Switching Quantum Reference Frames for Quantum Mea-
surement
Jianhao M. Yang
Qualcomm, San Diego, CA 92121, USA
Physical observation is made relative to
a reference frame. A reference frame is
essentially a quantum system given the
universal validity of quantum mechanics.
Thus, a quantum system must be de-
scribed relative to a quantum reference
frame (QRF). Further requirements on
QRF include using only relational observ-
ables and not assuming the existence of ex-
ternal reference frame. To address these
requirements, two approaches are pro-
posed in the literature. The first one is
an operational approach (F. Giacomini, et
al, Nat. Comm. 10:494, 2019) which fo-
cuses on the quantization of transforma-
tion between QRFs. The second approach
attempts to derive the quantum transfor-
mation between QRFs from first principles
(A. Vanrietvelde, et al, Quantum 4:225,
2020). Such first principle approach de-
scribes physical systems as symmetry in-
duced constrained Hamiltonian systems.
The Dirac quantization of such systems be-
fore removing redundancy is interpreted
as perspective-neutral description. Then,
a systematic redundancy reduction proce-
dure is introduced to derive description
from perspective of a QRF. The first prin-
ciple approach recovers some of the results
from the operational approach, but not yet
include an important part of a quantum
theory - the measurement theory. This pa-
per is intended to bridge the gap. We show
that the von Neumann quantum measure-
ment theory can be embedded into the
perspective-neutral framework. This al-
lows us to successfully recover the results
found in the operational approach, with
the advantage that the transformation op-
erator can be derived from the first prin-
ciple. In addition, the formulation pre-
Jianhao M. Yang: jianhao.yang@alumni.utoronto.ca
sented here reveals several interesting con-
ceptual insights. For instance, the projec-
tion operation in measurement needs to
be performed after redundancy reduction,
and the projection operator must be trans-
formed accordingly when switching QRFs.
These results represent one step forward in
understanding how quantum measurement
should be formulated when the reference
frame is also a quantum system.
1 Introduction
The idea that a physical system or a physical
phenomenon must be described relative to a ref-
erence frame is a well-accepted principle in the
relativity theory. Abandoning the concept of ab-
solute spacetime is a foundation of the general
relativity where the laws of physics are invari-
ant when changing reference systems. A refer-
ence frame essentially is also a quantum system,
if we agree that quantum mechanics is universally
valid. Thus, a physical system or a physical phe-
nomenon must be described relative to another
quantum system. This statement is not only ap-
plied to describe a relativity event, but also appli-
cable to descriptions of all quantum phenomena.
The implication here is that a more fundamental
theory should describe a physical system relative
to a quantum reference frame (QRF), and address
how such descriptions are transformed from one
to another when switching the QRFs.
There are extensive research literature related
to QRFs [1–16]. Here we are only interested to
those theories that satisfy two criteria: 1.) Com-
pletely abandoning the concept of classical sys-
tem. All reference systems are quantum systems
instead of some kinds of abstract entities. Treat-
ing a reference frame as a classical system, such
as how the relativity theory does, should be con-
sidered as an approximation of a more funda-
mental theory that is based on QRF. 2.) Com-
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arXiv:1911.04903v4 [quant-ph] 13 Jun 2020
pletely abandoning any external reference sys-
tem and the concept of absolute state. Physical
description is constructed using relational vari-
ables from the very beginning. These criteria
are at the heart of the relational quantum me-
chanics (RQM) [17–24]
1
. We may call approaches
based on these two criteria as RQM approaches.
This second criteria is crucial to differentiate the
RQM approaches from other important research
literature on QRF, especially the resource the-
ory [14, 15]. A comprehensive review of differ-
ence between resource theory and the RQM ap-
proaches is provided in Ref. [25].
Given these criteria, two RQM approaches
stand out as of importance. The first one is an
operational approach [25] that focuses on how the
dynamical physical laws are transformed when
switching QRFs. By operational approach, we
mean it assumes that every QRF is equipped with
hypothetical measuring apparatus that can per-
form measurement to justify a state assignment.
The theory has produced several interesting re-
sults. For instance, it allows us to derive the op-
erational meaning of spin of a relativistic parti-
cle; It shows that entanglement among quantum
systems depends on the choice of QRF; It also
shows how the measurement outcomes from per-
spectives of two different QRFs can be related.
The limitation of the operational approach is that
it assumes the transformation between two QRFs
is known beforehand. The operator associated
with the transformation is derived primarily from
intuition.
The second approach [26], which we call it
a first principle approach, attempts to address
the limitation in the operational approach. The
starting point in this approach is the same, that
is, quantum systems should be described with
relational variables only. What is novel in this
approach is that it implements such idea using
the tools and concepts of constraint Hamilton
1
In the context of relational quantum mechanics [17–
20], a quantum system needs to be described relative to
another quantum system [17]. RQM discards the separa-
tion of classical system and quantum system and assumes
all systems are quantum systems, including macroscopic
systems. The relational properties between two systems
are more basic than the independent properties of a sys-
tem. An implementation of RQM is constructed such that
quantum mechanics can be reformulated with relational
properties as starting point [21, 22].
systems [27, 28]
2
. The constraint conditions en-
code the corresponding gauge symmetries. For
instance, the translational symmetry results in
a constraint that total momentum to be zero.
This implies the system should be described in a
constraint surface in the phase space. There are
two commonly used methods to canonically quan-
tize the constrained systems, namely, the reduced
quantization, and the Dirac quantization. In the
reduced quantization method, one solves the con-
straints first in the context of classical mechan-
ics, then quantizes the reduced theory. Choosing
a perspective of a particular reference frame is
interpreted as imposing a gauge fixing condition.
Thus this method derives the quantum theory di-
rectly for a particular quantum reference frame.
On the other hand, in the Dirac quantization
method, one quantizes the system first without
considering the constraints, then solves the con-
straints in the quantum theory. The quantized
theory before removing the symmetry related re-
dundancy is interpreted as a perspective-neutral
theory and it does not admit immediate oper-
ational interpretation. It is essentially a global
description of physics prior to having chosen a
reference frame relative to which the physics is
described. When the redundancy is removed, the
theory is reduced to the perspective from a par-
ticular QRF. The reduced theory then admits an
operation interpretation. Removing the symme-
try is equivalent to fixing the gauge in the case of
classical mechanics. However, there is a subtlety
that in some generic systems such as the N-body
three dimensional systems, there is no globally
valid gauge fixing condition [31]. Regardless of
the interpretation, the essence of the procedure
is to remove the symmetry related redundancy
after the Dirac quantization. The procedure con-
firms that the Dirac quantization theory can be
consistently mapped to the reduced quantization
theory in the simple one-dimensional toy model.
The procedure can be applied to different QRFs,
resulting reduced quantum theory for different
QRFs, this allows us to derive the transformation
formulation between QRFs.
The first principle approach has successfully
2
The theory of constraint Hamilton system is also
key in the canonical formulation of general relativity and
quantum gravity [29, 30]. Thus, the first principle ap-
proach inherits such advantage and is potentially an im-
portant step towards a successful quantum gravity theory.
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recovered some of the results in the operational
approach, with the advantage that the transfor-
mation operators are naturally derived. These
include the quantum state transformation when
switching QRFs, the Schrödinger equation, and
the conclusion that entanglement among quan-
tum systems depends on the chosen QRF. How-
ever, an important part of a quantum theory, the
measurement theory, is not yet studied.
The goal of this paper is to investigate how the
measurement theory can be embedded into the
first-principle approach and how to recover the re-
sults related to measurements in the operational
approach. We focus on the von Neumann quan-
tum measurement theory [32], where the mea-
surement stages are separated into two sub pro-
cesses, namely, Process 1 and Process 2. In Pro-
cess 2, the measured system and the measuring
apparatus interact, but the combined systems can
be described as unitary process and determined
by the Schrodinger Equation. At the end of the
interaction, the combined systems are entangled,
and multiple outcomes are possible. Each of the
outcome is associated with a specific value of a
pointer variable of the apparatus and a proba-
bility. Then, in Process 1, which is described
by a projection operation, a specific outcome is
singled out. In Section 3, we show that the
von Neumann measurement theory can be suc-
cessfully embedded into the perspective-neutral
framework. We are able to derive the measure-
ment formulations relative to different QRFs and
the transformation between them. The results
are consistent with the results in the operational
approach [25] but with the advantage that they
are based on a generic first principle approach.
Thus the results can be applied to more com-
plex systems such as three-dimensional many-
body system.
