Oct 2025).

5. Operator Generalizations and Inequalities

Petz-Tsallis relative entropy extends to a broader operator-theoretic context, such as accretive operators, via operator means and power means (Raïssouli et al., 2017, Furuichi et al., 2020). The crucial features include:

• Integral and functional calculus representations capturing non-Hermitian and non-commutative settings.
• Norm and quadratic-form bounds, order-monotonicity under the real-part map, and operator convexity.
• Hierarchies of inequalities, e.g., $$S(A|B) \le T_v(A|B) \le E_v(A|B) \le E_{1-v}(A|B) \le S(A|B)$$ for $0<v\le1$, with reversals for $-1<v<0$ (Furuichi et al., 2020).

6. Applications in Quantum Information Processing

The Petz-Tsallis relative entropy underpins diverse quantum information applications:

• State Certification: Tolerant quantum state certification under Hellinger distance is achievable with $\widetilde{O}(r^{3.5})$ samples or $\widetilde{O}(r^{1.5})$ purified queries—exponentially outperforming standard tomography in polynomial-rank settings (Bao et al., 1 Oct 2025).
• Resource Quantification: Serves as the basis for five coherence measures (including convex-roof variants), three discord measures, and two correlation measures. Explicit formulas are provided for pure states (via Schmidt decomposition), and tight upper/lower bounds for mixed states are established (Vershynina, 2019).
• Inequalities for Entropies: Pinsker-, Fannes-, and Fano-type bounds provide quantitative continuity, stability, and robustness guarantees. For example, continuity bounds in the trace-norm and error-propagation under quantum channels (Rastegin, 2011).

7. Discussion and Relations to Broader Theories

Petz-Tsallis relative entropy integrates into Petz's quasi-entropy framework, ensuring that properties such as monotonicity under CPTP maps, joint convexity, and variational principles continue to hold (Shi et al., 2019). The operator generalizations support applications in matrix analysis, PDEs, and dynamics involving non-Hermitian evolutions (Raïssouli et al., 2017). In quantum resources, the construction allows for precise separation between monotonicity and strong monotonicity conditions, motivating various coherence and discord quantifiers (Vershynina, 2019).

The variational and sandwich operator inequalities, continuity and error bounds, and algorithmic estimation protocols collectively establish the Petz-Tsallis framework as a central tool in contemporary quantum information and operator analysis research.