A possible mechanism of time is formulated by developing an idea of time replaced by quantum correlations, with the aid of modern quantum information theory. We invent a microscopic model, where correlations of a closed system are steadily read out as internal, quantum clocks that define time via their relative phases. The model could realize emergent time evolutions which exhibit unitarity of quantum theory, while its underlying process is driven entropically. The key quantity turns out to be the amount of accessible information about the clocks recording past events. By postulating the so-called data-processing inequality (or strong subadditivity of entropy) as a fundamental, physical limitation about how information decays, we propose that conditional entropy about this past information should be constrained to be a positive constant. The proposal implies a holographic property of this conditional entropy in an analogous manner with the area law of entanglement entropy.
A possible mechanism of time is formulated by developing an idea of time replaced by quantum correlations, with the aid of modern quantum information theory. We invent a microscopic model, where correlations of a closed system are steadily read out as internal, quantum clocks that define time via their relative phases. The model could realize emergent time evolutions which exhibit unitarity of quantum theory, while its underlying process is driven entropically. The key quantity turns out to be the amount of accessible information about the clocks recording past events. By postulating the so-called data-processing inequality (or strong subadditivity of entropy) as a fundamental, physical limitation about how information decays, we propose that conditional entropy about this past information should be constrained to be a positive constant. The proposal implies a holographic property of this conditional entropy in an analogous manner with the area law of entanglement entropy.
Even though our microscopic physical theories, such as classical or quantum mechanics, are time-reversal invariant, time appears to march forward irreversibly. It is widely believed that effective entropy increase due to the second law of thermodynamics is at the origin of this emergent arrow of time. Any system that is out of thermal equilibrium and produces entropy can thus be used as a clock, be it a melting ice cube or a sophisticated atomic clock. The understanding of entropy in physics has significantly advanced with the advent of information theory in the last century. It has also led to the insight that the amount of entropy a clock produces is linked to how precisely the clock works, and thereby, how much information the clock reveals about the passage of time. In this thesis, we ask: what fundamental limits does the underlying physical theory impose on the precision of clocks within that theory? To rigorously answer this question, we devise a minimal clock model within quantum theory. By consistently modeling all clock constituents as quantum systems, we can ensure we account for all the resources used by the clock. This includes, in particular, the readout. Classically, one can obtain this for free, but quantum mechanically, this has a resource cost associated with it. To this end, we consolidate existing frameworks that describe clocks in both classical and quantum theory. Revising fluctuation theorems from quantum thermodynamics, we demonstrate how such theorems impose limits on the attainable precision of clocks. These precision bounds turn out to be more stringent for classical systems than for quantum ones. Quantum clocks can harness coherence and correlations to suppress fluctuations, allowing them to exponentially surpass precision bounds that hold for classical clocks. We show this separation between classical and quantum clocks by constructing an explicit example that uses quantum dynamics to saturate the quantum precision bounds. Finally, we close the thesis by exploring implications of these timekeeping bounds beyond fundamental science in quantum technologies. We show how quantum clocks can be used to quantify the energetic cost of quantum computation by linking computational fidelity to thermodynamic entropy production.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
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.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
We argue that (1) our perception of time through change and (2) the gap between reality and our observation of it are at the heart of both quantum mechanics and the dynamical mechanism of physical systems. We suggest that the origin of quantum uncertainty lies with the absence of infinities or infinitesimals in observational data and that our concept of time derives from observing changing data (events). We argue that the fundamentally important content of the Superposition Principle is not the "probability amplitude" of posterior state observation but future state availability conditional only on prior information. Since event detection also implies posterior conditions (e.g. a specific type of detectable event occurred) as well as prior conditions, the probabilities of detected outcomes are also conditional on properties of the posterior properties of the observation. Such posterior conditions cannot affect the prior state availabilities and this implies violation of counter-factual definiteness. A component of a quantum system may be chosen to represent a clock and changes in other components can then be expected to be correlated with clocks with which they are entangled. Instead of traditional time-dependent equations of motion we provide a specific mechanism whereby evolution of data is instead quasi-causally related to the relative \availability\ of states and equations of motion are expressed in terms of quantized clock variables. We also suggest that time-reversal symmetry-breaking in weak interactions is an artifice of a conventional choice of co-ordinate time-function. Analysis of a "free" particle suggests that conventional co-ordinate space-time emerges from how we measure the separation of objects and events.
What would be a reasonable definition of the conditional entropy of bipartite quantum processes, and what novel insight would it provide? We develop this notion using four information-theoretic axioms and define the corresponding quantitative formulas. Our definitions of the conditional entropies of channels are based on the generalized state and channel divergences, for instance, quantum relative entropy. We find that the conditional entropy of quantum channels has potential to reveal insights for quantum processes that aren't already captured by the existing entropic functions, entropy or conditional entropy, of the states and channels. The von Neumann conditional entropy S[A∣B]N of the channel NA′B′→AB is based on the quantum relative entropy, with system pairs A′,A and B′,B being nonconditioning and conditioning systems, respectively. We identify a connection between the underlying causal structure of a bipartite channel and its conditional entropy. In particular, we provide a necessary and sufficient condition for a bipartite quantum channel NA′B′→AB in terms of its von Neumann conditional entropy S[A∣B]N, to have no causal influence from A′ to B. As a consequence, if S[A∣B]N<−log∣A∣ then the channel necessarily has causal influence (signaling) from A′ to B. Our definition of the conditional entropy establishes the strong subadditivity of the entropy for quantum channels. We also study the total amount of correlations possible due to quantum processes by defining the multipartite mutual information of quantum channels.