Bloch sphere representations for multi-qubit quantum systems

Question

Just a short mixed quantum state representation question. Given a single qubit density matrix $\rho$, since the Pauli matrices form a basis for 2x2 complex matrices, the Bloch sphere representation can be given as

$$\rho = I + \vec{r} \cdot \vec{\sigma},$$

where $\vec{r} = (r_x,r_y,r_z)$ and $|\vec{r}| \leq 1$. To generalize this to multiple qubits for some density matrix $\rho$, it seems like a crude way would be to consider the reduced density matrix for each qubit and then average the $r_{x}, r_{y}, r_{z}$ to produce a Bloch sphere representation.

Question: Firstly am I correct in stating that $r_{x}, r_{y}, r_{z}$ coefficients are respectively the expectation values of the observables $\hat{x}, \hat{y}, \hat{z}$? Lastly, is there merit to the crude suggestion or is there a more standard/useful suggestion regarding similar type representations of multi-qubit mixed density matrices?

Answer 1

Let's denote the operators you wrote as $\hat{x}$, $\hat{y}$ and $\hat{z}$ as $\sigma_x$, $\sigma_y$ and $\sigma_z$, respectively. Then $\rho=\frac{1}{2}(I+\sum_{i=x,y,z}r_i\sigma_i)$, as I recall an additional $\frac{1}{2}$ factor in the formula you wrote. Now: $$ \begin{align} \langle\sigma_j\rangle & =\text{tr}(\rho\sigma_j) \\ & =\frac{1}{2}\text{tr}(\sigma_j+\sum_{i=x,y,z}r_i\sigma_j\sigma_i) \\ & =\frac{1}{2}\text{tr}(\sigma_j)+\frac{1}{2}\sum_{i=x,y,z}r_i\text{tr}(\sigma_j\sigma_i)\\ & =0+\frac{1}{2}\sum_{i=x,y,z}2r_i\delta_{ij}\\ & =r_j, \end{align} $$

so yes, you are right when thinking of $r_j$ as the expectation value of the respective Pauli matrix $\sigma_j$.

Regarding the second part of your question: as the above expression for $\rho$ makes clear, you parametrize the space of all single-qubit density matrices, i.e. all hermitian, positive semi-definite, $2\times2$ complex matrices with trace 1, by means of three real numbers, which allow for a three-dimensional visualization as by means of a Bloch sphere. Therefore, multi-qubit systems could only be visualized on higher-dimensional Bloch hyperspheres. However, as you already guessed, it's possible to use several Bloch spheres, not necessarily $n$ as well, at the same time to describe $n$-qubit states in many different ways depending on some properties of the given state, which may be pure or mixed as well as entangled or separable. For instance, it is even possible to make up a single 2-qubit Bloch sphere as long as your two-qubit state is pure.