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The Bloch sphere is an excellent way to visualize the state-space available to a single qubit, both for pure and mixed states. Aside from its connection to physical orientation of spin in a spin-1/2 particle (as given by the Bloch vector), it's also easy to derive from first principles mathematically, using the Pauli spin operators $$\mathcal S = \bigl\{\mathbf 1, X,Y,Z \bigr\} = \Bigl\{\mathbf 1,\; \sigma^{(x)}\!,\; \sigma^{(y)}\!,\; \sigma^{(z)}\Bigr\}$$ as an operator basis, and using only the facts that $\mathbf 1$ is the only element in $\mathcal S$ with non-zero trace, and that $\def\tr{\mathop{\mathrm{tr}}}\tr(\rho) = 1$ and $\tr(\rho^2) \leqslant 1$ for density operators $\rho$.

Question. Using either the Pauli operators as a matrix basis, or some other decomposition of Hermitian operators / positive semidefinite operators / unit trace operators on $\mathbb C^4 \cong \mathbb C^2 \otimes \mathbb C^2$, is there a simple presentation of the state-space of a two-qubit system — or more generally, a spin-3/2 system in which we do not recognise any tensor-product decomposition of the state-space (but might use some other decomposition of the state-space)?

My interest here is that the representation be simple.

  • The representation doesn't have to be presented visually in three dimensions; I mean that the constraints of the parameter space can be succinctly described, and specifically can be presented in algebraic terms which are easy to formulate and verify (as with the norm-squared of the Bloch vector being at most 1, with equality if and only if the state is pure).

  • The representation should also be practical for describing/computing relationships between states. For instance, with the Bloch representation, I can easily tell when two pure states are orthogonal, when two bases are mutually unbiased, or when one state is a mixture of two or more others, because they can be presented in terms of colinearity/coplanarity or orthogonality relationships. Essentially, it should be a representation in which linear (super-)operations and relationships on states should correspond to very simple transformations or relationships of the representation; ideally linear ones.

  • The representation should be unable to represent some Hermitian operators which are not density operators. For instance, there is no way to represent operators whose trace is not 1 in the Bloch representation (though non-positive operators can be represented by Bloch vectors with norm greater than 1). In fact, the Bloch representation essentially is that of the affine space of unit trace in the space of Hermitian matrices, centered on $\mathbf 1/2$. A simple and concise geometric description of density operators as a subset of the affine plane of unit-trace Hermitian operators centered on $\mathbf 1/2 \otimes \mathbf 1/2$, i.e. a generalization of the Bloch sphere representation (but not necessarily using the Pauli spin basis), would be ideal.

If there is such a representation, does it generalize? How would one construct a similar representation, for instance, for qutrits (spin-1 systems) or three-qubit states (spin-5/2 systems)? However, these questions should be understood to be secondary to the question for two qubits.

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One can represent a mixed qubit state on a Bloch sphere because $su(2)=so(3)$ and the manifold dimension is 3.

The relevant group for the double qubit is $su(4)=so(6)$, and the manifold has dimension 15. Apart from the description as the set of complex psd $4\times 4$ matrices of trace 1, one probably has a description as the set of real orthogonal $6\times 6$ matrices of determinant 1. This is not really simpler, but satisfies your final condition.

For the other cases, there is no exceptional isomorphism, and the description as the set of psd matrices of trace 1 is the only simple one.

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I would like somehow to unpack this description of the symmetries of the space, to obtain a more direct account of the space itself. For instance, from the fact that the unitary transformations of a single qubit is isomorphic to $so(3)$, I can infer that the state-space is the convex closure of a set whose symmetries are $so(3)$-symmetric, i.e. a sphere. But extra work is required to determine that it is the unit sphere (i.e. that the radial co-ordinate corresponds to measurement outcome bias), and it does not allow me to infer how else to use the representation. –  Niel de Beaudrap Oct 19 '12 at 14:33
    
@NieldeBeaudrap: It is a compact manifold of dimension 3, hence a sphere. It can have an arbitrary radius. Taking it to be a unit sphere is just a normalization of the transformation. The meaning of the coordinates can be found out by looking at what happens when particular generators are applied. –  Arnold Neumaier Oct 19 '12 at 14:45
    
It is also not immediately evident how properties such as orthogonality, mutually unbiased bases, etc. translate into a bloch-like representation through an isomorphism such as $su(4) = so(6)$. Perhaps it is just a matter of calculation, but at least for spin-1/2, the direct approach using the Pauli basis makes it quite manifest. Furthermore: while it's obvious that the two-qubit operators are embeddable in a space of dimension 15 (= $4^2 - 1$), it's not clear how to apply the symmetry group of $\mathrm{SO}(6)$, which are the rotations of the sphere in $\mathbb R^6$. –  Niel de Beaudrap Oct 19 '12 at 14:55
    
@NieldeBeaudrap: Sure. The exceptional isomorphisms themselves are not immediately evident, and one needs some time to discover all their implications. - I am not even sure whether the so(6) manifold I gave is the right one. You need to work it out on your own, starting form an explicit isomorphism, if you are interested, or search the literature haystack for the needle that contains an answer to your question. –  Arnold Neumaier Oct 19 '12 at 15:04
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In any case, you have very few choices for nice, natural descriptions, and for $n$-level systems with $n\ne 2,4$ you don't really have any alternative to the density matrix. But this is a very tractable natural description for any $n$ and makes for good algorithms, so there is no real need for alternatives. The case $n=2$ is special as the sphere is far better known than the matrix alternative. –  Arnold Neumaier Oct 19 '12 at 15:08
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