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İki Kübitli dolanık karışım durumlarının dönüşümleri.

Alkuş, Ümit

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In this thesis, the entangled mixed states of two qubits are considered. In the case where the matrix rank of the corresponding density matrix is 2, such a state can be purified to a pure state of 3 qubits. By utilizing this representation, the classification of such states of two qubits by stochastic local operations assisted by classical communication (SLOCC) is obtained. Also for such states, the optimal ensemble that appears in the computation of the concurrence and entanglement of formation is obtained.

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In this thesis, the entangled mixed states of two qubits are considered. In the case where the matrix rank of the corresponding density matrix is 2, such a state can be purified to a pure state of 3 qubits. By utilizing this representation, the classification of such states of two qubits by stochastic local operations assisted by classical communication (SLOCC) is obtained. Also for such states, the optimal ensemble that appears in the computation of the concurrence and entanglement of formation is obtained.
We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |ℓ,n> with ℓ=1,…,[n/2] are inequivalent under SLOCC.
We consider a single copy of a pure four-partite state of qubits and investigate its behavior under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to Greenberger-Horne-Zeilinger-like entanglement. The other ones contain essentially two-or three-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of four qubits, giving rise to a seven-parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single-copy distillation protocol.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
We introduce an operational entanglement classification of symmetric mixed states for an arbitrary number of qubits based on stochastic local operations assisted with classical communication (SLOCC operations). We define families of SLOCC entanglement classes successively embedded into each other, we prove that they are of nonzero measure, and we construct witness operators to distinguish them. Moreover, we discuss how arbitrary symmetric mixed states can be realized in the laboratory via a one-to-one correspondence between well-defined sets of controllable parameters and the corresponding entanglement families.
Classification of the Entangled States of 2×L×M×N2 \times L \times M \times N2×L×M×N
We present a practical entanglement classification scheme for pure state in form of 2×L×M×N2\times L\times M\times N2×L×M×N under the stochastic local operation and classical communication (SLOCC), where every inequivalent class of the entangled quantum states may be sorted out according to its standard form and the corresponding transformation matrix. This provides a practical method for determining the interconverting matrix between two SLOCC equivalent entangled states, and classification examples for some 2×4×M×N2\times 4\times M\times N2×4×M×N systems are also presented.
We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type ${\ensuremath{\rho}}^{\ensuremath{'}}\ensuremath{\sim}(A\ensuremath{\bigotimes}B)\ensuremath{\rho}(A\ensuremath{\bigotimes}{B)}^{\ifmmode\dagger\else\textdagger\fi{}}.$ We consider a real matrix parametrization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filtering operations for concentrating entanglement. Furthermore, we give a complete characterization of the mixed states that can be purified arbitrarily close to a Bell state. Finally, we obtain a new way of calculating the entanglement of formation.