On Quantum Correlations and Positive Maps (original) (raw)

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.

07 03 02 2 v 4 2 9 O ct 2 00 9 Completely positive maps and classical correlations

2009

We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We outline the implications of our results to quantum process tomography experiments.

Completely positive maps and classical correlations

2008

On entanglement of states and quantum correlations

Eprint Arxiv Math Ph 0202030, 2002

In this paper we present the novel qualities of entanglement of formation for general (so also infinite dimensional) quantum systems and we introduce the notion of coefficient of quantum correlations. Our presentation stems from rigorous description of entanglement of formation.

Remarks on entanglement measures and non-local state distinguishability

2002

We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.

Entanglement preserving maps and the universality of finite time disentanglement

arXiv: Quantum Physics, 2015

In this paper we investigate how common is the phenomenon of Finite Time Disentanglement (FTD) with respect to the set of quantum dynamics of bipartite quantum states with finite dimensional Hilbert spaces. Considering a quantum dynamics from a general sense, as just a continuous family of Completely Positive Trace Preserving maps (parametrized by the time variable) acting on the space of the bipartite systems, we conjecture that FTD does not happen only when all maps of the family are induced by local unitary operations. We prove that is the case for dynamics where all maps are induced by unitaries and, for pairs of qubits, where all maps are unital. Moreover, we prove some results about unitaries/CPTP maps preserving product/pure states.

Interpolating between Positive and Completely Positive Maps: A New Hierarchy of Entangled States

Entropy, 2021

A new class of positive maps is introduced. It interpolates between positive and completely positive maps. It is shown that this class gives rise to a new characterization of entangled states. Additionally, it provides a refinement of the well-known classes of entangled states characterized in terms of the Schmidt number. The analysis is illustrated with examples of qubit maps.

On Entanglement and Separability

We propose two necessary sufficient (NS) criteria to decide the separability of quantum states. They follow from two independent ideas: i) the Bloch-sphere-like-representation of states and ii) the proportionality of lines (rows, columns etc.) of certain multimatrix [1] associated with states. The second criterion proposes a natural way to determine the possible partial (or total, when possible) factorization of given multipartite state and in a sense can be used to determine the structure of the entanglement. We also introduce three entanglement measures based on the proposed new characterizations of entanglement. At last we discuss the second criterion mentioned above in the language of density matrix which is an inevitable language especially for mixed states.

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.

Bipartite entanglement-annihilating maps: Necessary and sufficient conditions

We fully characterize bipartite entanglement-annihilating (EA) channels that destroy entanglement of any state shared by subsystems and, thus, should be avoided in any entanglement-enabled experiment. Our approach relies on extending the problem to EA positive maps, the cone of which remains invariant under concatenation with partially positive maps. Due to this invariancy, positive EA maps adopt a well characterization and their intersection with completely positive tracepreserving maps results in the set of EA channels. In addition to a general description, we also provide sufficient operational criteria revealing EA channels. They have a clear physical meaning since the processes involved contain stages of classical information transfer for subsystems. We demonstrate the applicability of derived criteria for local and global depolarizing noises, and specify corresponding noise levels beyond which any initial state becomes disentangled after passing the channel. The robustness of some entangled states is discussed.