Table Of ContentA relational approach to quantum reference frames for spins
Jacques Pienaar
In the literature on quantum reference frames, the internal (relative) properties of a system are
defined as those which are preserved under an arbitrary change of reference frame. For a system
of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory.
However,thisapproachdoesnotaccountforthehypotheticalpossibilityofthelaboratorybecoming
entangled to the system, as described by a second laboratory (the ‘Wigner’s friend’ scenario), in
which case the relationship between the two laboratories is not a rotation, but is fundamentally
quantum. Toovercomethislimitation,were-definethereferenceframetransformations tobethose
that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct
symmetry group for transformations of a system of N spin-half particles. Next, we propose that
6
systems having the same internal properties should be regarded as physically equivalent in the
1
absenceofanexternalframe. Remarkably,thisimpliesthatasinglespininasuperpositionrelative
0
to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin.
2
Wediscuss theimplications of this result for the Wigner’s friend paradox.
t
c
O
I. INTRODUCTION spiredbyRovelli’srelationalinterpretation,accordingto
6 which different observers may give different accounts of
thesamephysicalsituation. Itfollowsthatanydisagree-
What is the meaning of a quantum state Ψ ? It can
] | i ments between them must be about external quantities.
h ultimately only have meaning in the context of a lab-
However,sincetheobserversthemselvesmustobeyquan-
p oratory in which it is possible to prepare and measure
- quantumsystems under controlledconditions. The state tum mechanics, their accounts must also be consistent
t with each other to some degree. The internal proper-
n represents a comparison between the state of the ob-
a ties are precisely those facts that all observers can agree
served system and the states of the measuring devices.
u upon. Our aim is to pin down what these properties
Ifweassumethephysicalstateofthelaboratoryisfixed,
q are, and to decide exactly what it means for two states
[ thenanychangeinthestateofthesystemmustbeinter-
to represent the ‘same physical situation’ in a relational
preted as a physical change in the system (and not the
2 quantum world.
laboratory). More generally, the laboratory can itself be
v It is clear that the distinction between external and
treatedasaphysicalsystemsubjecttochange,andhence
0 internalpropertiescannotbe fundamental,solongas we
changesinthestateofthesystemonlyrepresentchanges
2
give no fundamental significance to the attendant dis-
3 in the relationship between system and laboratory. This
tinction between the laboratory and the system. Never-
7 is called a relational interpretation of the state.
theless, the distinction is relevant to practicalconsidera-
0
Relational interpretations are not new; they include
. tions. Forexample,considertwolaboratoriesattempting
1 Everett’s‘relativestate interpretation’[1], Rovelli’srela-
to exchange quantum information, but whose measuring
0 tional interpretation [2], and Quantum Bayesianism [3].
devicesarenotmutuallyaligned. Sincethereceiverisig-
6
These approaches all agree that the quantum state de-
1 norantofthe sender’sreferenceframe, he mustsumover
scribes relative information, but disagree on whose in-
: all possibilities, in effect applying a random transforma-
v formation it is, and what the information is about. In
tion to the external degrees of freedom. This destroys
i all cases, however, the measurable properties of a sys-
X anyinformationencoded inthese degreesoffreedom, re-
tem can be split into two categories: those that change
r ducing the parties’ ability to communicate. This is the
a when the system is measured in a different laboratory, basis for the literature on quantum reference frames [6–
and those that are measured to be the same in all lab-
8], which provides an extensive toolbox of methods for
oratories. We call these properties external or internal,
performingquantumcommunication,cryptography,tele-
respectively [22].
portationand other information-theoretictasks in situa-
At one extreme, a subjective Bayesian (see eg. Ref. tions where the laboratories do not share a given refer-
[4])mightsaythatthequantumstaterepresentsonlythe ence frame.
beliefs of a rational agent. In this case all properties are The work on quantum reference frames proceeds from
external, since every state is a valid description of the theassumptionthattheseparationbetweeninternaland
beliefs of some agent, and the laws of physics describe externaldegreesoffreedomisknown. Usually,itisgiven
how these beliefs should be updated in light of measure- in the form of a symmetry group that is postulated to
ment outcomes. At the other extreme, a psi-ontologist preserve only the internal degreesGof freedom. The sym-
(see eg. Ref. [5]) would claim that two different states metry group is justified by appealing to some natural
representtwodifferentobjectiverealities,henceallprop- symmetry of the physical setting in question. For ex-
erties of the system are internal and do not depend on ample, if the systems to be communicated are spin-half
the observer. particles,itseemslogicalthatthemeasuringinstruments
In this work, we wish to explore a middle ground, in- inthe laboratoriesareStern-Gerlachmagnetsorsimilar.
