--- title: Petz-Tsallis Relative Entropy url: https://www.emergentmind.com/topics/petz-tsallis-relative-entropy type: topic --- # Petz-Tsallis Relative Entropy 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 [2510.00752][1102.5154]. 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 $A,B$ [1705.07042]. ## 2. Core Properties and Operator Inequalities The following properties are central: - **Nonnegativity and Equality Condition:** $D_\alpha(\rho \| \sigma) \geq 0$, with equality iff $\rho = \sigma$ [2510.00752][1912.05426]. - **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 $0<\alpha\leq 2$ [2510.00752][1102.5154][1912.05426]. - **Joint Convexity:** $(\rho, \sigma) \mapsto D_\alpha(\rho \| \sigma)$ is jointly convex [1912.05426][1705.07042]. - **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 $g(t) = 1 - \sqrt{1-t^2}$ [2510.00752][1102.5154]. - **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 $Q$ [1102.5154]. - **Golden-Thompson Inequality:** Deformed q-exponential traces satisfy a generalized Golden–Thompson inequality for $q \in [0,1)$: \[ \operatorname{Tr} \exp_q(A+B) \le \operatorname{Tr} \exp_q(A)^{2-q}\left[A(q-1)+\exp_q B\right] \] [1901.06807]. Operator inequalities, e.g., sandwich two-sided tangent inequalities, bound $T_v(A|B)$ by expressions involving operator means and logarithmic terms [2001.01342]. ## 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 [1901.06807]. 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 $\epsilon$ [2510.00752]. - **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 [2510.00752]. ## 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 [1705.07042][2001.01342]. 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