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Grigori G. Amosov

Publications and source records attributed to Grigori G. Amosov.

14 recordsLinked to original sources
arXiv2026

Towards Schrödinger Cat States in the Second Harmonic Generation

We investigate the quantum evolution of the pump field in second-harmonic generation under strong pump depletion. Starting from a coherent state, the pump develops a nonclassical phase-space structure resembling a Schrödinger cat state. This behavior originates from phase instability induced by vacuum fluctuations of the harmonic mode. A rigorous quantum analysis has been performed for mean photon numbers up to $\langle \hat n \rangle = 100$ in pump mode. For larger photon numbers, up to $\langle \hat n \rangle = 10^{7}$, the dynamics have been analyzed using a classical trajectory method with sampled initial conditions that reproduces the main features of the quantum evolution. The results indicate that nonlinear frequency conversion can generate macroscopic superposition-like states of the pump field. Although the resulting state is not pure due to correlations with the second-harmonic wave, it remains non-classical with negative zones of Wigner function. These results indicate that strongly nonlinear frequency conversion can provide a scalable route toward macroscopic nonclassical states of light.

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arXiv2016

Spectral properties of reduced fermionic density operators and parity superselection rule

We consider pure fermionic states with a varying number of quasiparticles and analyze two types of reduced density operators: one is obtained via tracing out modes, the other is obtained via tracing out particles. We demonstrate that spectra of mode-reduced states are not identical in general and fully characterize pure states with equispectral mode-reduced states. Such states are related via local unitary operations with states satisfying the parity superselection rule. Thus, valid purifications for fermionic density operators are found. To get particle-reduced operators for a general system, we introduce the operation $Φ(\varrho) = \sum_i a_i \varrho a_i^†$. We conjecture that spectra of $Φ^p(\varrho)$ and conventional $p$-particle reduced density matrix $\varrho_p$ coincide. Nontrivial generalized Pauli constraints are derived for states satisfying the parity superselection rule.

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arXiv2015

On the structutre of the algebra generated by the non-commutative operator graph demonstrating superactivation for a zero-error capacity

Recently M.E. Shirokov introduced the non-commutative operator graph depending on the complex parameter $θ$ to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We study the algebraic structure of this graph. The relations for the algebra generated by the graph are derived. In the limiting case $θ=\pm1$ the graph becomes abelian and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein group.

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arXiv2015

Towards a tomographic representation of quantum mechanics on the plane

On the base of symplectic quantum tomogram we define a probability distribution on the plane. The dual map transfers all observables which are polynomials of the position and momentum operators to the set of polynomials of two variables. In this representation the average values of observables can be calculated by means of integration over all the plane.

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arXiv2014

Estimating the output entropy of a tensor product of two quantum channels

In this paper we find, for a class of bipartite quantum states, a nontrivial lower bound on the entropy gain resulting from the action of a tensor product of identity channel with an arbitrary channel. By means of that we then estimate (from below) the output entropy of the tensor product of dephasing channel with an arbitrary channel. Finally, we provide a characterization of all phase-damping channels resulting as particular cases of dephasing channels.

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arXiv2011

On calculating the mean values of quantum observables in the optical tomography representation

Given a density operator $\hat ρ$ the optical tomography map defines a one-parameter set of probability distributions $w_{\hat ρ}(X,ϕ),\ ϕ\in [0,2π),$ on the real line allowing to reconstruct $\hat ρ$. We introduce a dual map from the special class $\mathcal A$ of quantum observables $\hat a$ to a special class of generalized functions $a(X,ϕ)$ such that the mean value $<\hat a>_{\hat ρ} =Tr(\hat ρ\hat a)$ is given by the formula $<\hat a>_{\hat ρ}= \int \limits_{0}^{2π}\int \limits_{-\infty}^{+\infty}w_{\hat ρ}(X,ϕ)a(X,ϕ)dXdϕ$. The class $\mathcal A$ includes all the symmetrized polynomials of canonical variables $\hat q$ and $\hat p$.

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arXiv2009

A use of the central limit theorem to obtain a classical limit for the center-of-mass tomogram

We investigate the dependence of the center-of-mass tomogram of a system with many degrees of freedom $N$ on the Planck constant $\hbar $. It is shown that to use the central limit theorem under taking the limit $N\to +\infty $ one should keep the energy of the system to be constant. In the case, the resulting distribution is Gaussian if the initial distribution is a product of independent excited states of a quantum oscillator or even and odd coherent states either. Then, if one turns the Planck constant $\hbar \to 0$ we get $δ$-function associated with the distribution concentrated in zero with the probability equal to one.

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arXiv2008

The decreasing property of relative entropy and the strong superadditivity of quantum channels

We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.

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arXiv2007

On the information completeness of quantum tomograms

We address the problem of information completeness of quantum measuremets in connection to quantum state tomography and with particular concern to quantum symplectic tomography. We put forward some non-trivial situations where informationally incomplete set of tomograms allows as well the state reconstruction provided to have some a priori information on the state or its dynamics. We then introduce a measure of information completeness and apply it to symplectic quantum tomograms.

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arXiv2007

The strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension

Given a quantum channel $Φ$ in a Hilbert space $H$ put $\hat H_Φ(ρ)=\min \limits_{ρ_{av}=ρ}Σ_{j=1}^{k}π_{j}S(Φ(ρ_{j}))$, where $ρ_{av}=Σ_{j=1}^{k}π_{j}ρ_{j}$, the minimum is taken over all probability distributions $π=\{π_{j}\}$ and states $ρ_{j}$ in $H$, $S(ρ)=-Trρ\logρ$ is the von Neumann entropy of a state $ρ$. The strong superadditivity conjecture states that $\hat H_{Φ\otimes Ψ}(ρ)\ge \hat H_Φ(Tr_{K}(ρ))+\hat H_Ψ(Tr_{H}(ρ))$ for two channels $Φ$ and $Ψ$ in Hilbert spaces $H$ and $K$, respectively. We have proved the strong superadditivity conjecture for the quantum depolarizing channel in any dimensions.

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arXiv2002

Stationary quantum stochastic processes from the cohomological point of view

Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra $\tilde K=\oplus_{i=1}^{\infty}K_i$ into a ring of (abelian) cohomologies of the one-parameter group $α$ consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from $K_i$ is translated into an additive $i-α$-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The $E_0$-semigroup $\tilde β$ on the von Neumann algebra $\cal N$ associated with the markovian perturbation of K-flow j posseses the restriction $\tilde β|_{{\cal N}_0}, {\cal N}_0\subset {\cal N}$, which is conjugate to the flow of Powers shifts $β$ associated with j. It yields for $\tilde β$ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space $L^2(\mathbb R)$ all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.

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