Entanglement or separability: the choice of how to factorize the algebra of a density matrix (original) (raw)
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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.
Separability and Entanglement of Composite Quantum Systems
Physical Review Letters, 1998
We show that all density operators of 2×N-dimensional quantum systems that remain invariant after partial transposition with respect to the first system are separable. Based on this criterion, we derive a sufficient separability condition for general density operators in such quantum systems. We also give a simple proof of the separability criterion in 2 × 2dimensional systems [A.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions, in which the density matrices are decomposed into various terms representing certain 1-qubit, 2-qubits…, n-qubits measurements. The present method is more general than previous methods for bipartite systems, as it can be used for quantification of entanglement for large n-qubits systems (3 n ≥). We demonstrate the use of the present method by analyzing 3-qubits GHZ states and 3-qubits general Bell-states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of these systems gives only classical correlations. Quantification of entanglement, for these systems, is given by the use of HS parameters.
2007
We investigate the geometric characterization of pure state bipartite entanglement of (2 × D)-and (3 × D)-dimensional composite quantum systems. To this aim, we analyze the relationship between states and their images under the action of particular classes of local unitary operations. We find that invariance of states under the action of single-qubit and single-qutrit transformations is a necessary and sufficient condition for separability. We demonstrate that in the (2×D)-dimensional case the von Neumann entropy of entanglement is a monotonic function of the minimum squared Euclidean distance between states and their images over the set of single qubit unitary transformations. Moreover, both in the (2×D)-and in the (3×D)-dimensional cases the minimum squared Euclidean distance exactly coincides with the linear entropy (and thus as well with the tangle measure of entanglement in the (2 × D)-dimensional case). These results provide a geometric characterization of entanglement measures originally established in informational frameworks. Consequences and applications of the formalism to quantum critical phenomena in spin systems are discussed.
The Separability versus Entanglement Problem
Lectures on Quantum Information
VI. Bell inequalities A. Detection of entanglement by Bell inequality VII. Classification of bipartite states with respect to quantum dense coding A. The Holevo bound B. Capacity of quantum dense coding VIII. Further reading: Multipartite states IX. Problems References
Entanglement transformations using separable operations
Physical Review A, 2007
We study conditions for the deterministic transformation |ψ −→ |φ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of |ψ and are the same as those of |φ . The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of |ψ are the same as those of |φ , we show that the Kraus operators of the separable transformation restricted to the supports of |ψ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states |ψj to pure states |φj .
Entanglement or Separability an introduction
2012
Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the ’Quantum Information’ field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for s...
Entanglement in continuous-variable systems: recent advances and current perspectives
Journal of Physics A: Mathematical and Theoretical, 2007
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures, and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of twomode Gaussian states according to different measures of entanglement, the unitary (reversible) localization, and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in threemode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states, and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non Gaussian states, a field of research that is to a large extent yet to be explored.
On the infeasibility of entanglement generation in Gaussian quantum systems via classical control
2011
This paper uses a system theoretic approach to show that classical linear time invariant controllers cannot generate steady state entanglement in a bipartite Gaussian quantum system which is initialized in a Gaussian state. The paper also shows that the use of classical linear controllers cannot generate entanglement in a finite time from a bipartite system initialized in a separable Gaussian state. The approach reveals connections between system theoretic concepts and the well known physical principle that local operations and classical communications cannot generate entangled states starting from separable states.
Interference and entanglement: an intrinsic approach
Journal of Physics A: Mathematical and General, 2002
An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matrix and the tensor product of partial traces. Entanglement for arbitrary quantum observables for multipartite systems is discussed. Star-product kernels are used to map the formulation of the addition rule of density operators onto the addition rule of symbols of the operators. Entanglement and nonlocalization of the pure state projector and allied operators are discussed. Tomographic and Weyl symbols (tomograms and Wigner functions) are considered as examples. The squeezed-states and some spin-states (two qubits) are studied to illustrate the formalism.