Geodesic
$\operatorname{SU}(6)$ Casimir invariants and $\operatorname{SU}(2)\otimes\operatorname{SU}(3)$ scalars for a mixed qubit-qutrit states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 102-121 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the present paper few steps are undertaken towards the description of the “qubit-qutrit” pair – quantum bipartite system composed of two and three level subsystems. Calculations of the Molien functions and Poincaré series for the qubit-qubit and qubit-qutrit “local unitary invariants” are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the enveloping algebra $\mathfrak{su}(6)$. 
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     title = {$\operatorname{SU}(6)$ {Casimir} invariants and $\operatorname{SU}(2)\otimes\operatorname{SU}(3)$ scalars for a~mixed qubit-qutrit states},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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V. Gerdt; D. Mladenov; Yu. Palii; A. Khvedelidze. $\operatorname{SU}(6)$ Casimir invariants and $\operatorname{SU}(2)\otimes\operatorname{SU}(3)$ scalars for a mixed qubit-qutrit states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 102-121. http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a3/

[1] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000 | MR | Zbl

[2] V. Vedral, Introduction to Quantum Information Science, Oxford University Press, New York, 2006 | Zbl

[3] H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, 1939 | MR

[4] V. L. Popov, E. B. Vinberg, “Invariant theory”, Algebraic Geometry, v. IV, Encycl. Math. Sci., 55, Springer-Verlag, 1994, 123–273

[5] N. Linden, S. Popescu, “On multi-particle entanglement”, Fortschr. Phys., 46 (1998), 567–578 | 3.0.CO;2-H class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[6] M. Grassl, M. Rötteler, T. Beth, “Computing local invariants of qubit systems”, Phys. Rev. A, 58 (1998), 1833–1859 | DOI | MR

[7] R. C. King, T. A. Welsh, P. D. Jarvis, “The mixed two-qubit system and the structure of its ring of local invariants”, J. Phys. A Math. Theor., 40 (2007), 10083–10108 | DOI | MR | Zbl

[8] V. Gerdt, Yu. Palii, A. Khvedelidze, “On the ring of local polynomial invariants for a pair of entangled qubits”, J. Math. Sci., 168 (2010), 368–378 | DOI | MR | Zbl

[9] M. Kus, K. Życzkowski, “Geometry of entangled states”, Phys. Rev. A, 63 (2001), 032307, 13 pp. | DOI | MR

[10] I. Bengtsson, K. Życzkowski, Geometry of Quantum States. An Introduction to Quantum Entanglement, Cambridge University Press, 2006 | MR | Zbl

[11] F. T. Hioe, J. H. Eberly, “$N$-Level Coherence Vector and Higher Conservation Laws in Quantum Optics and Quantum Mechanics”, Phys. Rev. Lett., 47 (1981), 838–841 | DOI | MR

[12] N. Jacobson, Lie Algebras, Wiley-Interscience, New-York–London, 1962 | MR | Zbl

[13] I. M. Gel'fand, “The center of the infinitesimal group ring”, Matematicheskii sbornik (English version: Sbornik: Mathematics), 26(68):1 (1950), 103–112 | MR | Zbl

[14] D. P. Zhelobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs, 40, AMS, 1978

[15] L. C. Biedenharn, “On the representations of the semisimple Lie groups. The explicit construction of invariants for the unimodular unitary group in $N$ dimensions”, J. Math. Phys., 4 (1963), 436–445 | DOI | MR | Zbl

[16] R. H. Dalitz, “Constraints on the statistical tensor for low-spin particles produced in strong interaction processes”, Nucl. Phys., 87 (1966), 89–99 | DOI

[17] P. Minnaert, “Spin-density analysis. Positivity conditions and Eberhard-Good theorem”, Phys. Rev., 151 (1966), 1306–1318 | DOI

[18] S. M. Deen, P. K. Kabir, G. Karl, “Positivity constraints on density matrices”, Phys. Rev. D, 4 (1971), 1662–1666 | DOI

[19] G. Kimura, “The Bloch vector for $N$-level systems”, Phys. Lett. A, 314 (2003), 339–349 | DOI | MR | Zbl

[20] M. S. Byrd, N. Khaneja, “Characterization of the positivity of the density matrix in terms of the coherence vector representation”, Phys. Rev. A, 68 (2003), 062322, 13 pp. | DOI | MR

[21] V. Gerdt, A. Khvedelidze, Yu. Palii, “Constraints on $\mathrm{SU}(2)\otimes\mathrm{SU}(2)$ invariant polynomials for a pair of entangled qubits”, Phys. Atom. Nucl. (to appear)

[22] M. Hochster, J. Roberts, “Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay”, Advances in Mathematic, 13, 1974, 125–175 | MR

[23] D. Djokovic, Poincaré series for local unitary invariants of mixed states of the qubit-qutrit system, 2006, 5 pp., arXiv: quant-ph/0605018v1

[24] A. J. Macfarlane, H. Pfeiffer, “On characteristic equations, trace identities and Casimir operators of simple Lie algebras”, J. Math. Phys., 41 (2000), 3192–3225 | DOI | MR | Zbl

[25] V. I. Ogievetsky, I. V. Polubarinov, “Eightfold-way formalism in $\mathrm{SU}(3)$ and 10- and 27-plets”, Yad. Fiz., 4 (1965), 853–861

[26] A. P. Bukhvostov, The algebra of $3\times 3$ matrices in 8-dimensional vector representation, Preprint of St-Petersburg INP-1821/-1822, 1992