Doubly Minimized Petz–Rényi Mutual Information

• Doubly minimized Petz–Rényi mutual information is a quantum correlation measure defined by optimizing the Petz Rényi divergence over all product reference states.
• It exhibits structural properties such as symmetry, additivity for α in [1/2,2], and operational significance in binary quantum hypothesis testing.
• Numerical computation via alternating minimization offers convergent algorithms with distinct linear or sublinear rates depending on the order α.

The doubly minimized Petz–Rényi mutual information is a bipartite quantum correlation measure defined by optimizing the Petz Rényi divergence over all product reference states. For a state $\rho_{AB}\in\mathcal S(AB)$ and order $\alpha\in[0,\infty)$, it is

$$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$

where

$$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$

for $\alpha\in(0,1)\cup(1,\infty)$, with the usual support conditions, and the limits at $\alpha=0,1$ taken by continuity. This quantity coincides with ordinary quantum mutual information at $\alpha=1$, but for $\alpha\neq 1$ it differs from both the non-minimized and singly minimized Petz–Rényi mutual informations. Recent work has established its symmetry, data processing, additivity for $\alpha\in[\tfrac12,2]$, fixed-point structure, operational meaning in binary quantum hypothesis testing, an exact identification with min-reflected entropy at $\alpha=\tfrac12$, and convergent numerical methods based on alternating minimization (Burri, 2024, Burri, 25 Feb 2025, Burri, 7 Jul 2025).

1. Definition and placement among Rényi mutual informations

The doubly minimized quantity belongs to a three-member family of Petz–Rényi mutual informations obtained by changing the reference product state against which $\alpha\in[0,\infty)$0 is compared. The paper introducing its systematic study also uses the shorthand

$\alpha\in[0,\infty)$1

so that $\alpha\in[0,\infty)$2 (Burri, 2024).

Variant	Definition	Comment
$\alpha\in[0,\infty)$3	$\alpha\in[0,\infty)$4	non-minimized
$\alpha\in[0,\infty)$5	$\alpha\in[0,\infty)$6	singly minimized
$\alpha\in[0,\infty)$7	$\alpha\in[0,\infty)$8	doubly minimized

At $\alpha\in[0,\infty)$9, the minimizer is $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$0, so all three reduce to the usual mutual information. For $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$1, the doubly minimized and singly minimized quantities are not generally equal, and explicit examples show inequivalence. The doubly minimized version is symmetric under exchanging $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$2 and $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$3, while the singly minimized one is not generally symmetric when $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$4. On classical-classical states, the doubly minimized quantum quantity reduces to the classical doubly minimized Rényi mutual information (Burri, 2024).

A recurrent source of confusion is the use of “Petz Rényi mutual information” for the non-minimized quantity $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$5. In quantum field theory, this non-minimized version has been advocated as a genuine correlation measure that is nonnegative, monotone under local operations, UV finite, and continuum well-defined; that construction is distinct from the doubly minimized measure, even though both are built from the same Petz divergence (Kudler-Flam, 2022).

2. Structural properties and variational theory

The doubly minimized Petz–Rényi mutual information satisfies a broad list of structural properties. It is symmetric for all $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$6, invariant under local isometries for all $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$7, and monotone under local CPTP maps for $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$8: $$I_\alpha^{\downarrow\downarrow}(A:B)_\rho \coloneqq \inf_{\substack{\sigma_A\in \mathcal S(A),\ \tau_B\in \mathcal S(B)}} D_\alpha(\rho_{AB}\| \sigma_A\otimes \tau_B),$$9 One of the main results is additivity: $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$0 for $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$1. The same work proves existence of minimizers for all $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$2, uniqueness for $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$3, and a fixed-point characterization of minimizers for $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$4 (Burri, 2024).

