
dependence on this reference frame by applying a G
twirl (a group averaging over all possible configura-
tions of the external reference frame). In some cases,
one can refactor the total Hilbert space into global
and relational subsystems and trace out the global
degrees of freedom [9, 10, 12, 13]. The main empha-
sis of these standard approaches is often to obtain
the physically meaningful (or reference frame inde-
pendent) quantities, in a similar fashion to identifying
noise free subsystems in error correction. In the work
of [3–7, 14, 15] however emphasis is given on the rela-
tional nature of the description (always starting from
a state that is given from the viewpoint of one of the
subsystems) and the main object of study is the re-
lation between different accounts. We make a similar
emphasis in the present work. We abstract the for-
malism of [6] and introduce an approach which makes
heavy use of the inherently group theoretic nature of
quantum reference frames. This allows us to gener-
alise the known results beyond the translation and ro-
tation groups to arbitrary finite and locally compact
groups (including non-Abelian groups).
In Section 2 we outline the relational approach to
quantum theory embraced in the present work as well
as give a simple example of a change of reference frame
for classical bits and an example of a change of quan-
tum reference frame for qubits. In Section 3 we define
the notion of a reference frame in terms of reference
systems and coordinate systems, as well as give a full
account of active and passive transformations as left
and right regular group actions. Combining these we
define changes of reference frame under a group G
for classical systems with configuration space X
∼
=
G.
In Section 4 we extend the classical change of refer-
ence frame to quantum systems L
2
(G) following the
principle of coherent change of reference system; and
define a general unitary operator which implements
this change of reference system. We prove a theorem
stating that only systems carrying a regular repre-
sentation of G can serve as reference frame, subject
to the principle of coherent change of reference sys-
tem. Following this we extend the change of quantum
reference frame operator between L
2
(G) systems de-
scribing systems which do not carry the right regular
representation of G. In Section 5 we define irreversible
changes of reference frames for groups G = N o P
and G = N × P via a truncation procedure. In Sec-
tion 6 we extend this change of reference frame to
quantum reference frames using the principle of co-
herent change of reference system once more. In Sec-
tion 7 we apply the tools developed in the preceding
sections to the Wigner’s friend thought experiment,
providing an explicit change of reference frame from
Wigner’s description to the friend’s. We discuss re-
lated work in Section 8 and discuss implications of the
present work as well as suggestions for future work.
In Section 9 we give some concluding remarks.
2 Relational approach to quantum
theory
In the construction of a relational formalism of quan-
tum mechanics, an essential task is to write quantum
states of systems relative to a specified reference sys-
tem. We introduce the following notation: |ψi
A
B
indi-
cates the state of system B relative to system A. In
contrast to the approach of [6], we assign a Hilbert
space to the system whose perspective is adopted and
assign to it the trivial state, corresponding to the
identity element of the group. Hence, by convention,
system A is in a default ‘zero-state’ relative to itself.
Once we introduce the notion of symmetry groups and
how they enter into the formalism, we will see that
this default zero-state corresponds to the identity ele-
ment of the group that describes the transformations
of the system. Thus, to be more precise, one can write
|0i
A
A
⊗ |ψi
A
B
. (1)
The upper index refers to the system relative to which
the state is given while the lower index refers to the
system that is being described (similarly to the per-
spectival approach of [16]). This description does not
make use of any external abstract reference frame nor
does it assume the existence of absolute, observer-
dependent values of physical observables. We ob-
serve that since system A can only ever assign itself
a single state there are no state self-assignment para-
doxes [17, 18].
A natural question to address on the relational ap-
proach to quantum theory is how to change reference
systems. Namely if the state of B relative to A is
|ψi
A
B
= |0i
A
A
⊗ |ψi
A
B
, what is the state |ψi
B
A
of A rela-
tive to B? This is the problem which will be addressed
in the present work.
Before introducing the general framework we will
be using, we give two simple examples of changes of
reference frame for relational states. The first is classi-
cal and the second its quantum generalisation. These
should hopefully provide the reader with an intuitive
picture of the general mechanisms at play.
Example 1 (Z
2
change of classical reference frame).
Let us consider the case where systems can be in two
states ↑ or ↓. Every system considers themselves to be
in the state ↑ (for example an observer free floating
in empty space would always consider the up direc-
tion to be aligned from their feet to their head). Con-
sider classical systems where the state relative to A is
↑
A
A
↑
A
B
↓
A
C
. Since A sees B in the state ↑ relative to itself,
B also sees A in the state ↑ relative to itself. The state
relative to B is ↑
B
B
↑
B
A
↓
B
C
. If the state relative to A was
instead ↑
A
A
↓
A
B
↓
A
C
then since A views B in the ↓ state,
this implies that B views things ‘upside down’ relative
to A. The change of perspective would give ↑
B
B
↓
B
A
↑
B
C
.
In the next example we give a quantum generali-
sation of the Z
2
change of reference frame. This is
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 2