We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.
We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.
We introduce a new approach to evaluating entangled quantum networks using information geometry. Quantum computing is powerful because of the enhanced correlations from quantum entanglement. For example, larger entangled networks can enhance quantum key distribution (QKD). Each network we examine is an n-photon quantum state with a degree of entanglement. We analyze such a state within the space of measured data from repeated experiments made by n observers over a set of identically-prepared quantum states – a quantum state interrogation in the space of measurements. Each observer records a 1 if their detector triggers, otherwise they record a 0. This generates a string of 1’s and 0’s at each detector, and each observer can define a binary random variable from this sequence. We use a well-known information geometry-based measure of distance that applies to these binary strings of measurement outcomes,1–3 and we introduce a generalization of this length to area, volume and higher-dimensional volumes.4 These geometric equations are defined using the familiar Shannon expression for joint and mutual entropy.5 We apply our approach to three distinct tripartite quantum states: the |GHZi state, the |Wi state, and a separable state |Pi. We generalize a well-known information geometry analysis of a bipartite state to a tripartite state. This approach provides a novel way to characterize quantum states, and it may have favorable scaling with increased number of photons.
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.
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.
We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher's singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher-dimensional volumes for multipartite quantum systems. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enables us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well-motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach, and in particular to a random sample of a thousand density matrices.
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.
It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localization of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides an interesting geometric characterization of the latter quantity. Applications include unified volume-based derivations of results in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a means for defining classical and quantum localization for arbitrary evolution processes, and a proposed definition for the spot size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.