Anna Vershynina

University of Houston | U of H, UH · Department of Mathematics

We prove Lieb-Robinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with time-dependent generators that satisfy a suitable decay condition in space.

We prove upper bounds on the rate, called "mixing rate", at which the von Neumann entropy of the expected density operator of a given ensemble of states changes under non-local unitary evolution. For an ensemble consisting of two states, with probabilities of p and 1-p, we prove that the mixing rate is bounded above by 4\sqrt{p(1-p)} for any Hamilt...

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary a...

We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Renyi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commuta...

Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or Rényi entropies, would it define good coherence measures? In other words, we define a difference between Tsallis...

Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or R\'enyi entropies, would it define good coherence measures? We define a Tsallis coherence monotone as a differenc...

We present a genuine coherence measure based on a quasi-relative entropy as a difference between quasi-entropies of the dephased and the original states. The measure satisfies non-negativity and monotonicity under genuine incoherent operations (GIO). It is strongly monotone under GIO in two and three dimensions, or for pure states in any dimension,...

In this article the operator trace function Λr,s(A)[K,M]:=tr(K⁎ArMArK)s is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data processing inequalities of various relat...

In this article the operator trace function $ \Lambda_{r,s}(A)[K, M] := {\operatorname{tr}}(K^*A^r M A^r K)^s$ is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data p...

We present a genuine coherence measure based on a quasi-relative entropy as a difference between quasi-entropies of the dephased and the original states. The measure satisfies non-negativity and monotonicity under genuine incoherent operations (GIO). It is strongly monotone under GIO in two- and three-dimensions, or for pure states in any dimension...

It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the...

It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. We ask a similar question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the dist...

It has been shown that the $\alpha-z$ R{\'e}nyi Relative Entropy satisfies the Data Processing Inequality (DPI) for a certain range of $\alpha$'s and $z$'s. Moreover, the range is completely characterized by Zhang in `20. We prove necessary and algebraically sufficient conditions to saturate the DPI for the $\alpha-z$ R{\'e}nyi Relative Entropy whe...

Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modific...

Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modific...

We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for any operator monotone decreasing function and either mixed qubit or classical states. Moreover, we derive an upper bound for the Umegaki and Tsallis relative entropies in the case of any finite-dimensional states. The bound for the relat...

We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for several types of the function, as well for any operator monotone decreasing function and mixed qubit states. We apply the result to the Umegaki relative entropy and the q-entropy.

We consider a quantum quasi-relative entropy $S_f^K$ for an operator $K$ and an operator convex function $f$. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the $f$-divergences (i.e. $K=I$). We also provide an error term for a class of operator inequalities, that generalize o...

We consider a quantum quasi-relative entropy for an operator K and an operator convex function f. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the f-divergences (i.e. K=I). We also provide an error term for a class of operator inequalities, that generalize operator strong s...

Let be a finite dimensional von Neumann algebra and a von Neumann subalgebra of it. For states and on , let and be the corresponding states induced on . The data processing inequality implies that where is the relative entropy. Petz proved that there is equality if and only if , where is the Petz recovery map. We prove a quantitative version of Pet...

We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to...

We provide upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide...

We present a new criterion that determines whether a fermionic state is a convex combination of pure Gaussian states. This criterion is complete and characterizes the set of convex-Gaussian states from the inside in the form of a sequence of solvable semidefinite programs. If a state passes a program it is a convex-Gaussian state. This criterion is...

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique groundstate by map...

We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. T...

In the study of quantum systems such as quantum optics, quantum information theory, atomic physics and condensed-matter physics it is important to know that there is a finite speed with which information can propagate. The existence of such a finite speed was discovered mathematically by Lieb and Robinson, (1972). It turns the locality properties o...

We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that t...

In this article, we define the transport dimension of probability measures on $\mathbb{R}^m$ using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensi...

Let A be a bounded linear operator on a Banach space and let g a be vector-valued function that is analytic in a neighborhood of the origin of ℝ. We obtain conditions of the existence of analytic solutions for the Cauchy problem $ \left\{ {\begin{array}{l} {\frac{{\partial u}}{{\partial t}} = A^2 \frac{{\partial ^2 u}}{{\partial x^2 }},} \\ {u\le...

Let A be a bounded operator on a Hilbert space and g a vector-valued function, which is holomorphic in a neighborhood of zero. The question about exis- tence of holomorphic solutions of the Cauchy problem 8