Symmetric Petz–Rényi Relative Entropy

• Symmetric Petz–Rényi relative entropy is defined as the average of the two directional Petz–Rényi divergences, creating a balanced quantum state distinguishability measure that interpolates between fidelity and relative entropy.
• It establishes tight bounds with trace distance and refines Pinsker-type inequalities, thereby providing operational significance across different quantum settings.
• The framework extends to Gaussian states, operator algebras, and uncertainty relations, underlining its utility in quantum thermodynamics and recovery processes.

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 $\alpha=1$ (Salazar, 2024). 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 (Salazar, 2024, Androulakis et al., 2022).

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 $\alpha\in(0,1)\cup(1,\infty)$0, it admits the spectral expression

$\alpha\in(0,1)\cup(1,\infty)$1

for spectral decompositions $\alpha\in(0,1)\cup(1,\infty)$2, $\alpha\in(0,1)\cup(1,\infty)$3 (Androulakis et al., 2022). 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 $\alpha\in(0,1)\cup(1,\infty)$4 weighted by $\alpha\in(0,1)\cup(1,\infty)$5 (Androulakis et al., 2022).

Within this landscape, the symmetric Petz–Rényi relative entropy is obtained by averaging the two directional divergences. The 2024 treatment adopts exactly

$\alpha\in(0,1)\cup(1,\infty)$6

for $\alpha\in(0,1)\cup(1,\infty)$7, and extends it continuously to $\alpha\in(0,1)\cup(1,\infty)$8 (Salazar, 2024). This symmetrization preserves the Petz-type dependence on $\alpha\in(0,1)\cup(1,\infty)$9 while removing the ordering of the two arguments.

A related structural identity for the directed divergence is

$$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$0

which implies in particular that $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$1 (Androulakis et al., 2022). Thus the Petz Rényi divergence is already symmetric at $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$2, and the explicit symmetrization becomes nontrivial away from this point.

2. Distinguished parameter values and basic structure

The family has two distinguished points. At $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$3, the symmetric Petz–Rényi relative entropy coincides with the directed one and is related to Holevo’s fidelity

$$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$4

through

$$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$5

(Salazar, 2024). At $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$6, continuity yields

$$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$7

the symmetric quantum relative entropy (Salazar, 2024).

Non-negativity follows directly from the non-negativity of the directed Petz divergence. The 2024 analysis states that $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$8, with equality if and only if $$D_\alpha(\rho\|\sigma)=\frac{1}{\alpha-1}\log \operatorname{tr}(\rho^\alpha \sigma^{1-\alpha}),$$9 under the standard support assumptions (Salazar, 2024). For the directed quantity, positivity and equality conditions are available in the broader infinite-dimensional treatment as well: $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$0 for $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$1, and for $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$2, $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$3 if and only if $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$4 (Androulakis et al., 2022).

In the commuting case, both the directed and symmetric quantities reduce to their classical counterparts. If $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$5, then in a common eigenbasis $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$6, $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$7,

$$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$8

and $$\tilde D_\alpha(\rho,\sigma):=\frac12\bigl(D_\alpha(\rho\|\sigma)+D_\alpha(\sigma\|\rho)\bigr),$$9 becomes the symmetrized classical Rényi divergence (Salazar, 2024, Androulakis et al., 2022).

Support conditions remain directional. For $\alpha=1$0, the directed divergence is infinite when $\alpha=1$1 (Androulakis et al., 2022). 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 $\alpha=1$2 to the trace distance

$\alpha=1$3

The symmetric Petz–Rényi relative entropies satisfy

$\alpha=1$4

for all density matrices $\alpha=1$5 and all $\alpha=1$6 (Salazar, 2024). The same work refers to this as the generalized Holevo inequality.

This bound improves the Pinsker-type inequality

$\alpha=1$7

which the paper explicitly calls loose for this family (Salazar, 2024). The new bound is tighter and reproduces Holevo’s inequality at $\alpha=1$8. Indeed, substituting $\alpha=1$9 gives

$\alpha=\tfrac12$0

which, using $\alpha=\tfrac12$1, is equivalent to

$\alpha=\tfrac12$2

(Salazar, 2024).

The proof strategy passes through a classical inequality involving the triangular discrimination

$\alpha=\tfrac12$3

followed by a Nussbaum–Szkoła reduction from the quantum pair $\alpha=\tfrac12$4 to classical distributions $\alpha=\tfrac12$5 satisfying

$\alpha=\tfrac12$6

(Salazar, 2024). 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 (Salazar, 2024).

4. Uncertainty relation and thermodynamic interpretations

The generalized Holevo inequality emerges as a corollary of a more general uncertainty relation. For any Hermitian operator $\alpha=\tfrac12$7, define

$\alpha=\tfrac12$8

for $\alpha=\tfrac12$9, and

$\alpha=1$0

The symmetric Petz–Rényi uncertainty relation is

$\alpha=1$1

(Salazar, 2024).

