The separable mixed 2-qubit X-states are classified in accordance with degeneracies in the spectrum of density matrices. It is shown that there are four classes of separable X-states, among them: one 4D family, a pair of 2D family and a single, zero-dimensional maximally mixed state.
We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W, and Greenberger-Horne-Zeilinger states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W class. Finally, we present a new family of PPTES "edge" states with maximal ranks.
We give a complete, hierarchic classification for arbitrary multiqubit mixed states based on the separability properties of certain partitions. We introduce a family of N-qubit states to which any arbitrary state can be depolarized. This family can be viewed as the generalization of Werner states to multiqubit systems. We fully classify those states with respect to their separability and distillability properties. This provides sufficient conditions for nonseparability and distillability for arbitrary states.
Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.
The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.
We study the classification of entanglement in tripartite systems by using Bell-type inequalities and principal basis. By using Bell unctions and the generalized three dimensional Pauli operators, we present a set of Bell inequalities which classifies the entanglement of triqutrit fully separable and bi-separable mixed states. By using the correlation tensors in the principal basis representation of density matrices, we obtain separability criteria for fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example is given to illustrate our criteria in classifying the tripartite entanglement.
The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\ensuremath{\rho}$ with the property that ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ is separable for all unitary matrices U. This problem has been solved when the local dimensions m and n satisfy m=2 and $n\ensuremath{\le}3$. We solve all remaining qubit-qudit cases (i.e., when m=2 and $n\ensuremath{\ge}4$ is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ has positive partial transpose for all unitary matrices U. This equivalence is in stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.