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What is the meaning of writing a state $|psirangle$ in its Bloch representation. Would I be correct in saying it's just writing out its Bloch vector?

The Bloch sphere formalism is used for geometrically representing pure and mixed states of two-dimensional quantum systems a.k.a qubits. Any pure state $|Psirangle$ of a qubit can be written in the form:

$$|Psirangle = cosfractheta2|0rangle + e^iphisinfractheta2|1rangle$$ where $0leq thetaleq pi$ and $0leq phileq 2pi$. This $|Psirangle$ can be represented on the Bloch sphere as:

The Bloch vector $vecain Bbb R^3$ is basically $(sintheta cosphi, sinthetasinphi, cos theta) = (a_1,a_2,a_3)$.

To represent mixed states you need to consider the corresponding density operator $rho$. the set of states of a single qubit can be described in terms of $2times 2$ density matrices and as $I,X,Y,Z$ forms a basis for the vector space of $2times 2$ Hermitian matrices, you can write the density operator as $$rho = a_0I+a_1X+a_2Y+a_3Z = frac12beginpmatrix1+a_3 & a_1-ia_2 \ a_1+ia_2 & 1-a_3endpmatrix.$$ As density matrices always have trace $1$, and here $mathrmtr(rho)=2a_0$, so $a_0$ is necessarily $frac12$. So from here you can find out the Bloch coordinates of the any mixed state i.e. $(a_1,a_2,a_3)$ after performing the Pauli decompostion of the density matrix. If you're wondering what ensures that $|veca|leq 1$, it's the positive semidefiniteness! The two eigenvalues of $rho$ are $frac12(1+|veca|)$ and $frac12(1-|veca|)$. Thus, to ensure that the second eigenvalue is non-negative, $|veca|leq 1$. The three properties of density matrices which you should drill into your brain are: self-adjointness, positive-semidefiniteness and unit trace.

Once you determine the values $a_1,a_2$ and $a_3$ from the density operator, you can easily find the location of the qubit state $(sintheta cosphi, sinthetasinphi, cos theta)$ inside the Bloch sphere. Let me emphasize on this point: pure states lie on the Bloch sphere (i.e. $|veca|=1$) whereas mixed states lie inside the Bloch sphere (i.e. $|veca|<1$); prove this as an exercise. If you're mathematically inclined, you'll also love to think about the Bloch sphere in terms of stereographic projections; it's excellently summarized in this Physics SE answer. I'll attach the image from there, which is originally from this blogpost (the article is in French, sorry :).

Here are a few "further readings" for you:

• Density matrices for pure states and mixed states


• Why do Bloch sphere wavefunctions have half angles?



• Homeomorphism or stereographic projection corresponding to the set of mixed states within the Bloch sphere




• Purity of mixed states as a function of radial distance from origin of Bloch ball




• Can the Bloch sphere be generalized to two qubits?




• Why is an entangled qubit shown at the origin of a Bloch sphere?

Essentially, go through the bloch-sphere tag; you'll find several interesting questions and answers, which should clarify most of your beginner confusions about the Bloch sphere formalism.