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 [3–7]. 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 [8–11], 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
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 1
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 [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
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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
n−1
),
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
n−1
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
n−1
. (5)
The state relative to the system i is:
s
i
=
g
i
0
, g
i
1
, g
i
2
, ..., g
i
n−1
. (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
n−1
. (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
n−1
) , (8)
φ
R
(g, s) = (x
0
g
−1
, x
1
g
−1
, ..., x
n−1
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
n−1
g
i
0
=
g
i
0
, g
i
1
, g
i
2
, ..., g
i
n−1
= 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
g∈G
|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
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 5
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
A→B
=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
A→B
=
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
n−1
i
n−1
. (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
n−1
n−1
, (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
n−1
n−1
. (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
x∈G
ψ(x) |gxidx =
R
x∈G
ψ(g
−1
x) |xidx and
R
x∈G
ψ(x)
xg
−1
dx =
R
x∈G
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
n−1
n−1
+ |ei
0
h
0
1
1
...
h
0
n−1
n−1
.
(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
n−1
g
i
0
n−1
+ |ei
i
h
i
0
0
h
0
1
h
i
0
1
...
h
0
n−1
h
i
0
n−1
. (24)
The operator which implements the coherent change
of reference systems 0 → i is:
U
0→i
=SWAP
0,i
◦
Z
g
0
i
∈G
g
i
0
g
0
i
i
⊗ 1
0
⊗ U
R
(g
0
i
)
⊗n−2
dg
0
i
.
(25)
The following lemmas are proven in Appendix B.
Lemma 1. U
0→i
is unitary.
Lemma 2.
U
0→i
†
= U
i→0
Lemma 3. U
i→j
U
k→i
= U
k→j
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,...,n−1
= 1
0
⊗ Z
0
1,...,n−1
then system i describes
the observable as Z
i
0,1,...,n−1
= U
0→i
Z
0
0,1,...,n−1
U
i→0
.
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
dθ
i
dθ
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
A→B
=SWAP
A,B
◦
Z
dθdθ
0
|−θihθ|
B
⊗ 1
A
⊗ |θ
0
− θihθ
0
|
C
=SWAP
A,B
◦
Z
dθdθ
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
n−1
)
n−1
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
n−1
)
n−1
7→
|ψ(e)i
j
ψ(g
j
0
)
E
0
...
ψ(g
j
i
)
E
i
...
ψ(g
j
n−1
)
E
n−1
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 n−m 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
⊗n−m
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
0→i
= SWAP
0,i
◦
Z
g
0
i
∈G
g
i
0
g
0
i
i
⊗ 1
0
⊗ U
R
(g
0
i
)
⊗m−2
⊗ V
R
(g
0
i
)
⊗n−m
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
A→B
= SWAP
A,B
◦
Z
dθ |−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, ..., d−1} 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.
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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
k−1
), (g
k
, ..., g
l−1
)) , (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 N ⊂ G is given by n 7→ ne. We
define the truncation map:
T : G → N ,
g = np 7→ n , (46)
and the map R
i
G
:
R
i
G
: (g
0
, ..., g
l−1
) 7→ (g
i
0
, ..., g
i
l−1
) . (47)
Then, the relative state s
0
= R
0
G
(s) is:
s
0
=
(e, g
0
1
, ..., g
0
k−1
), (g
0
k
, ..., g
0
l−1
)
=
(e, n
0
1
p
0
1
, ..., n
0
k−1
p
0
k−1
), (n
0
k
p
0
k
, ..., n
0
l−1
p
0
l−1
)
,
(48)
where n
j
i
p
j
i
= g
j
i
.
Let us consider the case of j ∈ {k, . . . , l − 1}, i.e.
where g
j
= n
j
. Then we have that g
0
j
= g
j
g
−1
0
=
n
j
g
−1
0
. Now let us observe that for all g there is a
unique decomposition into g = np, and so we write
g
0
j
= n
0
j
p
0
j
. We now work out this n
0
j
p
0
j
in terms of n
j
and g
0
= n
0
p
0
using the two following equalities:
g
0
j
= n
j
g
−1
0
= n
j
p
−1
0
n
−1
0
,
g
0
j
= n
0
j
p
0
j
.
Combining these gives:
n
j
p
−1
0
n
−1
0
= n
0
j
p
0
j
.
Now let us introduce an identity p
0
p
−1
0
on the RHS:
n
j
p
−1
0
n
−1
0
p
0
p
−1
0
= n
0
j
p
0
j
,
and observe that gng
−1
∈ N for all g ∈ G, which
implies that p
−1
0
n
−1
0
p
0
∈ N, in turn implying that
n
j
p
−1
0
n
−1
0
p
0
∈ N. Since the decomposition of g
0
j
into
n
0
j
p
0
j
is unique, this implies that n
0
j
= n
j
p
−1
0
n
−1
0
p
0
and p
0
j
= p
−1
0
for all j ∈ {k, . . . , l − 1}. Therefore
s
0
=
(e, g
0
1
, ..., g
0
k−1
), (g
0
k
, ..., g
0
l−1
)
=
(e, n
0
1
p
0
1
, ..., n
0
k−1
p
0
k−1
), (n
0
k
p
C
, ..., n
0
l−1
p
C
)
,
(49)
where p
C
= p
−1
0
.
For particle j with configuration space N, the state
s
j
is:
s
j
=
(n
j
0
, n
j
1
, ..., n
j
k−1
), (n
j
k
, ..., n
j
l−1
)
, (50)
which is obtained from s
0
by the map (Γ
R
(n
0
j
) ◦ T).
Here Γ
R
(n) is just shorthand for the right regular
action: Γ
R
(n)(g
0
, ..., g
l−1
) = φ
R
(n, (g
0
, ..., g
l−1
)) =
(g
0
n
−1
, ..., g
l−1
n
−1
). We have n
0
i
n
j
0
= n
j
i
.
We observe that all states s
0
of the form
(n
0
0
p
0
0
, n
0
1
p
0
1
, ..., n
0
k−1
p
0
k−1
), (n
0
k
p
0
k
, ..., n
0
l−1
p
0
l−1
)
for all p
0
i
∈ P give the same s
j
. The change of
reference frame s
0
7→ s
j
is an irreversible change of
reference frame.
5.2.1 R
∼
=
Z o U(1)
An example of such a truncation is the ‘modular trun-
cation’ of the translation group: R
∼
=
Z o U(1). In-
stead of distinguishing all position states on the real
line, we consider a reference frame that essentially
consists of a classical ruler. All points that lie within
an interval of length L are mapped to the same point
on the ruler (for simplicity, one can choose L = 1).
Hence, the map identifies a subset of elements of
G = R with the same element of N = Z:
T : R → Z ,
x 7→ n , (51)
where x = nL + p, p ∈ [0, L[, n =
x
L
∈ Z.
Physically, this truncation can be viewed as coarse-
graining; in the sense of resolution, this means we
transform from a finer to a coarser resolution.
Consider now three particles on the real line at po-
sitions x
1
1
, x
1
2
and x
1
3
relative to particle 1, where
particle 1 and 2 have configuration space R and par-
ticle 3 has configuration space Z. Let us write the
state s
1
= (0, n
1
2
L + p
1
2
, n
1
3
L + p
1
3
), then the state s
3
is obtained by Λ
1→3
T (s
1
)
= Λ
1→3
(0, n
1
2
L, n
1
3
L) =
(−n
1
3
L, (n
1
2
−n
1
3
)L, 0). If for instance T (x
1
1
) = T (x
1
2
),
i.e. n
1
1
= n
1
2
, particle 3 assigns the same state to
particles 1 and 2.
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G = R
N = Z
L
Figure 7: Modular encoding of the real line.
5.2.2 R
3
∼
=
R
2
× R
Another example is the truncation map R
3
→ R
2
. If
we have a symmetry group R
2
in a three-dimensional
space, all reference frames along a one-dimensional
line are identified with each other. In this case, when
applying the truncation map, all group elements in
R
3
are projected to the associated group elements in
R
2
. Consider three systems of which the first two
have configuration space R
3
and the last has config-
uration space R
2
. A general state of the systems is
s = (g
1
, g
2
, g
3
) where g
1
, g
2
∈ R
3
and g
3
∈ R
2
. Rel-
ative to particle 1, the state is s
1
= (e, g
1
2
, g
1
3
) =
(e, n
1
2
p
1
2
, n
1
3
p
C
) where p
1
2
, p
C
∈ R and n
1
2
, n
1
3
∈ R
2
.
First, we truncate the state: ˜s = (e = n
1
1
, n
1
2
, n
1
3
).
Then, we change to the state relative to system 3:
s
3
= (n
3
1
, n
3
2
, e). This can be understood as project-
ing the points in R
3
to a plane in R
2
. If all reference
frames along the z-axis are identified with each other,
this corresponds to projecting on the x-y−plane.
5.3 Inconsistency using the truncation method
for G = N o P
In Appendix D we prove that the state s
i
for a system
i with configuration space N obtained from state s
0
by truncating and changing reference system is not
equivalent to the state ˜s
i
obtained from the state s
by first truncating to obtain ˜s, then finding ˜s
0
and
then changing reference system to obtain ˜s
i
. Let us
write the change of reference system 0 → i for a group
G as Λ
0→i
G
s
0
= s
i
:
Λ
0→i
G
(g
0
0
, ..., g
0
l−1
) = Γ
R
(g
0
i
)(g
0
0
, ..., g
0
l−1
) , (52)
where Γ
R
(g
0
i
)s
0
is φ
R
(g
0
i
, s
0
), i.e. the right regular
action of g
0
i
on s
0
.
Theorem 2. Let s = ((g
0
, g
1
, ..., g
k−1
), (g
k
, ..., g
l−1
))
for G = N o P with g
j
= n
j
for j ∈
{k, . . . , l − 1} and g
i
= n
i
p
i
otherwise. Then
Λ
0→i
N
T (R
0
G
(s))
6= Λ
0→i
N
R
0
N
(T (s))
. Let
Λ
0→i
N
T (R
0
G
(s))
= ((n
0
, n
1
, ..., n
k−1
), (n
k
, ..., n
l−1
))
and Λ
0→i
N
R
0
N
(T (s))
=
((m
0
, m
1
, ..., m
k−1
), (m
k
, ..., m
l−1
)), then n
j
= m
j
for j ∈ {k, . . . , l − 1}. For j ∈ {0, . . . , k − 1} this is
not always the case.
The above theorem shows that depending on our
prior commitment to a well defined state s and our
interpretation of relative states s
i
, the truncation
method of obtaining relative states may not be de-
sirable. One may prefer an averaging procedure or
a method based on finding invariants. These two ap-
proaches are described in Appendix D using the exam-
ple of SO(3) o T (R
3
). We make use of the truncating
procedure in the present work since it can easily be
extended to a quantum version using the principle of
coherent change of reference system. We describe this
quantum generalisation in the next section.
6 Irreversible changes of quantum ref-
erence frame
The irreversible change of reference frame of the pre-
vious section can be extended to the quantum case by
applying the principle of coherent change of reference
frame, as in the reversible case. We describe this in
more detail in the following. We note that we only
define this change of reference frame for the specific
cases G = N o P and G = N × P .
6.1 G = N o P and G = N × P
Let us consider the change of reference frame from
system i with Hilbert space L
2
(G) to a system j with
Hilbert space L
2
(N). Let us consider the first k sys-
tems as being L
2
(G) and the last l − k systems as
being L
2
(N).
A generic product basis state
ψ
0
=
|ei
0
n
0
1
p
0
1
1
...
n
0
k−1
p
0
k−1
k−1
n
0
k
p
0
k
k
...
n
0
l−1
p
0
l−1
l−1
(where p
0
j
= e for j ∈ {k, . . . , l − 1}) maps
to
ψ
i
= |ei
i
n
i
0
0
n
i
1
1
...
n
i
l−1
l−1
for
i ∈ {k, . . . , l − 1}. Applying the coherent change
of reference system we have that the action on a
superposition state:
ψ
0
= |ei
0
O
r6=0
n
0
r
p
0
r
r
+ |ei
0
O
s6=0
m
0
s
q
0
s
s
, (53)
with m
0
s
∈ N, q
0
s
∈ P and p
0
r
= p
0
s
for all r, s ∈
{k, ..., l − 1} (and similarly for q
0
s
) maps to:
ψ
i
= |ei
i
O
r6=i
n
0
r
n
i
0
r
+ |ei
i
O
s6=i
m
0
s
m
i
0
s
. (54)
First define the truncation map T : L
2
(G) →
L
2
(N).
T =
Z
n∈N
Z
p∈P
|nihnp|dpdn , (55)
and the map:
U
i→j
N
=SWAP
i,j
◦
Z
n
i
j
∈N
n
j
i
ED
n
i
j
j
⊗ 1
i
⊗ U
R
(n
i
j
)
⊗l−2
dn
i
j
.
(56)
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The change of reference frame is given by:
V
i→j
= U
i→j
N
◦
T
⊗k
⊗ 1
⊗l−k
. (57)
6.1.1 R
∼
=
Z o U(1)
Consider again the example of the modular truncation
of the translation group: R
∼
=
Z o U(1). Take three
(quantum) systems A, B and C where A has configu-
ration space R but states relative to C are encoded in
LZ. Changing reference frame from A to C requires
|0i
A
|x
0
i
B
|x
1
i
C
= |0i
A
|n
0
L + p
0
i
B
|n
1
Li
C
7→ |0i
C
|−n
1
Li
A
|(n
0
− n
1
)Li
B
. (58)
The truncation operator is:
T =
X
n∈Z
Z
p∈U(1)
dp |nLihnL + p|, (59)
and the change of reference frame operator for L
2
(N)
is:
U
A→C
N
= SWAP
A,C
◦
Z
n∈N
n
−1
n
C
⊗ 1
A
⊗ U
R
(n)
B
dn.
