--- title: Symmetric Petz–Rényi Relative Entropy url: https://www.emergentmind.com/topics/symmetric-petz-renyi-relative-entropy type: topic --- # Symmetric Petz–Rényi Relative Entropy Symmetric Petz–Rényi relative entropy is the symmetrized form of the Petz Rényi divergence. For density matrices \(\rho,\sigma\) and \(\alpha\in(0,1)\cup(1,\infty)\), the directed Petz Rényi divergence is \[ D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}), \] and the symmetric quantity considered explicitly in recent work is \[ \tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr), \] extended by continuity at \(\alpha=1\) [2402.01390]. In this form it is a symmetric one-parameter family of distinguishability measures that interpolates between Holevo’s fidelity at \(\alpha=\tfrac12\) and the symmetric quantum relative entropy at \(\alpha=1\), while remaining anchored in the broader Petz \(f\)-divergence and quasi-entropy frameworks [2402.01390][2203.01964]. ## 1. Definition within the Petz–Rényi family The directed Petz Rényi divergence belongs to the standard quantum \(f\)-divergence formalism. In the finite- or infinite-dimensional setting, it can be written as \[ D_{\alpha}(\rho\Vert\sigma)=\frac{1}{\alpha-1}\log D_{f_\alpha}(\rho\Vert\sigma),\qquad f_\alpha(\lambda)=\lambda^\alpha, \] and, via the Nussbaum–Szkoła distributions \(P,Q\), it admits the spectral expression \[ D_{\alpha}(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\sum_{i,j} r_i^\alpha s_j^{1-\alpha}|\langle u_i|v_j\rangle|^2 \] for spectral decompositions \(\rho=\sum_i r_i|u_i\rangle\langle u_i|\), \(\sigma=\sum_j s_j|v_j\rangle\langle v_j|\) [2203.01964]. The same paper shows that this representation is valid in both finite and infinite dimensions and provides a pushforward-measure formula in terms of the distribution of the observable \(\sigma^{1-\alpha}\) weighted by \(\rho^\alpha\) [2203.01964]. Within this landscape, the symmetric Petz–Rényi relative entropy is obtained by averaging the two directional divergences. The 2024 treatment adopts exactly \[ \tilde{D}_\alpha(\rho,\sigma) := \frac{1}{2}\bigl(D_\alpha(\rho\|\sigma)+ D_\alpha(\sigma\|\rho)\bigr), \] for \(\alpha\in(0,1)\cup(1,\infty)\), and extends it continuously to \(\alpha=1\) [2402.01390]. This symmetrization preserves the Petz-type dependence on \(\rho^\alpha\sigma^{1-\alpha}\) while removing the ordering of the two arguments. A related structural identity for the directed divergence is \[ D_\alpha(\rho\Vert\sigma)=\frac{\alpha}{1-\alpha}\,D_{1-\alpha}(\sigma\Vert\rho),\qquad 0<\alpha<1, \] which implies in particular that \(D_{1/2}(\rho\Vert\sigma)=D_{1/2}(\sigma\Vert\rho)\) [2203.01964]. Thus the Petz Rényi divergence is already symmetric at \(\alpha=\tfrac12\), and the explicit symmetrization becomes nontrivial away from this point. ## 2. Distinguished parameter values and basic structure The family has two distinguished points. At \(\alpha=\tfrac12\), the symmetric Petz–Rényi relative entropy coincides with the directed one and is related to Holevo’s fidelity \[ F_H(\rho,\sigma):=[\operatorname{tr}(\sqrt{\rho}\sqrt{\sigma})]^2 \] through \[ \tilde{D}_{1/2}(\rho,\sigma)=D_{1/2}(\rho\|\sigma)=-\ln F_H(\rho,\sigma) \] [2402.01390]. At \(\alpha=1\), continuity yields \[ \tilde{D}_1(\rho,\sigma)=\tilde D(\rho,\sigma):=\frac12\bigl(D(\rho\|\sigma)+D(\sigma\|\rho)\bigr), \] the symmetric quantum relative entropy [2402.01390]. Non-negativity follows directly from the non-negativity of the directed Petz divergence. The 2024 analysis states that \(\tilde D_\alpha(\rho,\sigma)\ge 0\), with equality if and only if \(\rho=\sigma\) under