We denote by Mk the set of k by k matrices with complex entries. We consider quantum channels φL of the form: given a measurable function L: Mk → Mk and a measure µ on Mk we define the linear operator φL : Mk → Mk, by the law ρ → φL(ρ) = ∫ Mk L(v)ρL(v)† dµ(v).In a previous work, the authors show that for a fixed measure µ the Φ-Erg property is generic on the function L (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic processes: one takes values on the projective space P (ℂk) and the other on matrices in Mk. The Φ-Erg property and the purification condition are the nice hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on L, if ∫ |L(v)|2 log |L(v)| dµ(v) < ∞, the Lyapunov exponents ∞ > γ1 ≥ γ2 ≥ … ≥ γk ≥ −∞ are well defined.In a previous work, the concepts of entropy of a channel and Gibbs channel were presented; and also an example (associated to a stationary Markov chain) in which this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the mentioned example and we show that it is equal to −1/2 h, where h is the entropy of the associated Markov invariant probability.
We denote by Mk the set of k by k matrices with complex entries. We consider quantum channels φL of the form: given a measurable function L: Mk → Mk and a measure µ on Mk we define the linear operator φL : Mk → Mk, by the law ρ → φL(ρ) = ∫ Mk L(v)ρL(v)† dµ(v).In a previous work, the authors show that for a fixed measure µ the Φ-Erg property is generic on the function L (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic processes: one takes values on the projective space P (ℂk) and the other on matrices in Mk. The Φ-Erg property and the purification condition are the nice hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on L, if ∫ |L(v)|2 log |L(v)| dµ(v) < ∞, the Lyapunov exponents ∞ > γ1 ≥ γ2 ≥ … ≥ γk ≥ −∞ are well defined.In a previous work, the concepts of entropy of a channel and Gibbs channel were presented; and also an example (associated to a stationary Markov chain) in which this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the mentioned example and we show that it is equal to −1/2 h, where h is the entropy of the associated Markov invariant probability.
Denote Mk the set of complex k by k matrices. We will analyze here quantum channels ϕL of the following kind: given a measurable function L:Mk→Mk and the measure μ on Mk we define the linear operator ϕL:Mk→Mk, via the expression $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho L(v)^\dagger \, \dm(v).$
A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where L was the identity.
Under some assumptions on the quantum channel ϕL we analyze the eigenvalue property for ϕL and we also define entropy for the channel ϕL. For a fixed μ (the {\it a priori} measure) given an Hamiltonian H:Mk→Mk (which defines ϕH) we present a variational principle of pressure and relate it to the eigenvalue problem. We introduce the concept of Gibbs channel.
We describe a related process Xn, n∈N, taking values on the projective space $ P(\C^k)$ and analyze the question of existence of invariant probabilities.
We also consider an associated process ρn, n∈N, with values on Dk (Dk is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator fixed for ϕL.
The von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the “relative entropy distance” between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the “relative entropy distance” between N and the completely depolarizing channel. We establish that this definition satisfies all of the axioms, recently put forward in [Dour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
A short quantum Markov chain is a tripartite state such that system A can be recovered perfectly by acting on system C of the reduced state Such states have conditional mutual information equal to zero and are the only states with this property. A quantum channel is sufficient for two states ρ and σ if there exists a recovery channel using which one can perfectly recover ρ from and σ from The relative entropy difference is equal to zero if and only if is sufficient for ρ and σ . In this paper, we show that these properties extend to Rényi generalizations of these information measures which were proposed in (Berta et al 2015 J. Math. Phys. 56 022205 ; Seshadreesan et al 2015 J. Phys. A: Math. Theor. 48 395303 ), thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Rényi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.
In this exposition we prove an existence theorem for a quantum mechanical dynamical entropy based on von-Neumann’s measurement theory. To that end we introduced a Shannon type of information associated with a quantum channel or measurement based on Araki’s relative entropy. This is an invariance for the dynamics which generalizes Kolmogorov-Sinai ’s notion of dynamical entropy of a measure preserving transformation. In this context we also introduce a natural notion of channel capacity as a generalization of Shannon’s capacity of a channel and finds its relation with A.S. Holevo’s notion of capacity. 2 A measurement in classical dynamics gives rise to a measurable partition ζ = (ζi) of the configuration space ( measure space) (M, B, µ). For a par-µ(E∩ζi) tition ζ, we can associate a family of measure µζi(E) =
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).