Monthly Archives: April 2016

A somewhat coherent post on a robust idea

The word “coherence” has different meaning for different people. Most people may think of the notion of being logical and consistent, be it in speaking or in acting. Actually, we all hope to deal with people — especially politicians(!) — who exhibit coherence between what they say and what they do. And we all hope that the next major blockbuster movie is coherent, with no major plot holes that make you grind your teeth in your seat, unable to fully enjoy your popcorn.

Nonetheless, to a physicist, coherence is also a notion associated with wave behaviour. More precisely, it is associated with the possibility of seeing the effects of superposition, which is the coherent(!) combination of different physical possibilities. For example, the superposition of sounds waves is what allows people to listen to music in the background, while pleasantly chatting.

Among the effects of superposition more affected by coherence (or lack thereof) are phenomena of interference, be it constructive or destructive, like the ones that you can experience with noise-cancelling headphones, for sound, or by looking at the colours of a soap bubble, for light. The recent detection of gravitational waves was possible exactly using the fact that light is a wave, and as such it can be used to detect tiny variations in length within an “interferometer”. Without coherence, neither constructive nor destructive interference would be possible, because both kinds of interference would be “washed out” and inexistent in practice.

The importance of coherence becomes enormous, both conceptually and practically, when we realize that in quantum mechanics everything is also a wave, including what would normally — or, rather, “classically” — be considered “particles”, like electrons and atoms. Mathematically speaking, what we do is to associate a wave — the wave-function — to any physical system or compound of physical systems, more precisely to the state of the system. The evolution in time of the state of the object is given by the evolution of such a wave, described by the famous Schrödinger equation. Then, predictions of what one can observe, and with what probability, can be computed from the knowledge of the wave at a given time.

In the case of information, this wave-like property of objects leads to the consideration of the quantum bit, or qubit, where one can have the superposition of the standard values assumed by a bit, 0 and 1. While in the classical realm the latter would be considered alternative and mutually exclusive options, they can coexist — in the sense of superposition — in the quantum case. This is at the basis of the computational power of future quantum computers.


A coherent superposition is like a controlled combination of ingredients.

In a more realistic setting, and taking into account issues like ignorance(!), the (unwanted) interactions with an environment, and all kinds of “noise”, the state of an object is associated not with a wave, but rather with a so-called density matrix. The latter can be thought of as the incoherent combination of several waves, leading to the decrease and potentially disappearance of interference. One could compare coherent and incoherent mixing respectively to, on one hand, expert cooking, where many flavours combine nicely, either reinforcing or contrasting each other, and, on the other hand, blending everything in a mixer, making often a tasteless combination out of even the most delicious ingredients.

An incoherent mixture of foods may lead to a tasteless result; so can the incoherent mixture of waves and quantum states states.

An incoherent mixture of foods may lead to a tasteless result; so can the incoherent mixture of waves, or of quantum states. (Photo: Tim Patterson (CC BY-SA 2.0))

In the density matrix formalism, (the surviving) coherence is often equated with the presence of off-diagonal elements in the matrix representation of a quantum state. Such off-diagonal elements are the “fingerprint” of the quantum superposition of the (classically) mutually exclusive properties associated with the basis in which the matrix is written;  the latter, although in principle arbitrary, is typically singled-out by the physics, for example by the consideration of what are the various possible energy states of the system. Most importantly, interesting effects — like oscillations — can occur when, and only when, there are off-diagonal elements in the energy representation of a quantum state.

Somewhat surprisingly, the purposeful and focused study of coherence in the matrix formalism has been initiated only recently, leading to an explosion of interest and of works on the topic. Researchers are trying to develop a full consistent theory of coherence, which can be considered like a resource to be characterized, quantified, and manipulated.

In [C. Napoli et al., Phys. Rev. Lett. 116, 150502 (2016)], together with collaborators from the University of Nottingham in UK and the Mount Allison University in New Brunswick, Canada, I put forward a quantifier of coherence, the robustness of coherence, that has many appealing properties, including the possibility of efficiently calculating it when the density matrix is known, of directly measuring it in the lab, and of associating it with practical tasks. Indeed, we find that the robustness of coherence of a quantum state sets an ultimate limit for usefulness of the involved physical system for metrological tasks.

We prove that the robustness of coherence and the robustness of asymmetry quantify the usefulness of the corresponding quantum state for the sake of metrological tasks, like establishing which particular rotation among a set of possible rotations was actually applied.

We prove that the robustness of coherence and the robustness of asymmetry quantify the usefulness of the corresponding quantum state for the sake of metrological tasks, like establishing which particular rotation among a set of possible rotations was actually applied.

In the companion paper [M. Piani et al., Phys. Rev. A 93, 042107] we expand on these ideas, using the fact that coherence, despite being such a fundamental concept, can also be seen as “just” a special case of “asymmetry”, a word that may also mean different things to different people. Nonetheless, in this case, it is easy to grasp that the asymmetry of an object is associated with how different it looks when, let us say, we rotate it or flip it. It should be clear that a sphere is a very symmetric object; for example, it looks the same from whatever direction we look at it, e.g., even if we look at it while standing on our hands, rather than on our feet. On the other hand, say, a face, albeit typically symmetric with respect to a left-right flip, is not symmetric with respect to an upside-down flip. This means that we can realize that we are standing on our hands by noticing that the faces of the bystanders around us are upside-down themselves — this even disregarding the puzzlement or amusement that could transpire from the same faces.

In [M. Piani et al., Phys. Rev. A 93, 042107] we introduce the robustness of asymmetry as a quantifier of the asymmetry of a quantum state with respect to a set of transformations that form a group; that means, in particular, with respect to a set of transformations such that, if you combine two transformations, one followed by the other, you obtain again a transformation that is part of the group, and such that any transformation can be undone by another transformation in the group. Again, think of rotations of an object, and of how they can be combined and undone. We prove that also the robustness of asymmetry of a quantum state can be easily calculated, that it can be measured directly experimentally, and that it sets an ultimate limit to the usefulness of the system prepared in said state for the sake of telling apart the transformations of the group — another metrological task.

You might still wonder where the name “robustness” comes from. Well, it comes from the fact that the property of interest — coherence, or asymmetry — is quantified by the noise that it takes to destroy it; that is, literally, by how robust it is. What our works point out is that this already operational interpretation of the quantifier is precisely associated with how useful the coherence or asymmetry present in the quantum system are. That is, independently of whether you have a positive attitude (“what is the best use I can make of the resource?”) or you’d rather prepare for the worse (“how much noise can our system tolerate?”), robustness is your answer.

[This post is cross-posted on Quanta Rei]