## A Short History of Quantum Reference Frames, Part 1

I was recently at the Relativistic Quantum Information – North (RQI-N) meeting at Dartmouth College where I presented some of the work I have been doing on quantum reference frames and how to use them to overcome certain superselection rules (SSRs), specifically the SSR associated with CPT symmetry. I received a lot of terrific comments and numerous discussions have been spawned from the presentation. In particular, I seem to have introduced a number of people to the concept of a quantum reference frame for the first time. In fact I’m meeting back up with a few folks next week to discuss some of these issues. In preparing for next week’s discussions, I decided to do a little historical research to find out where the concept originated and how it got to its current form. I thought a blog post on the topic might be somewhat useful to some people out there and it would also provide me with a permanent record of some of the background papers in the field.

So what *is* a quantum reference frame? The idea appears to have first been proposed by Marco Toller in 1977 in a paper entitled “An operational analysis of the space-time structure” that appeared in *Il Nuovo Cimento B*. Here I quote the abstract:

We discuss the concepts related to space-time in a quantum-relativistic theory by means of the analysis of the physical procedures used to construct a new frame of reference starting from a pre-existent frame (transformation procedures). The physical objects which form a frame of reference are allowed to interact with the other physical objects and follow the laws of quantum physics. We suggest that there are conceptual limitations which do not permit the exact realization of a transformation of the Poincaré group by means of physical procedures. We remark also that the operations performed in order to construct a frame of reference perturb the surrounding physical objects and are influenced by them. We propose some general theoretical schemes which take these facts into account and permit the separation of the geometrical effects of a transformation procedure from the physical ones. Finally we find the conditions which permit the construction of a Poincaré-invariant theory of the usual kind by means of the introduction of some ideal concepts which have no direct operational meaning.

In other words, the most general definition of a ‘frame of reference’ is as some material object that is of the same nature as the objects that form the system under investigation as well as the measuring instruments themselves (Bohr’s classical-quantum contrast not withstanding). This idea was further developed by Aharonov and Kaufherr in 1984 in which they extended the principle of equivalence to quantum reference frames, and in a pair of articles written in 1991 by Carlo Rovelli (see here and here) which appear to have played some role in inspiring his relational interpretation of quantum mechanics. In this way, these ideas bear a striking resemblance to work attempted by Eddington in the 1930s and early 1940s (a topic I will leave for another blog post, but that served as the core topic of my long-forgotten PhD thesis).

Anyway, these ideas are clearly operational (Toller even uses the term in his original paper). They were, however, not necessarily *informational*, at least initially. However, in his 1982 book *Probabilistic and Statistical Aspects of Quantum Theory*, Alexander Holevo (who was just announced as the 2016 winner of the Claude E. Shannon award by the IEEE Information Theory Society) addressed the following question: can a system of N elementary spins (i.e. qubits, which weren’t yet named in 1982) be used to communicate, in a single transmission, the orientation of three mutually orthogonal unit vectors, i.e. a Cartesian reference frame? Holevo concluded that if the system had a well-defined total spin angular momentum ** J** then, at best, only

*one*of the three vectors could be communicated. A way around this limitation was found nearly two decades later by Bagan, Baig, and Muñoz-Tapia and, around the same time, Peres and Scudo found a way to do it with a single hydrogen atom. The idea was to allow two distant parties (i.e. our old friends Alice and Bob) to establish a common

*Cartesian*reference frame simply using a quantum channel. Thus these papers, while informational in their focus, used the less general definition of a reference frame as a Cartesian coordinate system. In fact it is not entirely clear that any of these authors (or others working on similar ideas – see the previously mentioned paper by Bagan, et. al. for additional references) was aware of the more general definition of the reference frame originally proposed by Toller.

One of the key ideas in the early information-related papers was that the Cartesian frame, i.e. the concept of a *spatial direction*, could be encoded in a particle’s spin state. Somewhere along the line (it’s not quite clear to me yet exactly when) someone put these two ideas together and the more general concept of a quantum reference frame was born. It appears that somewhere around 2002 or 2003 someone realized that a spatial direction is an example of a degree of freedom. Of course even classical physicists – even many engineers – know that there are more general and abstract spaces that have more than three degrees of freedom (e.g. phase space). For decomposable systems, a distinction can be made between what might be called ‘collective’ degrees of freedom, i.e. those between a system and something external to it, and ‘relative’ degrees of freedom, i.e. those between the systems constituent parts. Several authors (including John Preskill, who was at RQI-N) recognized that encoding information into the collective degrees of freedom posed a number of problems. Beginning, to the best of my knowledge, in 1997 with a paper by Zanardi and Rasetti, encoding information into the relative degrees of freedom of a system was shown to be more optimal in some situations. Hopefully, you can see where this is headed. The relational degrees of freedom harken back to the general frame of reference á là Rovelli and his relational interpretation of QM. For example, take a look at this early paper by Bartlett, Rudolph, and Spekkens. The first few paragraphs offer a fairly nice summary of some of the work that had just recently come out on relative quantum information, though the paper itself still primarily deals with something spatial.

As early as 1996, Toller himself recognized that limitations in representations of the Poincaré group necessitated taking “internal” degrees of freedom into account when working with quantum reference frames. An example of such an internal degree of freedom is electrical charge. In fact, in our first paper in *PRL*, we introduce a new quantum number that represents *all* of the *universally conserved* internal quantum degrees of freedom (which happen to only be electrical charge and the *difference* between baryon number and lepton number), though we were unaware of Toller’s paper at that point (in fact I was unaware of it until I started working on this blog post). It may well be, in fact, that we are the first to have considered internal degrees of freedom in such a manner.

At any rate, in Part 2 of this short history, I will attempt to nail down exactly who first suggested using a generalized reference frame in the manner of Toller in an information communication scheme. I will then discuss the relation to SSRs which play a vitally important role in this story.

July 18, 2015 at 2:38 pm

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