Ten results for deeper study

Where to begin.

For a theorem–proof–example study programme, prioritise these ten results.

PRIORITY 1

$[A,\rho(g)]=0\ \forall g\Rightarrow A=\lambda I$

For an irreducible complex representation, any operator commuting with every group element is scalar.

PRIORITY 2

$[X,P]\neq i\hbar I\ \text{for finite matrices}$

Taking the trace of $[X,P]=i\hbar I_d$ gives $0=i\hbar d$, a contradiction.

PRIORITY 3

$A=A^\dagger\Rightarrow A=\sum_i a_i|i\rangle\langle i|$

Every finite-dimensional Hermitian operator has real eigenvalues and an orthonormal eigenbasis.

PRIORITY 4

$\nexists\,U:\ U|\psi\rangle|0\rangle=|\psi\rangle|\psi\rangle\ \forall\psi$

Linearity of quantum mechanics forbids copying an unknown state.

PRIORITY 5

$S(ABC)+S(B)\le S(AB)+S(BC)$

Deep Lieb-Ruskai theorem, equivalent to many other entropy inequalities; it underpins nearly all of quantum information theory.

PRIORITY 6

$D(\mathcal N(\rho)\lVert\mathcal N(\sigma))\le D(\rho\lVert\sigma)$

Applying a channel can only make states less distinguishable.

PRIORITY 7

$\Theta^2=-I\Rightarrow E\ \text{levels doubly degenerate}$

For odd numbers of spin-1/2 particles with time-reversal symmetry, every level has a partner; magnetic fields break it.

PRIORITY 8

$PE_a^\dagger E_bP=c_{ab}P$

A code with projector $P$ corrects error set $\{E_a\}$ iff this holds with $c$ Hermitian: errors must not reveal logical information.

PRIORITY 9

$\text{Clifford + Pauli measurements}\in\mathrm{P}$

Stabilizer circuits can be simulated classically in polynomial time despite generating entanglement.

PRIORITY 10

$T\gg\hbar/\Delta_{gap}\Rightarrow\text{state follows eigenspace}$

If the Hamiltonian changes slowly relative to the gap, a system stays in its instantaneous eigenstate.

ψWhere to begin.