Abstract
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres–Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich’s squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.
Highlights
The states of quantum systems are determined by wave functions [1,2] or density matrices [3,4]
We review how the Bloch sphere geometry of qubit states is mapped onto a triangle geometry of qubit and qutrit states
We start with a qubit density matrix ρ = ρ†, Tr(ρ) = 1 satisfying the nonnegativity condition of its eigenvalues, i.e
Summary
The states of quantum systems are determined by wave functions [1,2] (pure states) or density matrices [3,4]. Entropy 2018, 20, 630 is identified with the set of three probability distributions of spin projections on three perpendicular directions in the space This description of the qubit state was studied and illustrated by the triangle geometry of the system, using the so-called Malevich square representation [16] known as quantum suprematism approach (after the Russian painter Kazmir Malevich (1879–1935), founder of suprematism, an art movement started around 1913 focused on basic geometric figures). The superposition principle for spin-1/2 state vectors was presented in explicit form as the nonlinear superposition of the classical probability distributions determining the qubit states in [19,36,37] This superposition was illustrated geometrically in the quantum suprematism approach as a superposition of squares.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
