von Neumann entropy

In plain terms

Entropy measures missing information, meaning how much you'd still need to learn to know the state of something completely. For ordinary things it behaves sensibly, so if you know everything about a machine you know everything about its parts, and quantum mechanics breaks this sensible behaviour in a specific, quantifiable way.

Two entangled particles can be in a jointly definite state, with entropy zero and nothing left to learn about the pair, while each particle on its own is in the most uncertain state possible, with maximum entropy. The information isn't hiding in one particle or the other, as it lives in the relationship between them, and no inspection of the parts separately can find it.

Von Neumann's entropy is the quantity that makes this precise. The inequalities it obeys, which say how the entropy of a whole constrains the entropies of its parts, are the working arithmetic of quantum information, and they also turn out to be most of thermodynamics.

In CSD

The corpus uses entropy for two different jobs. The first is to measure the irreversibility created when a pure state, one that's completely known, is coarse- grained into a record, so that many detailed configurations are lumped under one outcome. The substrate conserves entropy exactly and the record doesn't, and the gap between those two statements is where the second law of thermodynamics, and arguably the arrow of time, enters this programme.

The second job is as load-bearing machinery. Subadditivity, the inequality that says the entropy of a whole is at most the sum of the entropies of its parts, is the specific step that turns an entropy statement into the Landauer heat bound, and if you remove it that derivation doesn't close. The corpus proves subadditivity under a stated technical condition on the states of the two parts, that each is of full rank, so no possibility is assigned probability exactly zero. It carries the much harder three-part inequality, strong subadditivity, separately, derived from an explicitly stated data-processing assumption rather than proved from scratch, and both are scope decisions recorded in the corpus's ledger rather than something a reader has to discover.

The corpus also assigns an entropy to a preparation on the constraint surface itself, and not only to a matrix handed in from outside. A preparation is a probability measure on the surface, its density operator is the average of the rank- one projectors of the rays it occupies, and the entropy of the preparation is the entropy of that operator. Three things are proved about it. It's zero exactly when the preparation sits on a single ray, so purity and zero entropy are the same condition on the surface. Coarse-graining a preparation, meaning mixing several of them with weights, never lowers it, with no side condition on the states involved. And along a volume-preserving flow it's conserved, while the coarse-graining a measurement performs on the system alone can only raise it, which is the second law read on the surface, produced by the environment being ignored rather than by any loss in the dynamics.

Mathematically

For a bipartite density operator, the entropy of the whole is at most the sum of the entropies of the marginals, proved here at positive-definite marginal scope, which for pure global states means full Schmidt rank. The standard equality condition, that the bound is attained exactly on product states, is stated in the module as the reason the inequality isn't vacuous, and it isn't itself formalised, so don't cite it as proved.

Strong subadditivity, the three-party inequality of Lieb and Ruskai from which most structural results of quantum information follow, isn't re-proved from scratch, as it's derived from an explicit data-processing hypothesis, and the ledger says so.

On the surface, preparationDensity is the Gleason witness of a preparation and preparationDensity_eq_barycenter identifies it with the Bochner integral of the outer products of a unit section along the projective law (LF2/PreparationBarycenter.lean), while preparationEntropy_eq_zero_iff (LF2/PreparationPurity.lean) says the entropy vanishes exactly when the projective law is a Dirac mass at one ray, through the matrix-level vonNeumannEntropy_eq_zero_iff. Concavity, vonNeumannEntropy_mixture_ge (Mathlib/QuantumInfo/ConcavityFull.lean), holds for arbitrary density matrices, as the positive-definite case is Klein's inequality averaged and the support hypothesis is removed by mixing with the maximally mixed state, whose entropy is explicit, and passing to the closed set of mixing parameters on which the inequality holds. The second law on the surface is vonNeumannEntropy_le_pinching_flow (Thermo/SigmaSecondLaw.lean), which says the flowed preparation has the same entropy and the de-isolation channel, the pinching of the second-law module, doesn't lower it.

The name

John von Neumann (1903-1957) was born Neumann Janos in Budapest, into an ennobled banking family. He took a chemistry degree in Zurich and a mathematics doctorate in Budapest more or less simultaneously, then moved through Berlin and Hamburg to Princeton in 1930 and to the Institute for Advanced Study at its founding. The entropy appears in his 1932 book, the one that gave quantum mechanics its Hilbert- space form, fifteen years before Shannon.

The naming story is the good part. When Shannon later needed a name for his own uncertainty quantity, von Neumann reportedly told him to call it entropy for two reasons, as the formula already existed in statistical mechanics under that name, and nobody knew what entropy really was, so in a debate Shannon would always have the advantage. Shannon took the advice, and the joke has been paying dividends ever since.

Related entries

• 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.
• Klein's inequalityOne quantum state is never negatively distinguishable from another, as the relative entropy of a state with respect to a second is at least zero, and that single inequality drives the second law, subadditivity, concavity and the Holevo bound in the corpus.
• 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.
• Canonical typicalityA small piece of a large quantum system looks thermal not because anything prepared it that way, but because almost every state of the whole makes it look that way.