Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of a quantum system as being relational, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, Available as a pdf at https://arxiv.org/abs/hep-th/9403015. See also Rovelli (1996) below, which appears to be a later version of the same text. and has since been expanded upon by a number of theorists. It is inspired by the key idea behind special relativity, that the details of an observation depend on the reference frame of the observer, and Wheeler's idea that information theory would make sense of quantum mechanics.Wheeler (1990): pg. 3
The physical content of the theory has not to do with objects themselves, but the relations between them. As Rovelli puts it:
"Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world".Rovelli, C. (1996), "Relational quantum mechanics", International Journal of Theoretical Physics, 35: 1637–1678.
The essential idea behind RQM is that different observers may give different accurate accounts of the same system. For example, to one observer, a system is in a single, "collapsed" eigenstate. To a second observer, the same system is in a superposition of two or more states and the first observer is in a correlated superposition of two or more states. RQM argues that this is a complete picture of the world because the notion of "state" is always relative to some observer. There is no privileged, "real" account. The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. The terms "observer" and "observed" apply to any arbitrary system, microscopic or macroscopic. The classical limit is a consequence of aggregate systems of very highly correlated subsystems. A "measurement event" is thus described as an ordinary physical interaction where two systems become correlated to some degree with respect to each other.
Rovelli criticizes describing this as a form of "observer-dependence" which suggests reality depends upon the presence of a conscious observer, when his point is instead that reality is relational and thus the state of a system can be described even in relation to any physical object and not necessarily a human observer.
The proponents of the relational interpretation argue that this approach resolves some of the traditional interpretational difficulties with quantum mechanics. By giving up our preconception of a global privileged state, issues around the measurement problem and local realism are resolved.
History and development Relational quantum mechanics arose from a comparison of the quandaries posed by the interpretations of quantum mechanics with those resulting from Lorentz transformations prior to the development of special relativity. Rovelli suggested that just as pre-relativistic interpretations of Lorentz's equations were complicated by incorrectly assuming an observer-independent time exists, a similarly incorrect assumption frustrates attempts to make sense of the quantum formalism. The assumption rejected by relational quantum mechanics is the existence of an observer-independent state of a system.Rovelli (1996): pg. 2
The idea has been expanded upon by Lee SmolinSmolin (1995) and Louis Crane,Crane (1993) who have both applied the concept to quantum cosmology, and the interpretation has been applied to the EPR paradox, revealing not only a peaceful co-existence between quantum mechanics and special relativity, but a formal indication of a completely local character to reality.Laudisa (2001)Rovelli & Smerlak (2006)
In 2020, Rovelli published an account of the main ideas of the relational interpretation in his pop-science book Helgoland.
In 2023, Rovelli published a paper with Emily Adlam that presented a significant revision to relational quantum mechanics. Adlam and Rovelli introduced a new axiom, positing the existence of "cross-perspective links", which has been seen as a major move away from relationalism.
The problem of the observer and the observed This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider observer , measuring the state of the quantum system . We assume that has complete information on the system, and that can write down the wavefunction describing it. At the same time, there is another observer , who is interested in the state of the entire - system, and likewise has complete information.
To analyse this system formally, we consider a system which may take one of two states, which we shall designate and , ket vectors in the Hilbert space . Now, the observer wishes to make a measurement on the system. At time , this observer may characterize the system as follows:
where and are probabilities of finding the system in the respective states, and these add up to 1. For our purposes here, we can assume that in a single experiment, the outcome is the eigenstate (but this can be substituted throughout, without loss of generality, by ). So, we may represent the sequence of events in this experiment, with observer doing the observing, as follows:
This is the description of the measurement event given by observer . Now, any measurement is also a physical interaction between two or more systems. Accordingly, we can consider the tensor product Hilbert space , where is the Hilbert space inhabited by state vectors describing . If the initial state of is , some degrees of freedom in become correlated with the state of after the measurement, and this correlation can take one of two values: or where the direction of the arrows in the subscripts corresponds to the outcome of the measurement that has made on .
