When averaging quantum shifts produces singular states

Quantum channels describe changes to quantum states, including changes caused by averaging over uncertain transformations. Usually, states are represented by density operators: mathematical objects that encode the probabilities of measurement outcomes. In infinite-dimensional settings, however, a broader definition of a state becomes possible.

In an arXiv preprint, Sviatoslav V. Dzhenzher constructs channels that send every input state into this broader territory under specified conditions. The outputs are called singular states. At the same time, the channels preserve a chosen level of probability additivity. This is a mathematical construction and proof, not an experiment or a demonstrated device.

What makes a state singular?

The paper defines a state through the expectation values it assigns to a selected collection of observables, the mathematical objects representing measurable quantities. These assignments must be positive and give the identity observable an expectation value of one.

A normal state has a density-operator representation. A singular state, in the paper’s setting, instead assigns zero to every compact operator. Compact operators can be approximated in operator norm by finite-rank operators. In particular, a singular state gives zero to every projection onto a finite-dimensional subspace, while still giving one to the identity.

That sounds contradictory only if one assumes that probabilities must add correctly across every possible family of mutually exclusive alternatives. In a sufficiently large infinite-dimensional space, preserving a more limited addition rule does not necessarily impose that stronger requirement.

How much additivity survives?

The paper describes this distinction using the symbol κ, pronounced “kappa,” which specifies a size of family. A κ-additive state adds probabilities correctly for families of mutually orthogonal projections containing at most κ members, whenever the sum is among the observables under consideration.

For countably infinite κ, this includes finite families and countable sequences. Larger values impose stronger requirements. Complete additivity, by contrast, covers arbitrary families and characterizes normal positive states in the framework used here.

The central question is therefore precise: can averaging produce singular states while preserving κ-additivity? The preprint proves that it can, subject to its assumptions.

Building a channel from shifts

Dzhenzher starts with labels forming an abelian group: a set with a reversible addition operation whose order does not matter. Each label corresponds to an orthogonal basis vector in a Hilbert space. Shifts move these labels around and act as reversible unitary transformations.

The construction uses observables generated by those shifts and by multiplication with bounded measurable functions. It does not simply assume access to every bounded operator on the Hilbert space.

A probability measure specifies how to average the shifts. That measure must itself be κ-additive. The measurable sets must satisfy suitable closure conditions, and group addition must be jointly measurable. These technical conditions ensure that the averages actually make mathematical sense.

The author first averages transformed observables, producing a unital completely positive map. Its adjoint then defines the channel on the full space of state functionals. For κ-additive input states, the paper also proves an integral formula expressing the output as an average of shifted states.

Why every output can become singular

The decisive extra condition is that each individual shift has probability zero. Because the measure is at least countably additive, any countable collection of shifts also has probability zero.

Vectors in the Hilbert space used here have at most countably many nonzero coordinates. This lets the proof show that averaging annihilates projections onto finite sets of basis labels, and consequently every compact operator. Every output state is therefore singular, including outputs from normal inputs.

Meanwhile, a generalized convergence argument establishes that κ-additivity survives. An appendix develops the corresponding product measures and Fubini theorem, allowing the proof to interchange the relevant integrations.

What the result does not establish

The choice of observables is essential. The paper leaves open when these channels or output states extend to the full bounded-operator algebra while preserving the same additivity. It also identifies channel semigroups, continuity and long-time behavior as future questions.

There is a clear boundary: if the number of basis labels is no greater than κ, κ-additivity already forces normality. The singularising probability measure cannot then exist under the stated assumptions.

The result thus concerns the mathematical limits of quantum-state descriptions—not a practical recipe for creating unusual laboratory states.