2
These instruments indicate a preferred Cartesian refer- seen by Wigner and his friend. Although we ultimately
ence frame for the laboratory, representing the labelling do not succeedin this task, we are able to find an exam-
of the (x,y,z) co-ordinates in the lab. Since the Carte- ple that transcends the usual system-apparatus division
sian reference frames of the laboratoriesare related by a in the absence of an external reference frame, indicating
properrotation,theassociatedsymmetrygroupisSO(3). that future efforts along similar lines might bear fruit.
Forquantumparticles,wetypicallyseekaunitaryrepre- The outline ofthe paperis asfollows. InSectionII we
sentation of the rotation group, hence =SU(2). considera system composedofspins andask whatprop-
G
Most of the literature to date has restricted attention ertiesshouldbethesameforallobservers. Wearguethat
to transformations that preserve the separation between all observers should be subject to the same constraints
system and laboratory. That is to say, if one laboratory ontheir ability to distinguishbetween subsystems of the
performs a measurement on the system, the state is up- total system. Taking the quantum fidelity between sub-
dated according to the outcome of this measurement for systems as a measure of their distinguishability, we ob-
all laboratories, notjustthelaboratoryinquestion. This tainU(2)asthesymmetrygroupthatpreservesthem. In
assumption excludes the possibility of treating the labo- SectionIII we examinethe bearingthis resulthasonthe
ratories themselves as quantum systems relative to one microscopic / macroscopic distinction and the Wigner’s
another. Forexample,considerthefollowingthoughtex- friend thought experiment. Section IV contains our con-
perimentdue toWigner[9]: anexperimentercalledFran clusions and outlook.
(a friend of Wigner) performs a measurement of the sys-
tem and obtains an outcome. It should then be pos-
sible for another experimenter in a different laboratory II. THE INTERNAL PROPERTIES OF SPINS
(Wigner)to treatFranandher laboratoryas aquantum
system, and the measurement as a physical interaction
Our task is to define the internal properties of general
that entangles the state of Fran to the state of the sys-
pure states Ψ of N spins, namely, the attributes of the
tem. This physical interaction is a transformation that | i
system that do not depend on the observer’s laboratory.
appliesnotjusttothesystemalone,butalsotothewhole
Inwhatfollows,weassumethatalllaboratoriescanagree
laboratoryofFran,therebyincludingthislaboratoryand
onalabelingoftheN spins,sothatifoneobserverspec-
Franherselfaspartofthe‘system’. Thus,Fran’slabora-
ifies a subsystem of spins, say “spins numbered 7 12”,
toryisinasuperpositionrelativetoWigner’slaboratory. −
alltheobserversagreeonwhichsubsystemthisrefersto.
So far, there have been few attempts to apply the tools
Measurementsofspin performedwithin a laboratoryare
of quantum reference frames to physical settings of this
relativetothe(arbitrary)orientationofthatlaboratory’s
more general type.
spatial axes. The procedure for measuring ‘spin up’ in a
We note the following exceptions. Palmer et. al. [10] given direction is assumed to be the same for all spins,
have considered the task of changing reference frames hence labeling the ‘up’ direction for one spin fixes the
in a setting where the reference frame is an explicitly labelfor allspins in the same lab. Finally, we assumeall
quantum object; however they do not extend their anal- observers can measure the number of spins in any sub-
ysis to ‘Wigner’s friend’-type scenarios. Angelo et. al. system without difficulty. Other than that, the devices
[11, 12], and Pereira and Angelo [13], following work by in the laboratories are not calibrated with each other.