(60)
The following operator performs the change A → C:
V
A→C
= U
A→C
N
◦ (1
A
⊗ T
B
⊗ T
C
). (61)
When applying the change of reference frame from
A to C to a state in which system C is in a superposi-
tion state relative to A, the state becomes entangled
relative to C:
V
A→C
|0i
A
|x
0
i
B
(|x
1
i
C
+ |x
0
1
i
C
)
=V
A→C
|0i
A
|n
0
L + p
0
i
B
(|n
1
Li
C
+ |n
0
1
Li
C
)
= |0i
C
(|−n
1
Li
A
|(n
0
− n
1
)Li
B
+ |−n
0
1
Li
A
|(n
0
− n
0
1
)Li
B
) .
(62)
However, if x
1
and x
0
1
are located in the same in-
terval relative to C (with configuration space LZ), the
entanglement vanishes. Here, we recognize the depen-
dence of entanglement on the reference frame relative
to which it is described. This was already pointed out
in [6]. Note that we observe entanglement only if the
uncertainty in the position of C with respect to A (i.e.
the difference between x
1
and x
0
1
) is larger than the
resolution L of the configuration space of reference
system C.
6.1.2 R
3
∼
=
R
2
× R
As mentioned before, the truncation map R
3
→ R
2
essentially consists of a projection from a three-
dimensional configuration space to a two-dimensional
one. Here, we project all points in R
3
onto the x-y-
plane.
R
3
→ R
2
~
P
:
= (x, y, z) 7→ ~p
:
= (x, y) . (63)
On the level of Hilbert spaces, we assign the state
~
P
E
relative to a three-dimensional configuration space
while |~pi denotes a state in H
∼
=
R
2
. Hence, to change
from the state relative to A with configuration space
R
3
to the state relative to C with configuration space
R
2
, we apply the operator
V
A→C
= SWAP
A,C
◦
ZZ
d
~
P d
~
Q
−~p
ED
~
P
C
⊗
~q − ~p
ED
~
Q
B
,
(64)
where
~
Q
:
= (x
0
, y
0
, z
0
) 7→ ~q
:
= (x
0
, y
0
).
7 Wigner’s friend experiment
In this section we consider the Wigner’s friend exper-
iment, introduced by E. Wigner [23] in 1961. The
thought experiment consists of two observers, Wigner
and his friend, and a two-level quantum system.
While Wigner is an external observer of the exper-
iment, his friend is located inside an isolated box, to-
gether with the system S in the quantum state |ψi.
Once the experiment is initiated, the friend measures
the system in a certain basis. According to the projec-
tion postulate of standard quantum theory, the state
of the system collapses to one of the eigenstates of the
measurement operator. On the other hand, Wigner
describes this process from the outside and would
assign a unitary evolution. After the measurement
of the system by the friend, the outcome of which
Wigner does not know, Wigner would assign an en-
tangled state to the joint system of the friend and the
system. These two seemingly contradicting prescrip-
tions of standard quantum mechanics are at the core
of the so-called Wigner’s friend paradox.
Before proceeding we observe that there is no logi-
cal contradiction in the above two descriptions. The
state after applying the projection postulate is a state
of the system S alone, of the form |ψi
S
. The entangled
state assigned by Wigner is a state on S+F (where F is
the friend) of the form |φi
SF
. Since these are states of
two different objects (S versus S + F) there is no log-
ical contradiction within the postulates of quantum
theory.
We will now apply the relational formalism intro-
duced previously to this apparent paradox. Let us
consider W, short for Wigner, describing his friend
F who measures the system S in the {|↑i, |↓i} ba-
sis. The friend is located inside a perfectly isolated
box. We model everything using two-dimensional sys-
tems, since we are interested in two degrees of free-
dom alone. The change of reference frame will there-
fore be for Z
2
, first described in Example 2. We con-
sider the ready state of the friend to just be |↑i and
the state of the system before the measurement to be
|ψi = α |↑i+β |↓i. We first describe the measurement
interaction from the point of view of W starting in the
case where S is in an eigenstate of the measurement
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 13
operator:
|↑i
W
F
|↑i
W
S
7→ |↑i
W
F
|↑i
W
S
, (65)
|↑i
W
F
|↓i
W
S
7→ |↓i
W
F
|↓i
W
S
. (66)
The state of the friend depends on the measurement
outcome, hence the record of the outcome can be seen
as being stored in the state of the friend. The state
|↑i
F
is the state ‘the friend sees up’, and similarly for
|↓i
F
and ‘the friend sees down’.
The change of reference frame W 7→ F for the final
states gives:
U
W→F
|↑i
W
F
|↑i
W
S
= |↑i
F
W
|↑i
F
S
, (67)
U
W→F
|↓i
W
F
|↓i
W
S
= |↓i
F
W
|↑i
F
S
, (68)
where we observe that in both cases the state of S
relative to F is |↑i
F
S
. This state encodes the fact that
the friend and the system are perfectly correlated in
both cases; |↑i
F
S
tells us that the state of the friend
and the system are related by the identity element.
For an arbitrary superposition state of the system the
unitary measurement interaction gives the following
evolution:
|↑i
W
F
(α |↑i
W
S
+ β |↓i
W
S
) 7→ α |↑i
W
F
|↑i
W
S
+ β |↓i
W
F
|↓i
W
S
.
(69)
Now, we want to apply the change of reference
frame to switch to the perspective of F. If we ap-
ply the operator U
W→F
to the initial state, using the
principle of coherent change of reference system, we
obtain:
U
W→F
|↑i
W
F
(α |↑i
W
S
+ β |↓i
W
S
) = |↑i
F
W
(α |↑i
F
S
+ β |↓i
F
S
) .
(70)
If we apply the change of reference frame U
W→F
to
the final state we get:
U
W→F
α |↑i
W
F
|↑i
W
S
+ β |↓i
W
F
|↓i
W
S
= (α |↑i
F
W
+ β |↓i
F
W
) |↑i
F
S
.
(71)
If we interpret this result as the state seen by the
friend, this would imply that the friend always sees
the system as correlated with herself. This is consis-
tent with the initial description from Wigner’s per-
spective, where the state of the friend and system
are correlated in both terms of the entangled state
α |↑i
W
F
|↑i
W
S
+ β |↓i
W
F
|↓i
W
S
.
Now, we want to compare this result to the state
we get if we simply start from the perspective of the
friend. She describes the initial state as:
|↑i
F
W
(α |↑i
F
S
+ β |↓i
F
S
) , (72)
where Wigner is in the ready state. Applying the
projection postulate to the system gives:
p
↑
= |α|
2
: |↑i
F
W
|↑i
F
S
, (73)
p
↓
= |β|
2
: |↑i
F
W
|↓i
F
S
. (74)
Whatever outcome the friend observes, she would
always describe the system definitely being in either
one of the basis states and Wigner in the state |↑i
F
W
.
This seems very different to the description we obtain
by taking Wigner’s perspective and changing to the
friend’s reference frame. Note however that this does
not necessarily imply a contradiction. More precisely,
one should be careful when interpreting the state in
Equation (71) as the final state seen by the friend.
Rather, one should interpret it as the state Wigner
infers (concludes) the friend would see. In fact, start-
ing from the state from Wigner’s perspective, we only
take into account the information that Wigner has at
his disposal. Simply changing reference frames by ap-
plying a unitary operator can by no means introduce
new information, such as the actual outcome observed
by the friend.
Hence, it is not a trivial assumption that the state
actually seen by the friend should be the same as
the state of the system relative to the friend, ob-
tained by changing perspective from Wigner’s view-
point. In fact, this is assumption C (consistency as-
sumption) of the Frauchiger-Renner no-go theorem
[24]. In the framework of relational quantum mechan-
ics, one should not assume the consistency condition
to hold a priori. Indeed in this work we show that
what Wigner infers about the friend’s state assign-
ments is given by the change of reference frame U
W→F
and not the seemingly straightforward assumption C.
We observe that the conclusion that what Wigner
can infer about the state of the system relative to the
friend is that they are correlated is the same as Rov-
elli’s treatment of the measurement process in rela-
tional quantum mechanics [2]. Here instead of reason-
ing using Wigner’s measurement operators we used an
explicit change of reference frame from Wigner to the
friend.
One may think that introducing an additional ref-
erence system R into the box containing F and S could
help resolve the issue with Wigner’s friend. We show
in Appendix E that this is not the case.
8 Discussion
8.1 Related work
The underlying approach of this work is based on the
relational quantum mechanics of [2]. Other work for-
mulating a fully relational quantum theory include [3–
7, 14, 15], the toy model of [25] and the systematic
treatment of quantum reference frames in [11, 26].
The main difference between [11, 26] and the present
treatment is that we begin from an explicitly rela-
tional state and emphasise the notion of changing
between reference frames as opposed to deriving re-
lational states from an external non-relational state.
Related approaches to relational quantum mechanics
include the perspectival quantum mechanics of [16]
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 14
and the Ithaca interpretation [27].
The notion of quantum reference frames first ap-
peared as part of the debate on the existence of
charge superselection rules [28]. Later, Aharonov and
Kaufherr gave the first explicit study of quantum ref-
erence frames [8]. Typically a description of a quan-
tum system is given relative to a classical measuring
device of infinite mass. It was shown that there is
a consistent description of quantum systems relative
to quantum reference frames of finite mass. This ad-
dresses the issue of universality of quantum theory;
namely quantum systems are usually described rela-
tive to an implicit classical reference frame. If quan-
tum theory is universal, then one would expect that
reference frames should correspond to quantum sys-
tems.
Given a quantum reference frame R and a quantum
system S the relational observables are observables of
R + S which are invariant under the symmetry group.
Depending on the state of R one may recover a re-
lational description of S which is equivalent to the
standard absolute description of S (i.e. relative to an
implicit classical reference frame). Equivalently, for
any absolute description of the state or the observ-
ables on S, one can find a system R such that there is
a gauge invariant description on S + R. This observa-
tion was important for resolving the ‘optical coherence
controversy’ [29]. Explicit maps between the absolute
and relativised descriptions can be found in [9, 11],
where an emphasis on the localisation of the reference
system is placed in [11, 26]. Other works discussing
relational observables include [3–5, 14, 15].
In [10, 30] a description of quantum systems rela-
tive to other quantum systems is given, in particular
for the translation group. The initial description of R
and S is given relative to an external classical refer-
ence frame (in the position basis), and the description
relative to R is obtained by refactoring into center of
mass and relative position. Tracing out the center of
mass partition then allows to remove all global degrees
of freedom and leaves us with a relational description
of the systems. Moreover the description of systems
is shown to be dependent on the mass of the quan-
tum reference system. Note that such global degrees
of freedom never enter into our formalism in the first
place. Instead, we give the states of systems relative
to a specified system from the outset.
The change of reference frame in this work gen-
eralises the known changes of reference frame L
2
(R)
and L
2
(R
3
) for spatial position of [6, 14], and L
2
(R)
for temporal degrees of freedom of [4, 5, 7, 15]. Ref-
erence frames for rotational degrees of freedom are
studied in the perspective-neutral approach in [14].
Brukner and Mikusch are independently working on
a treatment of rotational degrees of freedom of quan-
tum reference frames using large spin coherent states.
We observe that in both [6] and the present work
the consistency of the fully relational account with
a description which is initially given externally (and
from which the relative description is then obtained)
is not proven. However in [3] it is shown that there
is a ‘perspective-neutral’ framework which encom-
passes all perspectives for [6]. Note that this frame-
work does not describe an external structure as it
only encodes relative information. Extending this
perspective-neutral approach to the general cases in
the present paper could constitute an interesting di-
rection for future work.
In [31] an emphasis is placed on the notion that a
reference system in a superposition relative to another
system gives new coordinates, which are not related
by a classical coordinate transformation to the initial
ones. This is conceptually closer to the perspective
in the present work than that of [6]. Namely we do
not make the claim that the relative descriptions are
‘operational’ as in [6].
Prior work with an emphasis on changing per-
spectives between quantum reference frames is found
in [32] where this change of reference frame is medi-
ated by an external description, and uses tools such
as G-twirling.
In recent years there has been a significant interest
in the study of quantum reference frames as resources
for measurements, communication tasks and thermo-
dynamic exchanges amongst others [33–39]. This ap-
proach is not motivated by the same considerations
as the present paper and the link between the two ap-
proaches is not fully clear to the authors. It is possible
that the present exposition, carried out explicitly in
a group theoretic language, could be used to relate
the two. For instance the ‘perfect reference frames’ of
[9] are of the form L
2
(G), which also plays a promi-
nent role here. The link between imperfect reference
frames as standardly defined [9, 36, 38] and the trun-
cation based approach of the present work also re-
mains to be worked out.
8.2 General comments
8.2.1 Relational quantum mechanics
We have provided a formalism for relational quan-
tum mechanics which captures its spirit and recov-
ers the same conclusions for Wigner’s friend and the
Frauchiger-Renner theorem. Future work could in-
volve applying the formalism developed in this work
to other thought experiments, in order to provide an
explicit account of how relational quantum mechanics
addresses various apparent paradoxes.
8.2.2 Observer dependence of the symmetry group
One further contribution of this work is that sym-
metries are not only relative to the system being de-
scribed, but also to the system doing the describing.
A system with Hilbert space H has a symmetry group
G if it carries a representation of G, however it can
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 15
only be described as transforming under G by a sys-
tem which carries a regular representation of G. For
example although a qubit carries a representation of
SU(2), two qubits ‘describing’ each other and a third
qubit would use the subgroup Z
2
, since they can only
carry a regular representation of Z
2
and not the full
SU(2). This is reminiscent of the argument by Pen-
rose in [40].
