# Standard Notation

Should we agree on some standard notations? This should make it much easier to understand new posts once a notation is established.

In particular, I think it would be helpful to agree on symbols for common gates beforehand. This applies to formulas and circuit diagrams.

I suggest we start a list of notation here, then we can refer to this thread to conveniently define the symbols we use and to encourage new users to follow our convention. Use one answer for a single symbol or a group of symbols that logically belong together. Give the symbol and the description in a header, and give context, explanation, references below it.

Different notations for the same entity might be acceptable, and the votes will indicate the most popular variant. We can also discuss it in comments, of course.

• Consider adding the meta-faq tag to this question. – Sanchayan Dutta Mar 15 '18 at 9:26
• a great idea. I'm sure this will prevent confusion :D – ItamarG3 Mar 21 '18 at 10:30

Definition/notation: $n$-qubit ($2^n$-dimensional) Hadamard denoted by $H^{\otimes n}$, with $H^{\otimes 0} = 1$ and the single qubit (2 dimensional) Hadamard $H = H^{\otimes 1}$

$$H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix},$$ $$H^{\otimes n} = H\otimes H^{\otimes \left(n-1\right)}$$

Mathjax: $H$, $H^{\otimes n}$

# $\operatorname{tr}$: trace

Definition: $$\operatorname{tr}A= \sum_k \langle k | A |k\rangle = \sum_k A_{kk}$$
MathJax: \operatorname{tr}

# $X$, $Y$, $Z$: Pauli matrices

Definition: $$\begin{array}{ccccc} X=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right), &&Y=\left(\begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array}\right), &&Z=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right) \end{array}$$ MathJax: $X$, $Y$, $Z$

# $^{\mathsf{T}}$, $^\dagger$: transpose, conjugate transpose

Definition: The transpose of a matrix $M$, denoted by $M^{\mathsf{T}}$, has elements $M^{\mathsf{T}}_{ij} = M_{ji}$. The conjugate transpose additionally performs a complex conjugation $^*$: $M^{\dagger}_{ij} = M_{ji}^*$.

MathJax: $^{\mathsf{T}}$, $^\dagger$

• You could add also ordinary transpose here, and mention that it is preferably not italicized as it is not a variable. – Kiro Mar 23 '18 at 15:52
• @Kiro: like this? somehow the T is pretty large, but I guess we also don't want to use very long MathJax for it... – M. Stern Mar 24 '18 at 0:07
• Looks good to me. – Kiro Mar 24 '18 at 14:56

# $\rho$: Density operator

Definition: $\rho$ is a positive semidefinite, normalized operator ($\rho\geq 0$, $\mathrm{tr } \rho = 1$)
MathJax: $\rho$

• It is not only hermitian and normalized, but also a positive operator, i.e., its eigenvalues are non-negative real numbers. – Zoltan Zimboras Mar 15 '18 at 11:05
• thanks for changing it. Btw, I would still use the phrase "positive operator" (and not "positive semidefinite", as you write it now). See the definitions: quantiki.org/wiki/positive-operator and encyclopediaofmath.org/index.php/Positive_operator , – Zoltan Zimboras Mar 21 '18 at 13:25
• @ZoltanZimboras or non-negative-definite operator. :) All three versions are used. – M. Stern Mar 22 '18 at 19:26

# $\mathbb{I}$: Identity operator

Definition: $$\mathbb{I}=\left( \begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 1 & & 0 \\ \vdots & & \ddots & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$ MathJax: \mathbb{I}

• There are numerous conventions for the identity. A math-bold $I$ is one of them, but on this site has the unfortunate feature of being sans-serif, which therefore looks like a hollow rectangular box in a way that likely won't play well with Dirac notation. $\mathbf I$ (\mathbf I), $\mathbf 1$ (\mathbf 1), $\mathbb 1$ (\mathbb 1), or just plain $I$ are preferable in my opinion. – Niel de Beaudrap Mar 28 '18 at 23:35
• For me $\mathbf 1$ has every components equal to 1! :) Actually I prefer \mathds{1} for the identity operator, but it's not supported. I'm totally willing to go with whatever the majority prefers – M. Stern Mar 29 '18 at 13:25

# $C(U)_{ij}$: controlled-Unitary

Definition: controlled Unitary, where $i$ is the control and $j$ is the target.
Mathjax: C(U)_{ij}
circuit diagram (created with overleaf):

Special cases:
$U=X$ (CNOT gate)

$U=Z$ (controlled-Phase gate)