Representations of two-qubit and ququart states via discrete Wigner functions

Abstract

By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU}(\mathrm{N})$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU}(\mathrm{N})$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU}(2)\otimes \mathrm{SU}(2)$ and $\mathrm{SU}(4)$ Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.

33 pages, 16 figures, minor changes

References in corpus (12)

• Notes on Some Entanglement Properties of Quantum Field Theory
• Beating the channel capacity limit for superdense coding with entangled ququarts
• Discrete Wigner functions and quantum computational speedup
• Experimental optimal cloning of four-dimensional quantum states of photons
• Alternative fidelity measure for quantum states
• Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities
• Entanglement universality of two-qubit X-states
• Quantifying entanglement of a two-qubit system via measurable and invariant moments of its partially transposed density matrix
• Maximally entangled mixed states for qubit-qutrit systems
• History states of systems and operators
• Analytical formula connecting entangled state and the closest disentangled state
• Bounds on bipartite entanglement from fixed marginals