Petz–Rényi Relative Entropy in Quantum Information

• Petz–Rényi relative entropy is a quantum divergence that generalizes classical Rényi divergence to noncommuting density operators and recovers Umegaki entropy as α approaches 1.
• It bridges quantum and classical settings via Nussbaum–Szkoła distributions, enabling exact classical reductions and closed-form expressions in Gaussian and thermal state frameworks.
• The measure is crucial for diverse applications such as quantum hypothesis testing, state discrimination, and resource theories in quantum information and quantum field theory.

Petz–Rényi relative entropy is a Rényi-type quantum divergence that generalizes classical Rényi divergence to noncommuting states and converges to Umegaki relative entropy in the limit $\alpha\to1$. For density operators $\rho$ and $\sigma$, and parameter $\alpha\in(0,1)\cup(1,\infty)$, it is defined by

$D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$

with the support condition being essential for $\alpha>1$. The quantity appears throughout quantum information theory, continuous-variable theory, and quantum field theory, and admits both exact classical reductions and model-specific closed forms (Androulakis et al., 2022, Androulakis et al., 2023, Fröb et al., 2024).

1. Definition and classical reduction

A useful structural feature of Petz–Rényi relative entropy is that it can be written exactly as a classical Rényi divergence after passing to the Nussbaum–Szkoła distributions. If

$$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$

then the associated classical distributions on the index set $(i,j)$ are

$$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$

One then has

$$D_\alpha^{P}(\rho\Vert\sigma) = \frac{1}{\alpha-1} \log\sum_{i,j} r_i^\alpha s_j^{1-\alpha} |\langle u_i|v_j\rangle|^2 = D_\alpha(P\Vert Q).$$

This identity remains valid in both finite and infinite dimensions and supplies a direct bridge between quantum and classical Rényi theory (Androulakis et al., 2022).

In the commuting case, where $\rho$0 and $\rho$1 are simultaneously diagonalizable, the formula reduces to the ordinary classical Rényi divergence

$\rho$2

This reduction clarifies that the genuinely quantum content of the Petz quantity is encoded in the overlap matrix $\rho$3, rather than in the eigenvalues alone (Androulakis et al., 2022).

2. Domain, limits, and fundamental properties

Several foundational properties are known in full generality. The map $\rho$4 is nondecreasing on $\rho$5, $\rho$6, and for $\rho$7 equality holds iff $\rho$8. It is continuous on

$\rho$9

and obeys the skew-symmetry relation

$\sigma$0

Its principal limits are

$\sigma$1

$\sigma$2

and

$\sigma$3

For $\sigma$4, finiteness is equivalent to $\sigma$5; for $\sigma$6, finiteness requires $\sigma$7 together with finiteness of the spectral sum $\sigma$8 (Androulakis et al., 2022).

The data-processing inequality is known in the standard Petz regime $\sigma$9, with the concave range $\alpha\in(0,1)\cup(1,\infty)$0 and the operator-convex range $\alpha\in(0,1)\cup(1,\infty)$1 treated by the $\alpha\in(0,1)\cup(1,\infty)$2-divergence formalism. Petz’s quantum $\alpha\in(0,1)\cup(1,\infty)$3-divergence and the later optimized quantum $\alpha\in(0,1)\cup(1,\infty)$4-divergence place Petz–Rényi and sandwiched Rényi divergences into a common operator-Jensen framework. In this setting, Petz–Rényi monotonicity under CPTP maps holds for $\alpha\in(0,1)\cup(1,\infty)$5, while the sandwiched divergence has a broader known monotonicity range, and the two coincide in the commuting case (Wilde, 2021).

