Experimentally feasible measures of distance between quantum operations

We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for distance measure between quantum processes. We also argue that they can be especially useful as diagnostic measures to get preliminary knowledge about imperfections in an experimental setup. In particular we provide quantum circuit which can be used to measure the superfidelity between quantum processes. As the behavior of the superfidelity between quantum processes is crucial for the properties of the introduced measures, we study its behavior for several families of quantum channels. We calculate superfidelity between arbitrary one-qubit channels using affine parametrization and superfidelity between generalized Pauli channels in arbitrary dimensions. Statistical behavior of the proposed quantities for the ensembles of quantum operations in low dimensions indicates that the proposed measures can be indeed used to distinguish quantum processes.Comment: 9 pages, 4 figure

Nonconvex minimization related to quadratic double-well energy - approximation by convex problemsenergy – approximation by convex problems

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied taking into account minimization of the corresponding integral functional. Such integral, as being not sequentially weakly lower semicontinuous, does not admit classical minimizers. To derive the relaxation formula for the infimum, the appropriate minimizing sequence is constructed. It consists of solutions of some approximating convex problems involving characteristic functions related to the phase energies. The weak limit of this sequence and the weak limit of the sequence of solutions of dual problems combined with the weak-star limits of the characteristic functions related to the phase energies allow to establish the final relaxation formula. It is also shown that infimum can be expressed by the Young measure associated with constructed minimizing sequence. An explicit form of Young measure in some regions of the involved domain is calculated

On Simulation of the Young Measures – Comparison of Random-Number Generators

"Young measure" is an abstract notion from mathematical measure theory.  Originally, the notion appeared in the context of some variational problems related to the analysis of sequences of “fast” oscillating of functions.  From the formal point of view the Young measure  may be treated as a continuous linear functional defined on the space of Carathéodory integrands satisfying certain regularity conditions. Calculating an explicit form of specific Young measure is a very important task.  However, from a strictly mathematical standpoint  it is a very difficult problem not solved as yet in general. Even more difficult would be the problem of calculating Lebasque’s integrals with respect to such measures. Based on known formal results it can be done only in the most simple cases.  On the other hand in many real-world applications it would be enough to learn only some of the most important probabilistic  characteristics  of the Young distribution or learn only approximate values of the appropriate integrals. In such a case a possible solution is to adopt Monte Carlo techniques. In the presentation we propose three different algorithms designed for simulating random variables distributed according to the Young measures  associated with piecewise functions.  Next with the help of computer simulations we compare their statistical performance via some benchmarking problems. In this study we focus on the accurateness of the distribution of the generated sample

On Simulation of the Young Measures – Comparison of Random-Number Generators

"Young measure" is an abstract notion from mathematical measure theory. Originally, the notion appeared in the context of some variational problems related to the analysis of sequences of “fast” oscillating of functions. From the formal point of view the Young measure may be treated as a continuous linear functional defined on the space of Carathéodory integrands satisfying certain regularity conditions. Calculating an explicit form of specific Young measure is a very important task. However, from a strictly mathematical standpoint it is a very difficult problem not solved as yet in general. Even more difficult would be the problem of calculating Lebasque’s integrals with respect to such measures. Based on known formal results it can be done only in the most simple cases. On the other hand in many real-world applications it would be enough to learn only some of the most important probabilistic characteristics of the Young distribution or learn only approximate values of the appropriate integrals. In such a case a possible solution is to adopt Monte Carlo techniques. In the presentation we propose three different algorithms designed for simulating random variables distributed according to the Young measures associated with piecewise functions. Next with the help of computer simulations we compare their statistical performance via some benchmarking problems. In this study we focus on the accurateness of the distribution of the generated sample

Majorization uncertainty relations for mixed quantum states

Majorization uncertainty relations are generalized for an arbitrary mixed quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies