We introduce a formal notion of observer aperture, representing the measurement interfaceavailable to an observer interacting with a quantum system. Given a restricted family of POVMsM, we define the aperture-restricted distinguishability between two quantum states as thesupremum of classical relative entropy between the induced outcome distributions. This quantitycan be viewed as a measurement-restricted variant of measured quantum relative entropy. Weprove monotonicity under aperture expansion and establish a data-processing inequality underquantum channels via Heisenberg pullback of measurement families. We further show thatthe quantity is subadditive across independent systems. The resulting framework provides anoperational interpretation of observer bandwidth connecting measurement constraints, coarse-graining, and accessible information within standard quantum information theory.
We introduce a formal notion of observer aperture, representing the measurement interfaceavailable to an observer interacting with a quantum system. Given a restricted family of POVMsM, we define the aperture-restricted distinguishability between two quantum states as thesupremum of classical relative entropy between the induced outcome distributions. This quantitycan be viewed as a measurement-restricted variant of measured quantum relative entropy. Weprove monotonicity under aperture expansion and establish a data-processing inequality underquantum channels via Heisenberg pullback of measurement families. We further show thatthe quantity is subadditive across independent systems. The resulting framework provides anoperational interpretation of observer bandwidth connecting measurement constraints, coarse-graining, and accessible information within standard quantum information theory.
Quantum hypothesis testing studies the distinguishability of quantum states using measurements.Standard formulations assume access to arbitrary POVMs. In realistic experimentalsettings, however, observers are restricted to specific measurement interfaces. In this work westudy quantum state discrimination under such measurement constraints.Given a measurement family M representing an observer aperture, we analyze the optimaldiscrimination of two quantum states using only measurements from M. We show that theasymptotic error exponent is governed by the aperture-restricted relative entropy introduced inour previous work. This provides an operational interpretation of observer aperture as a limiton achievable discrimination rates.
Abstract Observational entropy provides a general notion of quantum entropy that appropriately interpolates between Boltzmann’s and Gibbs’ entropies, and has recently been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. Here we study the mathematical properties of observational entropy from an information-theoretic viewpoint, making use of recently strengthened forms of the monotonicity property of quantum relative entropy. We present new bounds on observational entropy applying in general, as well as bounds and identities related to sequential and post-processed measurements. A central role in this work is played by what we call the ‘coarse-grained’ state, which emerges from the measurement’s statistics by Bayesian retrodiction, without presuming any knowledge about the ‘true’ underlying state being measured. The degree of distinguishability between such a coarse-grained state and the true (but generally unobservable) one is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies.
Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including Rényi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.
We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, ⟨O1⟩ and ⟨O2⟩, but may not have access to ⟨O1O2⟩. This problem is relevant for the study of localized quantum information in gravity since the set of approximately local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a ``little Hilbert space,'' which parametrizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to ``coarse-grained'' and ``fine-grained'' subregion dualities in AdS/CFT and provide a few sample calculations to illustrate our formalism.
This work presents a conceptual and mathematical framework addressing the longstanding tension between microscopic reversibility in quantum mechanics and macroscopic irreversibility in thermodynamics. Rather than modifying unitary quantum dynamics or invoking ad hoc coarse-graining procedures, the paper proposes that the thermodynamic arrow of time arises as an information-theoretic consequence of the physical limitations of observers. The central hypothesis, termed the Holographic Shannon–Landauer (HSL) framework, models the observer as a finite-bandwidth quantum channel (a Completely Positive Trace-Preserving map). Within this framework, entropy is redefined not as an intrinsic property of matter, but as the quantum relative entropy between the true global quantum state and the state accessible to a bandwidth-limited observer. By formalizing observation as a channel-capacity–constrained physical process, the work derives a Bandwidth–Entropy Monotonicity Theorem, demonstrating that reduced observational resolution necessarily leads to increased observed entropy via the data processing inequality. The Second Law of Thermodynamics emerges naturally from the dynamical spreading of entanglement into degrees of freedom inaccessible to the observer, rather than from any fundamental time asymmetry in the underlying laws. This manuscript is intended as a foundational and unifying perspective, connecting quantum thermodynamics, information theory, and observer-dependent physics. It is archived here as a citable preprint to establish priority and provide a stable reference point for subsequent technical developments and potential journal submissions.
Traditional measures of entropy, like the Von Neumann entropy, while fundamental in quantum information theory, are insufficient when interpreted as thermodynamic entropy due to their invariance under unitary transformations, which contradicts observed entropy increases in isolated systems. Recognizing this limitations of existing measures for thermodynamic entropy, recent research has focused on observational entropy (OE) as a promising alternative, offering practical applicability and theoretical insights. In this work, we extend the scope of observational entropy by generalizing it to a parameterized version called α-Observational entropy (α-OE). α-OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The α-OE reduces to OE under α→1. We prove various properties of the α-OE, which are the generalization of the properties of OE, including the monotonically increasing of α-OE as a function of refinement of coarse-graining. We further explore the role of α-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.