Petz-Rényi Mutual Information

• Petz–Rényi mutual information is a one-parameter family of quantum correlation measures defined by replacing Umegaki’s relative entropy with the Petz–Rényi divergence.
• It retains key properties like non-negativity, monotonicity under local operations, and additivity, and converges to standard quantum mutual information as the order approaches one.
• Operational interpretations span hypothesis testing, channel coding, and quantum field theory, with numerical methods such as alternating minimization addressing computational challenges.

Petz–Rényi mutual information is a one-parameter family of correlation measures obtained by replacing the Umegaki relative entropy in the identity $$I(A\!:\!B)_\rho=D(\rho_{AB}\Vert \rho_A\otimes\rho_B)$$ with the Petz Rényi divergence. In its most common form, for $\alpha\in(0,1)\cup(1,\infty)$, it is

$I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$

with the usual support conditions for finiteness; in the limit $\alpha\to1$ it recovers the ordinary quantum mutual information. The same Petz divergence also underlies singly minimized and doubly minimized variants, which play distinct roles in hypothesis testing, channel coding, tensor-network computation, many-body physics, and quantum field theory (Berta et al., 2015, Burri, 2024, Kudler-Flam, 2022).

1. Definitions, variants, and notation

The Petz–Rényi divergence between positive operators $\rho$ and $\sigma$ is

$D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$

For $0<\alpha<1$, one requires that $\rho$ and $\sigma$ are not perfectly orthogonal; for $\alpha\in(0,1)\cup(1,\infty)$0, one requires $\alpha\in(0,1)\cup(1,\infty)$1, otherwise the divergence is set to $\alpha\in(0,1)\cup(1,\infty)$2 (Hayashi et al., 2014).

For a bipartite state $\alpha\in(0,1)\cup(1,\infty)$3, the standard Petz–Rényi mutual information is

$\alpha\in(0,1)\cup(1,\infty)$4

An equivalent characterization minimizes only over one marginal,

$\alpha\in(0,1)\cup(1,\infty)$5

and the unique minimizer is $\alpha\in(0,1)\cup(1,\infty)$6 (Berta et al., 2015).

A distinct construction is the doubly minimized Petz–Rényi mutual information,

$\alpha\in(0,1)\cup(1,\infty)$7

sometimes called the fully minimized form. Recent work treats this quantity as an autonomous correlation measure rather than a mere reformulation of the standard definition. For $\alpha\in(0,1)\cup(1,\infty)$8, the non-minimized, singly minimized, and doubly minimized Petz-based mutual informations are in general different; only at $\alpha\in(0,1)\cup(1,\infty)$9 do they all reduce to the von Neumann mutual information (Burri, 2024).

The Petz case also sits inside the two-parameter $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$0-$I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$1 Rényi relative entropies: $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$2 with the Petz choice corresponding to $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$3 and the sandwiched choice to $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$4. In that notation,

$I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$5

This locates Petz–Rényi mutual information within a broader family of Rényi correlation measures derived from Rényi relative entropies (Kudler-Flam et al., 2023).

A recurrent source of confusion is the “naïve” Rényi mutual information $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$6. That expression need not even be non-negative for $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$7, and it fails data-processing. By contrast, the Petz construction is built from relative entropy and retains the structural features expected of a correlation measure. In the commuting case, however, the Petz expression reduces to the entropy combination, so the two notions coincide for fully classical or fully decohered states (Berta et al., 2015).

2. Structural properties

Petz–Rényi mutual information is non-negative, and equality holds if and only if $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$8. The same product-state characterization holds for the doubly minimized version: $I_\alpha^P(A\!:\!B)_\rho = D_\alpha^P(\rho_{AB}\Vert \rho_A\otimes\rho_B) = \frac{1}{\alpha-1}\log \Tr\!\bigl[\rho_{AB}^{\,\alpha}(\rho_A\otimes\rho_B)^{\,1-\alpha}\bigr],$9 if and only if $\alpha\to1$0 is a product state (Scalet et al., 2021, Burri, 2024).

A central property is data-processing under local quantum operations. For $\alpha\to1$1, the Petz divergence is monotone under CPTP maps, and therefore the corresponding Petz–Rényi mutual information cannot increase under local channels on $\alpha\to1$2 or $\alpha\to1$3. The doubly minimized variant is likewise non-increasing under local CPTP maps for $\alpha\to1$4, and it is invariant under local isometries (Kudler-Flam, 2022, Burri, 2024).

The map $\alpha\to1$5 is non-decreasing for fixed $\alpha\to1$6, so $\alpha\to1$7 is also non-decreasing. The limit $\alpha\to1$8 recovers the Umegaki mutual information, while $\alpha\to1$9 yields a max-type mutual information obtained by minimizing $\rho$0 over product states (Scalet et al., 2021, Burri, 2024).

Additivity is another important feature. For the standard Petz quantity,

$\rho$1

because the Petz divergence is additive on tensor products. For the doubly minimized quantity, additivity has been established for $\rho$2: $\rho$3 On $\rho$4, the doubly minimized function is continuously differentiable in $\rho$5, and $\rho$6 is convex on $\rho$7 (Berta et al., 2015, Burri, 2024).

Within the $\rho$8-$\rho$9 family, the Petz and sandwiched versions are ordered by

$\sigma$0

This makes the Petz quantity a natural upper envelope of the sandwiched mutual information in the relevant region (Kudler-Flam et al., 2023).

3. Operational interpretations

One operational interpretation arises in composite hypothesis testing. Hayashi and Tomamichel consider a binary test with null hypothesis $\sigma$1 and composite alternative $\sigma$2, where $\sigma$3 is fixed and $\sigma$4 is arbitrary. For the associated trade-off function $\sigma$5, the direct Hoeffding exponent for $\sigma$6 is

$\sigma$7

whenever $\sigma$8. In this setting, Petz–Rényi mutual information quantifies the optimal type-II versus type-I trade-off in correlation detection (Hayashi et al., 2014).

For the doubly minimized quantity, Burri studies binary discrimination between $\sigma$9 and the composite alternative consisting of all product states on $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$0. For $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$1, the direct exponent is determined by the doubly minimized Petz–Rényi mutual information: $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$2 for $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$3, with the right-hand side strictly positive if and only if $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$4, and equal to zero for $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$5. This gives the fully minimized quantity a direct-exponent interpretation against arbitrary product alternatives (Burri, 2024).

In classical–quantum channel coding, the Petz–Rényi information of order $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$6 for an input distribution $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$7 and a CQ channel $D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$8 is

$D_\alpha^P(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\!\bigl[\rho^\alpha \sigma^{1-\alpha}\bigr].$9

As $0<\alpha<1$0, this becomes the Holevo information. For $0<\alpha<1$1, the maximized quantity

$0<\alpha<1$2

is the Petz–Augustin capacity of order $0<\alpha<1$3, which characterizes the random-coding error exponent for CQ channel coding (Chu et al., 10 Jan 2026).

These operational roles clarify a common misconception. Petz–Rényi mutual information is not merely a formal deformation of $0<\alpha<1$4; in several regimes it is the exponent governing explicit decision-theoretic or coding problems. A plausible implication is that different minimization conventions are best viewed as different operational relaxations of “distance from product structure,” rather than as notationally interchangeable objects.

4. Computation and algorithms

For the doubly minimized Petz–Rényi mutual information, no closed-form expression is known in general, and no global analytic solution is known even in the classical–classical special case when $0<\alpha<1$5. This motivates numerical methods (Burri, 7 Jul 2025).

A natural method is alternating minimization over the two local marginals. Starting from $0<\alpha<1$6, one updates

$0<\alpha<1$7

and then

$0<\alpha<1$8

Each block subproblem is solved in one step by the quantum Sibson identity, and if

$0<\alpha<1$9

then $\rho$0 and $\rho$1 as $\rho$2 (Burri, 7 Jul 2025).