If we now consider the description of the measurement event by the other observer, , who describes the combined system, but does not interact with it, the following gives the description of the measurement event according to , from the linearity inherent in the quantum formalism:
Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers and give different but equally correct accounts of the events .
Note that the above scenario is directly linked to the Wigner's Friend thought experiment, which serves as a prime example when understanding different interpretations of quantum theory.
Central principles
Observer-dependence of state According to , at , the system is in a determinate state, namely spin up. And if quantum mechanics is complete, then so is this description. But, for , is not uniquely determinate, but is rather entangled with the state of note that his description of the situation at is not factorisable no matter what basis chosen. But, if quantum mechanics is complete, then the description that gives is also complete.
Thus the standard mathematical formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty. It could be described as an epistemic limitation observers with full knowledge of the system, we might say, could give a complete and equivalent description of the state of affairs, but obtaining this knowledge is impossible in practice. But which observer? What makes 's description better than that of , or vice versa? Alternatively, we could claim that quantum mechanics is not a complete theory, and that by adding more structure we could arrive at a universal description (the troubled hidden variables approach). Yet another option is to give a preferred status to a particular observer or type of observer, and assign the epithet of correctness to their description alone. This has the disadvantage of being ad hoc, since there are no clearly defined or physically intuitive criteria by which this super-observer ("who can observe all possible sets of observations by all observers over the entire universe"Page, Don N., "Insufficiency of the quantum state for deducing observational probabilities", Physics Letters B, Volume 678, Issue 1, 6 July 2009, 41–44.) ought to be chosen.
RQM, however, takes the point illustrated by this problem at face value. Instead of trying to modify quantum mechanics to make it fit with prior assumptions that we might have about the world, Rovelli suggest to modify our view of the world to conform to what amounts to our best physical theory of motion.Rovelli (1996): pg. 16 Just as forsaking the notion of absolute simultaneity helped clear up the problems associated with the interpretation of the Lorentz transformations, so many of the conundrums associated with quantum mechanics dissolve, provided that the state of a system is assumed to be observer-dependent like simultaneity in special relativity. This insight follows logically from the two main hypotheses which inform this interpretation:
Hypothesis 1: the equivalence of systems. There is no a priori distinction that should be drawn between quantum and macroscopic systems. All systems are, fundamentally, quantum systems. Hypothesis 2: the completeness of quantum mechanics. There are no hidden variables or other factors which may be appropriately added to quantum mechanics, in light of current experimental evidence.
Thus, if a state is to be observer-dependent, then a description of a system would follow the form "system is in state with reference to observer " or similar constructions, much like in relativity theory. In RQM it is meaningless to refer to the absolute, observer-independent state of any system.
Information and correlation It is generally well established that any quantum mechanical measurement can be reduced to a set of yes–no questions or bits that are either 1 or 0. RQM makes use of this fact to formulate the state of a quantum system (relative to a given observer!) in terms of the physical notion of information developed by Claude Shannon. Any yes/no question can be described as a single bit of information. This should not be confused with the idea of a qubit from quantum information theory, because a qubit can be in a superposition of values, whilst the "questions" of RQM are ordinary binary variables.
Any quantum measurement is fundamentally a physical interaction between the system being measured and some form of measuring apparatus. By extension, any physical interaction may be seen to be a form of quantum measurement, as all systems are seen as quantum systems in RQM. A physical interaction is seen by other observers unaware of the result, as establishing a correlation between the system and the observer, and this correlation is what is described and predicted by the quantum formalism.
But, Rovelli points out, this form of correlation is precisely the same as the definition of information in Shannon's theory. Specifically, an observer O observing a system S will, after measurement, have some degrees of freedom correlated with those of S, as described by another observer unaware of the result. The amount of this correlation is given by log2k bits, where k is the number of possible values which this correlation may take the number of "options" there are, as described by the other observer.
Note that if the other observer is aware of the measurement result, there is only one possible value for the correlation, so they will not regard the (first observer's) measurement as producing any information, as expected.
All systems are quantum systems