While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give
several natural definitions of a universal information metric, based on length of shortest programs for either ordinary computations or reversible (dissipationless) computations. It turns out that these definitions are equivalent up to an additive logarithmic term. We show that the information distance is a universal cognitive similarity distance. We investigate the maximal correlation of the shortest programs involved, the maximal uncorrelation of programs (a generalization of the Slepian-Wolf theorem of classical information theory), and the density properties of the discrete metric spaces induced by the information distances. A related distance measures the amount of nonreversibility of a computation. Using the physical theory of reversible computation, we give an appropriate (universal, anti-symmetric, and transitive) measure of the thermodynamic work required to transform one object in another object by the most efficient process. Information distance between individual objects is needed in pattern recognition where one wants to express effective notions of pattern similarity or cognitive similarity between individual objects and in thermodynamics of computation where one wants to analyse the energy dissipation of a computation from a particular input to a particular output.
Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one
noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical IID sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e. simulations in which the sender retains what would escape into the environment in an ordinary simulation), on non-tensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.