The convergence rate depends on the Rényi order. For $\rho$3, each block update is a contraction in Hilbert’s projective metric with coefficient $\rho$4, which yields geometric convergence of the objective: $\rho$5 For $\rho$6, the argument instead uses joint convexity, uniqueness of the partial minimizers, uniqueness of the global minimizer, and a recursion lemma to obtain

$\rho$7

The paper also reports that for $\rho$8 the error decays roughly geometrically, whereas for $\rho$9 it decays roughly like $\sigma$0 for moderate $\sigma$1 (Burri, 7 Jul 2025).

For CQ channels, computing the Petz–Augustin capacity via the Petz–Rényi information leads to a different algorithmic problem: maximizing $\sigma$2 over the simplex. For $\sigma$3, the exponentiated objective

$\sigma$4

is convex and Hölder smooth with respect to the $\sigma$5-norm, so Nesterov’s universal fast gradient method with negative Shannon entropy prox applies. The resulting approximation error obeys

$\sigma$6

This is presented as the first non-asymptotic algorithm for computing the Petz–Augustin capacity via the Rényi-information route (Chu et al., 10 Jan 2026).

For many-body states represented as matrix-product operators, the direct power-trace form

$\sigma$7

is computationally useful. For integer $\sigma$8, the quantity can be expressed as the contraction of an MPO of manageable bond dimension, and one may also combine computable upper bounds from $\sigma$9 or the geometric divergence with lower bounds from measured-divergence constructions (Scalet et al., 2021).

5. Quantum field theory and many-body physics

In quantum field theory, the Petz definition supplies a “proper” Rényi mutual information in the sense that it is non-negative, monotone under local operations, UV finite, and well-defined in the continuum limit. Kudler-Flam develops a replica path-integral representation by computing $\alpha\in(0,1)\cup(1,\infty)$00 for integer $\alpha\in(0,1)\cup(1,\infty)$01, introducing $\alpha\in(0,1)\cup(1,\infty)$02 “double-slit” sheets for $\alpha\in(0,1)\cup(1,\infty)$03 and $\alpha\in(0,1)\cup(1,\infty)$04 “single-slit” sheets for $\alpha\in(0,1)\cup(1,\infty)$05 and $\alpha\in(0,1)\cup(1,\infty)$06, and analytically continuing $\alpha\in(0,1)\cup(1,\infty)$07. In $\alpha\in(0,1)\cup(1,\infty)$08 dimensions this reduces to twist-field correlators (Kudler-Flam, 2022).

For two disjoint intervals $\alpha\in(0,1)\cup(1,\infty)$09 and $\alpha\in(0,1)\cup(1,\infty)$10, the replica representation gives

$\alpha\in(0,1)\cup(1,\infty)$11

followed by analytic continuation $\alpha\in(0,1)\cup(1,\infty)$12. In a $\alpha\in(0,1)\cup(1,\infty)$13D CFT of central charge $\alpha\in(0,1)\cup(1,\infty)$14, the leading universal behavior depends on the cross-ratio

$\alpha\in(0,1)\cup(1,\infty)$15

and takes the form

$\alpha\in(0,1)\cup(1,\infty)$16

For adjacent intervals, one recovers a single-log divergence proportional to $\alpha\in(0,1)\cup(1,\infty)$17, and finite temperature is obtained by placing the twist correlator on the cylinder (Kudler-Flam, 2022).

Kudler-Flam, in the broader $\alpha\in(0,1)\cup(1,\infty)$18-$\alpha\in(0,1)\cup(1,\infty)$19 framework, also gives adjacent-interval and finite-temperature formulas that specialize to the Petz case $\alpha\in(0,1)\cup(1,\infty)$20. For adjacent intervals in the vacuum of a $\alpha\in(0,1)\cup(1,\infty)$21D CFT,

$\alpha\in(0,1)\cup(1,\infty)$22

and a corresponding finite-temperature expression is obtained by replacing the interval factor with a $\alpha\in(0,1)\cup(1,\infty)$23 term (Kudler-Flam et al., 2023).

Petz–Rényi mutual information also bounds connected correlators. One form is

$\alpha\in(0,1)\cup(1,\infty)$24

and at $\alpha\in(0,1)\cup(1,\infty)$25 a stronger logarithmic bound in terms of the connected correlator is available. In the $\alpha\in(0,1)\cup(1,\infty)$26-$\alpha\in(0,1)\cup(1,\infty)$27 framework, analogous lower and upper bounds are given in terms of operator norms and operator-Schmidt decompositions (Kudler-Flam, 2022, Kudler-Flam et al., 2023).

