9
$\begingroup$

I need some useful sources about the geometry of qutrit. Specifically related to the Gell-Mann matrix representation.

$\endgroup$
  • 4
    $\begingroup$ Hello! What specific information are you looking for, about the relationship between the Gell-Mann matrices and the geometry of qutrits? Would you be willing to expand on your question a little? $\endgroup$ – Niel de Beaudrap Sep 18 '18 at 15:36
  • $\begingroup$ arxiv.org/abs/1501.00054 page 9. If this matches the sort of thing you are looking for, I'll expand more. $\endgroup$ – AHusain Sep 19 '18 at 6:19
6
$\begingroup$

There are many ways to describe a qutrit or a general $N$ level system geometrically. There is also a large amount of references either explaining these geometries or applying them to various problems in quantum information. I'll try to explain here one quite general geometrical method, somewhat in detail.

This method is a generalization of the Bloch sphere of the qubit, however, the qubit case is degenerate because the Bloch sphere describes the parameter space of both pure and mixed qubits (but not the maximally mixed case), while in the general case, the geometry of the parameter space depends on the degeneracy structure of the eigenvalues of the density matrix.

The description is based on the diagonalization formula of the density matrix of a general $N$ level density matrix: $$ \rho = U \Lambda U^{-1}$$ Where $\Lambda$ is the eigenvalue matrix, which in the most general case has the form: $$\Lambda = \mathrm{diag}(\underbrace{\lambda_1, \lambda_1, …}_{N_1 \mathrm{times}}, \underbrace{\lambda_2, \lambda_2, …}_{N_2 \mathrm{times}}, ...., \underbrace{\lambda_k, \lambda_k, …}_{N_k \mathrm{times}})$$ The matrix $U$ is an $N$ dimensional unitary matrix, i.e., belonging to $N$-dimensional unitary group $U(N)$.

Of course, since we are diagonalizing a density matrix, we must have: $$\sum_{i=1}^{k}N_i \lambda_i = 1 \space\mathrm{and} \space \lambda_i \ge 0 \space \mathrm{for \space all}\space i$$ Inspecting the eigenvalue vector, we see that the action of a general $N_i$ unitary matrix belonging to a $U(N_i)$ subgroup keeps the eigenvalue matrix diagonal, therefore the space of the unitary transformations that do change the density matrix can be identified with the coset space: $$\frac {U(N)}{U(N_1) \times U(N_2) … \times U(N_k)}$$ The above spaces are called coadjoint orbits. They all admit explicit parametrizations in coordinates which allow actual work on specific cases. They are compact, homogeneous and Kähler i.e., they are compatibly complex and Riemannian. They are described in a rather elementary manner in the following work of Bernatska and Holod. Please see the following work by Loi, Mossa and Zuddas for explicit parametrization formulas for general cases.

However, even from the general form of the coset space we can extract some information of the parameter space, namely its dimension, which is in the general case: $$d = N^2-\sum_i N_i^2$$ In the following paragraph, I'll describe in more detail the case of a pure qutrit case. Here: $$\Lambda = \mathrm{diag}(1, 0, 0)$$ And the space parametrizing the pure single qutrit case is: $$\mathbb{C}P^2 = \frac {U(3)}{U(1) \times U(2)}$$ This is the two-dimensional complex projective space, (having a real dimension of $4$).

It is quite easy to parametrize this space since we know that we can parametrize (almost) every pure qutrit space by means of the unit vector: $$v = \frac{1}{\sqrt{1+|z_1|^2+|z_2|^2}}\begin{pmatrix}1 \\z_1 \\ z_2 \end{pmatrix}$$ The coordinates $(z_1, z_2)$ are the complex coordinates of $\mathbb{C}P^2$

We get the parametrization of the pure qutrit density matrix (which is a projector): $$ \rho(z_1, z_2, \bar{z}_1, \bar{z}_2) = v v^{\dagger} = \frac{1}{1+|z_1|^2+|z_2|^2}\begin{pmatrix}1 &\bar{z}_1 & \bar{z}_2 \\ z_1 &z_1 \bar{z}_1 & z_1 \bar{z}_2 \\ z_2 &z_2 \bar{z}_1 & z_2 \bar{z}_2 \end{pmatrix}$$ This space is symplectic, which characteristic to closed quantum system, its symplectic form sometimes called KKS (after Kirillov-Kostant-Souriau) is given by: $$\omega_{\alpha \bar{\beta}} = \mathrm{tr}\partial_{\alpha} \rho \bar{\partial}_{\beta} \rho$$ Being Kählerian, the symplectic form can be computed from a Kähler potential: $$\omega_{\alpha \bar{\beta}} = \partial_{\alpha} \bar{\partial}_{\beta} K$$ with $$K = \ln(1 + |z_1|^2 +|z_2|^2)$$ Given a set of Gell-Mann matrices $\mathbf{G}_i, \space i=1,…,8$, their expectation values in a general pure qubit state given by $$ G (z_1, z_2, \bar{z}_1, \bar{z}_2) = \mathrm{tr}(\rho(z_1, z_2, \bar{z}_1, \bar{z}_2) \mathbf{G}_i)$$