Although the works presented here extensively
uses the theory of the first principle approach,
our results reveal novel insights on the mea-
surement theory when switching QRFs that are
not presented in Ref. [26]. 1.) At the concept
level, we show that Process 2 can be described
in the perspective-neutral framework. However,
Process 1 is perspective and in general needs to be
implemented after a QRF is chosen, except in the
special case where the pointer variable is invari-
ant in the symmetry reduction procedure. 2.) At
the methodology level, we show how the symme-
try reduction procedure can be embedded in the
unitary formulation of Process 2, and then con-
sistently integrated with projective operation in
the reduced Hilbert space. There are ample cal-
culation examples in Section 3 and the appendix
to demonstrate this technique. 3.) At the ap-
plication level, we discuss in Section 4 the im-
portance of synchronizing the projection operator
when switching QRFs, which further confirms the
synchronization principle proposed in Ref. [22,43]
to resolve the extended Wigner’s friend paradox.
The projection operator needs to be transformed
properly in order to preserve the measurement
probability when switching QRFs.
In summary, inspired by the works started in
Ref. [25, 26], we make one step forward in the
understanding of how quantum measurement is
formulated when the reference frame itself is a
quantum system and when switching QRFs.
2 Switching QRFs via a Perspective-
Neutral Frameworks
This section briefly reviews the first principle ap-
proach developed in Ref. [26] for switching QRFs
via a perspective-neutral framework. For conve-
nience, we will adopt the same notations used in
Ref. [26]. At the core of the theory is the no-
tion that physics is purely relational. Physically
meaningful variables are relational observable,
they are invariant under certain gauge transfor-
mation. To achieve such description, the systems
under study are described as constrained Hamil-
tonian systems [27, 28].
2.1 A Toy Model
The theory is illustrated through a simple one di-
mensional toy model. We start to describe the toy
model in the context of classical mechanics. The
model consists N particles with unit mass and
canonical pairs {p
i
, q
i
}. These canonical pairs de-
fine a 2N phase space. To ensure only relational
observables are used to describe the systems, the
Lagrangian of such systems, in the context of clas-
sical mechanics, must be invariant under global
translation. This requirement leads to the fol-
lowing constraint
P =
N
X
i
p
i
≈ 0 (1)
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i.e., the momenta of the particles are not all inde-
pendent. The constraint reduces the phase space
dimension to be 2N −1. The above equation thus
defines a constraint surface in the original phase
space such that it only holds in that constraint
surface (the symbol ≈ is used to reflect this weak
equality). A Dirac observable O is defined as a
variable that is Poisson-commuting with P in the
constraint surface, i.e., {O, P } ≈ 0. Obviously
all momenta p
i
and the relative distance q
i
− q
j
are all Dirac observables. The total Hamiltonian
is read as [26]
H
tot
=
1
2
N
X
i
p
2
i
+ V ({q
i
− q
j
}) + λP (2)
where λ is the Lagrangian multiplier. It is de-
termined when choosing a reference frame thus
fixing the gauge. For instance, if one chooses par-
ticle A as reference frame, the gauge is fixed, and
it may be defined as choosing q
A
= 0 in the con-
straint surface. The momentum of A is solved
from (1):
p
A
≈ −
X
i6=A
p
i
. (3)
From the equations of motion and the gauge fix-
ing condition q
A
= 0, one can derive that λ =
−p
A
. Then, the corresponding reduced Hamilton
from A perspective can be computed as
H
red
A
=
X
i6=A
p
2
i
+
X
i6=j;i,j6=A
p
i
p
j
+ V ({q
i
}
i6=A
). (4)
Now we quantize this toy model. For simplic-
ity, only three particles A, B, C, are considered,
i.e., N = 3. First we consider the reduced quanti-
zation method. Supposed A is chosen as QRF, we
need to quantize the theory in the reduced phase
space from A’s perspective. The standard pro-
cedure is to promote {p
i
, q
i
, i 6= A} to operators
that satisfy the following commutation relations:
[ˆq
i
, ˆp
j
] = δ
ij
, [ˆq
i
, ˆq
j
] = [ˆp
i
, ˆp
j
] = 0, i, j 6= A.
(5)
This defines a 2(N − 1) dimensional phase space
in an infinite Hilbert space. In the N = 3 case,
the reduced Hamiltonian (4) is quantized as
ˆ
H
BC|A
= ˆp
2
B
+ ˆp
2
C
+ ˆp
B
ˆp
B
+ V (ˆq
B
, ˆq
B
). (6)
An arbitrary state vector for particle B and C,
from A’s perspective, can be written as
|ψi
BC|A
=
Z
dp
B
dp
c
ψ
BC|A
(p
B
, p
C
)|p
B
i|p
C
i.
(7)
2.2 Dirac Quantization
In the Dirac Quantization method, one first quan-
tizes the system without considering the con-
straints. This is achieved by promoting all {p
i
, q
i
}
to operators and with appropriate commutation
relations. This defines a 2N dimensional phase
space in an infinite kinematical Hilbert space,
H
kin
. Next, the momentum constraint (1) is
quantized in this Hilbert space. In the Dirac
quantization method, this is amounted to require
that the physical states of the systems are an-
nihilated by the momentum constraint operator,
i.e.,
ˆ
P |ψi
phys
= (ˆp
A
+ ˆp
B
+ ˆp
C
)|ψi
phys
= 0. (8)
To ensure proper inner product is well-defined for
the physical state, a physical Hilbert space H
phys
is constructed from H
kin
by an improper projec-
tor δ(
ˆ
P ) : H
kin
→ H
phys
. This is defined by the
following map [30],
|ψi
phys
= δ(
ˆ
P )|ψi
kin
= (
1
2π
Z
dse
is
ˆ
P
)|ψi
kin
.
(9)
With this definition, from an arbitrary state in
H
kin
,
|ψi
kin
=
Z
dp
A
dp
B
dp
C
ψ
kin
(p
A
, p
B
, p
C
)
× |p
A
i|p
B
i|p
C
i,
(10)
one can obtain the solution for |ψi
phys
, if solving
the constraint for particle A, as
|ψi
phys
=
Z
dp
B
dp
C
ψ
BC|A
(p
B
, p
C
)
× |− p
B
− p
C
i
A
|p
B
i
B
|p
C
i
C
,
(11)
where ψ
BC|A
(p
B
, p
C
) = ψ
kin
(−p
B
− p
C
, p
B
, p
C
).
It can be verified that (8) is satisfied. The sought-
after proper inner product for |ψi
phys
is
(ψ
phys
, φ
phys
) = hψ|δ(
ˆ
P )|φi
kin
. (12)
So far the quantization procedure follows the
standard Dirac quantization for constraint sys-
tem. The novelty in Ref [26] is to interpret the
results of the Dirac quantization as a perspective-
neutral framework for the systems. It takes all
perspectives of reference systems at once, be it
from particle A, or B, or C. To recover the relative
states from Dirac quantization to a reduced quan-
tum theory for a specific reference frame, Ref [26]
proposes the following procedure:
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Step 1, pick a reference system, say, particle A.
The physical state is then given in (11).
Step 2, apply a unitary transformation defined
in H
kin
ˆ
T
A,BC
= e
iˆq
A
(ˆp
B
+ˆp
C
)
. (13)
This effectively defines a new representation of
the physical Hilbert space. The physical state
(11) becomes
|ψi
A,BC
:=
ˆ
T
A,BC
|ψi
phys
= |p = 0i
A
⊗ (
Z
dp
B
dp
C
ψ(p
B
, p
C
)|p
B
i
B
|p
C
i
C
).
(14)
It is important to note that such transformation
is a global operation acting on all three particle
at once instead of just a local operation acting
on only one particle. Thus, it can change the
entanglement properties among the particles.
Step 3, apply a projection to remove the redun-
dancy of the degree of freedom from the reference
system. This also eliminates the self-reference
problem. This step is equivalent to project the
reference system onto the classical gauge fixing
condition. The reduced state from reference sys-
tem A, is
|ψi
BC|A
=
√
2πhq
A
= 0|ψi
A,BC
=
Z
dp
B
dp
C
ψ
BC|A
(p
B
, p
C
)|p
B
i
B
|p
C
i
C
.
(15)
This recovers the same reduced state as in (7).
The resulting Hilbert space is denoted as H
BC|A
.
2.3 Switching QRFs
If we repeat the three steps in the previous sub-
section but picking particle C as the reference sys-
tem, we obtain the reduced state
|ψi
AB|C
=
Z
dp
A
dp
B
ψ
AB|C
(p
A
, p
B
)|p
A
i
A
|p
B
i
B
,
(16)
where ψ
AB|C
(p
A
, p
B
) = ψ
kin
(p
A
, p
B
, −p
A
− p
B
),
and the resulting Hilbert space is denoted as
H
AB|C
. Switching QRF from particle A to C is
described by the following map:
ˆ
S
A→C
: H
BC|A
→ H
AB|C
. (17)
Such map can be derived by inverting the
transformation from A-perspective back to the
neutral-perspective framework and then apply-
ing the transformation to C-perspective. It con-
nects the two reduced states as
ˆ
S
A→C
|ψi
BC|A
=
|ψi
AB|C
. Ref [26] shows that this map is equiva-
lent to
ˆ
S
A→C
=
ˆ
P
CA
e
iˆq
C
ˆp
B
, (18)
where
ˆ
P
CA
is the parity-swap operator
3
, which
when acting on momentum eigenstate of C yields
ˆ
P
CA
|pi
C
= | −pi
A
. (19)
Thus, the theory developed in Ref [26] is equiva-
lent to that in Ref [25].