8.2.3 G = N o P and G = N ×P
For the imperfect reference frames we describe a very
specific case, though it seems natural since it is the
structure of well known space time groups. Group
extension tells us that for any group subgroup pair
(G, N) with N normal, the extension is of the form
G = N o P or G = N ×P . Hence our approach fully
covers imperfect reference frames for group subgroup
pairs (G, N) with N normal. However for a full ac-
count of imperfect reference frames one would need
to relax the assumption of N normal and see whether
one can define meaningful changes of reference frame.
9 Conclusion
The construction of a relational quantum theory re-
quires a description of systems and states relative to
other systems. In this work, we present a formalism
that allows to describe the relative states of systems
and to change between the descriptions of different
reference systems. Starting from the analysis of refer-
ence frames in the classical realm, we moved on to the
description of quantum states relative to quantum ref-
erence frames. Depending on the system whose view-
point is adopted, the description of physical phenom-
ena changes. We find that quantum properties such
as entanglement and superposition are not absolute
but depend on the reference frame relative to which
they are described. As such the conclusions of [6] are
shown to be generic to quantum reference frames for
arbitrary groups.
Previous work on quantum reference frames in the
relational paradigm restricted itself to the transla-
tion group (for time, space and momentum transla-
tions) and the Euclidean group in three dimensions
(including rotations) [14]. Here we have extended
this to arbitrary groups, including finite groups and
non-Abelian groups. Moreover we have highlighted
the key representation theoretic features needed for
reversible changes of quantum reference frame of a
group G, namely that the quantum reference sys-
tems have Hilbert space L
2
(G). Using this insight we
have developed a more abstract formulation directly
in terms of the relevant symmetry group.
We also extended the study of reference frames in
the relational approach to imperfect reference frames
using a ‘truncation’ approach, in contrast to group
averaging approaches more common in quantum in-
formation based approaches. Future work involves ex-
ploring imperfect reference frames for different cases
than (G, N) with N a normal subgroup. One bene-
fit of our approach is that it formulates the relational
approach of [6] in a group theoretic language which is
more common in the quantum information literature.
As such it may help in developing a fully rigorous ac-
count of the link between the two. Recent work [15]
in this direction has been exploring links between dif-
ferent approaches in the case of the translation group
and would be useful to pursue in this more general
case.
Our exploration of the Wigner’s friend scenario
shows how to apply our relational framework to an
important discussion point in the literature. Using
our approach we reach the same conclusion as Rovelli
in his relational treatment of the measurement pro-
cess in [2]: all that can be said by Wigner is that the
friend is correlated with the measurement outcome.
The novel aspect of this work is the use of an explic-
itly relational formalism (which embodies the philo-
sophical position of relational quantum mechanics) as
well as obtaining the friend’s perspective through an
explicit change of reference frame, as opposed to rea-
soning indirectly using measurement operators.
Acknowledgments
The authors thank
ˇ
Caslav Brukner, Esteban Castro-
Ruiz, Flaminia Giacomini, Matt Leifer, Leon
Loveridge, Pierre Martin-Dussaud, Carlo Rovelli,
David Schmid and Rob Spekkens for helpful dis-
cussions. The authors thank Philipp H¨ohn, Leon
Loveridge and Markus M¨uller for helpful comments
on a draft of this paper. This research was supported
by Perimeter Institute for Theoretical Physics. Re-
search at Perimeter Institute is supported in part by
the Government of Canada through the Department
of Innovation, Science and Economic Development
Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.
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A Background
A.1 Coordinate systems and coordinate charts
We distinguish two separate notions: coordinate sys-
tems and coordinate charts following the presenta-
tion in [19]. A coordinate system is an isomorphism
f : X → Y where X is the object of interest and Y
is some known object we are using to describe X. In
the terminology of Korzybski, X is the territory and
Y is the map.
However it is often the case (as in general relativ-
ity) that there is no isomorphism between the object
(space-time) and the description of the object (coor-
dinate chart). A coordinate chart is a map in the
opposite direction: f : Y → X where once more X is
the object of interest and Y is the known object be-
ing used to describe X, where the map f is no longer
an isomorphism (for instance for X a curved manifold
one needs multiple coordinate charts
∼
=
R
n
to describe
X fully).
In the present paper we consider coordinate systems
of a manifold X and not coordinate charts.
A.2 Coordinate systems on G-torsors
Given a space X
∼
=
G for some group G it is clear that
every g ∈ G gives rise to an isomorphism f : X → Y
(where Y
∼
=
G) via the map f : x 7→ gx. While e is
the identity element of the group G it is the origin of
the G-torsor. Different isomorphisms f : X → Y cor-
respond to different choices of origin. To quote [41]:
‘A torsor is like a group that has forgotten its iden-
tity.’ Since every g ∈ G leads to a different coordinate
system, the set of different choices of coordinate sys-
tems is also G. As noted in the main body in the
case where X
∼
=
G is a Lie group and we are not con-
cerned with the group structure but only the smooth
differentiable manifold structure one could find other
coordinate systems via the isomorphism f : x 7→ mx
where m is an element of Diff(X), the diffeomorphism
group on X. In the case where X
∼
=
R this would
consist of considering coordinate systems which are
related by more general transformations than trans-
lations, for instance re-scaling by a real scalar.
A.3 Reference frames
A reference frame is a coordinate system together
with a physical system whose configuration uniquely
determines the coordinate system. Since every choice
of coordinate system is in correspondence with an el-
ement g ∈ G for a space X
∼
=
G, a valid choice of
physical system used to form a reference frame is a
physical system with configuration space G.
B Proofs of Lemmas 1, 2 and 3
B.1 Proof of Lemma 1
We show that U
i→j
is unitary.
To change reference frame between systems i and
j, where i and j have state space L
2
(G), and n − 2
other systems carrying the right regular representa-
tion U
k
(g
i
j
)
ψ(g
i
k
)
k
=
ψ(g
i
k
g
j
i
)
E
k
=
ψ(g
j
k
)
E
k
, the
general operator is
U
i→j
= SWAP
i,j
◦
Z
g
i
j
∈G
dg
i
j
g
j
i
ED
g
i
j
j
⊗ 1
i
⊗
O
k6=i,j
U
k
(g
i
j
). (75)
We can show that this operator is unitary:
(U
i→j
)
†
=
Z
h
i
j
∈G
dh
i
j
h
i
j
ED
h
j
i
j
⊗ 1
i
⊗
O
k6=i,j
U
†
k
(h
i
j
) ◦ SWAP
†
i,j
. (76)
Hence,
(U
i→j
)
†
U
i→j
=
Z
h
i
j
∈G
dh
i
j
h
i
j
ED
h
j
i
j
⊗ 1
i
⊗
O
k6=i,j
U
†
k
(h
i
j
) ◦ SWAP
†
i,j
◦ SWAP
i,j
◦
Z
g
i
j
∈G
dg
i
j
g
j
i
ED
g
i
j
j
⊗ 1
i
⊗
O
k6=i,j
U
k
(g
i
j
)
=
Z
h
i
j
∈G
dh
i
j
Z
g
i
j
∈G
dg
i
j
h
i
j
D
h
j
i
g
j
i
E
g
i
j
j
⊗ 1
i
⊗
O
k6=i,j
U
†
k
(h
i
j
)U
k
(g
i
j
)
=
Z
g
i
j
∈G
dg
i
j
g
i
j
g
i
j
j
⊗ 1
i
⊗
O
k6=i,j
U
†
k
(g
i
j
)U
k
(g
i
j
)
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 18
= 1
j
⊗ 1
i
⊗
O
k6=i,j
U
k
((g
i
j
)
−1
g
i
j
)
= 1
j
⊗ 1
i
⊗
O
k6=i,j
1
k
= 1.
