Quantum κ-Entropy: A Quantum Computational Approach

A novel approach to the quantum version of κ-entropy that incorporates it into the conceptual, mathematical and operational framework of quantum computation is put forward. Various alternative expressions stemming from its definition emphasizing computational and algorithmic aspects are worked out: First, for the case of canonical Gibbs states, it is shown that κ-entropy is cast in the form of an expectation value for an observable that is determined. Also, an operational method named "the two-temperatures protocol" is introduced that provides a way to obtain the κ-entropy in terms of the partition functions of two auxiliary Gibbs states with temperatures κ-shifted above, the hot-system, and κ-shifted below, the cold-system, with respect to the original system temperature. That protocol provides physical procedures for evaluating entropy for any κ. Second, two novel additional ways of expressing the κ-entropy are further introduced. One determined by a non-negativity definite quantum channel, with Kraus-like operator sum representation and its extension to a unitary dilation via a qubit ancilla. Another given as a simulation of the κ-entropy via the quantum circuit of a generalized version of the Hadamard test. Third, a simple inter-relation of the von Neumann entropy and the quantum κ-entropy is worked out and a bound of their difference is evaluated and interpreted. Also the effect on the κ-entropy of quantum noise, implemented as a random unitary quantum channel acting in the system's density matrix, is addressed and a bound on the entropy, depending on the spectral properties of the noisy channel and the system's density matrix, is evaluated. The results obtained amount to a quantum computational tool-box for the κ-entropy that enhances its applicability in practical problems.