the standard support assumptions [2402.01390]. For the directed quantity, positivity and equality conditions are available in the broader infinite-dimensional treatment as well: \(D_\alpha(\rho\Vert\sigma)\ge 0\) for \(\alpha\in[0,\infty]\), and for \(\alpha>0\), \(D_\alpha(\rho\Vert\sigma)=0\) if and only if \(\rho=\sigma\) [2203.01964]. In the commuting case, both the directed and symmetric quantities reduce to their classical counterparts. If \([\rho,\sigma]=0\), then in a common eigenbasis \(\rho=\sum_i p_i|i\rangle\langle i|\), \(\sigma=\sum_i q_i|i\rangle\langle i|\), \[ D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log\sum_i p_i^\alpha q_i^{1-\alpha}, \] and \(\tilde D_\alpha(\rho,\sigma)\) becomes the symmetrized classical Rényi divergence [2402.01390][2203.01964]. Support conditions remain directional. For \(\alpha>1\), the directed divergence is infinite when \(\operatorname{supp}\rho\not\subseteq \operatorname{supp}\sigma\) [2203.01964]. A symmetric average therefore requires both directional terms to be finite. This suggests that, in applications, the symmetric quantity is naturally most regular on pairs of states for which both support inclusions hold. ## 3. Tight bounds in terms of trace distance A central result is a tight inequality relating \(\tilde D_\alpha(\rho,\sigma)\) to the trace distance \[ T(\rho,\sigma):=\frac12\|\rho-\sigma\|_1. \] The symmetric Petz–Rényi relative entropies satisfy \[ \tilde{D}_\alpha(\rho,\sigma) \ge \frac{1}{\alpha-1} \ln\frac{\cosh\bigl[(2\alpha-1)\operatorname{arctanh}(T(\rho,\sigma))\bigr]} {\cosh\bigl[\operatorname{arctanh}(T(\rho,\sigma))\bigr]}, \] for all density matrices \(\rho,\sigma\) and all \(\alpha\in(0,1)\cup(1,\infty)\) [2402.01390]. The same work refers to this as the generalized Holevo inequality. This bound improves the Pinsker-type inequality \[ \tilde{D}_\alpha(\rho,\sigma)\ge 2\min(\alpha,1)\,T(\rho,\sigma)^2, \] which the paper explicitly calls loose for this family [2402.01390]. The new bound is tighter and reproduces Holevo’s inequality at \(\alpha=\tfrac12\). Indeed, substituting \(\alpha=\tfrac12\) gives \[ \tilde{D}_{1/2}(\rho,\sigma)\ge \ln\frac{1}{1-T(\rho,\sigma)^2}, \] which, using \(\tilde D_{1/2}=-\ln F_H\), is equivalent to \[ T(\rho,\sigma)\le \sqrt{1-F_H(\rho,\sigma)} \] [2402.01390]. The proof strategy passes through a classical inequality involving the triangular discrimination \[ \delta(P,Q):=\sum_s \frac{(P_s-Q_s)^2}{P_s+Q_s}, \] followed by a Nussbaum–Szkoła reduction from the quantum pair \((\rho,\sigma)\) to classical distributions \(P,Q\) satisfying \[ D_\alpha(P\|Q)=D_\alpha(\rho\|\sigma),\qquad \tilde D_\alpha(P,Q)=\tilde D_\alpha(\rho,\sigma) \] [2402.01390]. The resulting bound is not merely asymptotic: the paper states that Holevo’s inequality is tight and exhibits a two-level example in which the generalized bounds are exactly achieved [2402.01390]. ## 4. Uncertainty relation and thermodynamic interpretations The generalized Holevo inequality emerges as a corollary of a more general uncertainty relation. For any Hermitian operator \(\hat\theta\), define \[ \langle \hat{\theta}\rangle_x := \operatorname{tr}(x\hat{\theta}),\qquad \langle\!\langle \hat{\theta}\rangle\!\rangle_x := \operatorname{tr}(x\hat{\theta}^2)-\operatorname{tr}(x\hat{\theta})^2, \] for \(x\in\{\rho,\sigma\}\), and \[ s(\rho,\sigma;\hat{\theta}) := \Bigg[ \frac{\frac12(\langle\hat{\theta}\rangle_\rho-\langle\hat{\theta}\rangle_\sigma)^2} {\langle\!\langle \hat{\theta}\rangle\!\rangle_\rho+\langle\!\langle \hat{\theta}\rangle\!