Aharonov et. al. [14], consider the problem of treating The problem can now be understood as follows. If
quantum systems as observers and changing frames be- a system of N spins is prepared according to a fixed
tweenthem. However,thatworkisconcernedwithlinear preparation procedure, it will nevertheless have a differ-
position and momentum degrees of freedom, whereas we ent state accordingto different laboratories. We wantto
considerangularmomentumandspin(sincethe rotation know: how much canthe states seen by two laboratories
group is compact, we can apply the standard tools of differ? Clearly, any differences can only be attributed to
quantumreferenceframes). Finally,Ho¨hnet. al. [15–17] a change of laboratory frame, since the preparation of
havemanagedtoreconstructquantummechanicsandits the system is invariant. We have assumedthat the labo-
keysymmetrygroupsforqubitsusinganoperationalap- ratoriesarerelatedby collective transformationsoftheir
proach that inspired the approach used in the present apparatuses that preserve the number of spins and the
work; however they also do not address the Wigner’s spin ordering, but otherwise the symmetry group that
friend scenario. relates the laboratories is unknown.
In this work, we make an incremental step towards Our first clue comes from asking how well the labora-
extending the formalism of quantum reference frames toriesareable todistinguishdifferentpreparationproce-
to more general scenarios of the kind applicable to the dures. Imagine that a machine prepares N-spin systems
Wigner’s friend thought experiment. Our main idea is according one of two preparation procedures, A or B.
to try and derivethe symmetry groupof reference frame In the first phase of the experiment, it prepares many
transformations from first principles, rather than pos- systems using procedure A and sends multiple copies to
tulating it from a classical limit, in the hopes that the eachlaboratoryforanalysis. Inthesecondphase,itdoes
resulting physical symmetry will permit us to draw an the same but only using procedure B. As a result, each
equivalence between the apparently irreconcilable states laboratoryisabletoperformtomographyonthesystems
3
theyreceivedanddeterminethestateobtainedfromeach copy of the state is provided, then the appropriate mea-
preparationprocedure relative to their owninstruments. sureofdistinguishabilityisthequantumerrorprobability.
In general, the pair of states ρ ,ρ seen by one lab will More generally,after randomly choosing whether to pre-
A B
not be the same as the pair ρ′ ,ρ′ in another lab. pare A or B, the machine could send a fixed number of
A B
In the third phase of the experiment, the machine copies of the system to each lab. In this case, there is
sendsonlyasinglecopyofthesystemtoeachlaboratory, a whole zoo of distinguishability measures that could be
and in each case it chooses randomly which preparation used (see eg. Ref [18]). We will adopt the quantum fi-
proceduretouse: withprobabilityp itappliesprocedure delity as the measure of interest in the remainder of this
A, and with probability (1 p) it applies B. Each lab is work, but we conjecture that the results obtained here
−
thenaskedtoguesswhetherthesystemtheyreceivedwas hold also for other measures of distinguishability. The
preparedusingprocedureAorB. Theycanuseonlythe quantum fidelity can be expressed as:
knowledge they have gained in the previous two phases,
plus the value of p. F(ρA,ρB):= min X pTr[ρAFk]Tr[ρBFk], (1)
Sincethepairofstatesbeingdistinguishedwillingen- {Fk} k
eraldiffer amonglaboratories,it isa priori possiblethat
one laboratory might consistently perform better at the wherethe minimisationisoverallpossiblePOVMs Fk
{ }
task than another laboratory (for instance, if the pair on the Hilbert space of the systems.
of states seen by one laboratory are more alike than the Why choose the fidelity? For spins, the fidelity can
states as seen by another laboratory). Hopefully, this be relatedto the relativespatial anglesbetween the spin
possibility strikes the reader as rather unnatural. After vectors. Specifically, if we restrict attention to a tensor
all, the laboratories are all trying to distinguish systems productofN purespinstates,thenthestatespaceisiso-
produced by the same pair of preparation procedures, morphic to a set of unit vectors in 3-dimensional space
and these procedures are frame-independent notions, so (Blochvectors). Inthatcase,anaturalcandidateforthe
we intuitively expect that no laboratory has any special internal properties of the system are the relative angles
advantage over another. Hence, the likelihood of suc- betweentheBlochvectors. Theangleθij betweentheith
ceeding should be independent of which laboratory (i.e. andjth vectorsis directly relatedto the overlap φj φi
|h | i|
observer) is asked to perform the task. between their respective states, which is just the fidelity
Without loss of generality, we can imagine that in this case. For more general (entangled) states, the fi-
the machine implements the two preparations by first delitiescanberelatedtoanglesbetweenBlochvectorsof
preparing a single system according to a fixed procedure the purifications of the spin states (Uhlmann’s theorem,
and then choosing one of two N-spin subsystems, Ref. [19]). The connectionbetweenthe fidelities andthe
designated A or B, as the output. For example, let relativeanglesbetweenspatialvectorsisone ofthe main
N = 3. Starting with an initially prepared state of inspirationsforusingthefidelityasthemeasureofdistin-
M N spins, the machine could choose either spins guishability. However,itismerelyaheuristicmotivation,
≥
5,11,14 (subsystem A) or spins 17,42,99 (subsystem as the Bloch vectorsof higher-dimensionalsystems obvi-
B), discardthe rest,and re-labelthe chosenspins 1,2,3. ouslydonotexistin3-dimensionalspace. Ultimately,we
In this way, the choice of preparation procedure reduces have chosen the fidelities mostly because they are math-
to a choice of subsystems sampled from a larger system ematically convenient and lead to interesting results.
with a fixed preparation. With this perspective, the An N spin system can be partitioned into subsys-
considerations of the previous paragraph amount to the tems of different numbers of spins. Obviously, if the
statement that all laboratories should be equally good subsystems have different sizes, they are perfectly
at distinguishing a given pair of subsystems of a total distinguishable by measuring the number of spins, so
system prepared by a fixed procedure. This forms our the only nontrivial cases arise when the subsystems
first postulate: have equal numbers of spins. This is important, because
the fidelity is defined only for subsystems of equal size.
Postulate 1: Let A and B label two (not necessarily Using the fidelity, we can now give a mathematical
disjoint) equal-sized subsystems of a quantum system version of Postulate 1 for spins:
prepared by a fixed procedure. We assume that all
observers have access to identically prepared copies of Postulate 1 for spins: Given an arbitrary pure
this totalsystemandits subsystems. Thenthe abilityof state Ψ consisting of N spins, the internal properties
| i
any observer to distinguish A from B (when presented arethe setoffidelities F(ρA,ρB)betweenallsubsystems
with a random sample of either one) must be the same ρA,ρB with equal numbers of spins.
for all observers.
We next show that these fidelities are preserved by
(Note that although we motivated this postulate from (and only by) actions of the unitary group U(2):
the example of spins, its scope is quite general). For a
given pair of states ρ and ρ , quantum mechanics sup- Theorem 1: The set of fidelity-preserving transfor-
A B
plies bounds on their distinguishability. If just a single mations acting on the N-spin Hilbert space is the image
4
of a unitary representation of the group U(2), namely: Although Theorem 1 states that fidelities are guaran-
teed to be preservedby all andonly those unitaries with
π :V U(2) V⊗N U(2N). (2) the form U = V⊗N, this does not imply that two states
∈ 7→ ∈
having the same fidelities must necessarily be related by
Proof: The action of a unitary U on Ψ induces the aunitarymapofthisform. Anexampleofthisintriguing
| i
following transformations on the reduced states: situation will be explored in the next section.
ρA Tr6=A(cid:2)U Ψ ΨU†(cid:3):= (ρA),
→ | ih | E
ρB Tr6=B(cid:2)U Ψ ΨU†(cid:3):= ′(ρB). (3)
→ | ih | E
III. PHYSICAL INTERPRETATION
It can be shown using Uhlmann’s theorem [19] that the
fidelity will be preserved only if the CPT maps are the
same, i.e. ′ = . We then make use of a theorem
In analogy with special relativity, let us assume that
E E
due to Molnar [20], which states that the fidelities are
transformations that preserve the internal properties
preservedifandonly if (ρ)=VρV† forsomeunitaryV
relate physically equivalent situations. That is, the
E
acting on the Hilbert space of the subsystems (this is a internal properties constitute the observer-independent
generalisation of a well-known theorem of Wigner [21]).
core of quantum mechanics:
It follows that U decomposes into U = U U U where
A B R
R labels the systems not in A or B, and U =U =V.
A B
Postulate 2: In the absence of a preferred external
Applying this to all pairs of single-spin subsystems
referenceframe,twoquantumstatesofthe samenumber
leads to the conclusion that U preserves the fidelities iff
U = V⊗N with V U(2). Under the map V V⊗N, of spins represent the same physical situation iff they
the image of U(2)∈in U(2N) is therefore prec7→isely the havethe sameinternalproperties(i.e. the samefidelities
between their subsystems).
set of fidelity preserving maps acting on the space of N
spins. (cid:3)
This leads to the following counter-intuitive result: a
While U(2) is the appropriate group for states in microscopic superposition as seen by one lab can appear
Hilbert space, one is typically concerned only with to be a macroscopic superposition as seen by a different
‘physical states’ in the form of projectors Ψ Ψ, which lab. Let := η1,η2,...ηM with η 0,1 be the
| ih | B {| i} ∈ { }
live in the projective Hilbert space. In that case, the computational basis for a set of M spins. Consider the
following corollary applies: following two states, expressed in the basis :
B
Corollary: Under the representation mentioned in
φ = 00...0 (α0 +β 1 )
Theorem 1, fidelity-preserving maps that act on the
| i | i | i | i
projective Hilbert space constitute the image of the := 0 1 Φ ,
| i | i
group SO(3). φ′ = (α00...0 +β 11...1 ) 0
| i | i | i | i
:= Φ′ 0 . (4)
M
Proof: Theorem 1 implies that the fidelity-preserving | i| i
unitaries acting on the projective space have the form
W⊗N, where W is any element of the 1-spin projective The first state φ might represent, for example, a large
| i
unitary group PU(2). This defines the representation magnet consisting of M 1 spins, with the Mth spin in
−
of PU(2) by fidelity-preserving unitaries acting on the a superposition relative to the magnet. The basis is
B
projective space, and since PU(2) is isomorphic to a natural or preferred choice of labeling in a laboratory
SO(3), the statement of the Corollary follows. (cid:3) wherethe physicalstates correspondingtoelements of
B
are easy to prepare, and where superpositions of these
The symmetry group of reference frame transforma- elements are easy to prepare whenever the correspond-
tionsisusuallytakenforgrantedintheliterature,butwe ing elements of are similar to each other. An example
B
have derived it using a novel assumption (Postulate 1). is the state φ , whose superposed states differ at only
| i
Note that since the fidelities are not in general indepen- a single spin in the basis ; we call φ a microscopic
B | i
dent,theyshouldbereducibletoasmallersetofindepen- superposition relative to such a lab. Conversely, if the
dentinternalproperties,whichwehavenotattemptedto two superposed states are very different when expressed
identify here. It is straightforward to do so using tools in the natural basis of the lab, the state represents a
from the literature on quantum reference frames; for ex- macroscopic superposition. Thus φ′ is macroscopicrel-
| i
ample, Eq. 3.20 in [7] presents the decomposition into ative to the lab for which is the natural basis. If we
B
collective and relative (i.e. internal) degrees of freedom have a preferred basis in which to prepare both these
B
for the group SU(2), and one sees that the internal de- states, we will find they appear physically different, the
grees of freedom spanthe degenerate subspaces of states state φ′ being significantly harder to prepare than the
| i
with definite total angular momentum. state φ . Wenowobservethatthesetwostateshavethe
| i
5
same internal quantities, namely: precise. In order to do so, it would be necessary to treat
Fran as a physical system (perhaps herself composed of
F(ρA,ρB) = 1, if A,B exclude spin M; spin-half particles) initially entangled to spin M. The
F(ρ ,ρ ) = α2 if M appears in one of A,B; task would then be to discover a state having the same
A B
| |4 fidelities as the original, but in which the spin M has a
F(ρ ,ρ ) = α if M appears at different
A B
| | definite value, in accordance with what we expect Fran
sites in both A,B; to see. The problem is that the equivalence of φ′ and
| i
F(ρA,ρB) = 1 if M appears at the same φ seems to be rather a special case: simple generaliza-
| i
site in both A,B. (5) tions, like adding more spins to the initially superposed
state, or even adding an extra magnet pointing in a dif-
Postulate 2 then asserts that, in the absence of a pre- ferent direction, lead to states whose fidelity-equivalent
ferred external reference frame, these states should be counterparts don’t seem to possess the desired proper-
physically equivalent. To see how this can occur, con- ties. For the present, it is left as an open problem to
sider a different basis ′ related to by a transforma- classify the sets of physically equivalent states induced
B B
tion that exchanges 0 1 in all elements except for by the fidelities via Postulate 2, but our efforts so far
↔
00...0 and 11...1 . For instance, for M = 3 the super- seem to indicate that these equivalence classes are not
| i | i
position(000 + 001 )/√2inthe basis correspondsto rich enough to capture the duality between Wigner and
(000 + 1|10 i)/√|2 inithe basis ′. NotBice that φ and Fran’s conflicting experiences.