The dependence on $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$5 is also controlled. The map $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$6 is continuous on $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$7, continuously differentiable on $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$8, and $$D_\alpha(\rho\|\sigma)\coloneqq\frac{1}{\alpha -1}\log \operatorname{tr}[\rho^\alpha \sigma^{1-\alpha}]$$9 is convex on $\alpha\in(0,1)\cup(1,\infty)$0. At $\alpha\in(0,1)\cup(1,\infty)$1,

$\alpha\in(0,1)\cup(1,\infty)$2

where $\alpha\in(0,1)\cup(1,\infty)$3 is the mutual information variance. The paper also derives the upper bound

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

for all $\alpha\in(0,1)\cup(1,\infty)$5, with a sharper characterization of equality for $\alpha\in(0,1)\cup(1,\infty)$6 in terms of flat support spectrum and the condition $\alpha\in(0,1)\cup(1,\infty)$7 (Burri, 2024).

The variational theory extends to many-copy settings. For $\alpha\in(0,1)\cup(1,\infty)$8,

$\alpha\in(0,1)\cup(1,\infty)$9

with $\alpha=0,1$0 and $\alpha=0,1$1 the universal permutation invariant states. For $\alpha=0,1$2, this strengthens to an exact $\alpha=0,1$3-copy variational formula over symmetric states on $\alpha=0,1$4 and $\alpha=0,1$5. This suggests that the measure is compatible with permutation-invariant asymptotics in a way that is unusually rigid among optimized Rényi quantities (Burri, 2024).

3. Operational interpretation through binary quantum hypothesis testing

The principal operational meaning currently known for $\alpha=0,1$6 concerns direct exponents in composite binary quantum state discrimination. The null hypothesis is

$\alpha=0,1$7

while the alternative consists of product states of the form

$\alpha=0,1$8

Writing $\alpha=0,1$9 for the minimum type-I error under type-II budget $\alpha=1$0, the direct exponent theorem states that if

$\alpha=1$1

then for any $\alpha=1$2,

$\alpha=1$3

The same formula holds for the i.i.d. alternative (Burri, 2024).

This identifies the doubly minimized Petz–Rényi mutual information of orders $\alpha=1$4 as the governing rate function for the direct exponent. The exponent is positive iff $\alpha=1$5 and vanishes for $\alpha=1$6. The result generalizes a previous classical statement to arbitrary finite-dimensional quantum states and is proved using a combination of universal permutation-invariant states, a quantum Hoeffding bound, and a converse based on the classical binary hypothesis testing method. The operational role is therefore not merely asymptotic monotonicity; it is an exact exponent formula in a composite testing problem against product alternatives (Burri, 2024).

4. Numerical computation and alternating minimization

No closed-form expression is known in general for the doubly minimized Petz–Rényi mutual information when $\alpha=1$7. This has led to the development of numerical methods that optimize over the product-state factors directly. The most complete analysis to date studies alternating minimization: fix $\alpha=1$8, optimize exactly over $\alpha=1$9; then fix that $\alpha\neq 1$0, optimize exactly over $\alpha\neq 1$1; and iterate. The update rules are expressed as maps $\alpha\neq 1$2 and $\alpha\neq 1$3, and the quantum Sibson identity shows that each update is the true minimization over one factor holding the other fixed, rather than a heuristic step (Burri, 7 Jul 2025).

For $\alpha\neq 1$4, each update map is a contraction in Hilbert’s projective metric with contraction factor

$\alpha\neq 1$5

If $\alpha\neq 1$6, the contraction improves to $\alpha\neq 1$7 for some $\alpha\neq 1$8. As a consequence, the objective values converge linearly, with rate

$\alpha\neq 1$9

For $\alpha\in[\tfrac12,2]$0, the convergence is sublinear: $\alpha\in[\tfrac12,2]$1 so the rate is $\alpha\in[\tfrac12,2]$2. At $\alpha\in[\tfrac12,2]$3, alternating minimization is trivial because the first update already yields the exact minimizer $\alpha\in[\tfrac12,2]$4. By contrast, for $\alpha\in[\tfrac12,2]$5, the method need not converge to the global minimizer; an explicit example shows that it can get stuck at a non-optimal fixed point (Burri, 7 Jul 2025).

