The nature of quantum entropies, and its use in Quantum Information Theory in the form of (i) total entropy, (ii) relative entropy and (iii) conditional entropy is revisited. In this ordering, we first show the correlations existing between the total q-entropy and entanglement, quantified in the form of entanglement of formation. Then, we revisit the use of the quantum relative entropy as a measure of entanglement, and we finally discuss some features of the quantum conditional q-entropies, which are used in turn as a separability criterion.
The nature of quantum entropies, and its use in Quantum Information Theory in the form of (i) total entropy, (ii) relative entropy and (iii) conditional entropy is revisited. In this ordering, we first show the correlations existing between the total q-entropy and entanglement, quantified in the form of entanglement of formation. Then, we revisit the use of the quantum relative entropy as a measure of entanglement, and we finally discuss some features of the quantum conditional q-entropies, which are used in turn as a separability criterion.
On the premises that total correlations in a bipartite quantum state are measured by the quantum mutual information, and that separation of total correlations into quantum and classical parts satisfies an intuitive dominance relation, we examine to what extent various entropic entanglement measures, such as the distillable entanglement, the relative entropy entanglement, the squashed entanglement, the entanglement cost, and the entanglement of formation, can be regarded as consistent measures of quantum correlations. We illustrate that the entanglement of formation often overestimates quantum correlations and thus is too big to be a genuine measure of quantum correlations. This indicates that the entanglement of formation does not quantify the quantum correlations intrinsic to a quantum state, but rather characterizes the pure entanglement needed to build the quantum state via local operations and classical communication. Furthermore, it has the consequence that, if the additive conjecture for the entanglement of formation is true (as is widely believed), then the entanglement cost, which is an operationally defined measure of entanglement with significant physical meaning, cannot be a consistent measure of quantum correlations in the sense that it may exceed total correlations. Alternatively, if the entanglement cost is dominated by total correlations, as our intuition suggests, then we can immediately disprove the additive conjecture. Both scenarios have their counterintuitive and appealing aspects, and a natural challenge arising in this context is to prove or disprove that the entanglement cost is dominated by the quantum mutual information.
Regarding the strange properties of quantum entropy and entanglement, e.g., the negative quantum conditional entropy, we revisited the foundations of quantum entropy, namely, von Neumann entropy, and raised the new method of quantum fine-grained entropy. With the applications in entanglement theory, quantum information processing, and quantum thermodynamics, we demonstrated the capability of quantum fine-grained entropy to resolve some notable confusions and problems, including the measure of entanglement and quantumness, the additivity conjecture of entanglement of formation etc, and the definition of temperature for single quantum system.
This paper introduces a novel approach to quantifying quantum entanglement using entanglement entropy. Traditional entanglement measures, such as concurrence and entanglement of formation, often struggle to fully capture the intricate correlations present in entangled states. We propose a new definition of entanglement entropy based on the density matrix of the entangled system, explicitly accounting for the correlations between subsystems. This approach leverages information-theoretic entropy calculations to provide a more precise and comprehensive measure of entanglement strength and the degree of correlation. We demonstrate that this entropy-based metric offers advantages over existing measures, particularly in characterizing complex entangled states and distinguishing between different levels of entanglement. The core claim is that entanglement entropy can serve as a more accurate indicator of entanglement than existing metrics. The underlying mechanism centers around a density matrix-based entropy calculation, recognizing the interconnectedness within entangled states. The novelty lies in moving beyond purely state-based descriptions to incorporate a measure of subsystem correlations.
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
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.