Tumula Information: Reversed Quantum Correlations

• Tumula information is a bipartite correlation measure defined as the doubly minimized reversed quantum relative entropy that vanishes on product states.
• It forms a hierarchical framework with umlaut and lautum information, providing key operational significance in asymmetric quantum hypothesis testing.
• Extensions to quantum channels reveal super-additivity and variational structures, highlighting both numerical challenges and insights into channel capacities.

Searching arXiv for the primary and related papers to ground the article. arXiv search: tumula information (Schmitt et al., 17 Mar 2026) Tumula information is a bipartite correlation measure introduced as the doubly minimized reversed analogue of mutual information. For a finite-dimensional bipartite quantum state $\rho_{AB}$, it is defined by

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$

where $D(\cdot\|\cdot)$ is the quantum relative entropy. In the same framework, the lautum information is $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$, and the umlaut information is $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$. Tumula information therefore arises by reversing the arguments of relative entropy and then minimizing over both factors of the product benchmark. Its main significance is operational: in composite asymmetric quantum hypothesis testing against product alternatives, the Sanov exponent is exactly the tumula information, while the reverse direct exponent is governed by its Petz–Rényi generalization (Schmitt et al., 17 Mar 2026).

1. Definition and placement among reversed correlation measures

The standard mutual information of a bipartite state $\rho_{AB}$ has the relative-entropy form

$$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$

The reversed form is the lautum information,

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$

and tumula information is the doubly minimized reversed version,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$

In the classical case, for a joint distribution $P_{XY}$,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$0

The usual support condition for quantum relative entropy applies: $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$1 is finite only if $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$2, equivalently $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$3 (Schmitt et al., 17 Mar 2026).

The three reversed correlation measures form a hierarchy because the optimization domains are nested:

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$4

Hence

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$5

This establishes tumula information as the least constrained member of the reversed family.

Measure	Definition	Relation
Lautum $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$6	$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$7	Largest in the reversed hierarchy
Umlaut $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$8	$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$9	$D(\cdot\|\cdot)$0
Tumula $D(\cdot\|\cdot)$1	$D(\cdot\|\cdot)$2	Vanishes exactly on product states

Non-negativity is immediate: $D(\cdot\|\cdot)$3. More strongly, for any $D(\cdot\|\cdot)$4 in the Rényi extension described below,

$D(\cdot\|\cdot)$5

In particular, tumula information vanishes exactly on product states (Schmitt et al., 17 Mar 2026).

2. Petz–Rényi formulation and variational structure

The Petz divergence of order $D(\cdot\|\cdot)$6 is

$D(\cdot\|\cdot)$7

For $D(\cdot\|\cdot)$8, it is finite if $D(\cdot\|\cdot)$9; for $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$0, it is finite if $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$1. It is extended by limits at $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$2 and $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$3. A central identity for $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$4 is

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$5

Using this divergence, the paper introduces three Petz–Rényi lautum-information variants:

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$6

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$7

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$8

The last quantity is the doubly minimized Petz–Rényi lautum information, abbreviated PRLI (Schmitt et al., 17 Mar 2026).

The reverse identity connects PRLI directly to Petz–Rényi mutual information:

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\|\rho_{AB})$$9

$$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$0

$$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$1

The limits at $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$2 recover the relative-entropy quantities:

$$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$3

The optimization problem admits explicit partial minimizers. For $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$4, if $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$5 is fixed and $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$6, then

$$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$7

and symmetrically

$$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$8

At $$U(A\!:\!B)_\rho := \inf_{\tau_B} D(\rho_A\otimes \tau_B\|\rho_{AB})$$9, the corresponding partial minimizers become exponential maps:

$\rho_{AB}$0

$\rho_{AB}$1

These formulas provide the basic fixed-point structure for numerical evaluation (Schmitt et al., 17 Mar 2026).

3. Structural properties

Tumula information and its PRLI extension satisfy a broad list of information-theoretic properties. For $\rho_{AB}$2 and local channels $\rho_{AB}$3, $\rho_{AB}$4,

$\rho_{AB}$5

At $\rho_{AB}$6, this yields the local data-processing inequality

$\rho_{AB}$7

Additivity also holds. For $\rho_{AB}$8,

$\rho_{AB}$9

and at $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$0,

$$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$1

Thus the measure behaves extensively on tensor-product states (Schmitt et al., 17 Mar 2026).

The Rényi dependence is regular in several senses. The map $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$2 is monotone nondecreasing and continuous on $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$3. On $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$4, $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$5 is convex. For $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$6, $$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$7 is continuously differentiable, and its derivative can be written in terms of doubly minimized PRMI:

$$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$8

Several special values are explicit:

$$I(A\!:\!B)_\rho = D(\rho_{AB}\,\|\,\rho_A\otimes \rho_B).$$9

For $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$0, the minimizing pair $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$1 is unique, and its supports satisfy

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$2

The behavior on special classes of states is sharply characterized. If $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$3 is pure, then $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$4 when the state is product, and $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$5 otherwise. If $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$6 is diagonal in product bases, corresponding to a classical-classical state with joint distribution $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$7, then for all $$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$8,

$$L(A\!:\!B)_\rho := D(\rho_A\otimes \rho_B\,\|\,\rho_{AB}),$$9

hence

$$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$0

In the classical setting,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$1

and this upper bound is tight (Schmitt et al., 17 Mar 2026).

An additional asymptotic characterization replaces the double minimization by universal permutation-invariant states $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$2 and $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$3. For $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$4,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$5

and at $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$6,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$7

4. Operational meaning in asymmetric hypothesis testing

The principal operational interpretation of tumula information is formulated in composite asymmetric binary quantum state discrimination. For a POVM $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$8 at blocklength $$T(A\!:\!B)_\rho := \inf_{\sigma_A,\tau_B} D(\sigma_A\otimes \tau_B\,\|\,\rho_{AB}).$$9, the worst-case type-I and type-II errors are

$P_{XY}$0

The reverse direct exponent quantifies the optimal decay of type-II error under a type-I decay constraint of the form $P_{XY}$1, while the Sanov exponent quantifies the optimal decay of type-I error under a fixed type-II constraint $P_{XY}$2 (Schmitt et al., 17 Mar 2026).

For the singly minimized setting, with

$P_{XY}$3

and alternatives of the form

$P_{XY}$4

the reverse direct exponent is

$P_{XY}$5

For the doubly minimized setting, with

$P_{XY}$6

and either

$P_{XY}$7

or

$P_{XY}$8

the reverse direct exponent is governed by the doubly minimized PRLI:

$P_{XY}$9

for

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$00

where

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$01

The same expression also holds when the supremum is taken over $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$02 (Schmitt et al., 17 Mar 2026).

The Sanov interpretation is more direct. Under the same doubly minimized product alternatives,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$03

hence

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$04

This identifies tumula information exactly with the optimal large-deviation rate for mistaking the true correlated source for any product source in the corresponding reverse-testing problem.

The zero-rate limits place mutual, lautum, umlaut, and tumula information in a single asymptotic scheme. In particular, when the alternatives are doubly optimized over product states and a technical threshold $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$05 vanishes,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$06

while in the reverse setting with the same alternative classes,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$07

5. Extension from states to channels

The paper extends tumula information from states to quantum channels. For a channel $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$08,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$09

where $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$10 and the supremum is over pure states $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$11. In Choi form,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$12

with $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$13 the unnormalized Choi operator (Schmitt et al., 17 Mar 2026).

This channel quantity is super-additive:

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$14

Consequently the regularized quantity

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$15

exists by Fekete’s lemma and equals $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$16.

For classical-to-quantum channels $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$17 with orthonormal basis $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$18, the paper gives two explicit variational forms. Defining

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$19

one has

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$20

and equivalently

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$21

The corresponding channel umlaut information is

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$22

By setting $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$23 one obtains

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$24

and the inequality is strict in general by Jensen (Schmitt et al., 17 Mar 2026).

For classical channels $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$25,

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$26

and the explicit forms are

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$27

as well as

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$28

Super-additivity persists:

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$29

A notable special case is the identity channel $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$30 on a classical alphabet $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$31:

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$32

Hence any classical channel $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$33 obeys

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$34

At the same time, the zero-rate unassisted error exponent $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$35 is infinite. The paper therefore concludes that tumula information, even after regularization, cannot serve as an assisted reliability exponent in the classical setting. For channel-level comparison, the work explicitly relates its findings to previous results on channel umlaut information (Girardi et al., 27 Mar 2025, Schmitt et al., 17 Mar 2026).

6. Examples, computation, and broader context

Several examples delimit the range of possible behavior. If $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$36, then

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$37

If $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$38 is pure and entangled, then $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$39; in particular, the Bell state has infinite tumula information. For classical-classical states, the problem reduces exactly to the classical optimization over product distributions, and the classical upper bound

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$40

is tight (Schmitt et al., 17 Mar 2026).

Closed-form formulas beyond product states, pure states, and classical-classical states are not given in the paper. For $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$41, numerical evaluation can proceed by alternating minimization using the Sibson-based partial minimizers

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$42

and for $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$43 by fixed-point iterations based on

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$44

For $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$45, uniqueness of the minimizer pair makes this optimization well posed.

Conceptually, tumula information occupies a specific position within the landscape of correlation measures. Mutual information governs forward-testing exponents, whereas tumula information governs reverse-testing exponents against product alternatives. At $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$46 the hierarchy

$$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$47

measures how the exponent changes as the product benchmark becomes less constrained. This suggests that tumula information is the natural reversed quantity when both factors of the null product structure are composite, rather than fixed by the marginals.

The broader information-theoretic context is complementary rather than overlapping. Recent work has shown that mutual information can be upper-bounded by Fisher information, both classically and quantumly, yielding bounds on Bayesian quadratic cost and on Holevo information (Górecki et al., 2024). Related work shows that, under a sub-Gaussian score assumption, processed Fisher information is bounded linearly by mutual information or by channel capacity, with applications to distributed estimation and strong data-processing inequalities (Barnes et al., 2021). Those results concern mutual information and Fisher-information control rather than tumula information directly, but they clarify the distinct operational niche occupied by the reversed measures.

The main open questions identified in the tumula-information paper concern channel interpretations in genuinely quantum settings and structural characterization beyond the explicitly solved cases. In the classical channel setting the paper gives a negative answer to an analogue of the meta-converse interpretation. In the quantum channel setting, by contrast, $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$48 can be infinite, for example for noiseless quantum channels, and whether $$T(A\!:\!B)_\rho := \inf_{\sigma_A\in S(A),\,\tau_B\in S(B)} D(\sigma_A\otimes \tau_B \,\|\, \rho_{AB}),$$49 or its PRLI variants characterize meaningful exponents or capacities in quantum-assisted scenarios remains open (Schmitt et al., 17 Mar 2026).