The amount of information that can be accessed via measurement of a quantum system prepared in different states is limited by the Kholevo bound. We present a simple proof of this theorem and its extension to sequential measurements based on the properties of quantum conditional and mutual entropies. The proof relies on a minimal physical model of the measurement which does not assume environmental decoherence, and has an intuitive diagrammatic representation.
The amount of information that can be accessed via measurement of a quantum system prepared in different states is limited by the Kholevo bound. We present a simple proof of this theorem and its extension to sequential measurements based on the properties of quantum conditional and mutual entropies. The proof relies on a minimal physical model of the measurement which does not assume environmental decoherence, and has an intuitive diagrammatic representation.
We prove a new result limiting the amount of accessible information in a quantum channel. This generalizes Kholevo's theorem and implies it as a simple corollary. Our proof uses the strong subadditivity of the von Neumann entropy functional $S(\ensuremath{\rho})$ and a specific physical analysis of the measurement process. The result presented here has application in information obtained from ``weak'' measurements, such as those sometimes considered in quantum cryptography.
Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.
If a is a quantum effect and ρ is a state, we define the ρ-entropy Sa(ρ) which gives the amount of uncertainty that a measurement of a provides about ρ. The smaller Sa(ρ) is, the more information a measurement of a gives about ρ. In Entropy for Effects, we provide bounds on Sa(ρ) and show that if a+b is an effect, then Sa+b(ρ)≥Sa(ρ)+Sb(ρ). We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their ρ-entropies. In Entropy of Observables and Instruments, we employ Sa(ρ) to define the ρ-entropy SA(ρ) for an observable A. We show that SA(ρ) directly provides the ρ-entropy SI(ρ) for an instrument I. We establish bounds for SA(ρ) and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider ρ-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.
Information plays an important role in our understanding of the physical world. Hence we propose an entropic measure of information for any physical theory that admits systems, states and measurements. In the quantum and classical worlds, our measure reduces to the von Neumann and Shannon entropies, respectively. It can even be used in a quantum or classical setting where we are only allowed to perform a limited set of operations. In a world that admits superstrong correlations in the form of non-local boxes, our measure can be used to analyze protocols such as superstrong random access encodings and the violation of 'information causality'. However, we also show that in such a world no entropic measure can exhibit all the properties we commonly accept in a quantum setting. For example, there exists no 'reasonable' measure of conditional entropy that is subadditive. Finally, we prove a coding theorem for some theories that is analogous to the quantum and classical settings, providing us with an appealing operational interpretation.
Redundancy-Limited Emergence of Classical Histories in Quantum Systems This work investigates how classical reality emerges from underlying quantum dynamics, focusing on the role of environmental information, noise, and fundamental entropy limits. In quantum theory, systems evolve into superpositions of many possible outcomes. However, in practice we observe a world with definite, classical histories. A widely accepted explanation is decoherence, in which interactions with the environment suppress interference between different quantum states. More recent developments, particularly in quantum Darwinism, emphasize that classical information becomes objective when it is redundantly recorded across many independent parts of the environment. This paper addresses a key unresolved question: how many distinct classical histories can physically exist? To answer this, we construct a minimal computational model in which a system’s history is encoded into multiple environmental fragments, each subject to noise. By computing how much information about the system can be recovered from subsets of these fragments, we identify a characteristic scale that determines how many fragments are required to reconstruct the system’s history. Our results reveal a simple and robust relationship: the ability to recover classical information depends on the entropy of the noise affecting environmental records. As noise increases, more fragments are required to reconstruct a history, reflecting a loss of distinguishability. This leads to the definition of a “recoverability factor,” which quantifies how efficiently entropy can be converted into usable, classical information. We then connect this information-theoretic result to fundamental bounds from gravitational physics. In particular, existing entropy bounds limit the total amount of information that can pass through a region of spacetime. We show that only a fraction of this entropy—determined by the recoverability factor—can contribute to distinguishable classical histories. This leads to a new principle: the number of classical histories is not limited only by total entropy, but by the efficiency with which that entropy can be encoded, preserved, and accessed. Applying this framework to cosmological entropy estimates, we derive explicit upper bounds on the number of distinguishable classical histories in the observable universe. While this number is extraordinarily large, it is nonetheless finite and strongly constrained by both entropy and noise. These results refine the standard picture of quantum branching. While quantum theory allows a vast number of possible outcomes, only a subset can become physically meaningful in the sense of being stable, redundantly recorded, and accessible to observers. More broadly, this work provides a quantitative bridge between decoherence theory, information theory, and gravitational entropy bounds. It suggests that classical reality is fundamentally constrained not just by physical laws governing entropy, but by the efficiency of information transfer within those constraints.
The accessible information obtained in combined and sequential quantum measurements is calculated for a binary pure quantum state signal. It is shown that the accessible information obtained in the combined measurement is equal to that obtained in the sequential measurement. The result indicates that the conjecture by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] is not valid, at least for a binary quantum state signal. Furthermore, the result is compared with that obtained by Brody and Meister [Phys. Rev. Lett. 76, 1 (1996)].