This concludes the overview of the first princi-
ple approach to switch QRFs via the perspective-
neutral framework. The goal of this paper is to
extend this theory to the quantum measurement
formulation. We wish to develop the theory for
quantum measurement starting from Dirac quan-
tization, and follow the same symmetry reduction
procedure to derive the reduced theory for a spe-
cific reference frame. It is also expected that by
swapping between two QRFs, the measurement
theory consistently recovers the results in the op-
erational approach.
3 Quantum Measurement
To investigate how the quantum measurement
process be embedded in a perspective-neutral
framework, we employ the von Neumann quan-
tum measurement theory, where a quantum mea-
surement is implemented by two sub processes,
Process 1 and Process 2. It is pointed out [22]
that Process 1 must be described explicitly as
observer-dependent
4
. Different observers may
have different descriptions of the same measure-
ment event, which is vividly manifested by the
Wigner’s Friend thought experiment [33, 34]. An
observer is associated with a specific QRF, this
means Process 1 must be described specifically
relative to a QRF. This is consistent with the fact
that Process 1 is operational. An operational pro-
cess should not be described in the perspective-
neutral framework as the perspective-neutral
framework does not admit an immediate oper-
ational interpretation [26]. Therefore, Process
1 needs to be described in the reduced Hilbert
space, e.g., H
AB|C
if particle C is the chosen
3
There is no physical meaning associated with the
parity-swap operator. Instead it should be regarded as a
mathematical tool for the consistency of the formulation.
4
This statement plays a crucial role in the resolution
of EPR paradox, as shown in [22].
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QRF. On the other hand, in this section we will
show that the unitary Process 2 can be con-
structed through the perspective-neutral frame-
work, i.e., from the Hilbert space H
phys
.
With these guidelines in mind, our strategy
is to formulate Process 2 in H
phys
, perform the
redundancy reduction (or, symmetry reduction)
procedure, and finally apply the projection oper-
ator representing Process 1 in the reduced Hilbert
space.
3.1 Measurement Theory for the Toy Model
The same toy model is used to develop the mea-
surement theory. Besides particle A, B and C,
another particle E is added as a measuring (or,
auxiliary) particle. In this setup, we assume there
is a pointer variable for particle E that is used to
measure particles A and B. Recall that a pointer
variable is an observable of the measuring appa-
ratus used to distinguish the outcomes at the end
of the measurement process. For simplicity, the
momentum of particle E is chosen as pointer vari-
able. We will derive the measurement formula-
tion by initially choosing particle C as the QRF,
then examine the theory by switching QRF from
particle C to particle A.
The momentum constraint (8) is extended to
include particle E, so that
ˆ
P |ψi
phys
= (ˆp
A
+ ˆp
B
+ ˆp
C
+ ˆp
E
)|ψi
phys
= 0. (20)
Supposed a complete set of measurement opera-
tors {
ˆ
Λ
p
E
} defined in H
kin
for particle A, B and
C are measured with initial state |ϕi
kin
ABC
, the
essential question here is that whether we can
construct a unitary operator to achieve Process
2. The measurement formulation typically starts
with the assumption that the initial state is a
product state between the measurement appara-
tus system and the rest of the system. In H
kin
,
the initial state is written as
|ψi
kin
= |ϕi
kin
ABC
|φ
0
i
E
, (21)
where we assume E is in an initial state of |φ
0
i
E
.
The Process 2, where particle E interacts with the
rest of system and becomes entangled with the
rest of system, is described as unitary process in
the von Neumann measurement theory. Denote
the unitary operator in H
kin
to describe Process
2 as
ˆ
U.
ˆ
U is constructed by the following map
5
ˆ
U|ψi
kin
=
ˆ
U|ϕi
kin
ABC
|φ
0
i
E
=
Z
dp
E
ˆ
Λ
p
E
|ϕi
kin
ABC
|p
E
i
E
,
(22)
which implies that
ˆ
Λ
p
E
=
E
hp
E
|
ˆ
U|φ
0
i
E
. The
proof that
ˆ
U defined by such map is a unitary
operator can be found in Ref. [36]. Note that
there are infinite number of
ˆ
Λ
p
E
and they are la-
beled with index p
E
. We need to examine how
this map is transformed when it is described in
H
phy
and the reduced Hilbert space, and whether
it can lead to the correct measurement formula-
tion in the reduced Hilbert space. First, we apply
the constraint map δ(
ˆ
P ) to map the Hilbert space
from H
kin
to H
phy
in both sides of (22)
6
,
|ψi
phys
:= δ(
ˆ
P )
ˆ
U|ψi
kin
= δ(
ˆ
P )
Z
dp
E
ˆ
Λ
p
E
|ϕi
kin
ABC
|p
E
i
E
.
(23)
Now we start the symmetry reduction procedure
by following the three steps described in previous
section. First, particle C is picked as the reference
frame. Next, a transformation operator, defined
below,
ˆ
T
C
= e
iˆq
C
(ˆp
A
+ˆp
B
+ˆp
E
)
, (24)
is applied to (22) after applying the constraint
map,
ˆ
T
C
|ψi
phys
=
ˆ
T
C
δ(
ˆ
P )
Z
dp
E
ˆ
Λ
p
E
|ϕi
kin
ABC
|p
E
i
E
.
(25)
In Appendix A, we show that
|ψi
C,ABE
:=
ˆ
T
C
|ψi
phys
= |p = 0i
C
⊗
Z
dp
E
Z
dp
A
dp
B
χ
p
E
(p
A
, p
B
)|p
A
i|p
B
i|p
E
i
E
.
(26)
where χ
p
E
(p
A
, p
B
) is defined in (63). If we further
assume that |ψi
kin
ABC
= |ϕi
AB
|ξi
C
is a product
state, it is shown in Appendix A that in this case,
|ψi
C,ABE
= |p = 0i
C
⊗
Z
dp
E
ˆ
Γ
p
E
|ϕi
AB
|p
E
i
E
.
(27)
5
Here we use the momentum eigenstate of particle E as
pointer state for illustration purpose. It is not operational
since such pointer state is maximally delocalized. A more
practical scenario is to use other internal degree of freedom
as pointer variable, which is discussed in Section 3.3.
6
Note this step of applying the constraint map δ(
ˆ
P )
and the step of applying
ˆ
U in (22) can be swapped, since
the two operators are commutative.
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where operator
ˆ
Γ
p
E
only acts on particle A and
B and is defined in (65). As the final step of the
reduction procedure, we discard the redundant
state for particle C.
|ψi
ABE|C
:=
√
2πhq
C
= 0|ψi
C,ABE
=
Z
dp
E
ˆ
Γ
p
E
|ϕi
AB|C
|p
E
i
E
.
(28)
At this point, the pointer variable p
E
is well-
defined in the reduced Hilbert space H
ABE|C
.
The Process 1 in von Neumann measurement the-
ory is described by applying the projection
ˆ
P
m
= |p
m
i
E
hp
m
| (29)
to (28), resulting
|ψi
m
ABE|C
:=
ˆ
P
m
|ψi
ABE|C
=
ˆ
Γ
p
m
|ϕi
AB|C
|p
m
i
E
.
(30)
Dropping the pointer state for particle E, and
defining
ˆ
M
m
=
ˆ
Γ
p
m
/
√
N, where N is a normal-
ization factor, we further rewrite the state vector
for particle A and B after measurement as the
well-known form,
|ϕi
m
AB|C
=
ˆ
M
m
|ϕi
AB|C
q
hϕ|
ˆ
M
†
m
ˆ
M
m
|ϕi
AB|C
. (31)
The probability of the measurement outcome for
m is
ρ
m|C
=
AB|C
hϕ|
ˆ
M
†
m
ˆ
M
m
|ϕi
AB|C
. (32)
Note that index m refers to p
m
which in our toy
model is a continuous real number p
m
∈ R for
particle E. Appendix C verifies that the set of
operators {
ˆ
M
m
} satisfy the completeness condi-
tion, i.e.,
Z
dp
m
ˆ
M
†
p
m
ˆ
M
p
m
=
ˆ
I
AB|C
, (33)
where
ˆ
I
AB|C
is a unit operator in Hilbert space
H
AB|C
.