\rangle_\sigma+\frac12(\langle\hat{\theta}\rangle_\rho-\langle\hat{\theta}\rangle_\sigma)^2} \Bigg]^{1/2}. \] The symmetric Petz–Rényi uncertainty relation is \[ \tilde{D}_\alpha(\rho,\sigma) \ge \frac{1}{\alpha-1} \ln\frac{\cosh\bigl[(2\alpha-1)\operatorname{arctanh}(s(\rho,\sigma;\hat{\theta}))\bigr]} {\cosh\bigl[\operatorname{arctanh}(s(\rho,\sigma;\hat{\theta}))\bigr]} \] [2402.01390]. Inverted, this yields lower bounds on variance-to-squared-mean ratios of observables in terms of \(\tilde D_\alpha\). The paper writes the result as \[ \frac{\langle\!\langle \hat{\theta}\rangle\!\rangle_\rho+\langle\!\langle \hat{\theta}\rangle\!\rangle_\sigma} {\frac12(\langle\hat{\theta}\rangle_\rho-\langle\hat{\theta}\rangle_\sigma)^2} \ge f\bigl(\alpha,\tilde D_\alpha(\rho,\sigma)\bigr), \] with \(f(\alpha,x)=1/[B^{-1}(\alpha,x)]^2-1\), where \(B(\alpha,\cdot)\) is the trace-distance bound function and \(B^{-1}\) its inverse [2402.01390]. Several known relations appear as special cases. At \(\alpha=\tfrac12\), the uncertainty relation becomes a Holevo-type bound involving \(F_H(\rho,\sigma)\). At \(\alpha=1\), it reduces to the quantum relative entropy uncertainty relation. In the classical commutative setting, the \(\alpha=1\) limit reproduces the tight hysteretic thermodynamic uncertainty relation, and an exchange-fluctuation-theorem thermodynamic uncertainty relation is recovered when \(p=P(\Gamma)\), \(q=P(\Gamma^\dagger)\), and \(\theta(\Gamma^\dagger)=-\theta(\Gamma)\) [2402.01390]. The paper’s perspective is that symmetric Petz–Rényi divergences can therefore function as generalized entropy productions in quantum and stochastic thermodynamics. ## 5. Broader analytical frameworks and extensions The symmetric construction sits inside a much larger analytical theory for directed Petz divergences. In infinite dimensions, the Petz Rényi divergence admits both spectral and pushforward-measure representations, and its finiteness can be characterized through integrability of \[ \int_0^\infty \lambda\,\mu^{\rho^\alpha,\sigma^{1-\alpha}}(d\lambda) \] together with the usual support condition for \(\alpha>1\) [2203.01964]. This places symmetric Petz–Rényi relative entropy on a firm footing whenever both directional terms are defined. On density spaces of \(C^*\)-algebras with faithful traces, the directed Petz-type Rényi relative entropy \[ \tau\text{-}D_\alpha^c(A\|B)=\frac{1}{\alpha-1}\log \tau(A^\alpha B^{1-\alpha}) \] has symmetry group given by standard-form maps implemented by a central positive invertible element and a Jordan *-isomorphism [1906.10412]. That paper does not define the symmetrized divergence explicitly, but it states that the same reasoning applies to natural symmetric combinations. This suggests that symmetric Petz–Rényi relative entropies inherit the same standard-form symmetry group on noncommutative density spaces [1906.10412]. Gaussian and operator-algebraic settings provide further extensions. For quantum Gaussian states, explicit formulas for directed Petz Rényi divergences are available in terms of mean vectors and covariance matrices, for both \(0<\alpha<1\) and the conditionally finite regime \(\alpha>1\) [1706.09885]. A pedagogical Gaussian derivation likewise gives a decomposition of the Umegaki limit into a classical part and a quantum part, and presents a generalized Petz–Rényi formula for Gaussian states with \(\alpha\uparrow 1\) convergence to relative entropy [2102.06708]. For displaced thermal states, the finiteness range is known exactly: \[ D_{\alpha}(\rho\Vert\sigma)<\infty \Leftrightarrow \alpha < \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text{ such that } r_j