3. Gaussian and continuous-variable formulations

For faithful quantum Gaussian states, closed forms for Petz–Rényi relative entropy can be expressed entirely in terms of mean vectors and covariance matrices. In the $\alpha\in(0,1)\cup(1,\infty)$6-mode setting, the formulas use the Gaussian mean vectors $\alpha\in(0,1)\cup(1,\infty)$7, covariance matrices $\alpha\in(0,1)\cup(1,\infty)$8, and the covariance transformation induced by taking normalized powers $\alpha\in(0,1)\cup(1,\infty)$9. These expressions cover $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$0 directly, and for $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$1 they yield faithful-state formulas under the sufficient condition

$D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$2

The same framework also gives a Gaussian formula for the max-relative entropy through symplectic eigenvalues of an auxiliary covariance matrix (Seshadreesan et al., 2017).

A particularly explicit subclass is furnished by thermal and displaced thermal states. For a single mode,

$D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$3

and an $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$4-mode thermal state is $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$5. A displaced thermal state has the form

$D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$6

where $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$7 is the Weyl displacement operator. Powers preserve thermality: $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$8 For thermal states with support inclusion, and for displaced faithful thermal states, the finiteness range for $D_{\alpha}^{P}(\rho\Vert\sigma) = \begin{cases} \displaystyle \frac{1}{\alpha-1}\log \operatorname{Tr}\!\big[\rho^{\alpha}\sigma^{1-\alpha}\big], & \operatorname{supp}\rho\subseteq \operatorname{supp}\sigma,\[1ex] +\infty, & \text{otherwise}, \end{cases}$9 is exact: $\alpha>1$0 with the convention $\alpha>1$1. In the displaced case, the value depends on overlap coefficients such as $\alpha>1$2, but the finiteness threshold is unchanged by displacement. In the same thermal-plus-displacement subclass, the condition is equivalent to the covariance inequality

$\alpha>1$3

establishing a special case of the Seshadreesan–Lami–Wilde conjecture (Androulakis et al., 2023).

A complementary symplectic derivation for $\alpha>1$4 decomposes Gaussian relative entropy into a classical part and a quantum part after bringing the reference state to thermal normal form. In that representation, the classical contribution is a weighted sum of Bernoulli relative entropies arising from thermal occupation parameters, while the quantum contribution depends on transformed one-mode means and covariance blocks; the same analysis proves $\alpha>1$5 in the Gaussian setting (Parthasarathy, 2021).

4. Symmetric variants and trace-norm bounds

A symmetric variant is obtained by averaging the two Petz divergences,

$\alpha>1$6

with $\alpha>1$7 defined by continuity from Umegaki relative entropy. The special value $\alpha>1$8 is

$\alpha>1$9

where $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$0 is Holevo’s symmetric fidelity (Salazar, 2024).

This symmetric family satisfies a tight lower bound in terms of trace distance

$$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$1

namely

$$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$2

At $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$3 this reproduces Holevo’s inequality, while at $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$4 it yields the symmetric Umegaki bound

$$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$5

The bound strictly improves the Pinsker-type estimate

$$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$6

It arises from a symmetric Petz–Rényi uncertainty relation formulated in terms of the means and variances of a Hermitian observable $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$7, and it is saturated by a binary two-level family with opposite biases (Salazar, 2024).

5. Modular theory and quantum field theory

In algebraic quantum field theory, Petz–Rényi relative entropy admits a modular-theoretic extension. For a von Neumann algebra $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$8 and normalized cyclic and separating vectors $$\rho=\sum_i r_i |u_i\rangle\!\langle u_i|,\qquad \sigma=\sum_j s_j |v_j\rangle\!\langle v_j|,$$9, let $(i,j)$0 be the relative modular operator. The Petz quasi-entropy is

$(i,j)$1

and the associated Petz–Rényi entropy is

$(i,j)$2

In this regime it is nonnegative, increasing in $(i,j)$3, and converges to the Araki–Uhlmann relative entropy as $(i,j)$4. For a free scalar field in the Minkowski wedge and for the free chiral current in a thermal state, explicit calculations show a qualitative difference from ordinary relative entropy: whereas the latter depends only on the symplectic form, the Petz–Rényi quantity also depends on the symmetric part of the two-point function and is therefore genuinely quantum (Fröb et al., 2024).