In lattice many-body physics, Petz–Rényi mutual information obeys thermal area laws. For a Gibbs state $\alpha\in(0,1)\cup(1,\infty)$28 with $\alpha\in(0,1)\cup(1,\infty)$29, one has

$\alpha\in(0,1)\cup(1,\infty)$30

where $\alpha\in(0,1)\cup(1,\infty)$31. In the commuting case this simplifies to

$\alpha\in(0,1)\cup(1,\infty)$32

while in high temperature and $\alpha\in(0,1)\cup(1,\infty)$33 dimensions,

$\alpha\in(0,1)\cup(1,\infty)$34

The same work shows

$\alpha\in(0,1)\cup(1,\infty)$35

for projected entangled-pair density operators with local purification bond dimension $\alpha\in(0,1)\cup(1,\infty)$36, and

$\alpha\in(0,1)\cup(1,\infty)$37

for classical Gibbs distributions with local dimension $\alpha\in(0,1)\cup(1,\infty)$38 (Scalet et al., 2021).

6. Special cases, equivalences, and comparison with related measures

At $\alpha\in(0,1)\cup(1,\infty)$39, the standard Petz divergence becomes fidelity-based: $\alpha\in(0,1)\cup(1,\infty)$40 with $\alpha\in(0,1)\cup(1,\infty)$41. Consequently,

$\alpha\in(0,1)\cup(1,\infty)$42

This makes the $\alpha\in(0,1)\cup(1,\infty)$43 case particularly close to state-overlap and discrimination bounds (Berta et al., 2015).

For the doubly minimized construction, the order-$\alpha\in(0,1)\cup(1,\infty)$44 case is exceptional. One has

$\alpha\in(0,1)\cup(1,\infty)$45

where $\alpha\in(0,1)\cup(1,\infty)$46 is the reduced state of the canonical purification. Thus the min-reflected entropy is exactly the doubly minimized Petz–Rényi mutual information of order $\alpha\in(0,1)\cup(1,\infty)$47. This identity yields the comparison chain

$\alpha\in(0,1)\cup(1,\infty)$48

and it implies that maximal reflected entropy is equivalent to maximal mutual information, hence characterizes maximally entangled states (Burri, 25 Feb 2025).

Classical specialization clarifies the relation to Sibson’s classical Rényi mutual information. If $\alpha\in(0,1)\cup(1,\infty)$49 is diagonal in a product basis, then

$\alpha\in(0,1)\cup(1,\infty)$50

reduces to the classical Rényi divergence $\alpha\in(0,1)\cup(1,\infty)$51, and the singly minimized Petz quantity becomes Sibson’s classical Rényi mutual information. For the doubly minimized version,

$\alpha\in(0,1)\cup(1,\infty)$52

for classical-classical states (Hayashi et al., 2014, Burri, 2024).

The pure-state case shows that the doubly minimized quantity can have a qualitatively different $\alpha\in(0,1)\cup(1,\infty)$53-dependence from the standard non-minimized one. For a pure bipartite state with Schmidt spectrum $\alpha\in(0,1)\cup(1,\infty)$54, Burri gives

$\alpha\in(0,1)\cup(1,\infty)$55

for the doubly minimized quantity. This suggests that extremal Rényi orders probe distinct spectral features of the underlying bipartite structure (Burri, 2024).

A final comparison concerns terminology. “Rényi mutual information” is used in the literature for several inequivalent objects: entropy-difference formulas, Petz-type divergences, sandwiched divergences, singly minimized forms, and doubly minimized forms. The Petz-based definitions are distinguished by positivity and monotonicity under local operations in the relevant $\alpha\in(0,1)\cup(1,\infty)$56-range, but they are not interchangeable. Any statement about “the” Rényi mutual information therefore depends on which divergence and which minimization convention is being used (Kudler-Flam et al., 2023, Burri, 2024).