Metainformation in Quantum Guessing Games

TL;DR

The paper investigates metainformation, a notion of information about information, within quantum guessing games where a receiver may gain side information after a measurement. It formalizes metainformation as distinct from standard side information and analyzes its impact across minimum-error and maximum-confidence discrimination tasks, showing that metainformation can, in some cases, elevate post-measurement performance to match pre-measurement capabilities, while in others it provides no benefit. Through a qubit-based basis-encoding scheme and concrete examples (including BB84-like setups), the work derives inequalities and explicit probabilities illustrating how the timing and form of information influence optimal strategies and attainable success probabilities. The results reveal a nuanced structure in the interplay between timing, information, and quantum measurements, with implications for the design of quantum information processing protocols and a roadmap for exploring metainformation in broader contexts.

Abstract

Quantum guessing games offer a structured approach to analyzing quantum information processing, where information is encoded in quantum states and extracted through measurement. An additional aspect of this framework is the influence of partial knowledge about the input on the optimal measurement strategies. This kind of side information can significantly influence the guessing strategy and earlier work has shown that the timing of such side information, whether revealed before or after the measurement, can affect the success probabilities. In this work, we go beyond this established distinction by introducing the concept of metainformation. Metainformation is information about information, and in our context it is knowledge that additional side information of certain type will become later available, even if it is not yet provided. We show that this seemingly subtle difference between having no expectation of further information versus knowing it will arrive can have operational consequences for the guessing task. Our results demonstrate that metainformation can, in certain scenarios, enhance the achievable success probability up to the point that post-measurement side information becomes as useful as prior-measurement side information, while in others it offers no benefit. By formally distinguishing metainformation from actual side information, we uncover a finer structure in the interplay between timing, information, and strategy, offering new insights into the capabilities of quantum systems in information processing tasks.

Figures (7)

• Figure 1: Four different scenarios that differ in the ways how classical information complements the transmission of quantum information. The components that can be optimized according to the classical information are gray.
• Figure 2: Four symbols are encoded in qubit states that form two bases. In these two cases the discrimination tasks are the same, but they differ in the form of side information. The partitions that define the form of side information are marked by the color (green or blue). The measurements that are optimal for post-measurement side information with metainformation are represented by purple dashed arrows.
• Figure 3: The success probabilities of different side information scenarios in Fig. \ref{['fig:2bases']} are plotted as functions of $\theta$. In (a), the side information is the value of the basis. For this encoding, there is a hierarchy $P_{\mathrm{pre}}>P_{\mathrm{meta}}>P_{\mathrm{post}}>P$ for all $0<\theta<\pi$. In (b), the side information is the value of parity. In this case, $P_{\mathrm{pre}}=P_{\mathrm{meta}}$ for all $0<\theta<\pi$.
• Figure 4: In the basis encoding scheme information is encoded into $k$ qubit bases. These can be depicted as central axes in the Bloch sphere. A label $i\pm$ is encoded to the quantum state $\varrho_{i\pm}$. The states $\varrho_{i+}$ and $\varrho_{i-}$ form an orthonormal basis.
• Figure 5: Two examples that illustrate different inequalities in success probabilities with side information which is marked by different colors. Anticipative measurements are depicted as purple dashed arrows. In (a), encodings are equally separated bases in a plane. Since the anticipative measurement coincides with a standard measurement, $P_{\mathrm{pre}}>P_{\mathrm{meta}}=P_{\mathrm{post}}$. In (b), the standard measurement is the same as the optimal measurement with pre-measurement information, and therefore $P_{\mathrm{pre}}=P_{\mathrm{meta}}=P_{\mathrm{post}}$.

Theorems & Definitions (7)

• Proposition 1: Prop.1 in carmeli2022quantum
• Proposition 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof