It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
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 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.
The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland, and Wootters [Phys. Rev. Lett. 76, 3452 (1996)] derived a bound that reduces to the Holevo bound for complete measurements, but that is tighter for incomplete measurements. The most general quantum operations may be both incomplete and inefficient. Here we show that the bound derived by SWW can be further extended to obtain one that is yet again tighter for inefficient measurements. This allows us, in addition, to obtain a generalization of a bound derived by Hall, and to show that the average reduction in the von Neumann entropy during a quantum operation is concave in the initial state, for all quantum operations. This is a quantum version of the concavity of the mutual information. We also show that both this average entropy reduction and the mutual information for pure state ensembles, are Schur concave for unitarily covariant measurements; that is, for these measurements, information gain increases with initial uncertainty.
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.
Abstract It is shown that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence topology). Some applications of this property are considered. New continuity conditions for the quantum mutual information and for the squashed entanglement in both bipartite and multipartite infinite-dimensional systems are obtained. It is proved, in particular, that the multipartite squashed entanglement of any countably indecomposable separable state with finite marginal entropies is equal to zero. Special continuity properties of the information gain of a quantum measurement with and without quantum side information are established that can be treated as robustness (stability) of these quantities with respect to perturbation of the measurement and the measured state.
When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers' average Shannon entropy for the encoding ensemble. This reduction, being the mutual information, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers' von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by rewriting a bound derived by Hall [Phys. Rev. A 55, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. 76, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy.