Quantum reference frames for general symmetry groups
Anne-Catherine de la Hamette
1,2
and Thomas D. Galley
2
1
Institute for Theoretical Physics, ETH Z
¨
urich, Wolfgang-Pauli-Str. 27, 8093 Z
¨
urich, Switzerland
2
Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
23 November 2020
A fully relational quantum theory necessar-
ily requires an account of changes of quan-
tum reference frames, where quantum refer-
ence frames are quantum systems relative to
which other systems are described. By in-
troducing a relational formalism which iden-
tifies coordinate systems with elements of a
symmetry group G, we define a general op-
erator for reversibly changing between quan-
tum reference frames associated to a group G.
This generalises the known operator for trans-
lations and boosts to arbitrary finite and lo-
cally compact groups, including non-Abelian
groups. We show under which conditions one
can uniquely assign coordinate choices to phys-
ical systems (to form reference frames) and
how to reversibly transform between them,
providing transformations between coordinate
systems which are ‘in a superposition’ of other
coordinate systems. We obtain the change of
quantum reference frame from the principles
of relational physics and of coherent change of
reference frame. We prove a theorem stating
that the change of quantum reference frame
consistent with these principles is unitary if
and only if the reference systems carry the left
and right regular representations of G. We
also define irreversible changes of reference
frame for classical and quantum systems in the
case where the symmetry group G is a semi-
direct product G = N o P or a direct prod-
uct G = N × P , providing multiple examples
of both reversible and irreversible changes of
quantum reference system along the way. Fi-
nally, we apply the relational formalism and
changes of reference frame developed in this
work to the Wigner’s friend scenario, finding
similar conclusions to those in relational quan-
tum mechanics using an explicit change of ref-
erence frame as opposed to indirect reasoning
using measurement operators.
Anne-Catherine de la Hamette:
annecatherine.delahamette@univie.ac.at
Thomas D. Galley: tgalley1@perimeterinstitute.ca
1 Introduction
In quantum mechanics, physical systems are implic-
itly described relative to some set of measurement de-
vices. When writing down the quantum state of a
system of interest, say a spin-1/2 system in the state
|↑
z
i, we mean that the state of the system is ‘up’ rel-
ative to a specified direction ˆz in the laboratory. In
practice, this direction will be associated to a macro-
scopic physical system in the lab. If we assume that
quantum mechanics is a universal theory and there-
fore applicable at all scales, the systems we make ref-
erence to to describe quantum systems should eventu-
ally be treated quantum mechanically as well. Refer-
ence systems that are themselves treated as quantum
systems are referred to as quantum reference frames.
Following the success of Einstein’s theory of relativ-
ity and its inherently relational nature, one may seek
to adopt a relational approach to quantum theory as
well. In such an approach, most physically mean-
ingful quantities are relational, i.e. they only take
on well defined values once we agree on the reference
system (or the observer) relative to which they are
described. In his papers [1, 2], Rovelli suggested that
quantum mechanics is a complete theory about the
description of physical systems relative to other phys-
ical systems. In his Relational Quantum Mechanics
(RQM) he rejected the idea of observer-independent
states of systems and values of observables. The im-
portance of changes of reference frame in special and
general relativity suggests the development of an ac-
count of changes of quantum reference frame in RQM.
Such an account is given in the present work.
Recently, there has been an increased interest in
analysing spatial and temporal quantum reference
frames and in establishing a formalism that allows
to switch between different perspectives [37]. The
present work is partially based on these approaches
which define changes of quantum reference frames for
systems that transform under the translation group
(in space and time) and the rotation group in three di-
mensional space. As opposed to other more standard
approaches [811], this formalism stresses the lack of
an external reference frame from the outset and de-
fines states of subsystems relative to another subsys-
tem. In standard approaches to quantum reference
frames [9, 10], one often starts from a description rel-
ative to an external reference frame and removes any
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arXiv:2004.14292v3 [quant-ph] 23 Nov 2020
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 [37, 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
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a specific instance of the general changes of quantum
reference frame defined in this work.
Example 2 (Z
2
change of quantum reference frame).
Let us consider the case with quantum systems C
2
with
basis {|↑i, |↓i}. Every system considers themselves to
be in the state |↑i. By embedding the classical sce-
nario above with the map 7→ |↑i and 7→ |↓i we
can reconstruct the classical example: if the state rel-
ative to A is |ψi
A
BC
= |↑i
A
A
|↑i
A
B
|↓i
A
C
then the state rel-
ative to B is |ψi
B
AC
= |↑i
B
B
|↑i
B
A
|↓i
B
C
. If the state was
|φi
A
BC
= |↑i
A
A
|↓i
A
B
|↓i
A
C
then the change of perspective
would give |φi
B
AC
= |↑i
B
B
|↓i
B
A
|↑i
B
C
.
Let us move to the quantum case with an ini-
tial state |τ i
A
BC
= |↑i
A
A
|↑i
A
B
+ |↓i
A
B
|↓i
A
C
. What is
the state |τi
B
AC
? First let us observe that |τi
A
BC
=
|ψi
A
BC
+ |φi
A
BC
, and let us assume that changes of
quantum reference frame are coherent (they observe
the superposition principle). Then the state |τi
B
AC
=
|↑i
B
B
|↑i
B
A
|↓i
B
C
+ |↓i
B
A
|↑i
B
C
which is an entangled state
of A and C.
The above example made use of the two guiding
principles of this work: the principle of relational
physics and the principle of coherent change of ref-
erence system. These are defined in Section 4.3.
Whenever we use phrases such as ‘from the view-
point of’ or ‘from the perspective of’, we simply mean
‘relative to’. Although these expressions might imply
that the state |ψi
A
B
indicates how system A perceives
system B, we do not make this interpretation here.
System A acts as the observer in this description but
we should note that there is nothing special about an
observer system. No interpretation is made as to what
the system sees. Rather a change of reference system
A B is a change of description from one where A is
at the origin to one where B is at the origin.
3 Classical changes of reference
frames associated to symmetry groups
3.1 Reference systems, coordinate choices and
changes of reference frame
A coordinate system is a purely mathematical object,
and need not in general be associated to a physical
system. A reference frame consists of a physical sys-
tem (known as a reference system), and a choice of
coordinates such that the reference system is at the
origin in that coordinate system. For full definitions
we refer the reader to Appendix A. In this section
we define changes of reference frames for classical sys-
tems where the configuration space is itself a group
G. In Section 5 we will consider cases where this is
no longer holds.
We begin by a simple example which illustrates
changes of reference frames and the use of group ele-
ments for relative coordinates.
Example 3 (Three particles on a line). Consider
three classical particles A, B and C on a line, with
state s = (x
A
, x
B
, x
C
) in some Cartesian coordinate
system (here we omit the velocities since we are just
interested in translations in space). The coordinate
system x
0
such that x
0
A
= 0 is said to be associated to
A. In this coordinate system the particles have state
s = (x
0
A
= 0, x
0
B
= x
B
x
A
, x
0
C
= x
C
x
A
). We observe
here that the relative coordinates (to A) x
0
A
, x
0
B
and x
0
C
uniquely identify the translation which maps system A
to systems A, B and C. Namely the relative distance
x
0
B
= x
B
x
A
is the distance needed to translate A to B.
The relative coordinates x
0
A
, x
0
B
and x
0
C
correspond to
the symmetry group transformations relating A, B and
C to A. If we label a translation of distance d by t
L
(d),
where t
L
(d)x = d + x, we have the state relative to A
as s
A
= (t
L
(0), t
L
(x
B
x
A
), t
L
(x
C
x
A
)). The state
relative to B is s
B
= (t
L
(x
A
x
B
), t
L
(0), t
L
(x
C
x
B
)).
These two relative states are themselves related by
the transformation s
B
= s
A
(x
B
x
A
). We de-
fine the right regular action of the translation group
T
R
(d) = x d. The change of reference frame A to
B is given by the right regular action T
R
(x
B
x
A
) of
the group element x
B
x
A
mapping A to B.
In the above example the configuration space R and
the symmetry group T = (R, +) acting on it are equiv-
alent as manifolds. This equivalence is essential for
the existence of a well defined reversible change of
quantum reference frame. In Sections 5 and 6 we
study scenarios where this is no longer the case, and
the changes of reference frame are irreversible. Since
the results in this paper also apply to finite groups we
cover a simple example.
Example 4 (Z
2
). Let us consider systems with con-
figuration space X = {↑, ↓}. The symmetry group
G = {I, F } = Z
2
consisting of the identity I() =
and the flip F () = is the symmetry group of X. A
state of four systems of the form s =, , , can be
expressed as s = I(), I(), F (), F (). By consider-
ing as the ‘coordinate system’ we have that the state
s has coefficients (I, I, F, F ). In the ‘coordinate sys-
tem’ the state s would have coefficients (F, F, I, I).
We observe that in the above examples group el-
ements of the global symmetry groups serve as rel-
ative coordinates. In the next subsection we make
this link more explicit. We also note the importance
of the one to one correspondence between states and
coordinate transformations. We observe that there
is always some conventionality in changes of coordi-
nates: one considers only translations on R for in-
stance, and not all diffeomorphisms of R as relating
different coordinates. In Appendix A we give explicit
definitions of coordinate systems on a manifold X,
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emphasising that a coordinate system is different to
a coordinate chart (typically used in general relativ-
ity), a distinction made in [19]. Roughly speaking a
coordinate system on a manifold X is an isomorphism
f : X Y (where Y a known mathematical object
used to describe X), whereas a coordinate chart is a
map from the mathematical object Y to the physi-
cal object X which need not be an isomorphism (for
instance multiple charts R
n
are used to describe a
curved n-dimensional manifold X in general relativ-
ity but there is no isomorphism from X to R
n
).
3.2 General treatment of reversible changes of
reference frame
Let us extract the general features of the above sce-
nario which allow for well defined reference frames
and reversible changes of reference frame. Consider a
configuration space X (which is typically a set with a
manifold structure) and a group G acting on X such
that there is a unique transformation g G relat-
ing any pair of points (the action is transitive and
free). This implies X
=
G (as sets/manifolds), and
the action of G on X
=
G is the group multiplica-
tion on itself: G × G G. This space is a principle
homogeneous space for G, sometimes called a G tor-
sor. We assume G locally compact and thus equipped
with a left Haar measure, denoted dg. Many groups
of interest in physics, such as the Poincar´e group,
the symmetric group, SU(d) and SO(d) are locally
compact (compact and finite groups are instances of
locally compact groups). One exception is the dif-
feomorphism group of some space-time manifold M,
which in general is not locally compact. We observe
that X
=
G follows from the requirement that there
exists a unique transformation g G relating any two
points in X. Note that we will later go beyond such
perfect reference frames and consider cases for which
the configuration space X differs from the group G.
We use the following example from [20] to introduce
active and passive transformations on G torsors.
Example 5 (Single observer and system on X
=
G).
Consider an observer 0 at location x
0
on X
=
G and
an object 1 at location x
1
. Then the unique transfor-
mation g such that gx
0
= x
1
is the relative location
of 1 relative to 0.
An active transformation is a transformation on the
object 1. A transformation h on the object 1 is given
by the left regular action x
1
7→ x
0
1
= hx
1
. The relative
location is now k where kx
0
= x
0
1
. Using gx
0
= x
1
and hx
1
= x
0
1
we find that k = hg: hgx
0
= hx
1
= x
0
1
.
Therefore an active transformation by h corresponds
to the left regular action of h on the relative location
g: g 7→ hg.
A passive transformation is a transformation on the
observer x
0
7→ hx
0
. This induces a transformation on
the relative location of 1 to 0 which we now outline.
Consider the case where gx
0
= x
1
and a passive trans-
formation h on 0 is applied while 1 is left unchanged
(at position x
1
). We have x
0
7→ x
0
0
= hx
0
. Then the
relative location of 1 to 0 is k where kx
0
0
= x
1
. Sub-
stituting in gx
0
= x
1
and hx
0
= x
0
0
gives khx
0
= gx
0
implying that that kh = g, and therefore k = gh
1
(where we remember that since X
=
G there is al-
ways a unique g G mapping a pair of points in
X). Writing in full gh
1
hx
0
= gh
1
x
0
0
= x
1
and so
the relative location of 1 relative to 0 is now gh
1
.
A passive transformation h on 0 corresponds to the
right regular action of h on the relative location g:
g 7→ gh
1
.
The left regular action and right regular action on
G
=
X are defined as follows:
φ
L
(g, x) = gx , (2)
φ
R
(g, x) = xg
1
. (3)
Both are defined using the group multiplication,
where x G. These two actions naturally commute,
and hence X
=
G carries an action of G × G, with
one factor typically being understood as the active
and the second as the passive transformations [21].
Although φ
R
acts ‘to the right’, it is a left group ac-
tion: φ(gh, x) = x(gh)
1
= xh
1
g
1
= φ(h, x)g
1
=
φ(g, φ(h, x)). Here we take φ
L
as active and φ
R
as
passive.
1
A given state of n systems is s = (x
0
, x
1
, ..., x
n1
),
where we omit the velocities ˙x
i
since we are defining
changes of reference frame for the ‘translation’ group
G on X
=
G. This can be expressed as:
s =
g
0
0
x
0
, g
0
1
x
0
, g
0
2
x
0
, ..., g
0
n1
x
0
, (4)
where g
i
j
is the unique g G such that g
i
j
x
i
= x
j
,
and e = g
i
i
the identity element. We observe that
g
j
k
g
i
j
= g
i
k
and (g
j
i
)
1
= g
i
j
. Then the state s
0
of the
n systems relative to system 0 is:
s
0
=
g
0
0
, g
0
1
, g
0
2
, ..., g
0
n1
. (5)
The state relative to the system i is:
s
i
=
g
i
0
, g
i
1
, g
i
2
, ..., g
i
n1
. (6)
We observe that we can also describe the state relative
to hypothetical systems (i.e. relative to a point x X
which is not occupied by a system). For instance in
the above consider a point x
n
X such that x
i
6=
x
n
i {0, ..., n 1}. Then we can write:
s
n
=
g
n
0
, g
n
1
, g
n
2
, ..., g
n
n1
. (7)
1
Active and passive transformations are typically defined as
either left actions on different spaces (states and coordinates)
or a left and a right action on the same space (typically coordi-
nates). In this case (X
=
G) they can be defined as left actions
on the same space.
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As in the examples given above, we see that a relative
state s
i
is given by all the symmetry transformations
g
i
j
from system i to system j for all j {0, ..., n 1}.
There is a unique relative state s
i
, which is such that
particle i is in state e.
A change of reference frame from system 0 to sys-
tem i is a map s
0
s
i
. Let us extend the left and
right regular actions of G on itself to states:
φ
L
(g, s) = (gx
0
, gx
1
, ..., gx
n1
) , (8)
φ
R
(g, s) = (x
0
g
1
, x
1
g
1
, ..., x
n1
g
1
). (9)
The transformation s
0
s
i
is given by the right
regular action of g
0
i
:
φ
R
(g
0
i
, s
0
) =
eg
i
0
, g
0
1
g
i
0
, g
0
2
g
i
0
, ..., g
0
n1
g
i
0
=
g
i
0
, g
i
1
, g
i
2
, ..., g
i
n1
= s
i
. (10)
We observe that this transformation cannot be
achieved using the left regular action: there is no el-
ements g G such that φ
L
(g, s
0
) = s
i
(unless G is
Abelian). The transformation s
0
s
i
is a passive
transformation.
1 2
0
g
1
0
g
1
2
=g
0
2
g
1
0
g
2
1
=g
0
1
g
2
0
g
2
0
g
0
2
g
0
1
Figure 1: Diagram capturing the relational states between
three systems. Each system i assigns the relative state along
the arrow point from i to j to system j (and the identity
to themselves). For instance system 0 assigns state e, g
0
1
, g
0
2
.
By the right regular action of g
0
1
, we obtain g
1
0
, e, g
0
2
g
1
0
= g
1
2
which is the relative state assigned by system 1.
4 Quantum reference frames associ-
ated to symmetry groups
We begin this section by reviewing changes of quan-
tum reference frame for three particles on the line.
We then define a quantum change of reference frame
operator for n identical systems L
2
(G) for arbitrary
G. This generalises the change of reference frame
in [6] beyond one parameter subgroups of the Galilean
group. Furthermore we show that it is only the L
2
(G)
system described so far for which a unitary reversible
change of reference system is possible. Finally, we de-
fine a change of reference frame operator for m iden-
tical L
2
(G) systems serving as reference frames and
n m systems which are not.
4.1 Comment on finite groups and notation
All our results apply for finite groups. In this case
L
2
(G) should be replaced by C[G]
=
C
|G|
and inte-
grals
R
gG
|gihg|dg by
P
i
|g
i
ihg
i
|. C[G] is the vector
space freely generated by the elements of G, i.e. for
which the elements of G form a basis.
In the cases where it is clear which system is the
reference system we sometimes omit the top label for
ease of reading. For instance the state |0i
A
A
|x
1
i
A
B
|x
2
i
A
C
is written as |0i
A
|x
1
i
B
|x
2
i
C
.
4.2 The example of L
2
(R) L
2
(R) L
2
(R)
Let us first rephrase the known case of the translation
group acting on three particles on the line [6] in the
formalism outlined above.
Take the translation group T = (R, +) and three
systems A, B and C whose joint state space is L
2
(R)
L
2
(R) L
2
(R). Let us for instance consider the state
|0i
A
|x
1
i
B
|x
3
i
C
, (11)
which is the state of three perfectly localised sys-
tems, described using a coordinate system centred
on system A. In standard quantum mechanics, when
changing from a classical, highly localised reference
frame at the position of A to another classical refer-
ence frame localised at B translated by an amount x
1
,
one simply applies the translation operator
ˆ
T (x
1
) =
e
ix
1
(ˆp
A
+ˆp
B
+ˆp
C
)
to the state of the three systems, where
ˆp
A
is the momentum operator for system A and simi-
larly for ˆp
B
, ˆp
C
and systems B and C. This shifts the
state to:
|−x
1
i
A
|0i
B
|x
3
x
1
i
C
. (12)
In the previous language we have g
A
B
= x
1
and g
A
C
=
x
3
. The action of
ˆ
T (x
1
) corresponds to the right
action of g
A
B
= x
1
.
The next step is to begin with a state of the follow-
ing form:
|0i
A
1
2
(|x
1
i + |x
2
i)
B
|x
3
i
C
, (13)
which is described by a coordinate system localised at
A. What is the change of reference frame A B in
this case? How can one describe classical coordinates
which assign state |0i
B
, when B is not localised rel-
ative to A? A standard translation of all states will
not work.
Following the reasoning presented in [6] we assume
that the change of perspective obeys the principle of
superposition. Namely the state of Equation (13)
is an equally weighted superposition of the states
|0i
A
|x
1
i
B
|x
3
i
C
and |0i
A
|x
2
i
B
|x
3
i
C
. The change of
reference frame for each of these states individually is
obtained by translating by x
1
and x
2
respectively.
Assuming that the superposition principle applies
to changes of reference systems, the state described in
coordinates ‘localised’ at system B is just the super-
position of the classical states obtained by translation
by x
1
and x
2
. This leads to the following state of
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the joint system from the viewpoint of B:
|0i
B
1
2
(|−x
1
i
A
|x
3
x
1
i
C
+ |−x
2
i
A
|x
3
x
2
i
C
).
(14)
When changing between the viewpoints of quantum
systems, we apply a weighted translation of the states
of systems, dependent on the state of the new refer-
ence frame whose viewpoint we are adopting. For the
state given above, this means applying a translation
for the state of B being |x
1
i and one for it being |x
2
i.
Figure 2: Example for translation group: In the upper sub-
figure, the state of the three systems is given from the per-
spective of system A. The lower subfigure shows the state
relative to B.
The states of A and C become entangled relative
to B. We see that to perform this change of refer-
ence frame, the state of A is mapped to the inverse
of the group element associated with the old state of
B. Also for each state of B, the state of C is shifted
respectively. Hence, for the translation group on the
real line, the reference frame change operator is
U
AB
=SWAP
A,B
Z
dx
i
dx
j
|−x
i
ihx
i
|
B
1
A
|x
j
x
i
ihx
j
|
C
.
(15)
This operator performs exactly the same reference
frame change as the operator given in [6]:
ˆ
S
AB
=
ˆ
P
AB
e
i/~ˆx
B
ˆp
C
, (16)
where
ˆ
P
AB
is the so-called parity-swap operator. It
acts as
ˆ
P
AB
ψ
B
(x) = ψ
A
(x). The proof of this is
given in Appendix F.
4.3 n identical systems L
2
(G)
Consider a configuration space X
=
G and n sys-
tems each with associated Hilbert space H
i
=
L
2
(G)
for G continuous (or C
|G|
for G finite):
G L
2
(G) ,
g
i
7→ |g
i
i. (17)
L
2
(G) is the space of square integrable functions G
C.
The left and right action of G onto itself induces
the left regular and right regular representation of G
on each H
i
. For a given H
i
this representation acts
on the basis {|gi} as:
U
L
(g
2
) : |g
1
i 7→ |g
2
g
1
i , (18)
U
R
(g
2
) : |g
1
i 7→
g
1
g
1
2
. (19)
An arbitrary basis state of the n systems is:
|ψi = |g
0
i
0
|g
1
i
1
... |g
n1
i
n1
. (20)
Following the classical case, the choice of coordinates
on G associated to H
0
is:
|ψi
0
= |ei
0
g
0
1
1
...
g
0
n1
n1
, (21)
where g
i
j
g
i
= g
j
. For general H
i
it is:
|ψi
i
=
g
i
0
0
g
i
1
1
... |ei
i
...
g
i
n1
n1
. (22)
The change of coordinate system |ψi
0
|ψi
i
is
given by U
R
(g
0
i
)
n
, when considering orthogonal ba-
sis states alone.
Let us observe that the left regular represen-
tation on the space of wave functions acts as
ψ(x) 7→ ψ(g
1
x) and the right regular rep-
resentation as ψ(x) 7→ ψ(xg). This follows
from
R
xG
ψ(x) |gxidx =
R
xG
ψ(g
1
x) |xidx and
R
xG
ψ(x)
xg
1
dx =
R
xG
ψ(xg) |xidx. We note
that for Lie groups G the objects |gi are not in L
2
(G)
and one should typically prefer the representation act-
ing on the wavefunctions. In the following however we
consider the representation acting on the elements |gi
in order to describe the continuous and discrete case
simultaneously.
Unlike some approaches to relational quantum dy-
namics [3, 14, 15] we do not assign a global state
|ψi H and then work out its expression relative to
a certain system. Rather, we begin from a state rela-
tive to a system and define changes of reference frame
to other systems. We formalise this in the following
principle:
Principle 1 (Relational physics). Given n systems,
states are defined to be relative to one of the systems.
A state relative to system i is a description of the
other n 1 systems, relative to i.
We observe that this principle does not preclude
the existence of a well defined global state of the n
systems.
A superposition state (relative to 0 in the G product
basis) is of the form:
|ψi
0
= |ei
0
g
0
1
1
...
g
0
n1
n1
+ |ei
0
h
0
1
1
...
h
0
n1
n1
.
(23)
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Since the state is in general not a basis state there is
no a priori well defined change given by an operator of
the form U
R
(g
0
i
)
n
. However, following the example
of [6] one can define a coherent change of reference
frame operator. We explicitly state this as a principle:
Principle 2 (Coherent change of reference system).
If |ψi
0
7→ |ψi
i
and |φi
0
7→ |φi
i
then α |ψi
0
+ β |φi
0
7→
α |ψi
i
+ β |φi
i
, α, β C.
This implies that |ψi
0
defined above changes to:
|ψi
i
= |ei
i
g
i
0
0
g
0
1
g
i
0
1
...
g
0
n1
g
i
0
n1
+ |ei
i
h
i
0
0
h
0
1
h
i
0
1
...
h
0
n1
h
i
0
n1
. (24)
The operator which implements the coherent change
of reference systems 0 i is:
U
0i
=SWAP
0,i
Z
g
0
i
G
g
i
0
g
0
i
i
1
0
U
R
(g
0
i
)
n2
dg
0
i
.
(25)
The following lemmas are proven in Appendix B.
Lemma 1. U
0i
is unitary.
Lemma 2.
U
0i
= U
i0
Lemma 3. U
ij
U
ki
= U
kj
4.3.1 Change of reference frame for observables
The change of reference frame operator also allows us
to transform between observables. Namely if system
0 describes an observable of systems 1, ..., n 1 as
Z
0
0,1,...,n1
= 1
0
Z
0
1,...,n1
then system i describes
the observable as Z
i
0,1,...,n1
= U
0i
Z
0
0,1,...,n1
U
i0
.
4.3.2 L
2
(U(1)) L
2
(U(1)) L
2
(U(1))
To illustrate the changes of reference frame described
previously we will give an example. Let us consider
the symmetry group U(1) and three particles A, B, C
on a circle with associated Hilbert space L
2
(U(1))
L
2
(U(1))L
2
(U(1)). In this case, the map from group
elements to elements of the Hilbert space is
U(1) L
2
(U(1)) ,
θ
i
7→ |θ
i
i , (26)
with θ
i
[0, 2π[ and hθ
i
|θ
j
i = δ(θ
i
θ
j
). The states
{|θ
i
i| θ
i
[0, 2π[} are states at all angular positions
of the unit circle and form a basis of the Hilbert space
L
2
(U(1)). A system consisting of three particles on
a circle could for instance be in the product state
|0i
A
π
2
B
|πi
C
relative to A.
An arbitrary state of the joint system relative to
particle A is given by
|0i
A
Z
i
j
ψ(θ
i
, θ
j
) |θ
i
i
B
|θ
j
i
C
, (27)
Figure 3: Basis states |0i , |θ
1
i and |θ
2
i of the state space
for L
2
(U(1)).
where |ei = |0i is the state associated to the identity
element of U(1). As a specific example, take the state
|0i
A
r
1
3
|θ
1
i +
r
2
3
|θ
2
i
!
B
|θ
3
i
C
. (28)
Relative to particle B, the state assigned to the joint
system would be
|0i
B
r
1
3
|−θ
1
i
A
|θ
3
θ
1
i
C
+
r
2
3
|−θ
2
i
A
|θ
3
θ
2
i
C
!
.
(29)
We see that the state of B is mapped to the state
corresponding to the inverse group element assigned
to the old state of B and the state of C is shifted
respectively. In the end, the labels of A and B are
swapped. The operator that performs this reference
frame change is
U
AB
=SWAP
A,B
Z
0
|−θihθ|
B
1
A
|θ
0
θihθ
0
|
C
=SWAP
A,B
Z
0
|−θihθ|
B
1
A
U
R
(θ)
C
.
(30)
Figure 4: Example of three particles on a circle.
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4.4 Unitarity of change of reference frame op-
erator
The change of reference frame defined previously is
highly constrained: it applies only to systems L
2
(G)
with symmetry group G. One could ask whether
one could define similar changes of reference frame
for systems H 6
=
L
2
(G) with states |ψ(g)i carry-
ing two representations: U
L
(h) |ψ(g)i = |ψ(hg)i and
U
R
(h) |ψ(g)i =
ψ(gh
1
)
.
First consider the symmetry group U(1). Our
results show that for a change of reference frame
to recreate our classical intuitions one needs sys-
tems L
2
(U(1)) which carry the right regular repre-
sentation. However one may wonder whether one
could use qubits with states along the X Y plane
|θi = cos(θ/2) |0i + sin(θ/2) |1i as reference systems
which transform in a manner which obeys the classical
change of reference frame.
We first provide an example to show that this
breaks linearity of the change of reference frame op-
erator for the case of rebits before proving a general
result.
Example 6 (Three qubits with U(1) group action).
Consider three qubits with states restricted to real val-
ued superpositions: |θi = cos(θ/2) |0i + sin(θ/2) |1i
(sometimes known as rebits). The space of pure states
of the three systems is U(1) U(1) U(1). We apply
our classical intuition of how a reference frame change
should act. Let the initial state of the three systems
relative to A be
|ψ(0)i
A
|ψ(θ)i
B
|ψ(θ
0
)i
C
. (31)
The map ψ takes the group element θ of U(1) to the
state in the two-dimensional Hilbert space:
ψ : U(1) H
θ 7→ cos(θ/2) |0i + sin(θ/2) |1i. (32)
We want this state to be mapped to the final state
relative to B:
|ψ(θ)i
A
|ψ(0)i
B
|ψ(θ
0
θ)i
C
. (33)
This corresponds to our intuition of what should hap-
pen when one changes from the viewpoint of A to the
viewpoint of B. Writing out this transformation in the
rebit basis corresponds to the map:
|0i
A
(cos(θ/2) |0i+ sin(θ/2) |1i)
B
(cos(θ
0
/2) |0i+ sin(θ
0
/2) |1i)
C
7→(cos(θ/2) |0i+ sin(θ/2) |1i)
A
|0i
B
(cos((θ
0
θ)/2) |0i+ sin((θ
0
θ)/2) |1i)
C
. (34)
On the other hand, considering the basis states of
the joint Hilbert spaces and assuming the map is lin-
ear, the following should hold:
|000i
ABC
7→ |000i
ABC
θ = θ
0
= 0 ,
|001i
ABC
7→ |001i
ABC
θ = 0, θ
0
= π ,
|010i
ABC
7→ |101i
ABC
θ = π, θ
0
= 0 ,
|011i
ABC
7→ |100i
ABC
θ = θ
0
= π. (35)
When comparing the coefficients in the map (34),
one sees that the reference frame change cannot be
linear. This means that the operator describing the
change from one rebit reference system to another one
is non-linear. As this non-linearity causes issues con-
cerning the invariance of probabilities under reference
frame change we conclude that rebits cannot serve as
reference frames that allow to reversibly transform be-
tween each other.
Given n classical systems with configuration space
X
=
G acted on by a symmetry group G we have
shown how to define states relative to these systems,
and to transform between them using the left and
right regular action of G on X.
The case L
2
(G) is a very specific ‘encoding’ of G
into a quantum system. It is a natural choice, in
that the classical states are embedded into orthogo-
nal states of the quantum system. However one could
have an injection G H, with g 7→ |ψ(g)i such that
the states |ψ(g)i are not all mutually orthogonal and
ask whether a change of reference system can be de-
fined. We require H to carry two unitary representa-
tions U
L
and U
R
, corresponding to active and passive
transformations, such that U
L
(h) |ψ(g)i = |ψ(hg)i
and U
R
(h) |ψ(g)i =
ψ(gh
1
)
, where |ψ(g)i =
|ψ(h)i g = h. Although one would naturally desire
them to commute (since active and passive transfor-
mations as usually defined act on different spaces and
therefore trivially commute), we do not impose this
here. The following theorem tells us that the change
of reference frame which acts as expected on prod-
uct states |ψ(e)i
i
ψ(g
i
0
)
0
...
ψ(g
i
j
)
j
...
ψ(g
i
n1
)
n1
and obeys the principle of coherent change of reference
frame is unitary exactly if the states |ψ(g)i it acts on
form an orthonormal basis of the Hilbert space.
Theorem 1. Take n identical systems with associated
Hilbert spaces H
i
each carrying two representations of
G: U
L
and U
R
such that U
L
(h) |ψ(g)i = |ψ(hg)i and
U
R
(h) |ψ(g)i =
ψ(gh
1
)
, where |ψ(g)i = |ψ(h)i
g = h. Then any operator U which performs the
change |ψ(e)i
i
ψ(g
i
0
)
0
...
ψ(g
i
j
)
j
...
ψ(g
i
n1
)
n1
7→
|ψ(e)i
j
ψ(g
j
0
)
E
0
...
ψ(g
j
i
)
E
i
...
ψ(g
j
n1
)
E
n1
and
obeys the principle of coherent change of reference
system is unitary if and only if the representations U
L
and U
R
are the left and right regular representations
acting on states |ψ(g)i which form an orthonormal
basis of H
i
(or a subspace thereof).
The proof can be found in Appendix C.
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4.5 m L
2
(G) systems describing nm systems
Let us consider the case where reference systems
L
2
(G) describe systems of a different type. The total
Hilbert space is L
2
(G)
m
H
nm
where for sim-
plicity we have assumed the n m systems to be of
the same type (but not L
2
(G)). The systems H are
such that there exists an injection φ:
φ : G H ,
g 7→ |ψ(g)i , (36)
and two representations V
L
and V
R
such that:
V
L
(g) |ψ(h)i = |ψ(gh)i , (37)
V
R
(g) |ψ(h)i =
ψ(hg
1
)
. (38)
To change from reference system 0 to i, where both
systems are assumed to be of the type L
2
(G), we ap-
ply the operator:
U
0i
= SWAP
0,i
Z
g
0
i
G
g
i
0
g
0
i
i
1
0
U
R
(g
0
i
)
m2
V
R
(g
0
i
)
nm
dg
0
i
,
(39)
where U
R
is the right regular representation acting on
the first m L
2
(G) systems.
We observe that not all systems H which carry a
representation of G will be such that there exists an
injective map φ : g 7→ |ψ(g)i. For instance the qubit
carries a representation of SU(2) but there is no in-
jection of φ : SU(2) PC
2
(where here we emphasise
that the pure states of a C
2
system form PC
2
the pro-
jective space of rays). Observe that for U(1) there is
an injection φ : U(1) PC
2
. We explore the exam-
ple of two L
2
(U(1)) systems describing a system C
2
carrying a representation of U(1).
4.5.1 L
2
(U(1)) L
2
(U(1)) C
2
Let us adapt the previous example of three particles
on a circle to the case in which the third system is
a qubit H
C
=
C
2
giving a total Hilbert space of the
joint system L
2
(U(1)) L
2
(U(1)) C
2
. H
C
carries a
representation of U(1) and an injection φ : U(1)
C
2
. The representation V
R
is given by:
V
R
(θ) =
cos(θ/2) sin(θ/2)
sin(θ/2) cos(θ/2)
(40)
in the {|0i, |1i} basis and acts by matrix multipli-
cation from the left. The injection is a map
φ : U(1) C
2
,
θ 7→ cos(θ/2) |0i + sin(θ/2) |1i. (41)
The operator that maps the state relative to A to
the state relative to B is
U
AB
= SWAP
A,B
Z
|−θihθ|
B
1
A
V
R
(θ)
C
.
(42)
As a specific example, consider the state
|0i
A
|πi
B
ψ(
π
2
)
E
C
(43)
relative to system A, where
ψ(
π
2
)
=
1
2
(|0i + |1i).
From the viewpoint of system B, the state is
|0i
B
|πi
A
ψ(
π
2
))
E
C
= |0i
B
|πi
A
1
2
(|0i |1i)
C
.
(44)
4.6 Changes of reference frame for arbitrary
identical systems
The above treatment shows that for any group G one
can define a change of quantum reference frame be-
tween n identical systems. However given n identical
systems can one always find a group allowing for a re-
versible change of quantum reference frame? Namely
for a configuration space X, can one always find a bi-
nary operation turning it into a group G
=
X? In the
case of finite systems C
d
one can pick an orthonormal
basis |xi, x {0, ..., d1} and choose the cyclic group
Z
d
acting on {0, ..., d 1}. In the case where X is a
countable set one has the group Z. In the case where
X is uncountable, the existence of a group G such
that G
=
X is equivalent to the axiom of choice [22].
We observe that if X has some additional structure
(such as being a manifold), then one may not be able
to find a group which is isomorphic as a manifold.
5 Irreversible changes of classical ref-
erence frame
In some cases one may not have access to a refer-
ence system which can distinguish all elements of the
symmetry group. Consider once more the case of par-
ticles on R acted on by the translation group. Given
a ruler with a set of marks corresponding only to
the subset of integers (i.e. with configuration space
X
=
Z), one would not be able to distinguish all
possible different configurations of the particles and
by extension all possible translations. Such imperfect
reference frames, with configuration space X which
is a coarse-graining of the group G, will lead to irre-
versible changes of reference frame as we will see in
this section.
For a given configuration space X all changes of co-
ordinates are related by a transformation g G. In
the case X
=
G there is a one to one correspondence
between points in X and coordinate systems. As such
one can identify coordinate systems as systems with
configuration space X. Namely if system i is in state
x
i
X then one assigns it the unique coordinate sys-
tem x
0
i
which maps x
i
7→ 0. However in situations
such as the one described previously one has a sym-
metry group G which is larger than X and there is
no unique element in G mapping a point x
i
to 0. We
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x
y
z
x
0
y
0
z
0
x
00
y
00
z
00
Figure 5: Reference frames associated to the same point in
R
3
.
consider an explicit example of this in the following
before describing the general case.
