The Geometry of Density States, Positive Maps and Tomograms (original) (raw)
Related papers
Positive maps of density matrix and a tomographic criterion of entanglement
Physics Letters A, 2004
The positive and not completely positive maps of density matrices are discussed. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Unitary U(∞)-group tomogram of photon state in Fock basis is constructed. Notion of tomographic purity of spin states is introduced. An entanglement criterion for multipartite spin-system is given in terms of a function depending on unitary group parameters and semigroup of positive map parameters. Some two-qubit and two-qutrit states are considered as examples of entangled states using depolarizing map semigroup.
2008
Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,mathbbR)IGL(n, \mathbb{R})IGL(n,mathbbR) and GL(n,mathbbR)GL(n, \mathbb{R})GL(n,mathbbR) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.
Journal of Russian Laser Research, 2003
The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying the adjoint representation. Positive maps of density operator are related to random matrices. The tomographic probability description of quantum states is used to formulate the problem of separability and entanglement as the condition for joint probability distribution of several random variables represented as the convex sum of products of probabilities of random variables describing the subsystems. The property is discussed as a possible criterion for separability or entanglement. The convenient criterion of positivity of finite and infinite matrix is obtained. The U(n)-tomogram of a multiparticle spin state is introduced. The entanglement measure is considered in terms of this tomogram.
On Positive Maps, Entanglement and Quantization
Open Systems & Information Dynamics (OSID), 2004
We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and we describe the quantization of correlations. A relation between these two areas is discussed.
On Quantum Correlations and Positive Maps
Letters in Mathematical Physics, 2004
We present a discussion on local quantum correlations and their relations with entanglement. We prove that vanishing coefficient of quantum correlations implies separability. The new results on locally decomposable maps which we obtain in the course of proof also seem to be of independent interest.
On positive maps in quantum information
Russian Journal of Mathematical Physics, 2014
Using Grothendieck approach to the tensor product of locally convex spaces we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented.
Geometry of quantum states: New construction of positive maps
Physics Letters A, 2009
We provide a new class of positive maps in matrix algebras. The construction is based on the family of balls living in the space of density matrices of n-level quantum system. This class generalizes the celebrated Choi map and provide a wide family of entanglement witnesses which define a basic tool for analyzing quantum entanglement.
Dynamical maps and density matrices
Journal of Physics: Conference Series, 2009
The relations between dynamical maps and quantum states of bipartite systems are analyzed from the perspective of quantum conditional probability. In particular, we explore new interesting relations between completely positive maps, which correspond to quantum channels, and states of bipartite systems which correspond to correlations between the initial and final states. The new connection emerges in a natural way from the generalisation of the classical concept of conditional probability. We develop applications of these relations which prove to be very useful in both directions, either for the classification of positive maps which are not completely positive, the classification of non-decomposable dynamical maps or for the classification of positive partial transpose and entangled states.
Positive tensor products of maps and n-tensor-stable positive qubit maps
We analyze positivity of a tensor product of two linear qubit maps, Φ1 ⊗ Φ2. Positivity of maps Φ1 and Φ2 is a necessary but not a sufficient condition for positivity of Φ1 ⊗ Φ2. We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both Φ1 and Φ2 are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n-tensor-stable positive qubit maps, i.e. such qubit maps Φ that Φ ⊗n is positive. Particular cases of 2-and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n-tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.
On the distribution of entanglement changes produced by unitary operations
Physics Letters A, 2003
We consider the change of entanglement of formation ∆E produced by a unitary transformation acting on a general (pure or mixed) state ρ describing a system of two qubits. We study numerically the probabilities of obtaining different values of ∆E, assuming that the initial state is randomly distributed in the space of all states according to the product measure recently introduced by Zyczkowski et al. [Phys. Rev. A 58 (1998) 883].