We can also write down the relation between
operators
ˆ
M
p
m
and
ˆ
Λ
p
m
as
ˆ
M
p
m
=
r
2π
N
hq
C
= 0|e
iˆq
C
(ˆp
A
+ˆp
B
+p
m
)
× δ(ˆp
A
+ ˆp
B
+ ˆp
B
+ p
m
)
ˆ
Λ
p
m
|ξi
C
.
(34)
The derivations process in this Section can be un-
derstood reversely. Suppose one wants to mea-
sure a Hermitian observable
ˆ
O
AB
in reduced
Hilbert space H
ABE|C
using ˆp
E
as pointer vari-
able.
ˆ
O
AB
can be first decomposed to a com-
plete set of operators {
ˆ
M
p
m
} through the eigen-
value decomposition such that
R
dp
m
ˆ
M
†
p
m
ˆ
M
p
m
=
ˆ
I
AB|C
. From (34), one can reversely derive a set
of operators {
ˆ
Λ
p
m
} in H
kin
, and then construct
the unitary operator through the map defined in
(22). Thus, we show that the measurement pro-
cess for observable
ˆ
O
AB
in H
ABE|C
can be imple-
mented by a projection operation in H
ABE|C
, and
a unitary process embedded in the perspective-
neutral framework. An important consequence is
that it allows us to derive the measurement for-
mulation when switching to a different quantum
reference frame.
3.2 Switching QRFs
Now we switch the QRF to be particle A
and examine how the measurement formulation
changes. Following the same derivation proce-
dure but not assuming a product state between
system AB and C as in deriving (31), we find the
final state for particle B and C after measurement
is
|ϕi
m
BC|A
=
1
√
ρ
m|A
Z
dp
B
dp
C
κ
p
m
(p
B
, p
C
)|p
B
i|p
C
i
=
1
√
ρ
m|A
|κi
p
m
BC|A
,
(35)
where
κ
p
m
(p
B
, p
C
) =
Z
dp
0
A
dp
0
B
dp
0
C
λ
p
m
ψ
ABC
(p
0
A
, p
0
B
, p
0
C
)
λ
p
m
=
Z
u(−p
B
− p
C
− p
m
, p
B
, p
C
, p
m
; p
0
A
, p
0
B
, p
0
C
, p
0
E
)
× φ
0
(p
0
E
)dp
0
E
ρ
m|A
=
BC|A
hκ|κi
p
m
BC|A
.
(36)
and matrix element u is defined in (60). From
(31) and (35), we can derive how the two state
vector |ϕi
m
BC|A
and |ϕi
m
AB|C
are connected. De-
fine a transformation operator
ˆ
S =
ˆ
P
AC
e
iˆq
A
(ˆp
B
+ˆp
E
)
, (37)
where
ˆ
P
AC
is the parity swap operator defined in
(19). In Appendix B, we prove that
|ϕi
m
BC|A
=
ˆ
S
m
|ϕi
m
AB|C
, (38)
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where
ˆ
S
m
=
ˆ
P
AC
e
iˆq
A
(ˆp
B
+p
m
)
.
As also shown in Appendix B, the probabil-
ity of measurement outcome m from the perspec-
tive of QRF A is conserved, i.e., ρ
m|A
= ρ
m|C
.
The conservation of the probabilities of the same
measurement outcome from perspectives of both
QRFs is a natural consequence of the formulation,
owning to the unitary property of operator
ˆ
S
m
.
This result is consistent with that in Ref [25].
It it important to note that the derivation of
(35) depends on two assumptions. First, the ob-
server associated with QRF A knows the same
measurement outcome m as the observer asso-
ciated with QRF C. This should not be taken
for granted if A and C are separated remotely.
The synchronization of the information regard-
ing the measurement outcome is necessary [22].
Second, the pointer variable projection
ˆ
P
m
, de-
fined in (29), is invariant under the switching of
QRF. One can easily verify that
ˆ
S
ˆ
P
m
ˆ
S
†
=
ˆ
P
m
. (39)
To demonstrate the effect of
ˆ
S
m
, we consider
a concrete example. Supposed the measurement
operator
ˆ
M
m
projects the state for particle A and
B into a product state from C perspective, i.e.,
ˆ
M
m
= |χ
m
i
A
|ϕ
m
i
B
⊗
A
hχ
m
|
B
hϕ
m
|. (40)
The final state after measurement, according to
(31), is
|φi
m
AB|C
= |χ
m
i
A
|ϕ
m
i
B
. (41)
Thus, from C perspective, A, B, and E are not en-
tangled with each other after measurement. Now
if we switch the QRF from C to A, from A per-
spective, the observed probability and the value
of pointer variable corresponding to m are the
same, but the quantum state for B and C after
measurement is
|φi
m
BC|A
=
ˆ
S
m
|χ
m
i
A
|ϕ
m
i
B
=
ˆ
S
m
Z
dp
A
dp
B
χ
m
(p
A
)ϕ
m
(p
B
)|p
A
i|p
B
i
=
Z
dp
B
dp
C
χ
m
(−p
B
− p
C
− p
m
)
× ϕ
m
(p
B
)|p
B
i|p
C
i.
(42)
It shows that B and C are entangled. This result
is similar to that in Ref [25], except that the aux-
iliary system E is not necessarily entangled with
B or C.
3.3 Measurement Apparatus as QRF
So far the formulation assumes the measuring sys-
tem (or, auxiliary system) and the QRF are two
different systems. However, there is an impor-
tant situation where we need to treat the mea-
suring system itself as the QRF. For instance,
supposed system A is the laboratory, systems B
and C are moving with a same speed relative to
A, and C is measuring B. It is more difficult to
describe the measurement process from A per-
spective, particularly when the pointer variable
depends on the speed. It is much easier to de-
scribe the measurement process first at the rest
reference frame and then transform to describe
the process from A perspective. In this case, it
means we take the measuring system C also as the
QRF, derive the measurement formulation from
C perspective, then transform back to the per-
spective of A as QRF
7
. In Ref. [25], such scenario
is considered where system C has both external
degree of freedom (i.e., the momentum degree of
freedom) that is discarded when A is taken as a
QRF, and the internal degree of freedom that is
acting as a pointer variable to measure system B.
In the toy model considered in this paper, par-
ticles A, B, and C are interacting and only con-
strained with a global translational invariance.
Relative speed is not considered in this simple
model. We assume particle C has both external
degree of freedom, and internal degree of freedom
that acts as a pointer variable. Thus, a general
state for particle C, before considering the con-
straint, is
|φi
kin
C
=
X
m
Z
dp
C
φ
m
(p
C
)|p
C
, σ
m
i
C
. (43)
An eigenstate associated with a particular value
of pointer variable σ
m
= σ
n
is
|φ
n
i
C
=
Z
dp
C
φ
n
(p
C
)|p
C
, σ
n
i
C
. (44)
The state vector satisfies the following orthogonal
identities:
hp
0
C
, σ
n
|p
C
, σ
m
i = δ
mn
δ(p
0
C
− p
C
),
hφ
n
|φ
m
i = δ
mn
.
(45)
The pointer variable (i.e., the internal degree of
freedom), which can be the energy level, or the
7
This technique has been proposed to derive the oper-
ation meaning of a relativistic spin system [35].
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 8
spin, in theory can depend on the momentum
variable. A good example is the spin of a rela-
tivistic particle that can depend on the velocity
of the particle. For simplicity we assume here
that the pointer variable is independent of the
momentum variable.
Similar to (21), we assume the initial state of
A, B, and C is a product state in H
kin
, written
as
|ψi
kin
= |ϕi
AB
|φ
0
i
C
. (46)
The Process 2, where particle C interacts with
B, is described as a unitary process. The unitary
operator
ˆ
U
kin
is associated with the Hamiltonian
ˆ
H
kin
defined in H
kin
.
ˆ
U
kin
|ψi
kin
=
ˆ
U
kin
|ϕi
AB
|φ
0
i
C
=
X
m
ˆ
Γ
kin
m
|ϕi
AB
|φ
m
i
C
,
(47)
where
ˆ
Γ
m
=
C
hφ
m
|
ˆ
U|φ
0
i
C
is an operator defined
in H
kin
and only acts on particle A and B. Next
we apply the constraint map δ(
ˆ
P ) to (47) and get
the physical state in Hilbert space H
phys
, (47) is
changed to
|ψi
phys
= δ(
ˆ
P )
ˆ
U|ψi
kin
= δ(
ˆ
P )
X
m
ˆ
Γ
m
|ϕi
AB
|φ
m
i
C
.
(48)
To proceed with the redundancy reduction proce-
dure, particle C is picked as the reference frame.
The transformation operator
ˆ
T
C
is simply defined
below
8
ˆ
T
C
= e
iˆq
C
(ˆp
A
+ˆp
B
)
. (49)
In Appendix D, we show that after applying
ˆ
T
C
to (48), we obtain
|ψi
C,AB
=
ˆ
T
C
|ψi
phys
=
X
m
|p = 0, σ
m
i
C
⊗
ˆ
M
m
|ϕi
AB
,
(50)
where operator
ˆ
M
m
is defined in (83). The last
step of removing the redundancy of external de-
gree of freedom of QRF C can be combined with
the projection of the Process 1 of the measure-
ment. Namely, for a measurement outcome cor-
responding to pointer variable of σ
n
, we can apply
8
Operator
ˆ
T
C
is independent of the pointer variable σ
due to the assumption that the pointer variable is invari-
ant when switching QRFs. In a more complicated model,
such assumption may not hold. See more discussion in
Section 4.1
√
2πhq
C
= 0, σ
n
|⊗ to (50), and obtain the final
state after the measurement
|ϕi
n
AB|C
=
√
2πhq
C
= 0, σ
n
|ψi
C,AB
=
ˆ
M
n
|ϕi
AB|C
.
(51)
Normalization of the above state vector gives
|˜ϕi
n
AB|C
=
ˆ
M
n
|ϕi
AB|C
q
hϕ|
ˆ
M
†
n
ˆ
M
n
|ϕi
AB|C
. (52)
Thus, we show that in the case of choosing
the measurement apparatus as QRF, the mea-
surement process in the reduced Hilbert space
H
ABE|C
can be implemented by a projection op-
eration in H
ABE|C
, and a unitary process embed-
ded in the perspective-neutral framework. Next
we want to know how the same measurement pro-
cess is perceived if we choose particle A as QRF.
In Appendix E, we show that starting from the
same decomposition (47), after carrying over the
redundancy reduction procedure with particle A
as the QRF, and with transformation operator
defined as
ˆ
T
A
= e
iˆq
A
(ˆp
B
+ˆp
C
)
, we get
|ϕi
BC|A
=
X
m
Z
dp
B
dp
C
φ
m
(p
C
)χ
m
(p
B
, p
C
)
× |p
B
i|p
C
, σ
m
i,
(53)
where χ
m
(p
B
, p
C
) is defined in (86). The question
here is that what projection operator for Process
1 we should apply at this point. Since we as-
sume the pointer variable does not transform un-
der translation, from A perspective, the pointer
variable is the same as from C perspective. Thus,
we define the projection operator from C perspec-
tive as
ˆ
P
n
= |σ
n
i
C
hσ
n
|, assuming that the pointer
variable σ
m
is separable from the momentum de-
gree of freedom p
C
, i.e., |p
C
, σ
m
i = |p
C
i|σ
m
i. Ap-
plying
ˆ
P
n
to (53), we obtain the final state after
measurement,
|ϕi
n
BC|A
=
ˆ
P
n
|ϕi
BC|A
=
Z
dp
B
dp
C
φ
n
(p
C
)χ
n
(p
B
, p
C
)|p
B
i|p
C
i|σ
n
i.
(54)
(54) shows that in the final state from A perspec-
tive, the degree of freedom of particle B is entan-
gled with external degree of freedom of C, but
disentangled with the internal degree of freedom
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 9
of C, i.e., the pointer variable of C. This repro-
duces the same conclusion in Ref [25]. Defining
transformation map
ˆ
S
n
=
ˆ
P
n
AC
e
iˆq
A
ˆp
B
where
ˆ
P
n
AC
|pi
A
= | −p, σ
n
i
C
.
(55)
We show in Appendix F that (54) and (51) can
be transformed similar to (38)
|ϕi
n
BC|A
=
ˆ
S
n
|ϕi
n
AB|C
, (56)
It follows immediately that the probability of
measurement outcome correspondent to pointer
variable value σ
n
is preserved from the perspec-
tive of either QRF.
4 Discussion and Conclusion
4.1 The Order of Reduction and Projection
The projection in the Process 1 of the von Neu-
mann quantum measurement theory is an oper-
ational process, i.e., a process that can be con-
firmed or observed through the pointer variable
of the measurement apparatus. The value of
the pointer variable is read and recorded with
respect to a specific QRF. Thus, it should be
described in the reduced Hilbert space with re-
spect to that particular QRF. This means we per-
form the symmetry reduction of the Dirac quan-
tization first, then the projection operation. As
a consequence, the operational meaning of the
pointer variable is well-defined in the reduced
Hilbert space. Such a procedure is consistent with
the expectation that a measurement event should
be operationally well-defined. An opposite pro-
cedure is to perform the projection first in the
perspective-neutral Hilbert space, then the sym-
metry reduction procedure. However, the pro-
jection operator can be transformed during the
reduction procedure, and the operational mean-
ing of that pointer variable becomes difficult to
define. For instance, let’s consider particle A, B,
C are relativistic particles with spins and mov-
ing with different speeds. Supposed the spin of
particle C is the pointer variable for a measure-
ment on particle B and C itself is also the QRF.
The operation meaning of spin is well defined in
a rest QRF with particle C. On the other hand,
it is a challenge to perform the projection op-
eration in the perspective-neutral formulation of
the physical Hilbert space. This is because the
spin of particle C depends on its momentum and
particle C can be in a superposition state of mo-
mentum. The symmetry reduction procedure in-
volves transformation of the momentum operator
which in turn transforms the spin. How to define
a projection operator in the perspective-neutral
Hilbert space that can be transformed to the re-
duced Hilbert space and yield the well-defined
spin projector is not yet considered and needs fur-
ther investigation
9
.
There is, however, a special situation when the
pointer variable is invariant in the reduction pro-
cedure. In this case, the order of reduction and
projection does not impact the formulation of
measurement. Considered the example in Sec-
tion 3.3, where we perform reduction first, and
projection later to derive the measurement for-
mulation. But in fact the order can be reserved
and yield the same results. Supposed we first ap-
ply the projection operator
ˆ
P
0
n
= |φ
n
i
C
hφ
n
| to
(47), and obtain
ˆ
P
0
n
ˆ
U
kin
|ψi
kin
=
ˆ
Γ
kin
n
|ϕi
AB
|φ
n
i
C
. (57)
Then we proceed the reduction procedure by
choosing C as QRF, and obtain (51). If choosing
A as QRF, the reduction process resulting with
|ϕi
n
BC|A
=
Z
dp
B
dp
C
φ
n
(p
C
)χ
n
(p
B
, p
C
)
× |p
B
i|p
C
, σ
n
i,
(58)
which is essentially the same as (54). The rea-
son for this is that the pointer variable, σ
m
, is
invariant in the symmetry reduction procedure,
and consequently also invariant when switching
QRFs. However, this is a special case and should
not be considered true in general.
4.2 Synchronization Among Different QRFs
When deriving (35) or (54), we assume that
the observer associated with QRF A knows the
measurement outcome that is inferred from the
pointer variable σ
n
associated with QRF C. In
9
Ref [35] provides a solution for defining the opera-
tional meaning of the spin of a relativistic particle when
swapping QRFs that are moving with a speed relative
to each other. However, Ref [35] is not based on the
perspective-neutral framework and of course does not
address the transformation from the perspective-neutral
Hilbert space to the reduced Hilbert space.
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 10
other words, we assume that the measurement
projection operator
ˆ
P
n
is known to both ob-
servers. This assumption should not be taken
for granted. Quantum measurement must be de-
scribed relative to the local observer. An observer
who does not access to the measurement results
will not have the complete information and can
only describe the system up to the level of previ-
ous knowledge that the observer has. To ensure
the descriptions of different observers are consis-
tent, different observers should synchronize in-
formation regarding the measurement results in
order to have consistent descriptions of a quan-
tum system [22, 43]. The results in this paper
shows that a further refinement on the informa-
tion synchronization is needed. Specifically, the
pointer variable needs to be transformed prop-
erly when switching QRFs. In the special case
that the pointer variable, consequently the pro-
jection operator
ˆ
P
n
, are invariant from either A
perspective or C perspective, we simply use the
same operator
ˆ
P
n
for the projecting operation.
This is what we have done when deriving (35) or
(56).
To illustrate this point more concretely, we go
back to the example in Section 3.3 where particle
C is the measurement apparatus and we choose
particle A as QRF. After the symmetry reduction
procedure we arrive at (53) and need to apply a
projection operator for particle C from A perspec-
tive. Now, instead of choosing
ˆ
P
n|A
= |σ
n
i
C
hσ
c
|,
one may attempt to assign the projection oper-
ator
ˆ
P
0
n|A
= |φ
n
i
C
hφ
n
|. One possible reason to
make this choice is that an arbitrary wave func-
tion for particle C that is associated with pointer
variable σ
n
is given by (44). Applying this pro-
jection operator to (53), we obtain a final state
after measurement,
|ϕ
0
i
n
BC|A
=
ˆ
P
0
n|A
|ϕi
BC|A
= (
Z
dp
B
χ
0
n
(p
B
)|p
B
i)|φ
n
i
C
,
χ
0
n
(p
B
) =
Z
dp
C
|φ
n
(p
C
)|
2
χ
n
(p
B
, p
C
).
(59)
Obviously, |ϕ
0
i
n
BC|A
is a product state. Particle B
is not entangled with either the external degree of
freedom or the internal pointer variable of particle
C. This result is different from (54) where particle
B is entangled with the external degree of freedom
of C. Further calculation shows that the probabil-
ity of the measurement resulting from
ˆ
P
0
n|A
is not
the same as the probability from C perspective.
The preservation of the measurement probability
from different QRFs appears to be a natural cri-
teria that should be satisfied. Thus, the choice of
projection operator
ˆ
P
0
n|A
is incorrect.
This observation can be further explained as
following. If from C perspective the projec-
tion operator is
ˆ
P
n|C
= |σ
n
i
C
hσ
c
|, then from A
perspective, the corresponding projection opera-
tor must be derived via proper transformation,
ˆ
P
n|A
=
ˆ
S
n
(
ˆ
P
n|C
)
ˆ
S
†
n
where
ˆ
S
n
is defined in (55).
Since
ˆ
S
n
has no effect on |σ
n
i
C
hσ
c
|, we obtain
ˆ
P
n|A
=
ˆ
P
n|C
. This rules out
ˆ
P
0
n|A
as the the cor-
rect projection operator.
We can summarize the implications of the re-
lational formulation of quantum measurement
based on Refs [22, 43] and this paper as follow-
ing.
1. Measured reality is relative. Information ob-
tained through quantum measurement is lo-
cal. Measurement must be described explic-
itly relative to the local observer.
2. Synchronization of local reality. Different ob-
servers should synchronize information re-
garding the measurement results in order to
have consistent descriptions of a quantum
system.
3. When the measurement information is syn-
chronized among different observers, proper
transformation on the projection operator
must be performed to ensure consistency.
Applying the first implication, we are able to
resolve the EPR paradox [22, 41]. In that resolu-
tion, a quantum measurement should be explic-
itly described as observer dependent. The idea
of observer-independent quantum state is aban-
doned since it depends on the assumption of Su-
per Observer. By recognizing that the element of
physical reality obtained from local measurement
is only valid relatively to the local observer, the
completeness of quantum mechanics and locality
can coexist [22]. Applying the second implica-
tion, we are able to resolve the Wigner’s friend
paradox and the extended version [42,43]. These
thought experiments provide clear example for
the need of information synchronization in order
to achieve a consistent description of a quantum
system by different observers. Ref [44] shows sim-
ilar idea that the assumption of observer indepen-
dent fact cannot resolve the Wigner’s friend type
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 11
of paradox. Latest experiment appears to confirm
that observer-independent description of a quan-
tum system must be rejected [40]. Lastly, the
third implication is a confirmation of the second
implication, and is just discussed in this section.
4.3 The Wigner-Araki-Yanase Theorem
One of the fundamental questions in quantum
measurement theory is whether a quantum mea-
surement procedure can be implemented for a
given Hermitian operator. When there is an addi-
tive conservation law, the Wigner-Araki-Yanase
(WAY) Theorem [37–39] imposes restriction on
the measurement procedure in that only observ-
ables that are commutative with the conserved
quantity can be implemented with a repeatable
measurement. More precisely, suppose a quan-
tum system S is measured with an apparatus
A and the correspondent Process 2 is described
by a unitary operator
ˆ
U. Suppose also there
is an additive conservation quantity, denoted as
ˆ
L =
ˆ
L
S
⊗
ˆ
I
A
+
ˆ
I
S
⊗
ˆ
L
A
where
ˆ
I is a unit operator,
such that [
ˆ
U,
ˆ
L] = 0. Then, the WAY theorem
states that the only observables
ˆ
O
S
for which it
is possible to implement the projective measure-
ment are those that are commutative with
ˆ
L
S
,
i.e. those that satisfy [
ˆ
O
S
,
ˆ
L
S
] = 0.
One may ask whether the WAY theorem im-
poses restriction on the derivation of measure-
ment formulation in this paper. We can exam-
ine the derivation in Section 3.1. The conser-
vative quantity in H
phys
is the total momentum
ˆ
P = (ˆp
A
+ ˆp
B
+ ˆp
C
) + ˆp
E
, which is the constraint
and satisfies [
ˆ
P ,
ˆ
U
phys
] = 0. We can consider
ˆ
L
S
= ˆp
A
+ ˆp
B
+ ˆp
C
and
ˆ
L
A
= ˆp
E
. However,
H
phys
is a perspective neutral Hilbert space and
as discussed earlier, one cannot consider the pro-
jective Process 1 in this Hilbert space. In other
words, it is not required that the pointer variable
states are orthogonal in H
phys
. Conceptually we
cannot implement a complete von Neumann mea-
surement procedure including both Process 1 and
2 in H
phys
. It is conceptually incorrect to apply
the WAY theory in H
phys
. On the other hand,
in the reduced Hilbert space, such as in H
ABE|C
,
the conservative quantity does not exist because
once taken particle C as the QRF, the transla-
tion symmetry is broken. Thus, the WAY theory
cannot be applicable in any reduced Hilbert space
either. We conclude that the WAY theorem is not
applicable to the formulation presented here.
However, the spirit behind the WAY theorem,
i.e., given a Hermitian operator and a constraint,
whether a unitary operator can be constructed
to implement Process 2 must be carefully exam-
ined. In the context of the toy model, this means
that given a Hermitian operator in the reduced
Hilbert space H
ABE|C
, we need to confirm that
there exists a unitary operator in H
kin
that can
implement Process 2. The last paragraph in Sec-
tion 3.1 shows this claim is indeed correct.
4.4 Limitation
The toy model used in this paper is relatively
simple. The systems in this model are one di-
mensional and only have constraint due to the
translational symmetry. The simple model allows
one to construct a globally valid gauge fixing con-
dition. Ref. [31] extends the method to three di-
mensional N-body systems that have both trans-
lational and rotational symmetries. It will be in-
teresting to confirm that the measurement theory
developed in this paper will be applicable to the
three dimensional N-body systems as well. How-
ever, as Ref. [31] points out, for three dimensional
N-body systems that have both translational and
rotational symmetries, one cannot find globally
valid gauge fixing conditions. Nevertheless, we
expect that the measurement formulation pre-
sented in this paper is applicable to three dimen-
sional N-body systems based on the formulation
in Ref. [31]. It is desirable to extend the theory to
even more complicated model that includes other
degree of freedom such as spin.
The assumption that the pointer variable is in-
variant during the reduction process is another
major limitation. This is particularly true in the
case that the measurement apparatus is also the
QRF where pointer variable ˆσ
m
is invariant when
switching QRF. If considered ˆσ
m
is spin of par-
ticle C, such assumption is reasonable in a non-
relativistic case. However in a relativistic case,
spin depends on momentum and changes when
switching QRFs or during the reduction process,
the transformation operator such as
ˆ
T
C
needs to
include the pointer variable, and the formulation
will be more complicated. In the present pa-
per, since we assume the pointer variable is in-
variant when switching QRFs, the transformation
of this pointer variable when switching QRFs is in
a sense a priori known. Thus, the promised of not
relying on a priori known transformation in the
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 12
first principle approach is not completely fulfilled.
Nevertheless, the transformation of other degree
of freedoms such as momentum of the QRF can
be derived. Once the dependency of pointer vari-
able ˆσ
m
is factored into
ˆ
T
C
during the reduction
process, such limitation can be overcome.
The projection process in the von Neumann
quantum measurement theory is a simple math-
ematical modeling of the actual measuring pro-
cess. It cannot explain the mechanism of “wave
function collapse”. Ref. [26] speculates that by
including the measurement interaction into the
perspective-neutral structure, it may possibly
lead to the “collapse” in the respective internal
perspective. Obviously our formulation here does
not achieve such goal. However, we do bring in
new conceptual implications discussed earlier.
4.5 Conclusions
Inspired by the novel approach of switching QRFs
via a perspective-neutral framework [26], this pa-
per extends the approach to the quantum mea-
surement process. Specifically, we show the
von Neumann quantum measurement theory can
be embedded in the perspective-neutral frame-
work. Based on the same simple toy model
as in Ref. [26], we show that Process 2 in
the von Neumann measurement theory, which
is a unitary process, can be formulated in the
perspective-neutral framework, while Process 1,
which is a projection, should be described after
the perspective-neutral structure is reduced to be
specifically relative to a QRF. In the special case
when the pointer variable is invariant to the re-
dundancy reduction procedure (hence invariant
when switching QRFs), the order of reduction
versus projection has no impact on the results.
Our results successfully reconstruct the measure-
ment formulation, as shown in Section 3.1 and
3.2, from perspectives of different QRFs. This al-
lows us to further derive the transformation oper-
ator for the measurement outcome when switch-
ing QRFs, given in (37). These results are consis-
tent with that in Ref. [25], with the advantage of
being derived from the first principles proposed
in Ref. [26].
Furthermore, when the measurement appara-
tus itself is considered as a QRF, our measure-
ment formulation provides additional conceptual
implications. In particularly, when switching
QRFs, the projection operator must be trans-
formed properly, otherwise it causes inconsis-
tency. For instance, the probabilities from differ-
ent observers for the same measurement outcome
can be different. Conceptually, this further con-
firms the synchronization principle [22,43] that is
used to resolve paradox for the extended Wigner’s
friend thought experiment.
In conclusion, the results presented in this pa-
per further confirm the validity of the first princi-
ple approach of switching QRFs via a perspective-
neutral framework [26]. Our formulations on the
measurement theory fills in the gap to recover the
transformation mechanism when switching QRFs
using the operational approach [25]. All these
research results together extend the understand-
ing on how quantum systems should be described
when the reference frame itself is a quantum sys-
tem and when switching QRFs.
Acknowledgements
I would like to acknowledge the contributions
from Philipp A. Höhn through many insightful
discussions that help to refine some of the ideas
in the paper, and sincerely thank him for point-
ing out technical errors during the preparation
of the manuscript. I would also like to thank the
anonymous referees for the valuable comments, in
particular, for bringing up the discussion of the
Wagner-Araki-Yanase theorem.
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Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 15
A Proof of Eq.(26)
In general, the unitary operator
ˆ
U can be decomposed as
ˆ
U =
Z
dp
A
dp
B
dp
C
d˜p
E
dp
0
A
dp
0
B
dp
0
C
d˜p
0
E
u(p
A
, p
B
, p
C
, ˜p
E
; p
0
A
, p
0
B
, p
0
C
, ˜p
0
E
)
× |p
A
i|p
B
i|p
C
i|˜p
E
ihp
0
A
|hp
0
B
|hp
0
C
|h˜p
0
E
|.
(60)
The decomposition of operator
ˆ
Λ
p
E
reads
ˆ
Λ
p
E
= hp
E
|
ˆ
U|φ
0
i
E
=
Z
dp
A
dp
B
dp
C
dp
0
A
dp
0
B
dp
0
C
λ
p
E
|p
A
i|p
B
i|p
C
i
C
hp
0
A
|hp
0
B
|hp
0
C
|,
λ
p
E
=
Z
u(p
A
, p
B
, p
C
, p
E
; p
0
A
, p
0
B
, p
0
C
, p
0
E
)φ
0
(p
0
E
)dp
0
E
.
(61)
Insert (61) to the R.H.S of (25), we obtain
|ψi
C,ABE
=
ˆ
T
C
Z
dp
A
dp
B
dp
C
dp
0
A
dp
0
B
dp
0
C
dp
E
δ(p
A
+ p
B
+ p
C
+ p
E
)λ
p
E
|p
A
i|p
B
i|p
C
i
C
|p
E
i
× hp
0
A
|hp
0
B
|hp
0
C
|ψi
ABC
= |p = 0i
C
⊗
Z
dp
E
|p
E
i
Z
dp
A
dp
B
dp
0
A
dp
0
B
dp
0
C
λ
p
E
|p
A
i|p
B
ihp
0
A
|hp
0
B
|hp
0
C
|ψi
ABC
= |p = 0i
C
⊗
Z
dp
E
dp
A
dp
B
χ
p
E
(p
A
, p
B
)|p
A
i|p
B
i|p
E
i,
(62)
where χ
p
E
(p
A
, p
B
) is defined as
χ
p
E
(p
A
, p
B
) =
Z
dp
0
A
dp
0
B
dp
0
C
λ
p
E
ψ
ABC
(p
0
A
, p
0
B
, p
0
C
)
λ
p
E
=
Z
u(p
A
, p
B
, −p
A
− p
B
− p
E
, p
E
; p
0
A
, p
0
B
, p
0
C
, p
0
E
)φ
0
(p
0
E
)dp
0
E
.
(63)
Now consider a special case when |ψi
ABC
= |ϕi
AB
|ξi
C
is a product state, the derivation of (62) is
simplified into
|ψi
C,ABE
= |p = 0i
C
⊗
Z
dp
E
dp
A
dp
B
dp
0
A
dp
0
B
dp
0
C
λ
p
E
|p
A
i|p
B
ihp
0
A
|hp
0
B
|hp
0
C
|ξi
C
|ϕi
AB
|p
E
i
= |p = 0i
C
⊗
Z
dp
E
ˆ
Γ
p
E
|ϕi
AB
|p
E
i,
(64)
where operator
ˆ
Γ
p
E
only acts on particles A and B, defined as
ˆ
Γ
p
E
=
Z
dp
A
dp
B
dp
0
A
dp
0
B
× γ
p
E
|p
A
i|p
B
ihp
0
A
|hp
0
B
|,
γ
p
E
=
Z
dp
0
C
λ
p
E
ξ(p
0
C
),
(65)
and λ
p
E
has been defined in (63).
B Proof of Eq.(38)
First we notice that (35) is derived without the assumption of |ψi
ABC
= |ϕi
AB
|ξi
C
. A similar form
for |ϕi
m
AB|C
can be derived from (62), which is more generic:
|ϕi
m
AB|C
=
1
√
ρ
m|C
Z
dp
A
dp
B
χ
p
m
(p
A
, p
B
)|p
A
i|p
B
i =
1
√
ρ
m|C
|χi
p
m
AB|C
,
ρ
m|C
=
AB|C
hχ|χi
p
m
AB|C
(66)
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 16
and χ
p
m
has been defined in (63). Next step is to evaluate
ˆ
S
m
|χi
p
m
AB|C
=
ˆ
P
AC
e
iˆq
A
(ˆp
B
+p
m
)
Z
dp
A
dp
B
χ
p
m
(p
A
, p
B
)|p
A
i|p
B
i
=
ˆ
P
AC
Z
dp
A
dp
B
χ
p
m
(p
A
, p
B
)|p
A
+ p
B
+ p
m
i|p
B
i.
(67)
Changing variable p
C
= −p
A
− p
B
− p
m
and applying the operator
ˆ
P
AC
, we get
ˆ
S
m
|χi
p
m
AB|C
=
Z
dp
B
dp
C
χ
p
m
(−p
B
− p
C
− p
m
, p
B
)|p
C
i|p
B
i (68)
From the definition in (63), χ
p
m
depends on λ
p
m
, which in turn depends on matrix element u. Explicitly,
λ
p
m
(−p
B
− p
C
− p
m
, p
B
) =
Z
φ
0
(p
0
E
)dp
0
E
u(−p
B
− p
C
− p
m
, p
B
, p
C
, p
m
; p
0
A
, p
0
B
, p
0
C
, p
0
E
). (69)
Compared (63), (69), and the definition of κ
p
m
in (35), one recognizes that χ
p
m
(−p
B
−p
C
−p
m
, p
B
) =
κ
p
m
(p
B
, p
C
). Inserting this identity to (68), we have
ˆ
S
m
|χi
p
m
AB|C
= |κi
p
m
BC|A
. (70)
This implies that the measurement probability
ρ
m|A
=
BC|A
hκ|κi
p
m
BC|A
=
AB|C
hχ|
ˆ
S
†
m
ˆ
S
m
|χi
p
m
AB|C
=
AB|C
hχ|χi
p
m
AB|C
= ρ
m|C
.
(71)
Given (70) and (71), and the definitions in (31) and (35), we obtain the desired identity
ˆ
S
m
|ϕi
p
m
AB|C
= |ϕi
p
m
BC|A
. (72)
(71) is derived without the assumption of product state condition |ψi
ABC
= |ϕi
AB
|ξi
C
. (38) certainly
holds true since it is for the special case with the product state condition.
C Completeness of {
ˆ
M
m
}
Note that index m refers to p
m
which is a continuous real number, p
m
∈ R for particle E. Changed
notation from p
m
to p
E
, the completeness is
Z
dp
E
ˆ
M
†
p
E
ˆ
M
p
E
=
ˆ
I
AB|C
. (73)
To verify that, we take the complex conjugate of (28), multiply to itself, and evaluation both sides:
L.H.S. =
ABE|C
hψ|ψi
ABE|C
= N
L
. (74)
N
L
is simply a constant. The inner product can be constructed from either the reduced Hilbert space
H
ABE|C
, or from H
kin
as described in Appendix C of Ref. [26]. To evaluate the R.H.S., we use the
property hp
0
E
|p
E
i = δ(p
E
− p
0
E
) and simplify R.H.S as
R.H.S. =
Z
dp
0
E
dp
E
(
AB|C
hϕ|
E
hp
E
|
ˆ
Γ
†
p
0
E
)
ˆ
Γ
p
E
|ϕi
AB|C
|p
E
i
E
=
Z
dp
E
(
AB|C
hϕ|
ˆ
Γ
†
p
E
ˆ
Γ
p
E
|ϕi
AB|C
)
=
AB|C
hϕ|(
Z
dp
E
ˆ
Γ
†
p
E
ˆ
Γ
p
E
)|ϕi
AB|C
.
