date: 2025.08.20

Quantum Computing and Quantum Information Cheat Sheet

Classical vs Quantum, Canonical structure

• Canonical commutation relation
  \([x,p]=i\hbar\)
• Heisenberg uncertainty (uncertainty relation)
  \(\Delta x\,\Delta p \ge \frac{\hbar}{2}\)
• Classical Hamiltonian
  \(H_{\rm cl}(x,p)=\frac{p^{2}}{2m}+V(x)\)

Dirac notation & Hilbert space

• State normalization
  \(\langle \psi|\psi\rangle=1\)
• Resolution of identity (orthonormal basis)
  \(\sum_i |i\rangle\langle i|=I\)
• Projector (outer product)
  \(P_\psi=|\psi\rangle\langle\psi|,\qquad P_\psi^2=P_\psi\)
• Expectation value
  \(\langle A\rangle_\psi=\langle\psi|A|\psi\rangle\)

Postulates: dynamics & measurement

• Schrödinger equation (time evolution)
  \(i\hbar\,\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle\)
• Unitary evolution; density operator
  \(U(t)=e^{-iHt/\hbar},\qquad \rho(t)=U\rho(0)U^\dagger\)
• Born rule (POVM measurement)
  \(p(j)=\mathrm{Tr}(M_j\,\rho),\quad M_j\ge0,\quad \sum_j M_j=I\)
• Kraus update (post-measurement state)
  \(\rho\ \to\ \frac{K_j\rho K_j^\dagger}{p(j)},\qquad M_j=K_j^\dagger K_j\)
• Two-outcome projective measurement
  \(M_0=P,\quad M_1=I-P\)

Qubit, Pauli, Bloch sphere, Single-qubit gates

• General pure qubit (Bloch angles)
  \(|\psi\rangle=\cos\frac{\theta}{2}\,|0\rangle+e^{i\phi}\sin\frac{\theta}{2}\,|1\rangle\)
• Bloch representation (Pauli vector)
  \(\rho=\tfrac12\big(I+\vec r\cdot\vec\sigma\big),\qquad |\vec r|\le 1\)
• Pauli matrices
  \(\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\ \sigma_y=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\ \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\)
• Single-qubit rotation (SU(2))
  \(R_{\hat n}(\phi)=e^{-i\frac{\phi}{2}\,\hat n\cdot\vec\sigma}\)
• Pauli expectation ↔ Bloch vector
  \(\langle \sigma_k\rangle=\mathrm{Tr}(\rho\,\sigma_k)=r_k\)
• Mixed state (convex mixture), purity
  \(\rho=\sum_k p_k\,|\psi_k\rangle\langle\psi_k|,\quad \rho\ge0,\ \mathrm{Tr}\rho=1,\quad \mathrm{Tr}(\rho^2)\le1\)

No-go & state discrimination

• No-cloning theorem
  \(\nexists\, U:\ U|\psi\rangle|0\rangle=|\psi\rangle|\psi\rangle\ \ \text{for all }|\psi\rangle\)
• Helstrom bound (minimum-error discrimination)
  \(P_{\rm err}^{\min}=\tfrac12\Big(1-\|p_0\rho_0-p_1\rho_1\|_1\Big),\qquad \|X\|_1=\mathrm{Tr}\sqrt{X^\dagger X}\)

Universal gates / logic

• Universal gate set (example)
  \(\{H,\ T=e^{i\pi\sigma_z/8},\ \mathrm{CNOT}\}\ \text{is universal}\)
• Classical universality (NAND)
  Boolean NAND is universal for classical circuits.

Physical models: Circuit / Adiabatic / MBQC

• Adiabatic interpolation
  \(H(s)=(1-s)H_0+sH_1,\qquad s=\frac{t}{T}\)
• Adiabatic condition (gap condition)
  \(T \gg \max_{s,m\ne n}\frac{\big|\langle m(s)|\partial_s H|n(s)\rangle\big|}{g(s)^2}\)
• Cluster state (MBQC resource)
  \(|C\rangle=\prod_{(i,j)\in E}\mathrm{CZ}_{ij}\,|+\rangle^{\otimes N},\qquad |+\rangle=\tfrac{|0\rangle+|1\rangle}{\sqrt2}\)

Deutsch algorithm (circuit & adiabatic)

• Oracle (phase kickback form)
  \(U_f|x,y\rangle=|x,\,y\oplus f(x)\rangle,\qquad f:\{0,1\}\to\{0,1\}\)

• One-query decision (constant vs balanced)
  Prepare $H\otimes H$ on $|1\rangle|1\rangle$, apply $U_f$, measure first qubit.

