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arXiv:1312.7413 · Journal of Physics A Mathematical and Theoretical

The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group $SL(3,{\mathbb {C}})$

Peter Jarvis

Published May 7, 2014• 11 citations

Abstract

We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.

DOI: 10.1088/1751-8113/47/21/215302

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The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group

SL(3,{\mathbb {C}})

We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an

N

-qubit entanglement monotone if and only if the homogeneous degree is not larger than 4. We then describe a common basis and formalism for the

N

-tangle and other known invariant polynomials of degree 4. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.

Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2) × SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.

We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SL(3,C)×3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.

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. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare 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 algebra su(6).

A family of N -qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the 3-tangle and some of the four-, five- and N -qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form with L = 2 N − n ⩾ l = 2 n . Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr ( L , l ) of l -planes in C L that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the Plücker embedding. The invariants are neatly expressed in terms of the Plücker coordinates of the Grassmannians.