We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(ρ)-f(σ)$ for a quantum state $ρ$ with finite $f(ρ)$ and any quantum state $σ$ $m$-partially majorized by the state $ρ$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$ and find simple sufficient conditions for vanishing this bound with $\,\min\{\varepsilon,1/m\}\to0\,$.
The obtained results are applied to the von Neumann entropy. The concept of $\varepsilon$-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator.
We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.
We construct for a Schur concave function f on the set of quantum states a tight upper bound on the difference f(ρ)−f(σ) for a quantum state ρ with finite f(ρ) and any quantum state σm-partially majorized by the state ρ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition 21∥ρ−σ∥1≤ε and find simple sufficient conditions for vanishing this bound with min{ε,1/m}→0.
The obtained results are applied to the von Neumann entropy. The concept of ε-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator.
We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.
We consider families of tight upper bounds on the difference S(ρ)−S(σ) with a rank/energy constraint imposed on the state ρ, which are valid provided that the state ρ partially majorizes the state σ and is close to the state σ with respect to the trace norm. The upper bounds within these families depend on the parameter m of partial majorization. The upper bounds corresponding to m=0 coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with a rank/energy constraint obtained in [1] and [2]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of ε-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and Shannon entropy are also considered. Bibliography: 20 titles.
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over ε-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance ε, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the α-Rényi entropy, for which we obtain novel bounds in the case α>1. We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.
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.
Entropy plays a fundamental role in several branches of physics. In the quantum setting, one usually considers the von Neumann entropy, but other useful quantities have been proposed in the literature; e.g., the Rényi and the Tsallis entropies. The evolution of an open quantum system, described by a semigroup of dynamical maps (in short, a dynamical semigroup), may decrease a quantum entropy, for some initial condition. We will discuss various characterizations of those dynamical semigroups that, for every initial condition, do not decrease a general class of quantum entropies, which is defined using the notion of Schur concavity of a function. We will not assume that such a dynamical semigroup be completely positive, the physical justification of this condition being controversial. Therefore, we will consider semigroups of trace-preserving, positive — but not necessarily completely positive — linear maps. We will next focus on a special class of (completely positive) dynamical semigroups, the twirling semigroups , having applications in quantum information science. We will argue that the whole class of dynamical semigroups that do not decrease a quantum entropy can be obtained as a suitable generalization of the twirling semigroups.
Given an arbitrary quantum state (σ), we obtain an explicit construction of a state ρε*(σ) [respectively, ρ*,ε(σ)] which has the maximum (respectively, minimum) entropy among all states which lie in a specified neighborhood (ε-ball) of σ. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of σ. The states ρε*(σ) and ρ*,ε(σ) depend only on the geometry of the ε-ball and are in fact optimizers for a larger class of entropies. These include the Rényi entropy and the minimum- and maximum-entropies, providing explicit formulas for certain smoothed quantities. This allows us to obtain local continuity bounds for these quantities as well. In obtaining this bound, we first derive a more general result which may be of independent interest, namely, a necessary and sufficient condition under which a state maximizes a concave and Gâteaux-differentiable function in an ε-ball around a given state σ. Examples of such a function include the von Neumann entropy and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat’s rule, and majorization theory.
Entropies play a fundamental role in classical and quantum information theory. There are a plethora of quantum entropies and they satisfy a host of interesting properties. The von Neumann entropy, the first quantum entropy to be defined and arguably the most important of them, satisfies a powerful property called strong subadditivity. It was proved by Lieb and Ruskai in 1973 and has turned out to be an essential tool in establishing various important results in quantum information theory. We briefly review this property and its applications. Our main focus, however, is on another desirable property of entropies, namely, continuity. We introduce a notion of majorization flow and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for Schur concave functions with a particular emphasis on entropic functions relevant in information theory. In particular, for the case of the α-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case α>1, thus resolving an open problem and providing a substantial improvement over previously known bounds.