The private quantum channel (PQC) maps any state to the maximally mixed for discrete as well bosonic Gaussian systems, and it has fundamental meaning on cryptographic tasks capacity problems. In this paper, we primally introduce a notion of approximate (ε-PQC) fermionic systems (i.e., ε-FPQC), construct its explicit form (Gaussian) channel. First all, suggest general structure ε-FPQC with respe...

In this paper, we fully characterize the duality mapping over space of matrices that are equipped with Schatten norms. Our approach is based on analysis saturation Hölder inequality for We prove in our main result that, p∈(1,∞), real-valued Schatten-p norm a continuous and single-valued function provide an explicit form its computation. For special case p = 1, set-valued; by adding rank constra...

In this paper we extend the classical Bohr’s inequality to setting of non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As consequence, obtain for operators in Neumann-Schatten class $\mathcal C_1$ and square matrices any finite order. Interestingly, establish that optimal bound $r$ above mentioned concerning is 1/3 whereas it 1/2 case $2\times 2$ reduces $\s...

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis, we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering spec...

The Schatten quasi-norm was introduced tobridge the gap between the trace norm andrank function. However, existing algorithmsare too slow or even impractical for large-scale problems. Motivated by the equivalencerelation between the trace norm and its bilin-ear spectral penalty, we define two tractableSchatten norms, i.e. the bi-trace and tri-tracenorms, and prov...

The goal of this review is to explain some recent results [5] regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).