Entanglement and quantum information
Schmidt decomposition, entropy inequalities, monotones under channels and the structure of entanglement.
Schmidt decomposition
Every bipartite pure state has orthonormal local bases in which it is a single sum, from the singular value decomposition of its coefficient matrix.
Entanglement ↗Schmidt rank criterion
Entangled pure states have Schmidt rank at least 2.
Entanglement ↗Reduced-state spectra
Both reduced states have eigenvalues $\lambda_i$, the squared Schmidt coefficients.
Entanglement ↗Pure-state entanglement entropy
The standard entanglement measure for pure bipartite states; zero iff separable, maximal $\log d$.
Entanglement ↗Entropy subadditivity
Joint uncertainty cannot exceed the sum of parts; equality holds for product states.
Entanglement ↗Araki-Lieb inequality
Unlike classical entropy, a joint quantum entropy can be smaller than a part's, which signals entanglement.
Entanglement ↗Strong subadditivity
Deep Lieb-Ruskai theorem, equivalent to many other entropy inequalities; it underpins nearly all of quantum information theory.
Entanglement ↗Quantum mutual information
Total correlations, classical plus quantum.
Entanglement ↗Conditional mutual information
Equivalent to strong subadditivity: given $B$, learning $C$ cannot lower correlations with $A$ below zero.
Entanglement ↗Data-processing inequality
Applying a channel can only make states less distinguishable.
Entanglement ↗Trace-distance contractivity
Noise cannot increase the optimal single-shot distinguishability of two states.
Entanglement ↗Fidelity monotonicity
Uhlmann fidelity, $F=\lVert\sqrt\rho\sqrt\sigma\rVert_1$, can only increase under channels.
Entanglement ↗Relative entropy positivity
Klein's inequality; zero iff $\rho=\sigma$.
Entanglement ↗Positive partial transpose
Peres-Horodecki criterion: negative partial transpose proves entanglement.
Entanglement ↗Entanglement monogamy
Coffman-Kundu-Wootters inequality for three qubits: strong entanglement with one party limits entanglement with others.
Entanglement ↗