• Adiabatic version (problem Hamiltonian)
  \(H_1=\sum_x f(x)\,|x\rangle\langle x|\)

Entanglement, witness, Bell

• Bell pair from circuit
  \(\mathrm{CNOT}(H\otimes I)|00\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}=|\Phi^+\rangle\)
• Separable state (definition)
  \(\rho_{AB}=\sum_k p_k\,\rho_k^{(A)}\otimes\rho_k^{(B)}\)
• Entanglement witness
  \(\mathrm{Tr}(W\rho_{\rm sep})\ge0\ \ \forall\rho_{\rm sep},\qquad \exists\rho_{\rm ent}:\ \mathrm{Tr}(W\rho_{\rm ent})<0\)
• CHSH operator & bounds
  \(S=\langle A_0\!\otimes\!B_0\rangle+\langle A_0\!\otimes\!B_1\rangle+\langle A_1\!\otimes\!B_0\rangle-\langle A_1\!\otimes\!B_1\rangle,\\ |S|\le2\ (\text{LHV}),\) \(|S|\le2\sqrt2\ (\text{quantum})\)

Quantum channel (CPTP), Stinespring, Choi

• Kraus form (CPTP map)
  \(\mathcal N(\rho)=\sum_k E_k\rho E_k^\dagger,\qquad \sum_k E_k^\dagger E_k=I\)
• Stinespring dilation
  \(\mathcal N(\rho)=\mathrm{Tr}_E\!\big[U\,(\rho\!\otimes\!|0\rangle\langle0|)\,U^\dagger\big]\)
• Choi–Jamiołkowski isomorphism
  \(J_{\mathcal N}=(\mathcal N\!\otimes\!\mathrm{Id})(|\Phi^+\rangle\langle\Phi^+|),\qquad |\Phi^+\rangle=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}|ii\rangle\)
• Complementary channel
  \(\mathcal N^c(\rho)=\mathrm{Tr}_B\!\big[U(\rho\!\otimes\!|0\rangle\langle0|)U^\dagger\big]\)

One-time pad & Teleportation (+ identities)

• Classical one-time pad (OTP)
  \(c=m\oplus k,\qquad m=c\oplus k\)
• Teleportation (Bell-basis decomposition)
  \(|\psi\rangle\otimes|\Phi^+\rangle=\frac12\sum_{\mu\in\{I,X,Y,Z\}} |\mathrm{Bell}_\mu\rangle\otimes \mu\,|\psi\rangle\)
• Transpose trick; trace identity
  \((A\otimes I)|\Phi^+\rangle=(I\otimes A^{\mathsf T})|\Phi^+\rangle,\\ \langle\Phi^+|(A\otimes B)|\Phi^+\rangle=\frac{1}{d}\,\mathrm{Tr}(A B^{\mathsf T})\)
• Swap trick / Pauli twirl (qubit)
  \(\mathrm{Tr}(AB)=\mathrm{Tr}\!\big[(A\otimes B^{\mathsf T})\,|\Phi^+\rangle\langle\Phi^+|\big],\qquad \frac{1}{4}\sum_{\sigma\in\{I,X,Y,Z\}}\sigma M \sigma = \frac{\mathrm{Tr}(M)}{2}\,I\)

LU equivalence, Schmidt, Partial transpose

• Schmidt decomposition
  \(|\psi\rangle=\sum_{i}\sqrt{\lambda_i}\,|i\rangle_A|i\rangle_B,\quad \lambda_i\ge0,\ \sum_i\lambda_i=1\)
• Local-unitary (LU) equivalence
  \(\rho'=(U_A\otimes U_B)\,\rho\,(U_A\otimes U_B)^\dagger\)
• Partial transpose (PPT criterion)
  \((|i\rangle\langle j|\otimes|k\rangle\langle \ell|)^{T_B}=|i\rangle\langle j|\otimes|\ell\rangle\langle k|,\\ \rho^{T_B}\ge0\ \text{(necessary; and sufficient for }2{\times}2,\,2{\times}3)\)

QKD (BB84/B92), security, parameter estimation

• BB84 states (mutually unbiased bases)
  \(\{|0\rangle,|1\rangle\},\quad \{|+\rangle,|-\rangle\},\qquad |{\pm}\rangle=\frac{|0\rangle\pm|1\rangle}{\sqrt2}\)
• Quantum bit error rate (QBER)
  \(q=\Pr(a\ne b)\)
• Asymptotic key rate (Devetak–Winter)
  \(K \ge I(A\!:\!B)-I(E\!:\!B)\) BB84 (symmetric, ideal devices)
  \(K \ge 1-2\,h_2(q)\)
• B92 (two-state protocol)
  Non-orthogonal states ${|\psi_0\rangle,|\psi_1\rangle}$ with $|\langle\psi_0|\psi_1\rangle|<1$ and unambiguous-discrimination-based sifting.

