Klein's inequality

In plain terms

Relative entropy measures how surprised you should be, on average, if you believe a system is in one state when it's really in another. It compares the true state with the assumed one, and it's the quantum version of a quantity that appears throughout statistics and information theory. Klein's inequality says this surprise is never negative and is zero only when the two states coincide, so believing something false about a system can never make its behaviour look more predictable than the truth would.

The statement sounds obvious and isn't. Quantum states are matrices, the logarithm of a matrix is defined through its eigenvalues, and two states generally have different eigenvectors, so the two logarithms don't simply line up. The proof expands one state in the eigenbasis of the other and uses the fact that the overlaps between the two sets of eigenvectors form a doubly stochastic pattern, one whose rows and columns each sum to one, which is enough to pull the ordinary convexity of the logarithm through the matrix algebra.

In CSD

The inequality is proved once in the corpus and spent many times. The thermodynamics track uses it for the H-theorem, the statement that coarse-graining a state onto a fixed basis can only raise its entropy, and that theorem is exactly what the de- isolation channel does to the measured system, so the second law on the constraint surface rests on it. Subadditivity of entropy, the Landauer bound that follows from subadditivity, the concavity of entropy under mixing and the Holevo bound on how much classical information a channel carries are all consequences, and in each case the inequality supplies the one step informal accounts tend to wave through.

Mathematically

klein_inequality (Mathlib/QuantumInfo/Subadditivity.lean) says that for rho positive semidefinite of unit trace and sigma positive definite, relEntropy is non-negative. The proof is the doubly stochastic overlap expansion in the two eigenbases with the convexity of x log x. The consumers are vonNeumannEntropy_subadditive and landauer_bound, vonNeumannEntropy_le_pinching (Thermo/SecondLaw.lean), which is the H-theorem, the concavity theorem vonNeumannEntropy_mixture_ge, obtained from the positive-definite case by mixing with the maximally mixed state and closedness in the mixing parameter, and holevoChi_nonneg. The equality case, that zero forces the two states to coincide, isn't formalised.

The name

Oskar Klein (1894-1977) was a Swedish physicist better known for the Kaluza-Klein fifth dimension, the Klein-Gordon equation and the Klein-Nishina formula. The inequality comes from a 1931 paper in the Zeitschrift fur Physik on the quantum- mechanical foundation of the second law of thermodynamics, written in Stockholm, where he had just taken the chair of theoretical physics. His aim was the H-theorem, to show that the entropy of a quantum system, once its fine-grained coherences are discarded, can only increase, so the inequality was the tool, and it outlived the argument it was built for.

Related entries

• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.
• Entropy subadditivityThe information missing from a whole never exceeds the sum of what's missing from its parts, and that's von Neumann's subadditivity inequality, proved at the matrix level with the operator-logarithm step informal proofs usually wave through.
• Holevo boundA quantum system can be prepared in infinitely many distinct ways, yet no reader can extract more classical information from it than from a classical system of the same size, and the measurement channel of CSD is proved to pass exactly one bit per qubit.
• Landauer's principleForgetting costs heat at a rate set by temperature alone, and that's the point at which information stops being an abstraction and acquires a physical price.
• Gibbs stateThe state a system settles into at a given temperature is the unique solution to a trade-off between energy and disorder.