Improving compressed sensing with the diamond norm

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio

Shallow-Depth Variational Quantum Hypothesis Testing

We present a variational quantum algorithm for differentiating several hypotheses encoded as quantum channels. Both state preparation and measurement are simultaneously optimized using success probability of single-shot discrimination as an objective function which can be calculated using localized measurements. Under constrained signal mode photon number quantum illumination we match the performance of known optimal 2-mode probes by simulating a bosonic circuit. Our results show that variational algorithms can prepare optimal states for binary hypothesis testing with resource constraints. Going beyond the binary hypothesis testing scenario, we also demonstrate that our variational algorithm can learn and discriminate between multiple hypotheses.Comment: Version 2, 12 pages, 8 figures, comments welcom

СУЧАСНІ СТЕГАНОГРАФІЧНІ МЕТОДИ ЗАХИСТУ ІНФОРМАЦІЇ

У роботі представлена систематизація та класифікація сучасних напрямів стеганографії, наведені переваги та недоліки конкретних стеганографічних методів. Виконане дослідження спрямоване на полегшення пошуку існуючих стеганографічних методів та засобів в науковій літературі з метою розробки нових ефективних систем захисту інформації

Computational Distinguishability of Quantum Channels

The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixed-unitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixed-unitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.Comment: Ph.D. Thesis, 178 pages, 35 figure