Tumula information and doubly minimized Petz Renyi lautum information

Abstract

We study a doubly minimized variant of the lautum information - a reversed analogue of the mutual information - defined as the minimum relative entropy between any product state and a fixed bipartite quantum state; we refer to this measure as the tumula information. In addition, we introduce the corresponding Petz Renyi version, which we call the doubly minimized Petz Renyi lautum information (PRLI). We derive several general properties of these correlation measures and provide an operational interpretation in the context of hypothesis testing. Specifically, we show that the reverse direct exponent of certain binary quantum state discrimination problems is quantified by the doubly minimized PRLI of order $α\in (0,1/2)$, and that the Sanov exponent is determined by the tumula information. Furthermore, we investigate the extension of the tumula information to channels and compare its properties with previous results on the channel umlaut information [Girardi et al., arXiv:2503.21479].

Figures (2)

• Figure 1: A pictorial representation of the channel tumula information $T(\pazocal{N})$, compared with the channel umlaut information $U(\pazocal{N})$, its regularization $U^\infty(\pazocal{N})$, and two relevant error exponents: the unassisted, zero-rate reliability function $E^{\emptyset}(0^+,\pazocal{N})$, and the activated, non-signalling assisted reliability function $E^{\rm NS,a}(0^+,\pazocal{N})$. For the definitions of these exponents, see Filippo25b.
• Figure 2: Regularized lautum information (yellow), umlaut information and lautum information (blue), tumula information (red), and zero-rate unassisted error exponent (green) of the binary symmetric channel $\pazocal{W}$ in terms of the crossover probability $\varepsilon$. Except for the tumula information, these quantities have been previously considered in Filippo25Lautum_08. The bifurcation point between the umlaut and tumula information lies at $\varepsilon = \frac{1}{1+e^2}\approx 0.1192$. For larger crossover probabilities, the two quantities coincide. For smaller crossover probabilities, a phase transition happens and the optimizer $q^\ast$ is no longer unique. In fact, in the limit of vanishing error, it approaches a deterministic distribution and one finds $\lim_{\varepsilon \to 0} T (\pazocal W) = \log 2\approx 0.69$.

Theorems & Definitions (64)

• Remark 1: (Classical setting)
• Theorem 2: (Properties of doubly minimized PRLI)
• proof
• Theorem 3: (Properties of the tumula information)
• proof
• Proposition 4: (Upper bound on classical tumula information)
• proof
• Theorem 5: (Reverse direct exponent, singly minimized PRLI)
• proof
• Theorem 6: (Reverse direct exponent, doubly minimized PRLI)