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 $X$ |

$X_{⊤}$ | Transpose of a matrix $X$ |

$X$ | complex conjugate of $X$. $X$ can be a matrix, a scalar, a function, etc |

$[⋅,⋅]$ | Commutator |

$E[⋅]$ | expectation operator |

$P[A]$ | probability of an event $A$ |

$L$ | Quantum Dynamical Semigroup generator |

$Tr[⋅]$ | Trace |

$Tr_{E}[⋅]$ | Partial Trace over a subspace $E$. This is usually referred to as tracing an operator over $E$. |

$f(x)∝g(x)$ | $f$ is proportional to $g$, meaning that $gf $ is a constant |

$H$ | The Hamiltonian of a system |

$H$ | A Hilbert Space |

$b≫a$ | $∃R≥a:∀b≥R$, “some condition” holds. It is informally read as “a condition is true for $b$ much greater than $a$“. |

$G$ | Groups, including Lie groups |

$g$ | Lie algebra |

$∫XdW$ | Stochastic integral under the Ito Calculus framework |

$∫X∘dW$ | Stochastic integral under the Stratonovich Calculus framework |

Info

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

# Hat operators

Hats over Operators, such as $ρ^ $ and $U^_{t}$, are often skipped in favour of $ρ$ 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 $K_{α}$ as it does $K_{α}$. Standardising indices as a sub script allows me to immediately know that $K_{α}$ is $K$ indexed by $α$, whereas $K_{α}$ is $K$ to the power of $α$. 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 $∫t_{2}dt$ and never $∫dtt_{2}$. This choice clarifies what the integrand is, for instance, $∫_{0}ρds$ is the integral with integrand $ρ(s)$, even though we decided to omit the parameterization of $ρ$ as a function of $s$.

# ”Identity” function

I often use the function $I_{A}={10 ifx∈Aifx∈/A $ 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: $θ(x)=I_{x≥0}$
- Kronecker delta: $δ_{ij}=I_{i=j}$
- step function on $[a,b]$ : $I_{[a,b]}$

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

$f(x)={g(x)h(x) ifx∈Bifx∈C $

where $B∩C=∅$ as $f(x)=g(x)I_{B}+h(x)I_{C}$. I find the latter easier to reason with during purely algebraic manipulations.