Become classical Hamiltonians on $\mathbb{C}P^2$, and their algebra closes to the Lie algebra $\mathfrak{su}(3)$ under the Poisson brackets:

$$\{G_i, G_j\} = \omega^{\alpha \bar{\beta}} (\partial_{\alpha} G_i \bar{\partial}_{\alpha} G_j - \partial_{\alpha} G_j\bar{\partial}_{\alpha} G_i )$$

Where $\omega$ with the upper indices is the inverse symplectic form.

This formulation of the qutrit allows many applications in quantum information theory, please see, for example, Hughston and Salamon , where they construct a SIC-POVM using this parametrization.

Another application by Chaturvedi, Ercolessi, Marmo, Morandi, Mukunda and Simon. Although they do not spell out the above parametrization, but they show that the connection: $$A_{\alpha} = (\partial_{\alpha} - \bar{\partial}_{\alpha })K$$ is a Berry connection which gives rise to Berry phases that can be used in holonomic quantum computation to generate gates for the qutrit system, Please see for example Boya, Perelomov and Santander and Khanna, Mukhopadhyay, Simon and Mukunda.

One very important property of the pure state parameter spaces is that there is a geometric interpretation of the measurement probabilities as follows: The complex projective spaces parametrizing the pure states are equipped with a metric called the Fubini-Study metric. The measurement probabilities of any observable, (for example one of the Gell-Mann matrices) is proportional to the geodesic length from the point representing the state to the point representing the observable eigenstate projector in $\mathbb{C}P^N$. Please see the important work by Ashtekar and Schilling. As far as I know a generalization of this property to mixed state cases has not been found.

In the case of $\mathbb{C}P^2$, the Fubini-Study metric is given by: $$g_{\alpha \bar{\beta}}= \frac{(1 + |z_1|^2 +|z_2|^2)\delta_{\alpha \beta}-z_{\alpha} \bar{z_{\beta}} }{(1 + |z_1|^2 +|z_2|^2)^2}$$

| improve this answer | |
$\endgroup$
  • $\begingroup$ The paper I linked in comment was an Ercolessi et al one about $\omega_{KKS}$ as it related to Fisher information $\endgroup$ – AHusain Sep 20 '18 at 13:32
  • 1
    $\begingroup$ @AHusain Thank you for the reference! and thank you for mentioning the relation of the metric to the Fisher information matrix. Actually, the first equation above for the KKS form is identical to the equation in their Proposition II.2. , expressed in complex coordinates. As I mentioned, these authors do not use the same complex parametrization. They rather prefer to work on the tangent space. This is possible since the manifold is homogeneous, however it is hard to evaluate in their parametrization global geometric quantities such as what is the volume of the state space. $\endgroup$ – David Bar Moshe Sep 20 '18 at 13:54
  • 1
    $\begingroup$ Yes it is hard. The same manifold with the Fisher information structure instead of the Kahler structure is much uglier. Authors of that paper and I have tried seeing what the difference is between the Berry phases of the two structures, but nothing pretty. $\endgroup$ – AHusain Sep 20 '18 at 16:45
3
$\begingroup$
I need some useful sources about the geometry of qutrit.

The most useful resource I know on the geometries of qutrits is the paper Geometry of the generalized Bloch sphere for qutrits.

Specifically related to the Gell-Mann matrix representation.

The eight Gell-Mann matrices, which form one of the generalizations of Pauli matrices to 3-level systems, are involved in what is sometimes called the "Bloch representation of a qutrit". This is described on Page 4 of the above linked paper.

If you are interested in the mathematics of the geometry of qutrits, the above resource is probably the best available. If you are more interested in the visualization of qutrits, the paper Three-dimensional visualization of a qutrit is the best resource I know. Keep in mind that generalizations of the Bloch sphere for higher dimensional qudits will never be as simple and elegant as the Bloch sphere is for 2-level systems, just as 4D hyper-spheres are not as easy to visualize as 3D spheres.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.