Device-independent QKD (DI-QKD), CHSH randomness

• CHSH winning probability
  \(p_{\rm win}=\tfrac12+\frac{S}{8},\qquad p_{\rm win}^{\rm max}=\tfrac12+\tfrac{1}{2\sqrt2}\)
• Min-entropy bound (one typical bound)
  \(H_{\min}(A|E)\ \ge\ 1-\log_2\!\Big(1+\sqrt{2-\tfrac{S^2}{4}}\Big)\)

LOCC convertibility (Nielsen’s majorization)

• Majorization criterion
  \(|\psi\rangle \xrightarrow{\rm LOCC} |\phi\rangle\ \ \text{iff}\ \ \boldsymbol\lambda^\psi \succ \boldsymbol\lambda^\phi: \quad \sum_{i=1}^k \lambda_i^\psi \ge \sum_{i=1}^k \lambda_i^\phi\ \ \forall k\)

Entanglement distillation

• Twirling to Werner/isotropic
  \(\mathcal T(\rho)=\int dU\,(U\!\otimes\!U^*)\,\rho\,(U\!\otimes\!U^*)^\dagger\)
• Werner state
  \(\rho_W=p\,|\Psi^-\rangle\langle\Psi^-|+(1-p)\,\frac{I}{4}\)
• Recurrence fidelity update (example)
  \(F'=\frac{F^2+\frac19(1-F)^2}{F^2+\frac23F(1-F)+\frac59(1-F)^2}\)

Entanglement measures

• Entropy of entanglement (pure state)
  \(E(|\psi\rangle)=S(\rho_A)=-\mathrm{Tr}(\rho_A\log\rho_A)\)
• Concurrence (two-qubit)
  \(C(\rho)=\max\!\big(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4\big),\\ \tilde\rho=(\sigma_y\!\otimes\!\sigma_y)\rho^*(\sigma_y\!\otimes\!\sigma_y),\ \{\lambda_i\}=\text{eig}\!\left(\sqrt{\sqrt{\rho}\,\tilde\rho\,\sqrt{\rho}}\right)\)
• Entanglement of formation (two-qubit)
  \(E_F(\rho)=h_2\!\Big(\frac{1+\sqrt{1-C(\rho)^2}}{2}\Big)\)
• Logarithmic negativity (PPT-based)
  \(E_N(\rho)=\log_2\|\rho^{T_B}\|_1\)
• Relative entropy of entanglement
  \(E_R(\rho)=\min_{\sigma\in\mathrm{SEP}} S(\rho\Vert\sigma)\)

GHZ game / Mermin inequality

• GHZ correlations (GHZ state $|\mathrm{GHZ}\rangle=\frac{|000\rangle+|111\rangle}{\sqrt2}$)
  \(\langle X\!\otimes\!X\!\otimes\!X\rangle=-1,\\ \langle X\!\otimes\!Y\!\otimes\!Y\rangle= \langle Y\!\otimes\!X\!\otimes\!Y\rangle= \langle Y\!\otimes\!Y\!\otimes\!X\rangle=1\)
• Winning rule (promise $x\oplus y\oplus z=0$)
  outputs $a,b,c$ must satisfy $a\oplus b\oplus c=(x\lor y\lor z)$.

CHSH game

• Correlators (correlation function)
  \(E_{xy}=\sum_{a,b\in\{\pm1\}} ab\,P(a,b|x,y)=\langle A_x\!\otimes\!B_y\rangle\)
• CHSH parameter & bounds
  \(S=E_{00}+E_{01}+E_{10}-E_{11},\qquad |S|\le2\ \text{(LHV)},\quad |S|\le2\sqrt2\ \text{(quantum)}\)
• Winning probability (uniform inputs)
  \(p_{\rm win}=\tfrac12+\frac{S}{8}\)

2-qubit gates & circuit identities

• CNOT (controlled-NOT)
  \(\mathrm{CNOT}=|0\rangle\langle0|\!\otimes\!I+|1\rangle\langle1|\!\otimes\!X\)
• CZ (controlled-Z)
  \(\mathrm{CZ}=|0\rangle\langle0|\!\otimes\!I+|1\rangle\langle1|\!\otimes\!Z\)
• Hadamard conjugation (basis swap)
  \(H X H=Z,\qquad H Z H=X,\qquad H Y H=-Y\)
• Control/target swap by local Hadamards
  \((H\!\otimes\!H)\,\mathrm{CNOT}\,(H\!\otimes\!H)=\mathrm{CNOT}_{\text{swapped}}\)

Glossary

• Hilbert space: complete complex inner-product space
• Projector / POVM: idempotent positive operator / set of positive operators summing to identity
• Density operator: positive semidefinite trace-one operator
• Unitary: $U^\dagger U=I$
• Kraus operator: operator in an operator-sum (CPTP) representation
• CPTP map: completely positive trace-preserving quantum channel
• Schmidt coefficients: singular values in bipartite pure-state decomposition
• PPT: positive partial transpose (Peres–Horodecki criterion)
• LOCC: local operations and classical communication
• Binary entropy: $h_2(p)=-p\log_2 p-(1-p)\log_2(1-p)$

---

Information Theory: Classical vs Quantum