Here, we used
D
h
j
i
g
j
i
E
= δ
gh
,
R
dg
i
j
g
i
j
g
i
j
j
= 1
j
and U
k
(e) = 1
k
.
B.2 Proof of Lemma 2
We show
U
0→i
†
= U
i→0
.
U
0→i
†
=
Z
g
0
i
∈G
dg
0
i
g
0
i
g
i
0
i
⊗ 1
0
⊗ U
†
R
(g
0
i
)
⊗n−2
◦ SWAP
†
0,i
= SWAP
i,0
◦
Z
g
0
i
∈G
dg
0
i
g
0
i
g
i
0
0
⊗ 1
i
⊗ U
†
R
(g
0
i
)
⊗n−2
= SWAP
i,0
◦
Z
g
i
0
∈G
dg
i
0
g
0
i
g
i
0
0
⊗ 1
i
⊗ U
R
(g
i
0
)
⊗n−2
= U
i→0
From the second to the third line, we commuted the
SWAP operator through to the left by changing the
labels of partitions 0 and i. From the third to the
fourth line, we used the fact that the integral over g
0
i
is the same as the integral over g
i
0
. Moreover, it holds
that U
†
R
(g
0
i
) = U
R
(g
i
0
) because U
R
(g
i
0
)U
R
(g
0
i
) |gi =
gg
i
0
g
0
i
= |gi, hence U
R
(g
i
0
) = U
†
R
(g
0
i
).
B.3 Proof of Lemma 3
We prove that U
i→j
U
k→i
= U
k→j
.
For this, we show how the operators act on an ar-
bitrary basis state:
U
i→j
U
k→i
|ei
k
g
k
0
0
...
g
k
i
i
...
g
k
j
j
...
g
k
n−1
n−1
=U
i→j
SWAP
k,i
|ei
k
g
k
0
g
i
k
0
...
g
k
i
i
...
g
k
j
g
i
k
j
...
g
k
n−1
g
i
k
n−1
= U
i→j
|ei
i
g
k
0
g
i
k
0
...
g
k
i
k
...
g
k
j
g
i
k
j
...
g
k
n−1
g
i
k
n−1
= SWAP
i,j
|ei
i
g
i
0
g
j
i
E
0
...
g
k
i
g
j
i
E
k
...
g
j
i
E
j
...
g
i
n−1
g
j
i
E
n−1
= |ei
j
g
j
0
E
0
...
g
j
k
E
k
...
g
j
i
E
i
...
g
j
n−1
E
n−1
= U
k→j
|ei
k
g
k
0
0
...
g
k
i
i
...
g
k
j
j
...
g
k
n−1
n−1
.
C On the unitarity of the change of
reference frame operator
Let us consider |ψ(g)i, with U
L
(k) |ψ(g)i = |ψ(kg)i.
Take two initial states defined relative to A:
ψ
1
= |ψ(e)i
A
|ψ(g
1
)i
B
|ψ(g
2
)i
C
, (77)
ψ
2
= |ψ(e)i
A
|ψ(h
1
)i
B
|ψ(g
2
)i
C
. (78)
The change of reference frame A → B:
ψ
i
7→
φ
i
gives:
φ
1
= |ψ(e)i
B
ψ(g
−1
1
)
A
ψ(g
2
g
−1
1
)
C
, (79)
φ
2
= |ψ(e)i
B
ψ(h
−1
1
)
A
ψ(g
2
h
−1
1
)
C
. (80)
For a superposition state
ψ
3
= α
ψ
1
+ β
ψ
2
(α, β 6= 0) the change of reference system gives:
ψ
3
= α
ψ
1
+ β
ψ
2
7→
φ
3
= α
φ
1
+ β
φ
2
,
(81)
where
φ
3
= |ψ(e)i
B
(α
ψ(g
−1
1
)
A
ψ(g
2
g
−1
1
)
C
+ β
ψ(h
−1
1
)
A
ψ(g
2
h
−1
1
)
C
) . (82)
Let us show that unitarity of the change of refer-
ence frame implies that the states |ψ(g)i are mutually
orthogonal. We compute the inner product of two ini-
tial states
ψ
1
and
ψ
3
:
ψ
1
ψ
3
= α + β hψ(g
1
)|ψ(h
1
)i
B
, (83)
and the inner product of the two final states
φ
1
and
φ
3
:
φ
1
φ
3
= α + β
φ
1
φ
2
, (84)
where
φ
1
φ
2
=
ψ(g
−1
1
)
ψ(h
−1
1
)
A
ψ(g
2
g
−1
1
)
ψ(g
2
h
−1
1
)
C
.
(85)
Since
ψ(g
2
g
−1
1
)
= U(g
2
)
ψ(g
−1
1
)
and
ψ(g
2
h
−1
1
)
= U(g
2
)
ψ(h
−1
1
)
we have that
ψ(g
2
g
−1
1
)
ψ(g
2
h
−1
1
)
C
=
ψ(g
−1
1
)
ψ(h
−1
1
)
C
. There-
fore:
φ
1
φ
3
= α + β(
ψ(g
−1
1
)
ψ(h
−1
1
)
2
). (86)
By assumption there are two commuting ac-
tions: U
L
(g) |ψ(h)i = |ψ(gh)i and U
R
(g) |ψ(h)i =
ψ(hg
−1
)
.
Now
ψ
1
ψ
3
=
φ
1
φ
3
requires hψ(g
1
)|ψ(h
1
)i =
ψ(g
−1
1
)
ψ(h
−1
1
)
2
. We assume this holds and show it
implies that hψ(g)|ψ(h)i = δ(h, g).
hψ(g
1
)|ψ(h
1
)i =
ψ(g
−1
1
)
ψ(h
−1
1
)
2
⇔ hψ(g
1
)|ψ(h
1
)i =
ψ(g
−1
1
)
ψ(h
−1
1
)
2
⇔
U
L
(g
−1
1
)ψ(g
1
)
U
L
(g
−1
1
)ψ(h
1
)
=
U
R
(h
−1
1
)ψ(h
−1
1
)
U
R
(h
−1
1
)ψ(g
−1
1
)
2
⇔
ψ(e)
ψ(g
−1
1
h
1
)
=
ψ(e)
ψ(g
−1
1
h
1
)
2
Which only holds when
ψ(g
−1
1
h
1
)
is orthogonal to
|ψ(e)i (when g
1
6= h
1
). Therefore for all g ∈ G it is
the case that hψ(e)|ψ(g)i = δ(g, e).
