Resource theory of quantum coherence

Short answer: The resource theory of coherence treats quantum coherence as a physical resource that can be quantified. A 2014 paper introduced a framework with defining conditions for coherence measures, a 2017 Reviews of Modern Physics colloquium reviews the field, and a 2019 Physical Review Letters paper generalizes the theory from fixed-basis (von Neumann) measurements to arbitrary positive-operator-valued measures (POVMs). All three are theory work.

Defining Coherence as a Resource

The 2017 colloquium says the development of a rigorous theory of quantum coherence as a physical resource was initiated only recently, despite the fundamental importance of coherence. Coherence represents a departure from the classical realm, combining coherent superposition with the quantization of observables. In many-body systems, it embodies the essence of entanglement and serves as an essential ingredient for phenomena in quantum optics, information, solid-state physics, and nanoscale thermodynamics.

To establish a sound footing for quantification, the 2014 authors adopt the viewpoint of coherence as a physical resource. This approach mirrors established frameworks for entanglement and reference frames. The core of the discussion involves characterizing incoherent states and defining incoherent operations. In standard coherence theory, incoherent states are diagonal with respect to a fixed orthonormal basis. These states can be seen as arising from a von Neumann measurement. Operations that map the set of incoherent states onto itself are termed incoherent operations. This operational definition allows for the identification of a maximally coherent state, which may serve as a unit for coherence.

Criteria for Valid Measures

A wide variety of coherence measures is in use, often based on the off-diagonal entries of a density matrix. Historically, such measures were justified principally on the grounds of physical intuition. The resource-theoretic approach puts these measures on a sound footing by establishing defining conditions for valid measures.

Prime among these conditions is the requirement of monotonicity under incoherent operations. By determining these defining conditions, the authors identify classes of functionals that satisfy them and other, at first glance natural quantities, that do not qualify as coherence measures. The framework distinguishes between quantum operations with and without sub-selection. These technical definitions lead to an operationally well-defined maximally coherent state. The authors describe the resulting measures as intuitive and easily computable, and end with an outline of questions that remain to be answered to complete the theory.

Generalization to POVMs

Standard coherence theory is defined with respect to von Neumann measurements. The 2019 paper introduces a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, specifically arbitrary positive-operator-valued measures (POVMs).

The POVM-based framework utilizes the Naimark construction. Quantum states are embedded to act on a higher-dimensional Hilbert space, known as the Naimark space. The POVM is extended to a projective measurement on this Naimark space, which defines a set of block-incoherent states. The POVM-coherence measure is defined as the distance between the embedded state and its projection onto these block-incoherent states. For the case of von Neumann measurements, the POVM-based coherence measures and POVM-incoherent operations coincide with their counterparts in standard coherence theory.

Operational Characterization and Interconversion

The POVM-based resource theory provides tools to characterize the interconversion properties of resource states. The 2019 authors provide a semidefinite program for this characterization.

The framework is exemplified by means of the qubit trine POVM. For this specific case, analytical results are shown. The study establishes POVM-based coherence measures and POVM-incoherent operations that align with standard theory in the projective limit. The 2017 colloquium describes the wider field as encompassing the characterization, quantification, manipulation, dynamical evolution and operational application of quantum coherence.

FAQ

What defines an incoherent state in standard theory?

In standard coherence theory, incoherent states are diagonal with respect to a fixed orthonormal basis. They are conceptually viewed as arising from a von Neumann measurement. This definition distinguishes them from coherent states, which possess off-diagonal elements in that basis.

Why are monotonicity conditions important for coherence measures?

Monotonicity under incoherent operations is a prime condition for any proper measure of coherence. It ensures that coherence does not increase under operations that preserve the set of incoherent states. Measures failing this condition are rejected by the resource-theoretic framework.

How does the POVM-based theory differ from standard theory?

The POVM-based theory generalizes coherence to arbitrary positive-operator-valued measures rather than just von Neumann measurements. It uses the Naimark construction to define block-incoherent states in a higher-dimensional space. For von Neumann measurements, the POVM-based measures coincide with standard coherence measures.

What is the role of the Naimark construction?

The Naimark construction embeds quantum states into a higher-dimensional Hilbert space to handle arbitrary POVMs. It extends the POVM to a projective measurement on this Naimark space. This defines the block-incoherent states used to calculate the POVM-coherence measure as a distance metric.

Are these measures computationally accessible?

The 2014 authors say they identify intuitive and easily computable measures of coherence, and the 2019 paper provides a semidefinite program to characterize interconversion properties of resource states.