Information and communication in polygon theories

Generalized probabilistic theories (GPT) provide a framework in which one can formulate physical theories that includes classical and quantum theories, but also many other alternative theories. In order to compare different GPTs, we advocate an approach in which one views a state in a GPT as a resource, and quantifies the cost of interconverting between different such resources. We illustrate this approach on polygon theories (Janotta et al. New J. Phys 13, 063024, 2011) that interpolate (as the number n of edges of the polygon increases) between a classical trit (when n=3) and a real quantum bit (when n=infinity). Our main results are that simulating the transmission of a single n-gon state requires more than one qubit, or more than log(log(n)) bits, and that n-gon states with n odd cannot be simulated by n'-gon states with n' even (for all n,n'). These results are obtained by showing that the classical capacity of a single n-gon state with n even is 1 bit, whereas it is larger than 1 bit when n is odd; by showing that transmitting a single n-gon state with n even violates information causality; and by showing studying the communication complexity cost of the nondeterministic not equal function using n-gon states.Comment: 18 page

Entangling capacity of operators

Given a unitary operator $U$ acting on a composite quantum system what is the entangling capacity of $U$? This question is investigated using a geometric approach. The entangling capacity, defined via metrics on the unitary groups, leads to a \emph{minimax} problem. The dual, a \emph{maximin} problem, is investigated in parallel and yields some familiar entanglement measures. A class of entangling operators, called generalized control operators is defined. The entangling capacities and other properties for this class of operators is studied.Comment: 18 page

Decoherence-free quantum information in the presence of dynamical evolution

We analyze decoherence-free (DF) quantum information in the presence of an arbitrary non-nearest-neighbor bath-induced system Hamiltonian using a Markovian master equation. We show that the most appropriate encoding for N qubits is probably contained within the ~(2/9) N excitation subspace. We give a timescale over which one would expect to apply other methods to correct for the system Hamiltonian. In order to remain applicable to experiment, we then focus on small systems, and present examples of DF quantum information for three and four qubits. We give an encoding for four qubits that, while quantum information remains in the two-excitation subspace, protects against an arbitrary bath-induced system Hamiltonian. Although our results are general to any system of qubits that satisfies our assumptions, throughout the paper we use dipole-coupled qubits as an example physical system.Comment: 8 pages, 4 figure

Consistent assignment of quantum probabilities

We pose and solve a problem concerning consistent assignment of quantum probabilities to a set of bases associated with maximal projective measurements. We show that our solution is optimal. We also consider some consequences of the main theorem in the paper in conjunction with Gleason's theorem. Some potential applications to state tomography and probabilistic quantum secret-sharing scheme are discussed.Comment: 19 page