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arXiv:1304.0253 · Annals of Physics

Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators

Marcelo A. Marchiolli, P.E.M.F. Mendonça

Published Jun 13, 2013• 16 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2013.05.009

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Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.
By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group SU(N)\mathrm{SU(N)}SU(N), it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete SU(N)\mathrm{SU(N)}SU(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 SU(2)⊗SU(2)\mathrm{SU(2)} \otimes \mathrm{SU(2)}SU(2)⊗SU(2) and SU(4)\mathrm{SU(4)}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.
We first show that partial transposition for pure and mixed two-particle states in a discrete NNN-dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter rrr of a Bell state ∣Φ+⟩|Φ^{+} \rangle∣Φ+⟩ and the completely incoherent state, ρ^r=r∣Φ+⟩⟨Φ+∣+(1−r)I^N2\hatρ_r = r |Φ^{+} \rangle \langle Φ^{+}| + (1-r)\frac{\hat{\mathbb{I}}}{N^2}ρ^​r​=r∣Φ+⟩⟨Φ+∣+(1−r)N2I^​: we find that for r0<r<1r_0 < r < 1r0​<r<1, where r0r_0r0​ is obtained as a function of the dimensionality NNN, the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.