We show that the space of density matrices for n-qubit states, considered as a (2^n)^2 dimensional real vector space, has a basis consisting of density matrices of stabilizer states. We describe an application of this result to automated verification of quantum protocols.
We present three different matrix bases that can be used to decompose density matrices of d -dimensional quantum systems, so-called qudits: the generalized Gell–Mann matrix basis , the polarization operator basis and the Weyl operator basis . Such a decomposition can be identified with a vector—the Bloch vector, i.e. a generalization of the well-known qubit case—and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic two-qudit state and decompose it according to each basis. In the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin-1 measurements, which is appropriate for an experimental realization.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of p particles for each value of p. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the N-representability problem is given, and two sufficient conditions and one necessary condition for N representability are given in terms of distances and norms.
Abstract Quantum nonlocality represents correlations between subsystems of a composite quantum system, usually including Bell nonlocality, steerability, and entanglement. According to the hypothesis of quantum mechanics, states of a quantum system Q described by a d -dimensional Hilbert space are denoted by density operators acting on . Under a basis e for the Hilbert space , every abstract density operator ρ of the system AB corresponds to a density matrix ρ e , which is a state of the complex Hilbert space . In this work, we discuss the consistency of quantum nonlocality of density operators ρ and their corresponding density matrices ρ e under the chosen basis e . It is proved that only when a basis e is a product one, a density operator ρ is entangled (respectively, Bell nonlocal, steerable) if and only if its density matrix ρ e is entangled (respectively, Bell nonlocal, steerable).
Local unitary stabilizer subgroups constitute powerful invariants for distinguishing various types of multipartite entanglement. In this paper, we show how stabilizers can be used as a basis for entanglement verification protocols on distributed quantum networks using minimal resources. As an example, we develop and analyze the performance of a protocol to verify membership in the space of Werner states, that is, multiqubit states that are invariant under the action of any one-qubit unitary applied to all the qubits.
We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.
We use polarization operators known from quantum theory of angular momentum to expand the N×N dimensional density operators. Thereby, we construct generalized Bloch vectors representing density matrices. We study their properties and derive positivity conditions for any N. We also apply the procedure to study Bloch vector space for a qubit and a qutrit.