A closely related construction in continuum QFT is the Petz-based Rényi mutual information

$(i,j)$5

This quantity is nonnegative, symmetric in $(i,j)$6 and $(i,j)$7, monotone under local CPTP maps for $(i,j)$8, and UV finite for separated regions. In $(i,j)$9-dimensional conformal field theory it admits a replica path-integral representation with twist fields, and for adjacent intervals the resulting expressions are universal functions of central charge and interval geometry. The same framework yields lower bounds on connected correlators through trace-distance and fidelity inequalities (Kudler-Flam, 2022).

In rational conformal field theories undergoing local quenches, Petz-type relative Rényi entropies can exhibit step-like temporal behavior. For half-line geometries they vanish before the quasiparticle reaches the subsystem and saturate to a constant afterward, with late-time values determined by finite-dimensional matrices built from descendant-state two-point coefficients. These results also show that relative entropy may fail to distinguish local operators when the subsystem receives the same information flux from them (Zhao et al., 2024).

6. Applications, algorithms, and ongoing developments

Operationally, Petz–Rényi relative entropy enters coding and source-encoding bounds, quantum hypothesis testing, and state discrimination. In Gaussian continuous-variable settings, the exact $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$0-thresholds for thermal and displaced thermal states determine when $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$1, and in particular $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$2, is well defined. This matters because $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$3 is the largest member of the Petz family with guaranteed data processing in the Gaussian discrimination regime emphasized in the literature (Androulakis et al., 2023).

Algorithmic work has recently moved beyond analytic formulas. A quantum algorithm for estimating Umegaki relative entropy and Petz–Rényi divergence from two unknown finite-dimensional states combines quadrature approximations, variational representations of quantum $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$4-divergences, and parameterized Hermitian polynomial operators. In the stated implementation range $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$5, the circuit size is at most $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$6 for $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$7-qubit inputs, and the method is directly applicable to distributed scenarios in which the two states reside on different devices (Lu et al., 13 Jan 2025).

For classical–quantum channels, the order-$$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$8 Petz–Rényi capacity in the regime $$P(i,j)=r_i\,|\langle u_i|v_j\rangle|^2,\qquad Q(i,j)=s_j\,|\langle u_i|v_j\rangle|^2.$$9 can be written as

$$D_\alpha^{P}(\rho\Vert\sigma) = \frac{1}{\alpha-1} \log\sum_{i,j} r_i^\alpha s_j^{1-\alpha} |\langle u_i|v_j\rangle|^2 = D_\alpha(P\Vert Q).$$0

A mirror-descent, exponentiated-gradient algorithm has been proposed for this optimization, with global sublinear convergence under relative smoothness and local linear convergence under a tangent-space nondegeneracy condition on a truncated simplex (Lai et al., 15 Jan 2026).

Resource-theoretic applications expose both utility and limitations. Petz–Rényi relative entropy induces coherence measures with closed forms and monotonicity under incoherent CPTP maps in the range $$D_\alpha^{P}(\rho\Vert\sigma) = \frac{1}{\alpha-1} \log\sum_{i,j} r_i^\alpha s_j^{1-\alpha} |\langle u_i|v_j\rangle|^2 = D_\alpha(P\Vert Q).$$1, but these measures can fail strong monotonicity under selective measurements. In multipartite entanglement theory, Petz- and sandwiched-Rényi relative entropies have been minimized over separable states to study monogamy, with state-dependent behavior across GHZ, W, star, Heisenberg, and transverse-field Ising families (Shao et al., 2016, Mannaï et al., 2024).

Taken together, these developments place Petz–Rényi relative entropy at a junction of operator theory, Gaussian analysis, modular methods, and algorithmic quantum information. Its central mathematical feature is the tractable but distinctly noncommutative quantity $$D_\alpha^{P}(\rho\Vert\sigma) = \frac{1}{\alpha-1} \log\sum_{i,j} r_i^\alpha s_j^{1-\alpha} |\langle u_i|v_j\rangle|^2 = D_\alpha(P\Vert Q).$$2; its central conceptual feature is that finiteness, monotonicity, and operational meaning depend delicately on the parameter regime, support structure, and physical model under consideration.