These results extend earlier convergence statements that were known only in the classical-classical case. The present theory covers arbitrary finite-dimensional quantum states, requires an initial point $\alpha\in[\tfrac12,2]$6 so that the update map is well-defined, and separates the full-rank case $\alpha\in[\tfrac12,2]$7 only when improved contraction constants are desired. This suggests that practical computation of $\alpha\in[\tfrac12,2]$8 is now on a substantially firmer footing in the parameter range $\alpha\in[\tfrac12,2]$9 (Burri, 7 Jul 2025).

5. The special point $\alpha=\tfrac12$0 and min-reflected entropy

At order $\alpha=\tfrac12$1, the doubly minimized Petz–Rényi mutual information admits an exact identification with a quantity from holography and reflected-entropy theory. For a bipartite state $\alpha=\tfrac12$2, let $\alpha=\tfrac12$3 denote its canonical purification and $\alpha=\tfrac12$4 the $\alpha=\tfrac12$5-marginal. The min-reflected entropy is

$\alpha=\tfrac12$6

The main theorem states

$\alpha=\tfrac12$7

Equivalently, the order-$\alpha=\tfrac12$8 optimization over product states can be rewritten as a largest-eigenvalue problem on the canonical-purification marginal (Burri, 25 Feb 2025).

The proof proceeds by expressing

$\alpha=\tfrac12$9

and then showing that the spectral maximization defining $\alpha\in[0,\infty)$00 can be restricted, without loss, to canonical-purification vectors derived from states $\alpha\in[0,\infty)$01. The key structural tool is the $\alpha\in[0,\infty)$02-invariance of $\alpha\in[0,\infty)$03, together with a representation theorem for $\alpha\in[0,\infty)$04-invariant unit vectors. This yields a closed spectral characterization: $\alpha\in[0,\infty)$05 For classical-classical states with $\alpha\in[0,\infty)$06, this reduces to

$\alpha\in[0,\infty)$07

For product states, $\alpha\in[0,\infty)$08, and for pure states,

$\alpha\in[0,\infty)$09

The identification also transfers operational meaning and inequalities to min-reflected entropy, including

$\alpha\in[0,\infty)$10

(Burri, 25 Feb 2025).

6. Relations to adjacent measures and recurring distinctions

The doubly minimized Petz–Rényi mutual information is best understood against two nearby but inequivalent constructions. The first is the non-minimized Petz–Rényi mutual information used in quantum field theory,

$\alpha\in[0,\infty)$11

which has been emphasized as a genuine correlation measure because it is nonnegative and monotone under local channels, and because it is UV finite and well-defined in the continuum limit. That QFT quantity is an upper bound on the minimized version mentioned in earlier literature, but it is not the doubly minimized measure, and the “double replica” method used to compute it is a computational device rather than a second minimization (Kudler-Flam, 2022).

The second nearby construction is the singly minimized Petz–Rényi mutual information

$\alpha\in[0,\infty)$12

Its order-$\alpha\in[0,\infty)$13 case has a distinct operational meaning: for one-shot entanglement transmission, the entanglement fidelity of the Petz decoder satisfies

$\alpha\in[0,\infty)$14

This is an exact statement about a singly minimized quantity on the complementary channel, not about $\alpha\in[0,\infty)$15 (Burri, 24 Feb 2025).

A further comparison concerns the doubly minimized sandwiched Rényi mutual information $\alpha\in[0,\infty)$16, obtained by replacing the Petz divergence with the sandwiched Rényi divergence. That quantity obeys its own duality relation for $\alpha\in[0,\infty)$17, is additive for $\alpha\in[0,\infty)$18, and governs strong converse exponents rather than direct exponents in composite binary quantum hypothesis testing. The two optimized Rényi mutual informations are ordered by

$\alpha\in[0,\infty)$19

and

$\alpha\in[0,\infty)$20

This suggests that “doubly minimized Rényi mutual information” is not a single object but a family whose behavior depends sensitively on the underlying divergence and on whether one studies direct or strong-converse asymptotics (Burri, 2024).