(75)
Equating both sides, we obtain
N
L
=
AB|C
hϕ|(
Z
dp
E
ˆ
Γ
†
p
E
ˆ
Γ
p
E
)|ϕi
AB|C
(76)
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 17
Since |ϕi
AB|C
is an arbitrary state vector, the above equation implies
Z
dp
E
ˆ
Γ
†
p
E
ˆ
Γ
p
E
= N ×
ˆ
I
AB|C
, (77)
where N is a constant. Inserting (77) into (76), we find
N =
N
L
AB|C
hϕ|ϕi
AB|C
. (78)
Defining
ˆ
M
p
E
=
ˆ
Γ
p
E
/
√
N, we obtain the identify (73).
D Proof of (50)
The unitary operator
ˆ
U in H
kin
is decomposed as
ˆ
U =
X
i,j
Z
dp
A
dp
B
dp
C
dp
0
A
dp
0
B
dp
0
C
u(p
A
, p
B
, p
C
, σ
i
; p
0
A
, p
0
B
, p
0
C
, σ
j
)|p
A
i|p
B
i|p
C
, σ
i
ihp
0
A
|hp
0
B
|hp
0
C
, σ
j
|.
(79)
Then, from definition of
ˆ
Γ
m
, we have
ˆ
Γ
m
:= hφ
m
|
ˆ
U|φ
0
i =
Z
d¯p
C
d˜p
C
φ
∗
m
(¯p
C
)φ
0
(˜p
C
)h¯p
C
, σ
m
|
ˆ
U|˜p
C
, σ
0
i. (80)
Inserting (79) into (80), and applying the orthogonal identities (45), we obtain
ˆ
Γ
m
=
X
i,j
Z
dp
A
dp
B
dp
C
dp
0
A
dp
0
B
dp
0
C
d¯p
C
d˜p
C
u(p
A
, p
B
, p
C
, σ
i
; p
0
A
, p
0
B
, p
0
C
, σ
j
)φ
∗
m
(¯p
C
)φ
0
(˜p
C
)
× h¯p
C
, σ
m
|p
A
i|p
B
i|p
C
, σ
m
ihp
0
A
|hp
0
B
|hp
0
C
, σ
0
|˜p
C
, σ
0
i
=
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
|p
A
i|p
B
ihp
0
A
|hp
0
B
|
γ
m
=
Z
d¯p
C
d˜p
C
φ
∗
m
(¯p
C
)φ
0
(˜p
C
)u(p
A
, p
B
, ¯p
C
, σ
m
; p
0
A
, p
0
B
, ˜p
C
, σ
0
).
(81)
Inserting (81) into (48),
|ψi
C,AB
=
ˆ
T
C
δ(
ˆ
P )
X
m
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
|p
A
i|p
B
ihp
0
A
|hp
0
B
|
×
Z
d˜p
A
d˜p
B
d˜p
C
ϕ(˜p
A
, ˜p
B
)φ
m
(˜p
C
)|˜p
A
i|˜p
B
i|˜p
C
, σ
m
i
=
ˆ
T
C
X
m
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
δ(p
A
+ p
B
+ ˜p
C
)|p
A
i|p
B
ihp
0
A
|hp
0
B
|
Z
d˜p
A
d˜p
B
d˜p
C
× ϕ(˜p
A
, ˜p
B
)φ
m
(˜p
C
)|˜p
A
i|˜p
B
i|˜p
C
, σ
m
i
=
ˆ
T
C
X
m
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
φ
m
(−p
A
− p
B
)| −p
A
− p
B
, σ
m
i
C
× |p
A
i|p
B
ihp
0
A
|hp
0
B
|
Z
d˜p
A
d˜p
B
ϕ(˜p
A
, ˜p
B
)|˜p
A
i|˜p
B
i
=
X
m
|p = 0, σ
m
i
C
⊗ {
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
φ
m
(−p
A
− p
B
)|p
A
i|p
B
ihp
0
A
|hp
0
B
|}|ϕi
AB
=
X
m
|p = 0, σ
m
i
C
⊗
ˆ
M
m
|ϕi
AB
.
(82)
where
ˆ
M
m
is a modified version of
ˆ
Γ
m
,
ˆ
M
m
=
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
φ
m
(−p
A
− p
B
)|p
A
i|p
B
ihp
0
A
|hp
0
B
|. (83)
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 18
E Proof of (53)
The derivation of (53) is simliar to (82), except this time we pick particle A as QRF.
|ψi
A,BC
=
ˆ
T
A
X
m
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
δ(p
A
+ p
B
+ p
C
)|p
A
i|p
B
ihp
0
A
|hp
0
B
|
×
Z
d˜p
A
d˜p
B
dp
C
ϕ(˜p
A
, ˜p
B
)φ
m
(p
C
)|˜p
A
i|˜p
B
i|p
C
, σ
m
i
=
ˆ
T
A
X
m
Z
dp
B
dp
0
A
dp
0
B
γ
m
| −p
B
− p
C
i
A
|p
B
ihp
0
A
|hp
0
B
|
×
Z
d˜p
A
d˜p
B
dp
C
ϕ(˜p
A
, ˜p
B
)φ
m
(p
C
)|˜p
A
i|˜p
B
i|p
C
, σ
m
i
= |p = 0i
A
⊗
X
m
Z
dp
B
dp
0
A
dp
0
B
γ
m
|p
B
ihp
0
A
|hp
0
B
|
×
Z
d˜p
A
d˜p
B
dp
C
ϕ(˜p
A
, ˜p
B
)φ
m
(p
C
)|˜p
A
i|˜p
B
i|p
C
, σ
m
i.
(84)
Removing the redundancy state of A by applying
√
2πhq
A
= 0|⊗, we have
|ϕi
BC|A
:=
√
2πhq
A
= 0|ψi
A,BC
=
X
m
Z
dp
B
dp
C
φ
m
(p
C
)|p
B
i|p
C
, σ
m
i
Z
d˜p
A
d˜p
B
ϕ(˜p
A
, ˜p
B
)γ
m
(−p
B
− p
C
, p
B
; ˜p
A
, ˜p
B
)
=
X
m
Z
dp
B
dp
C
φ
m
(p
C
)χ
m
(p
B
, p
C
)|p
B
i|p
C
, σ
m
i,
(85)
where
χ
m
(p
B
, p
C
) =
Z
d˜p
A
d˜p
B
ϕ(˜p
A
, ˜p
B
)γ
m
(−p
B
− p
C
, p
B
; ˜p
A
, ˜p
B
) =
A
h−p
B
− p
C
|hp
B
|
ˆ
Γ
m
|ϕi
AB
, (86)
and
ˆ
Γ
m
is defined in (81). (85) gives (53).
F Proof of (56)
From the definition of
ˆ
M
m
in (83), we have
ˆ
S
m
ˆ
M
m
|ϕi
AB|C
=
ˆ
P
n
AC
e
iˆq
A
ˆp
B
ˆ
M
m
|ϕi
AB|C
=
ˆ
P
n
AC
Z
dp
A
dp
B
dp
0
A
dp
0
B
γ
m
φ
m
(−p
A
− p
B
)|p
A
+ p
B
i|p
B
ihp
0
A
|hp
0
B
|ϕi
AB
.
(87)
Changing variable p
C
= −p
A
− p
B
, and noting hp
0
A
|hp
0
B
|ϕi
AB
= ϕ
AB
(p
0
A
, p
0
B
), we rewrite (87) as
ˆ
S
m
ˆ
M
m
|ϕi
AB|C
=
ˆ
P
m
AC
Z
dp
C
dp
B
dp
0
A
dp
0
B
φ
m
(p
C
)| −p
C
i
A
|p
B
i
B
ϕ
AB
(p
0
A
, p
0
B
)γ
m
(−p
B
− p
C
, p
B
; p
0
A
, p
0
B
)
=
Z
dp
C
dp
B
dp
0
A
dp
0
B
φ
m
(p
C
)γ
m
(−p
B
− p
C
, p
B
; p
0
A
, p
0
B
)|p
C
, σ
m
i
C
|p
B
i
B
ϕ
AB
(p
0
A
, p
0
B
)
=
Z
dp
B
dp
C
φ
m
(p
C
)χ
m
(p
B
, p
C
)|p
B
i|p
C
, σ
m
i
= |ϕi
m
BC|A
.
(88)
Accepted in Quantum 2020-06-08, click title to verify. Published under CC-BY 4.0. 19