Notation choices

There is variability in the way authors write their mathematics and physics. In this webpage we’ve made some choices:

Frequently used icons

Notation consistency removes the mental overhead of understanding notation when jumping through different contexts. I (try to) use the symbols below consistently across all notes.

Symbol	Meaning
$X^{†}$	Adjoint Operator of a Matrix $X$
$X^{\top }$	Transpose of a Matrix $X$
$\overline{X}$	complex conjugate of $X$. $X$ can be a Matrix, a scalar, a function, etc
$[\cdot ,\cdot ]$	Commutator
$\mathbb{E}[\cdot ]$	expectation operator
$\mathbb{P}[A]$	probability of an event $A$
$\mathbb{L}$	Quantum Dynamical Semigroup generator
$\mathrm{Tr}[\cdot ]$	Trace
${\mathrm{Tr}}^E[\cdot ]$	Partial Trace over a subspace $E$. This is usually referred to as tracing an operator over $E$.
$f(x)\propto g(x)$	$f$ is proportional to $g$, meaning that $\frac{f}{g}$ is a constant
$H$	The Hamiltonian of a system
$\mathcal{H}$	A Hilbert Space
$b\gg a$	$\exists R\ge a:\forall b\ge R$, “some condition” holds. It is informally read as “a condition is true for $b$ much greater than $a$“.
$G$	Groups, including Lie groups
$\mathfrak{g}$	Lie algebra
$\int XdW$	Stochastic integral under the Ito Calculus framework
$\int X\circ dW$	Stochastic integral under the Stratonovich Calculus framework
$\mathcal{F}$	Fisher Information or Filtration, dependent on context

Info

If you find incorrect usage of the above notation, do reach out.

Hat operators

Hats over Operators, such as $\hat{\rho }$ and ${\hat{U}}_t^{†}$, are often skipped in favour of $\rho$ and $U_t^{†}$. I try to make Operators clear from context, and writing the hat on top of every symbol is time consuming.

Super / sub scripts

Indices are subscript, powers are superscripts.

It takes as much work to write $K_{\alpha }$ as it does $K^{\alpha }$. Standardising indices as a sub script allows me to immediately know that $K_{\alpha }$ is $K$ indexed by $\alpha$, whereas $K^{\alpha }$ is $K$ to the power of $\alpha$. If we have multiple scripts, we can keep them ordered, for example $K_{ijkl}$ is indexed by 4 numbers.

Integrands

We always use $dt$ as one of the delimiters of the integrand, i.e. we always write $\int t^{2}dt$ and never $\int dt{t}^2$. This choice clarifies what the integrand is, for instance, ${\int }_0^t\rho ds$ is the integral with integrand $\rho (s)$, even though we decided to omit the parameterization of $\rho$ as a function of $s$.

”Identity” function

I often use the function ${\mathbb{I}}_A=\{\begin{array}{ll}1& \text{ if }x\in A\\ 0& \text{ if }x\notin A\end{array}$ in unusual places. This notation is very powerful as it is easy to memorise, and it replaces a lot of other functions, such as:

• heaviside step function: $\theta (x)={\mathbb{I}}_{x\ge 0}$
• Kronecker delta: ${\delta }_{ij}={\mathbb{I}}_{i=j}$
• step function on $[a,b]$ : ${\mathbb{I}}_{[a,b]}$

It also come very handy as a way of preventing the branching of equations: I can write any branched equation

$f(x)=\{\begin{array}{ll}g(x)& \text{ if }x\in B\\ h(x)& \text{ if }x\in C\end{array}$

where $B\cap C=\varnothing$ as $f(x)=g(x){\mathbb{I}}_B+h(x){\mathbb{I}}_C$. I find the latter easier to manipulate algebraically.