States and measurements
Density operators, the Born rule, and the limits on cloning, broadcasting, signalling and state discrimination.
Born rule
Probability of outcome $i$ for a POVM $\{E_i\}$ with $E_i\ge0$, $\sum_iE_i=I$.
States ↗Density operators
Describe pure and mixed states, including subsystems of entangled states.
States ↗Purity criterion
A pure state has a single nonzero eigenvalue equal to 1.
States ↗Purity bounds
The lower bound is attained by the maximally mixed state $I/d$.
States ↗Maximally mixed state
By Schur's lemma, only a multiple of the identity commutes with all unitaries.
States ↗Bloch-ball representation
Qubit states form a solid ball.
States ↗Pure qubit condition
Since $\mathrm{Tr}\rho^2=\tfrac12(1+|\vec r|^2)$, purity requires $|\vec r|=1$.
States ↗No-cloning theorem
Linearity of quantum mechanics forbids copying an unknown state.
States ↗No-deleting theorem
Time-reversed cousin of no-cloning: one cannot universally erase one of two identical unknown copies.
States ↗No-broadcasting theorem
Even mixed states cannot be shared across two systems with correct marginals unless they commute, i.e. are effectively classical.
States ↗No-signalling theorem
Local trace-preserving actions on one party do not alter the other party's reduced state, so entanglement cannot send messages.
States ↗Information-disturbance principle
Extracting information about nonorthogonal states necessarily disturbs them.
States ↗Perfect pure-state discrimination
One measurement can identify the state without error only for orthogonal states.
States ↗Nonorthogonal states
No single-copy measurement distinguishes them with certainty.
States ↗Helstrom bound
Optimal error probability for two states with equal priors, achieved by projecting onto the positive part of $\rho_0-\rho_1$.
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