Optimal measurement preserving qubit channels

We consider the problem of discriminating qubit states that are sent over a quantum channel and derive a necessary and sufficient condition for an optimal measurement to be preserved by the channel. We apply the result to the characterization of optimal measurement preserving (OMP) channels for a given qubit ensemble, e.g., a set of two states or a set of multiple qubit states with equal a priori probabilities. Conversely, we also characterize qubit ensembles for which a given channel is OMP, such as unitary and depolarization channels. Finally, we show how the sets of OMP channels for a given ensemble can be constructed.Comment: 12 pages, 2 figures. Corrected typo

Causal Asymmetry of Classical and Quantum Autonomous Agents

Why is it that a ticking clock typically becomes less accurate when subject to outside noise but rarely the reverse? Here, we formalize this phenomenon by introducing process causal asymmetry - a fundamental difference in the amount of past information an autonomous agent must track to transform one stochastic process to another over an agent that transforms in the opposite direction. We then illustrate that this asymmetry can paradoxically be reversed when agents possess a quantum memory. Thus, the spontaneous direction in which processes get 'simpler' may be different, depending on whether quantum information processing is allowed or not.Comment: 12 pages, 4 figure

Geometry of Uncertainty Relations for Linear Combinations of Position and Momentum

For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by N linear combinations of position and momentum observables. The bounds depend on their degree of incompatibility defined by the area of a parallelogram in an N-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle

Universality in Uncertainty Relations for a Quantum Particle

A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle's variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal: no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schrödinger. Our approach provides a unified perspective on existing uncertainty relations for a single continuous variable, and it leads to new inequalities for second moments which can be checked experimentally