Counterexample to an additivity conjecture for output purity of quantum channels

A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p>4.79. The same example (with p=infinity) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.Comment: 3 pages, 3 figures, revte

Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs

We prove multiplicativity of maximal output $p$ norm of classical noise channels and thermal noise channels of arbitrary modes for all $p>1$ under the assumption that the input signal states are Gaussian states. As a direct consequence, we also show the additivity of the minimal output entropy and that of the energy-constrained Holevo capacity for those Gaussian channels under Gaussian inputs. To the best of our knowledge, newly discovered majorization relation on symplectic eigenvalues, which is also of independent interest, plays a central role in the proof.Comment: 9 pages, no figures. Published Versio

Multiplicativity of maximal output purities of Gaussian channels under Gaussian inputs

We address the question of the multiplicativity of the maximal p-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the reduction of unitary dynamics in larger Hilbert spaces. It is shown that the maximal output purity of tensor products of single-mode channels under Gaussian inputs is multiplicative for any p>1 for products of arbitrary identical channels as well as for a large class of products of different channels. In the case of p=2 multiplicativity is shown to be true for arbitrary products of generic channels acting on any number of modes.Comment: 9 page

Qubit channels with small correlations

We introduce a class of quantum channels with correlations acting on pairs of qubits, where the correlation takes the form of a shift operator onto a maximally entangled state. We optimise the output purity and show that below a certain threshold the optimum is achieved by partially entangled states whose degree of entanglement increases monotonically with the correlation parameter. Above this threshold, the optimum is achieved by the maximally entangled state characterizing the shift. Although, a full analysis can only be done for the 2-norm, both numerical and heuristic arguments indicate that this behavior and the optimal inputs are independent of p>1 when the optimal output purity is measured using the p-norm.Comment: 11 pages, 4 figures, 2 table

Notes on multiplicativity of maximal output purity for completely positive qubit maps

A problem in quantum information theory that has received considerable attention in recent years is the question of multiplicativity of the so-called maximal output purity (MOP) of a quantum channel. This quantity is defined as the maximum value of the purity one can get at the output of a channel by varying over all physical input states, when purity is measured by the Schatten $q$-norm, and is denoted by $\nu_q$. The multiplicativity problem is the question whether two channels used in parallel have a combined $\nu_q$ that is the product of the $\nu_q$ of the two channels. A positive answer would imply a number of other additivity results in QIT. Very recently, P. Hayden has found counterexamples for every value of $q>1$. Nevertheless, these counterexamples require that the dimension of these channels increases with $1-q$ and therefore do not rule out multiplicativity for $q$ in intervals $[1,q_0)$ with $q_0$ depending on the channel dimension. I argue that this would be enough to prove additivity of entanglement of formation and of the classical capacity of quantum channels. More importantly, no counterexamples have as yet been found in the important special case where one of the channels is a qubit-channel, i.e. its input states are 2-dimensional. In this paper I focus attention to this qubit case and I rephrase the multiplicativity conjecture in the language of block matrices and prove the conjecture in a number of special cases.Comment: Manuscript for a talk presented at the SSPCM07 conference in Myczkowce, Poland, 10/09/2007. 12 page

Relativistic Doppler effect in quantum communication

When an electromagnetic signal propagates in vacuo, a polarization detector cannot be rigorously perpendicular to the wave vector because of diffraction effects. The vacuum behaves as a noisy channel, even if the detectors are perfect. The ``noise'' can however be reduced and nearly cancelled by a relative motion of the observer toward the source. The standard definition of a reduced density matrix fails for photon polarization, because the transversality condition behaves like a superselection rule. We can however define an effective reduced density matrix which corresponds to a restricted class of positive operator-valued measures. There are no pure photon qubits, and no exactly orthogonal qubit states.Comment: 10 pages LaTe

On Hastings' counterexamples to the minimum output entropy additivity conjecture

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.Comment: 17 pages + 1 lin

Complete hierarchies of efficient approximations to problems in entanglement theory

We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multi-party entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multi-party entanglement, (D) the test whether states are multi-party entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree three or less. We then apply very recently established known methods from the theory of semi-definite relaxations to the formulated optimization problems. By this construction we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of novel, efficiently decidable sufficient criteria for multi-particle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.Comment: 14 pages, 3 figures, tiny modifications, version to be published in Physical Review

Conceptual foundations of the development of the tourist complex of the region in the conditions of the COVID-19 coronavirus pandemic

For the tourism industry, the COVID-19 coronavirus pandemic has become the most serious challenge of its existence. The industry suffers serious losses and loses jobs, which generally negatively affects the unemployment rate in the country. Meanwhile, strict anti-epidemic measures introduced by various countries have accelerated the transformation of world tourism. High-tech companies with global ambitions, digital startups are appearing on the market, which leads to new opportunities for individual tours and formats of international cooperation. It is obvious that the global crisis related to the COVID-19 coronavirus pandemic will lead to a new, more sustainable format of the tourist complex in the future. The purpose of this article is to study the current state and measures to stimulate the tourism industry in the context of the COVID-19 coronavirus pandemic and to develop a conceptual model for the development of the tourist complex in such conditions. The following methods of scientific cognition are used in the work: abstraction, analysis, induction, synthesis. The article substantiates the relevance of the study. Some indicators of the functioning of domestic tourism during the pandemic in comparison with previous periods are considered, in particular, the tourist flow of Russia in view of the spread of coronavirus infection. Measures to support tourism in the current conditions are considered, relevant measures to support the industry are singled out separately. The main ideas of targeted federal projects are outlined, such as: the national project «Tourism and the hospitality industry», «Strategy for the development of tourism in Russia until 2035». A conceptual model of management of the tourist complex of the region in the conditions of the COVID-19 coronavirus pandemic is proposed, which assumes a systematic approach to the diagnosis of tourism development and allows formalizing managerial impacts on the tourist complex of solutions to ensure its sustainable development