φ|′ swi itc|hroileswhenexpressediBntermsofbasis |′.iThis Despite this disappointment, the example does show
m| eians that if they are both prepared in a lab foBr which that if we take internal properties as fundamental, and
′ is the preferred basis, φ′ will be microscopic (hence their symmetry transformations as merely derived, then
eBasier to prepare) and φ |wiill appear macroscopic. This the line between apparatus and measured system can be
example clearly shows|thiat if no preferred basis is speci- transgressed.
fied (i.e. there is no external lab to tell us which basis is
mostnatural)the states φ′ and φ may be regardedas
| i | i
representing the same physical situation.
IV. CONCLUSIONS
Comment: there is a certain intuition according to
whichonemightexpectanelectroninsuperpositionwith
respecttoalargemagnetto‘see’themagnetasbeingina WehaveshownthatitispossibletoderiveU(2)asthe
superposition. This procedure of ‘stepping into the elec- group that preserves the internal degrees of freedom of
tron’s shoes’ and talking about ‘what the electron sees’ spins by postulating (cf. Postulate 1) that the internal
seemstofitwiththetransformationfrom φ to φ′ . One degrees of freedom are the fidelities between subsystems
| i | i
mustbe carefulinmaking suchclaims,however,since as of spins. Furthermore, we proposed that states having
wehaveshownabove,thisphysicalequivalenceofframes the same internal quantities should represent the same
can only be maintained in the absence of a preferred ex- ‘physicalsituation’,differingonlyintermsofthedescrip-
ternal basis. In practice, there always exists a preferred tion relative to some laboratory (Postulate 2). This led
basis for the external laboratory,which is singled out by totheobservationthat,inabsenceofapreferredexternal
decoherenceofthesystemandmeasuringapparatusbya laboratory, microscopic and macroscopic superpositions
common environment. This is why, for example, placing become equivalent.
asingleparticleinaspatialsuperpositionoveralongdis- Theworkpresentedherewasinspiredbytheideathat,
tanceisconsidered‘macroscopic’: theenvironmenttends in certain circumstances, a single quantum particle can
to decoheresystems inthe positionbasis. Aninteresting be considered an observer. Indeed, the approach shows
taskforfutureworkwouldbetoincludetheenvironment promisefor reconciling the viewpoints of Wigner andhis
in the preceding analysis, to see explicitly how it breaks friend in the thought experiment, by utilizing the notion
the symmetry between the states φ′ and φ . ofanequivalenceclassofstatesallhavingthesameinter-
| i | i
Setting aside decoherence, one might still expect this nalproperties. Unfortunately,we wereunable to achieve
surprising equivalence of states to have some bearing on this goal using the fidelities as the relevant degrees of
the problem of Wigner’s friend. After all, it presents a freedom.
dualperspectivethatseemstocorrespondtotheconflict- The author thanks Cˇaslav Brukner, Philipp Ho¨hn,
ing experiences of the two parties: in one point of view Alexander Smith and an anonymous referee for helpful
(Wigner),theMth spinisinasuperposition,whileinthe comments. This work has been supported by the Euro-
other point of view (Fran) it’s spin has a definite value. pean Commission Project RAQUEL, the John Temple-
The fact that the two points of view are reconciled by ton Foundation, FQXi, and the Austrian Science Fund
Postulate 2 seems promising. Alas, further investigation (FWF)throughCoQuS,SFBFoQuS,andthe Individual
leads us to conclude that the analogy cannot be made Project 2462.
6
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