Inverted, this yields lower bounds on variance-to-squared-mean ratios of observables in terms of $\alpha=1$2. The paper writes the result as

$\alpha=1$3

with $\alpha=1$4, where $\alpha=1$5 is the trace-distance bound function and $\alpha=1$6 its inverse (Salazar, 2024).

Several known relations appear as special cases. At $\alpha=1$7, the uncertainty relation becomes a Holevo-type bound involving $\alpha=1$8. At $\alpha=1$9, it reduces to the quantum relative entropy uncertainty relation. In the classical commutative setting, the $f$0 limit reproduces the tight hysteretic thermodynamic uncertainty relation, and an exchange-fluctuation-theorem thermodynamic uncertainty relation is recovered when $f$1, $f$2, and $f$3 (Salazar, 2024). 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

$f$4

together with the usual support condition for $f$5 (Androulakis et al., 2022). This places symmetric Petz–Rényi relative entropy on a firm footing whenever both directional terms are defined.

On density spaces of $f$6-algebras with faithful traces, the directed Petz-type Rényi relative entropy

$f$7

has symmetry group given by standard-form maps implemented by a central positive invertible element and a Jordan *-isomorphism (Molnár, 2019). 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 (Molnár, 2019).

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 $f$8 and the conditionally finite regime $f$9 (Seshadreesan et al., 2017). 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 $f$0 convergence to relative entropy (Parthasarathy, 2021). For displaced thermal states, the finiteness range is known exactly: $f$1 with the minimum of the empty set defined to be infinity (Androulakis et al., 2023). In these Gaussian contexts, a symmetric Petz–Rényi quantity is naturally obtained by averaging the two directional formulas, although the symmetrized object is not always introduced explicitly (Seshadreesan et al., 2017, Androulakis et al., 2023, Parthasarathy, 2021).

A separate point of terminology arises in modular quantum field theory. There, Petz–Rényi relative entropy between cyclic and separating vectors is defined through the relative modular operator,

$f$2

and explicit free-field computations show dependence on the symmetric part of the two-point function as well as the symplectic form (Fröb et al., 2024). This is a different use of “symmetric” from the state symmetrization $f$3, and the distinction is conceptually important.

6. Structural consequences, related constructions, and computation

The Petz framework also interacts with preserver problems and recovery-based formulations. On positive operators in finite dimension, bijections preserving the Mosonyi–Ogawa Rényi-type divergence or maximal $f$4-divergences are characterized by unitary or antiunitary conjugations, with an additional global positive scaling in the former case (Gaál et al., 2017). That paper does not define symmetric Petz–Rényi relative entropy explicitly, but it states that typical symmetrizations can be constructed from Petz-type divergences and that the preserver theorems constrain such symmetric quantities as well. A plausible implication is that the rigidity of symmetrized Petz-type divergences is essentially the same as that of their ordered counterparts (Gaál et al., 2017).

In quantum information applications based on monotonicity and recovery, the same Petz sufficiency structure reappears. A relative-entropy formulation of symmetric discord uses

$f$5

and the corresponding analysis states that this structure can be transferred conceptually to Petz–Rényi relative entropies and their symmetric variants through Petz-type sufficiency and recovery maps (Zhang et al., 2012). This suggests a natural route from symmetric Petz–Rényi distinguishability to symmetric Petz–Rényi discord-like quantities, although the latter are not defined in the original paper.

Computation has also become explicit. A 2025 quantum algorithm estimates Umegaki relative entropy and Petz Rényi divergence of two unknown states by combining quadrature approximations, a variational representation of quantum $f$6-divergences, and parameterized Hermitian polynomial operators; the circuit size is at most $f$7 qubits for $f$8-qubit states (Lu et al., 13 Jan 2025). That work does not define a symmetric Petz–Rényi divergence, but it states that a natural symmetric version is obtained by estimating $f$9 and $$D_{\alpha}(\rho\Vert\sigma)=\frac{1}{\alpha-1}\log D_{f_\alpha}(\rho\Vert\sigma),\qquad f_\alpha(\lambda)=\lambda^\alpha,$$0 separately and averaging them. In this computational sense, the symmetric quantity is operationally no more difficult than two evaluations of the directed one (Lu et al., 13 Jan 2025).

Taken together, these results identify symmetric Petz–Rényi relative entropy as a symmetrized Petz divergence with three salient features: an exact interpolation between fidelity-like and relative-entropy-like regimes, a tight and explicitly computable relation to trace distance and observable fluctuations, and a broad compatibility with the analytical machinery developed for directed Petz divergences across finite-dimensional, Gaussian, $$D_{\alpha}(\rho\Vert\sigma)=\frac{1}{\alpha-1}\log D_{f_\alpha}(\rho\Vert\sigma),\qquad f_\alpha(\lambda)=\lambda^\alpha,$$1-algebraic, and modular-theoretic settings (Salazar, 2024, Androulakis et al., 2022).