Qubits, entanglement, density matrices, channels, entropy, nonlocality, teleportation, cryptography, and quantum networks. Expanded with original in-page technical chapters, mathematical proofs, independently checked calculation baselines and directly embedded credited original instructor videos where relevant.
Published syllabus
Qubits are rays, not tiny classical switches. A normalized state in C² is determined physically by two real Bloch coordinates once global phase is removed. A spin measurement along unit axis n has effects (I±n·σ)/2, so outcome probabilities depend on the projection of the Bloch vector onto that axis. The poles of the Bloch sphere represent Z eigenstates; other points represent genuine superpositions relative to Z.
Two registers combine through a tensor product, which multiplies Hilbert-space dimensions. The ordered basis |00>, |01>, |10>, |11> spans two qubits; most vectors in this space do not factor into individual qubit states. Local transformations have form U_A⊗I_B and I_A⊗U_B, while genuinely entangling operations cannot be decomposed this way. The ordering convention matters when constructing Kronecker products and partial traces.
Schmidt decomposition writes any bipartite pure vector as a sum of orthogonal product terms with nonnegative coefficients. The number of nonzero coefficients is the Schmidt rank. A pure state is separable exactly when that rank is one. For a Bell pair, both coefficients are 1/sqrt(2), and either marginal is maximally mixed. This bridges singular-value decomposition and entanglement without dependence on a particular computational basis.
A density operator is positive semidefinite, Hermitian, and trace one. It represents both proper classical mixing of preparations and the operational state of a subsystem after inaccessible degrees of freedom are traced out. Ensembles of distinct pure states can give the same density matrix, so an ensemble decomposition is not a directly observable property of the operator. Purity Tr(rho²) distinguishes an ideal pure state from mixtures but does not identify why the mixing arose.
A generalized measurement is a family of positive effects E_a summing to the identity. A quantum channel is a completely positive trace-preserving linear map whose Kraus operators satisfy a completeness relation. The POVM determines outcome probabilities; the associated measurement instrument determines back-action. Channels describe physical state transitions, and not every merely positive map remains positive when tensored with an untouched reference system.
Von Neumann entropy depends only on the eigenvalues of the density operator. It reduces to Shannon entropy for diagonal classical mixtures and equals zero for pure states. Relative entropy compares distinguishability of states, but its support condition matters when one state has zero eigenvalues. Conditional entropy in quantum information may be negative, unlike its classical counterpart, and that reflects quantum correlations rather than a negative number of physical bits.
The CHSH expression combines correlations from two independently chosen settings at each laboratory. Any local hidden-variable model satisfying the relevant measurement-independence and outcome assumptions obeys an absolute value at most two. A maximally entangled two-qubit state with suitable measurement axes reaches 2sqrt(2). The theorem does not predict a violation for every entangled state or every choice of axes; experimental statements require a precise sampling and loophole analysis.
The no-cloning theorem prohibits a single linear physical operation from copying an arbitrary unknown pure state while leaving it intact. Assume U|psi>|0>=|psi>|psi> and similarly for |phi>. Preservation of inner products would require <psi|phi>=<psi|phi>², impossible for generic distinct nonorthogonal states. Teleportation instead transfers an unknown state using prior entanglement, a Bell measurement, two classical bits, and a conditional correction. It consumes the original localized input state.
Superdense coding exploits a previously distributed Bell pair: applying one of four local Pauli operators generates four orthogonal Bell states, and sending one qubit allows a receiver to distinguish two classical bits when a full Bell measurement is available. Quantum key distribution, such as BB84, instead uses nonorthogonal preparation and basis reconciliation to detect certain adversarial disturbances under an explicit security model. These are different information tasks despite sharing qubit gates.
Resource theories begin by specifying which operations are considered free. Entanglement theory often treats local operations and classical communication as free and asks how many singlets are required or distillable under these constraints. Monotones cannot increase under free operations on average. Different resources, such as coherence, magic, asymmetry and thermodynamic athermality, require different free sets and measures.
Quantum network protocols must account for channel loss, finite memory lifetime, entanglement-generation probability, swapping fidelity and classical communication delay. Entanglement swapping correlates remote qubits through a Bell measurement on intermediate systems, but it neither eliminates exponential fiber attenuation automatically nor makes classical messages irrelevant. Repeaters improve scaling by segmenting channels, purifying states or using error correction, at the price of substantial hardware overhead.
A quantum information paper should be read by identifying its exact task, resource assumptions and error metric. Distinguish theorem claims from numerical evidence; look for an explicit model of noise, input ensemble, allowed operations and complexity scaling. In channel-discrimination research, diamond norm captures ancillary-assisted distinguishability and differs from an ordinary matrix norm. In algorithm papers, query complexity is not identical to full physical runtime.