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 19
Now consider hψ(g)|ψ(h)i for arbitrary g, h ∈
G. This is equal to
U
L
(g
−1
)ψ(g)
U
L
(g
−1
)ψ(h)
=
ψ(e)
ψ(g
−1
h)
= δ(e, g
−1
h) = δ(g, h). This entails
that the states |ψ(g)i form an orthonormal basis for
H
i
or a subspace of it if they do not span the full
space.
Hence, the operator which performs the required
change of reference frame and obeys the principle of
coherent change of reference frame is unitary if and
only if the states |ψ(g)i form an orthonormal basis for
H
i
or a subspace thereof.
D Imperfect reference frames
D.1 Proof of Theorem 2
s = ((g
0
, ..., g
k−1
), (g
k
, ..., g
l−1
)) (87)
= ((n
0
p
0
, ..., n
k−1
p
k−1
), (n
k
, ..., n
l−1
)) (88)
We first find the state ˜s
j
= Λ
0→j
R
0
N
(T (s))
. Let us
truncate s:
˜s = T (s) = ((n
0
, ..., n
k−1
), (n
k
, ..., n
l−1
)) . (89)
Then the state ˜s
0
is:
˜s
0
= R
0
N
(n
0
0
, ..., n
0
k−1
), (n
0
k
, ..., n
0
l−1
)
. (90)
Then the state ˜s
j
= Λ
0→j
˜s
0
is:
T (s)
j
=
(n
0
0
n
j
0
, ..., n
0
k−1
n
j
0
), (n
0
k
n
j
0
, ..., n
0
l−1
n
j
0
)
(91)
=
(n
j
0
, ..., n
j
k−1
), (n
j
k
, ..., n
j
l−1
)
(92)
where n
i
j
n
i
= n
j
and n
r
i
n
j
r
= n
j
i
.
Now we compute ˜s
j
= Λ
0→j
T (R
0
G
(s))
. s
0
= R
0
G
s
is:
s
0
=
(g
0
0
, ..., g
0
k−1
), (g
0
k
, ..., g
0
l−1
)
(93)
=
(m
0
0
q
0
0
, ..., m
0
k−1
q
0
k−1
), (m
0
k
q
0
k
, ..., m
0
l−1
q
0
l−1
)
,
(94)
where g
j
i
= m
j
i
q
j
i
is the decomposition into NP . Let
us consider the elements m
0
j
q
0
j
for j ∈ {k, . . . , l − 1}.
We have that g
0
j
g
0
= n
j
→ m
0
j
q
0
j
n
0
p
0
= n
j
. Now
observe that since m
0
j
and n
j
in N this implies that
q
0
j
n
0
p
0
= n
0
∈ N . Now there is a unique q
0
j
such that
this holds, and observe that p
−1
0
n
0
p
0
= m
0
. There
is a unique m ∈ N such that mm
0
= n
0
. Therefore
q
0
j
= mp
−1
0
. However since q
0
j
∈ P this implies m = e
and therefore q
0
j
= p
−1
0
.
Let us truncate:
T (s
0
) =
(m
0
0
, ..., m
0
k−1
), (m
0
k
, ..., m
0
l−1
)
. (95)
Then one can obtain T (s
0
)
j
= Λ
0→j
T (s
0
) =
Γ
R
(m
0
j
)T (s
0
):
T (s
0
)
j
=
(m
j
0
, ..., m
0
k−1
m
j
0
), (m
0
k
m
j
0
, ..., m
0
l−1
m
j
0
)
.
(96)
Does m
0
i
m
j
0
= n
j
i
, i.e. does m
0
i
m
j
0
n
j
= n
i
. There are
two cases: i ∈ {0, . . . , k − 1} and i ∈ {k, . . . , l − 1}.
Let us consider the second:
m
0
i
q
0
i
n
0
p
0
= n
i
(97)
m
0
i
= n
i
(p
−1
0
n
0
p
0
)
−1
(98)
since q
0
i
= p
−1
0
. Similarly m
0
j
= n
j
(p
−1
0
n
0
p
0
)
−1
.
Therefore
m
0
i
m
j
0
= m
0
i
(m
j
0
)
−1
= n
i
(p
−1
0
n
0
p
0
)
−1
(p
−1
0
n
0
p
0
)n
−1
j
,
= n
i
n
−1
j
. (99)
This is indeed n
j
i
since it maps n
j
to n
i
.
Now let us consider the case i ∈ {0, . . . , k − 1}:
m
0
i
q
0
i
n
0
p
0
= n
i
p
i
(100)
where q
0
i
is not necessarily equal to p
−1
0
. Then we
obtain
m
0
i
m
j
0
= m
0
i
(m
j
0
)
−1
= n
i
p
i
(q
0
i
n
0
p
0
)
−1
(p
−1
0
n
0
p
0
)n
−1
j
(101)
= n
i
p
i
p
−1
0
n
−1
0
(q
0
i
)
−1
p
−1
0
n
0
p
0
n
−1
j
(102)
which is not equal to n
j
i
in general. For a specific
example we can look at R = Z o S
1
. Consider
s = (n
0
L + x
0
, n
1
L + x
1
, n
2
L, n
3
L). (103)
Then T (s) = (n
0
L, n
1
L, n
2
L, n
3
L) and T (s)
2
= (n
0
−
n
2
, n
1
− n
2
, 0, n
3
− n
2
)L. For the other order we get
s
0
= (0, n
1
L+x
1
−(n
0
L+x
0
), n
2
L−(n
0
L+x
0
), n
3
L−
(n
0
L + x
0
)). Let us assume that x
1
− x
0
< 0, then
T (s
0
) = (0, n
1
−n
0
−1, n
2
−n
0
, n
3
−n
0
)L and T (s
0
)
2
=
(−(n
2
−n
0
), n
1
−n
0
−1 −(n
2
−n
0
), n
2
−n
0
−(n
2
−
n
0
), n
3
−n
0
−(n
2
−n
0
))L which gives T (s
0
)
2
= (n
0
−
n
2
, n
1
−n
2
−1, 0, n
3
−n
2
)L and does not equal T (s)
2
.
D.2 Averaging and invariants for the example
of SO(3) o T (R
3
)
Now let us return to the case where the configu-
ration space is R
3
but the transformation group is
E
+
(3)
∼
=
T (R
3
)oSO(3) and the case of three particles.