5.1 E
+
(3)
=
R
3
o SO(3)
Let us consider n particles in R
3
, with each parti-
cle i having state (x
i
, y
i
, z
i
) expressed in Cartesian
coordinates (x, y, z). The set of Cartesian coordi-
nates is acted on by the Euclidean group E
+
(3) =
R
3
o SO(3). A choice of coordinates (x
0
, y
0
, z
0
) such
that (x
0
i
, y
0
i
, z
0
i
) = (0, 0, 0) is said to be associated to
particle i if and only if it is the unique set of coor-
dinates with this property. There are infinitely many
such coordinate choices (for instance all coordinate
systems which are rotated relative to (x
0
, y
0
, z
0
) will
also assign state (0, 0, 0) to particle i). In this case
there is no obvious unique manner of associating a
coordinate system to a particle.
All Cartesian coordinate systems for R
3
are related
by an element g E
+
(3) where E
+
(3) is the Eu-
clidean group. The action of E
+
(3) on the set of
Cartesian coordinates is transitive and free. To put
it visually every element in E
+
(3) can be considered
as a translation followed by a rotation. Every choice
of Cartesian coordinates is associated to a set of or-
thogonal axes located at some point r R
3
with a
given orientation. These are all related to the Carte-
sian coordinates at (0, 0, 0) in a given orientation by a
rotation followed by a translation. We cannot assign a
unique coordinate system to each point in R
3
. For in-
stance take a coordinate system centred at the origin;
then any other coordinate system obtained by rota-
tion about the origin will also assign the state (0, 0, 0)
to the origin. See Figure 5.
5.1.1 Enlarging the space of states of the reference sys-
tems
There are multiple ways of addressing this issue. One
can say that systems with configuration spaces R
3
(i.e.
particles) are not good reference systems for E
+
(3).
Rather one should choose systems with a larger con-
figuration space. This is what is typically done, where
we choose solid bodies in R
3
as reference systems.
Since solid bodies have an orientation (unlike points),
which is to say that rotating a solid body changes
its state, they have configuration space E
+
(3). One
z
x
y
z
00
x
00
y
00
z
0
x
0
y
0
Figure 6: For each point in R
3
a representative member of
all reference frames centred at the point is chosen. Here the
representative member is chosen so that each representative
member has the same orientation. This ensures that the
closure of the set of transformations relating the different
reference frames is T (R
3
) and not a larger group.
can assign a unique coordinate system to every state
x X
=
E
+
(3) of a solid body. An example of a
solid body would be three physical orthogonal axes in
R
3
labelled 1, 2 and 3. For a given state x of these
three physical axes one can associate the coordinate
system which assigns +x, +y and +z to the axes 1, 2
and 3. Using this approach would allow us to make
use of the results of the previous section.
However one could also keep the reference systems
as having configuration space X but rather assign to
each state x the equivalence class of coordinate sys-
tems centred on x. One can either choose a represen-
tative member of the equivalence class (in the above
case one can fix all coordinate systems to have a given
orientation as in Figure 6) or one could average over
the possible elements of the equivalence class.
5.1.2 Representative element of each equivalence class
In the case where all the reference systems have con-
figuration space R
3
, it makes sense to assign to each
point x R
3
a unique coordinate system centred on
that point (from the equivalence class of coordinate
systems centred on that point).
Take K to be the subset of transformations which
relates these coordinate systems. We require that K is
a group in order for us to have a well defined change of
coordinate system. If K is not a group, then by com-
posing different elements in K we can obtain a group
G
0
(which is larger than K as a set). This symmetry
group will take coordinate systems we have selected
to coordinate systems which we have not chosen.
In order for all representative members (i.e. the
coordinate system we chose to be associated to each
point) to be related by a group K E
+
(3) and for the
representative members to be closed under the action
of K one can choose them to all have the same ori-
entation (i.e. be related by just translations). In this
case the symmetry group relating coordinate choices
becomes K
=
R
3
once more.
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 10
5.2 G = N o P and G = N × P : truncation
Let us consider the case where there are systems with
configuration space G and systems with configuration
space N, where G = N o P or G = N × P . In both
cases N is normal, and for every g G there is a
unique n N and p P such that g = np.
The configuration space N is embedded in G via an
embedding map E : N G, E : n 7→ np
C
for some
constant p
C
, where the choice of p
C
is conventional
and is typically chosen to be the identity. For a choice
p
C
, the points np
C
n N are related by transfor-
mations n N (acting to the left). As such the sym-
metry group of E(N) is K
=
N. If the map did not
fix a unique convention (for instance np 7→ np
0
(n))
where the image depends on which equivalence class
is chosen, then the set K of transformations between
the images E(n) would typically not be a group, and
its closure would not be isomorphic to N (in some
cases it would be the full group G).
Take k systems G and l k systems N. A general
state of the l systems is:
s = ((g
0
, g
1
, ..., g
k1
), (g
k
, ..., g
l1
)) , (45)
where g
i
G for i {0, . . . , k 1} and g
j
N for
j {k, . . . , l 1}. Moreover there is a unique n
i
N
and p
i
P such that g
i
= n
i
p
i
. Here p
j
= e for
systems j {k, . . . , l 1}. The description relative to
the first k systems and the transformations between
them is just the case described in Section 3. In the
following we describe how to change reference system
from a system with configuration space G to a system
with configuration space N.
The embedding of