Assessing dimensions from evolution

Using tools from classical signal processing, we show how to determine the dimensionality of a quantum system as well as the effective size of the environment's memory from observable dynamics in a model-independent way. We discuss the dependence on the number of conserved quantities, the relation to ergodicity and prove a converse showing that a Hilbert space of dimension D+2 is sufficient to describe every bounded sequence of measurements originating from any D-dimensional linear equations of motion. This is in sharp contrast to classical stochastic processes which are subject to more severe restrictions: a simple spectral analysis shows that the gap between the required dimensionality of a quantum and a classical description of an observed evolution can be arbitrary large.Comment: 5 page

Wiretap and Gelfand-Pinsker Channels Analogy and its Applications

An analogy framework between wiretap channels (WTCs) and state-dependent point-to-point channels with non-causal encoder channel state information (referred to as Gelfand-Pinker channels (GPCs)) is proposed. A good sequence of stealth-wiretap codes is shown to induce a good sequence of codes for a corresponding GPC. Consequently, the framework enables exploiting existing results for GPCs to produce converse proofs for their wiretap analogs. The analogy readily extends to multiuser broadcasting scenarios, encompassing broadcast channels (BCs) with deterministic components, degradation ordering between users, and BCs with cooperative receivers. Given a wiretap BC (WTBC) with two receivers and one eavesdropper, an analogous Gelfand-Pinsker BC (GPBC) is constructed by converting the eavesdropper's observation sequence into a state sequence with an appropriate product distribution (induced by the stealth-wiretap code for the WTBC), and non-causally revealing the states to the encoder. The transition matrix of the state-dependent GPBC is extracted from WTBC's transition law, with the eavesdropper's output playing the role of the channel state. Past capacity results for the semi-deterministic (SD) GPBC and the physically-degraded (PD) GPBC with an informed receiver are leveraged to furnish analogy-based converse proofs for the analogous WTBC setups. This characterizes the secrecy-capacity regions of the SD-WTBC and the PD-WTBC, in which the stronger receiver also observes the eavesdropper's channel output. These derivations exemplify how the wiretap-GP analogy enables translating results on one problem into advances in the study of the other

Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. III: potential function in local stochastic dynamics and in steady state of Boltzmann-Gibbs type distribution function

From a logic point of view this is the third in the series to solve the problem of absence of detailed balance. This paper will be denoted as SDS III. The existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial. Following an earlier explicit construction by one of us (Ao, J. Phys. {\bf A37}, L25 '04, arXiv:0803.4356, referred to as SDS II), in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences. The local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steady-state meaning via a novel and explicit form of Fokker-Planck equation, the zero mass limit. We also give a procedure to obtain the special stochastic differential equation for any given Fokker-Planck equation. No detailed balance condition is required in our demonstration. For the first time we obtain here a formula to describe the noise induced shift in drift force comparing to the steady state distribution, a phenomenon extensively observed in numerical studies. The comparison to two well known stochastic integration methods, Ito and Stratonovich, are made ready. Such comparison was made elsewhere (Ao, Phys. Life Rev. {\bf 2} (2005) 117. q-bio/0605020).Comment: latex. 13 page

The Blackwell relation defines no lattice

Blackwell's theorem shows the equivalence of two preorders on the set of information channels. Here, we restate, and slightly generalize, his result in terms of random variables. Furthermore, we prove that the corresponding partial order is not a lattice; that is, least upper bounds and greatest lower bounds do not exist.Comment: 5 pages, 1 figur