Here E
+
(3) is the special Euclidean group consisting
of translations followed by a rotation. Let us say that
particle 0 uses a coordinate system (x
0
, y
0
, z
0
) such
that it assigns itself the state (0, 0, 0) =
~
0. There
are infinitely many such coordinate systems corre-
sponding to all reference frames centred on parti-
cle 0, related by rotations O ∈ SO(3). We write
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 20
~v
0
i
= (x
0
i
, y
0
i
, z
0
i
) for the state of particle i in a refer-
ence frame centred at 0. The coordinates correspond-
ing to Cartesian reference frames are in one to one
correspondence with group elements in T (R
3
)oSO(3).
Each state ~v
0
i
= (x
0
i
, y
0
i
, z
0
i
) is stabilized by a SO(3)
subgroup: ~v
0
i
o O ∀O ∈ SO(3) . We can write
R
3
∼
=
E
+
(3)/SO(3).
The description of the state of the three particles
is s
0
0,1,2
=
~
0, ~v
0
1
, ~v
0
2
. What can particle 0 infer about
the description used by particle 1, knowing only that
the convention is such that particle 1 uses Cartesian
coordinates placed at its position? There are infinitely
many such choices of coordinates, related to the co-
ordinates (x
0
i
, y
0
i
, z
0
i
) by t
−v
0
1
o O for all O ∈ SO(3).
Hence the states s
1,O
0,1,2
=
−O
~
v
0
1
,
~
0, O
~v
0
2
−
~
v
0
1
for
any O ∈ SO(3) correspond to coordinate choices cen-
tred on particle 1. In order to describe the ignorance
about O one needs a probabilistic representation of
states: a state is now a measure on R
3
. The state
−~a is the Dirac measure δ
−~a
and the description that
particle 0 assigns to particle 1 is:
s
1,av
0,1,2
=
Z
O∈SO(3)
Oδ
−
~
v
0
1
, δ
~
0
, δ
~v
0
2
−
~
v
0
1
(104)
=
µ
S
2
,|
~
v
0
1
|
, δ
~
0
, µ
S
2
,|~v
0
2
−
~
v
0
1
|
, (105)
where µ
S
2
,|~a|
is the normalised Haar measure on the
sphere S
2
of radius |~a| centred at the origin (in par-
ticle 1’s coordinates). We observe that there is no
reversible transformation from s
1,average
0,1
representing
particle 0’s knowledge of particle 1’s description back
to the initial description of particle 0. Starting from
s
0
and mapping to s
2
one obtains:
s
2,av
0,1,2
=
Z
O∈SO(3)
Oδ
−~v
0
2
, δ
~v
0
1
−~v
0
2
, δ
0
(106)
=
µ
S
2
,|
~
v
0
2
|
, µ
S
2
,|~v
0
1
−
~
v
0
2
|
, δ
~
0
. (107)
We leave to future work the development of a full
account of changes of reference frame in this proba-
bilistic case (for instance it is not clear that there is
a well defined change s
1,av
0,1,2
7→ s
2,av
0,1,2
). Here we are
interested only in the single change from s
0
7→ s
1
.
We also note that when describing ignorance one
cannot integrate over the pure states, or configura-
tions. In the above example this would result in
s
1,average
0,1
being
~
0,
~
0
which is nonsensical. This is
similar to the case in quantum theory where inte-
grating over a group action on pure states does not
represent ignorance (rather it is a projection) but in-
tegrating over mixed states corresponds to ignorance.
One must move to a probabilistic description of states
in terms of measures on X in order to infer states
of other particles for these types of cases. Alterna-
tively one could stay at the pure state description,
and instead of finding the point of view that particle
0 infers particle 1 holds, we can search for what par-
ticle 0 knows holds true for all possible descriptions
s
1
0,1
=
O
~
−a,
~
0
for all O ∈ SO(3) from the point of
view of particle 1. In other words: what are the in-
variants? Here it is straightforward to see that any
f(|~a|) is an invariant quantity.
E Wigner’s friend with additional ref-
erence system
One could argue that introducing an additional refer-
ence system R into the box containing F and S might
help to resolve the paradox of Wigner’s friend. In
this case, we can describe the measurement interac-
tion from the point of view of W as follows:
|↑i
W
R
|↑i
W
F
|ψi
W
S
→ |↑i
W
R
(α |↑i
W
F
|↑i
W
S
+ β |↓i
W
F
|↓i
W
S
).
(108)
The reference system is not affected by the measure-
ment. Now, we want to apply the change of reference
frame to switch to the perspective of F. We apply the
operator U
W→F
AS
to the initial state:
U
W→F
RS
|↑i
W
R
|↑i
W
F
|ψi
W
S
= |↑i
F
W
|↑i
F
R
|ψi
F
S
, (109)
and the final state:
U
W→F
RS
|↑i
W
R
(α |↑i
W
F
|↑i
W
S
+ β |↓i
W
F
|↓i
W
S
)
= (α |↑i
F
W
|↑i
F
R
+ β |↓i
F
W
|↓i
F
R
) |↑i
F
S
. (110)
We see that, again, the state of the system is always
correlated with the state of the friend. Wigner and
the additional reference system end up being entan-
gled. We conclude that introducing an additional ref-
erence system into the laboratory does not provide a
resolution to the paradox.
F Comparison to RF change operator
of [6]
This is the formal proof that the operator given in [6]
for the change between two reference frames A and B
on the real line (one-dimensional translation group)
is equivalent to the operator given in Equation (15),
namely
U
A→B
= SWAP
A,B
◦
Z
dxdy |−xihx|
B
⊗ 1
A
⊗ |y − xihy|
C
.
(111)
Let us take the most general state of three particles
on the real line R relative to the first particle:
|Ψi
ABC
= |0i
A
⊗
Z
dx
Z
dy ψ(x, y) |xi
B
⊗
Z
dy |yi
C
.
(112)
In the formalism of [6], the reference system is not
explicitly included in the state of the joint system.
The equivalent state would be
|Ψi
BC
=
Z
dx
Z
dy ψ(x, y) |xi
B
⊗
Z
dy |yi
C
. (113)
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 21
When changing from reference system A to reference
system B, we apply the operator of [6]:
ˆ
S
A→B
=
ˆ
P
AB
e
i/~ˆx
B
ˆp
C
. (114)
The final state is then
ˆ
S
A→B
|Ψi
BC
=
Z
dx
Z
dy ψ(x, y) |−xi
A
⊗ |y − xi
C
.
(115)
When applying the operator in Equation (111) to the
initial state in Equation (112), we get the final state
U
A→B
|Ψi
ABC
= SWAP
A,B
◦
|0i
A
Z
dx
0
dy
0
dxdy ψ(x, y) |−x
0
ihx
0
||xi|y
0
− x
0
ihy
00
||yi
= |0i
B
Z
dx
Z
dy ψ(x, y) |−xi
A
⊗ |y − xi
C
. (116)
Hence, we see that both operators act in the same way
on the most general states of particles on the real line.
Accepted in Quantum 2020-11-16, click title to verify. Published under CC-BY 4.0. 22