Holevo bound

In plain terms

A single qubit, the quantum unit of information, can be prepared in a continuum of different states, and it's natural to hope this continuum could be used to store more than one bit. Holevo's bound says it can't. Suppose a sender chooses one of several states with known probabilities and hands the system over. The most a receiver can learn about which state was chosen, whatever measurement is performed, is bounded by the entropy of the average state, and entropy in turn is bounded by the logarithm of the number of perfectly distinguishable states the system has, so for a qubit that's one bit. The continuum of preparations is real, but the states in it aren't distinguishable from one another, and the bound measures exactly how much distinguishability is available.

The quantity in the bound is the difference between the entropy of the average state and the average of the entropies of the individual states. It's zero when the states are identical, it's largest when they're perfectly distinguishable, and it's never negative. The last fact isn't a triviality, as it's the statement that mixing states never lowers their average entropy, which is a concavity property of the entropy itself.

In CSD

The bound is proved in the corpus with no side conditions, for any finite collection of states and any weights, and it's then applied to a channel the programme produces rather than posits. The de-isolation channel, which is what the measurement flow looks like from the measured system alone, keeps the pointer basis intact and removes everything else. Its single-letter classical capacity, the largest amount of information any collection of input states can carry through it in one use, is proved to be exactly one bit for a qubit, as the bound caps every collection and the two pointer states attain it. That's the information-theoretic reading of a record, as a measurement extracts the pointer bit and nothing more, and the flow that creates the record is the same flow that limits what it can carry.

Mathematically

holevoChi p rho is S(sum p_i rho_i) minus sum p_i S(rho_i). holevoChi_nonneg is concavity of the von Neumann entropy without a support hypothesis (Mathlib/QuantumInfo/ConcavityFull.lean). holevoChi_le_vonNeumannEntropy and holevoChi_le_log_card are the bound and its dimension form, using only non- negativity of the components' entropies. holevoRange of a channel is the set of Holevo quantities of its outputs over all finite ensembles of density matrices, bounded by holevoRange_le_log_card, and deisolationChannel_holevoCapacity (LF6/DeisolationCapacity.lean) states that log 2 is the greatest element of the range of the de-isolation channel on a qubit. The regularised capacity over many uses isn't defined, and the converse direction, that the bound is asymptotically attainable, isn't proved.

The name

Alexander Holevo, born in 1943, has spent his career at the Steklov Mathematical Institute in Moscow. The bound appears in a 1973 paper in the Soviet journal Problems of Information Transmission, at a time when quantum information wasn't yet a field, and it stood for a quarter of a century as a limit before the matching achievability result, proved independently by Holevo and by Schumacher and Westmoreland in the late 1990s, turned it into the capacity formula for classical information over quantum channels. He received the Shannon Award of the IEEE Information Theory Society in 2016.

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.
• 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.
• De-isolationNothing collapses, as a measured system stops being isolated, and measurement is what de-isolation dynamics looks like from inside.
• Measurement flowA deterministic, volume-preserving flow carries out a measurement, and the volumes of its pointer blocks are the Born weights, for every preparation.
• Quantum channelThe most general thing that can happen to a quantum state is captured in CSD as a concrete structure, a family of operators with one completeness law, and every certificate of its physicality is proved rather than assumed.