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arXiv:1004.5252 · Journal of Physics A Mathematical and Theoretical

A composite parameterization of unitary groups, density matrices and subspaces

Christoph Spengler, Marcus Huber, Beatrix C. Hiesmayr

Published Aug 17, 2010• 47 citations
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Abstract

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way. We show explicitly how any element of can be composed of matrix exponential functions of generalized anti-symmetric σ-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank k can be formulated. Our construction can also be used to derive an orthonormal basis of any k -dimensional subspaces of with the minimal number of parameters. As an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the m -concurrence).

DOI
10.1088/1751-8113/43/38/385306

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Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way. We show explicitly how any element of can be composed of matrix exponential functions of generalized anti-symmetric σ-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank k can be formulated. Our construction can also be used to derive an orthonormal basis of any k -dimensional subspaces of with the minimal number of parameters. As an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the m -concurrence).
Abstract We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one‐particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group‐theoretic sense, for the group of unitary transformations of the set of one‐electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle‐number‐conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non‐particle‐number‐conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.
This paper presents a parametrization of a degenerate density matrix. The problem needs to be approached first with a diagonalized form (the spectral representation) to deal with degeneracy. Such a form is useful for this parametrization in that the conditions to be a density matrix from a Hermitian matrix are applied only to a diagonal eigenvalue matrix, not a unitary matrix. Those conditions can be satisfied by parametrizing eigenvalues with squared spherical coordinates in dimension of the matrix. Degeneracy in eigenvalues brings symmetries between a eigenvalue matrix and a unitary matrix, which are realized in a form of a commuting unitary matrix, called a commutant. The associated redundant parameters in a unitary matrix have to be eliminated. It is realized in this paper that degrees of degeneracies can be defined as the total number of possible pairs of the same eigenvalues and one degree of degeneracy corresponds to one phase and one two dimensional rotation in a unitary matrix or a commutant. In this way all the degeneracies are identified and assigned to one phase-one rotation block. Therefore, a unitary matrix or a commutant is a product of these blocks and a general diagonal phase matrix. In physics a unitary matrix is often parametrized by rotation and phase matrices, called an angular representation here. There are many possible different phase configurations. It is often not a trivial matter whether a given parametrized unitary matrix is general. A simple diagram will be introduced to illustrate how to transform one phase configuration to another.
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.
We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.
We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.