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A mixed state $\rho$ can be written as

$$\rho=\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)\qquad\left(\vec{r}:=\left(r_x,r_y,r_z\right)^T\in\mathbb{R}^3; ||\vec{r}||\leq 1\right)$$

according to Nielsen & Chuang in "Quantum Computation and Quantum Information" (from 2010) (p. 105).

Does anyone know a reference where this has been proven and not just stated? Thank you!

glS
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manuel459
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1 Answers1

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The state space of a qubit $\mathscr H\simeq \mathbb C^2$ is two-dimensional. The Pauli matrices $\{\sigma_i\}_{i=0,\ldots,3}$ form a basis for the real vector space of hermitian $2\times2$ matrices of dimension $2^2=4$, cf. e.g.this and the links therein. As such, any hermitian matrix can be expanded in this basis set. A density operator is a positive operator (and hence hermitian) of unit trace and you can thus expand it in the basis of Pauli matrices:

$$ \rho = \sum\limits_{i=0}^3 r_i \, \sigma_i \quad , \tag{1}$$ with $r_i \in \mathbb R .$

Now use the positivity and trace condition to find conditions for the expansion coefficients. For example, you can write $\rho$ in a the basis of eigenvectors of $\sigma_3$ and diagonalize it. Then you'll see that $\rho$ is a density matrix if and only if $$r_0=\frac{1}{2}\quad \text{and}\quad\sum\limits_{i=1}^3 r_i^2 \leq \frac{1}{4} \tag{2} $$holds.

Tobias Fünke
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  • By chance: do you know a reference (book) that has comments on the said basis property of pauli matrices? Thank you! – manuel459 Mar 15 '22 at 18:42
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    @manuel459 Sorry, no. But I think this is a fairly well-known fact, so there should be some things online, perhaps Wikipedia, yet I don't know. Does this help? – Tobias Fünke Mar 15 '22 at 18:53
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    I mean this fact is stated in some books (cf. Preskill's lecture notes) but not proven. – Tobias Fünke Mar 15 '22 at 18:58
  • Stated is enough in this minor case. I'll take a look at it! Do you know which number? As far as it is something I could "officially" cite it works perfect. Thank you again. – manuel459 Mar 15 '22 at 18:59
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    @manuel459 Chapter 2.3.2 in his lecture notes from 1998. Also in Wikipedia this fact is mentioned several times. You might have a look there. But they don't cite a source as far as I see. – Tobias Fünke Mar 15 '22 at 19:36
  • Thank you very much! – manuel459 Mar 15 '22 at 20:18