Quantum Superpositions Cannot be Epistemic

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by  John-Mark A. Allen

Quantum Superpositions Cannot be Epistemic
John-Mark A. Allen∗
Department of Computer Science, University of Oxford,
Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom.
(Dated: May 2015)
Quantumsuperposition statesarebehindmanyofthecuriousphenomenaexhibitedbyquantum
systems, including Bell non-locality, quantum interference, quantum computational speed-up, and
themeasurementproblem. Atthesametime,manyqualitativepropertiesofquantumsuperpositions
canalso beobservedinclassical probabilitydistributionsleadingtoasuspicion thatsuperpositions
may be explicable as probability distributions over less problematic states; that is, a suspicion
that superpositions are epistemic. Here, it is proved that, for any quantum system of dimension
6 d>3, this cannot be the case for almost all superpositions. Equivalently,any underlying ontology
1 must contain ontic superposition states. A related question concerns the more general possibility
0 that some pairs of non-orthogonal quantum states |ψi,|φi could be ontologically indistinct (there
2 are ontological states which fail to distinguish between these quantum states). A similar method
proves that if |hφ|ψi|2 ∈(0,1) then |ψi,|φi must approach ontological distinctness as d→∞. The
n 4
robustnessof these results to small experimentalerror is also discussed.
a
J

2
I. INTRODUCTION orthonormal basis (ONB). Superpositions are behind

quantum interference, the uncertainty principle, wave-
]
h particleduality,entanglement,Bellnon-locality[10],and
Is the quantum state ontic (a state of reality) or epi-
p the probable increasedcomputationalpower ofquantum
stemic (a state of knowledge)? This, rather old, ques-
-
theory[11]. Perhapsmostalarmingly,superpositionsgive
t tion is the subject of the now-famous PBR theorem [1],
n rise to the measurement problem, so captivatingly illus-
a whichprovesthatthe quantumstate ofa systemis ontic
trated by the “Schrödinger’s cat” thought experiment.
u given reasonable assumptions about the ontic structure
q of multi-partite systems. Whilst these assumptions ap- Schrödinger’s cat is set up to be in a superposition of
[ pear weak and well-motivated, they have also been fre- dead and alive quantum states. The epistemic real-
| i | i
2 quently challenged and, as a result, many recent papers ist (and probably the cat) would ideally prefer the ontic
v havesoughtto addressthe onticity ofthe quantumstate state of the cat to only ever be one of “dead” or “alive”
9 using only single-system arguments [2–7]. These theor- (viz.,onlyinonticstatesaccessibletoeitherthe dead or
| i
6 ems and discussions are reviewed in Ref. [8]. alive quantum states). In that case, the cat’s apparent
| i
9 quantumsuperpositionwouldbeepistemic—therewould
All of this work addresses the epistemic realist, who
5 be nothing ontic about the superposition state. Con-
0 assumesthataphysicalsystemisalwaysinsomedefinite
versely, if there are ontic states which can only obtain
. onticstate (realist)andhopesthatuncertaintyaboutthe
1 when the cat is in a quantum superposition (and never
onticstatemightexplaincertainfeaturesofquantumsys-
0 whenthecatisineitherquantum alive or dead states)
5 tems (epistemic). The features that the epistemic realist | i | i
thenthesuperpositionisunambiguouslyontic: thereare
1 mightliketoexplaininthiswayinclude: indistinguishab-
ontologicalfeatures which correspond to that superposi-
: ilityofnon-orthogonalstates,no-cloning,stochasticityof
v tionbutnottonon-superpositions,sothatsuperposition
measurement outcomes, and the exponential increase in
i is real.
X state complexity with increasing system size [9]. Pre-
Obviously quantum superpositions are different from
r paringsomequantumstate ψ mustresultinsomeontic
a stateλobtaining,sosomepr|obiabilitydistribution,called proper mixtures of basis states. The question here is
ratherwhether quantumsuperpositions overbasis states
a preparation distribution, must describe the probabilit-
canbeunderstoodasprobabilitydistributionsoversome
ies with which each λ obtains in that preparation. In
subset of underlying ontic states, where each such ontic
general, preparation distributions for some pair of non-
state is also accessible by preparing some basis state.
orthogonal quantum states might overlap—there might
be ontic states accessible by preparing either of those The epistemic realist perspective on the foundations
quantum states. The main strategy of the single-system of quantum theory is not only philosophically attract-
ontologyargumentsis to provethat, in orderto preserve ive but also appears to be tenable. Theories in which
quantum predictions, these overlaps must be unreason- thequantumstateisexplainedinanepistemicallyrealist
ably small—too small to explain any quantum features. mannerhavebeendemonstratedtoreproduceinteresting
subsets of quantum theory which include characteristic-
This paper initially concentrates on quantum super-
allyquantumfeatures[9,12–14]. Moreover,they include
position states defined with respect to some specified
theories where superpositions are not ontic in the sense
describedabove. Thequestionoftherealityofsuperpos-
itions in quantum theory is, therefore, very much open.
∗Electronicaddress: [email protected] For example, in Spekkens’ toy theory [9] the “toy-bit”
2
reproduces a subset of qubit behaviour. A toy bit con- pirically verifiable). Recall that a quantum system is
sists of four ontic states, a,b,c,d, and four possible pre- describedwith a d-dimensionalcomplex Hilbert space
parations, 0), 1), +), ), which are analogous to the with ( )d=ef ψ : ψ = 1, ψ eiθ ψ as thHe
correspondi|ngly| na|med|−qubit states. Each preparation set ofPdiHstinct{p|uire∈qHuantukmkstates|.2iQ∼uant|umi} super-
corresponds to a uniform probabilistic distribution over positions are defined with respect to some ONB of
exactlytwoonticstates: 0)isadistributionoveraandb; and are simply those ψ ( ) for which ψ B. H
1)adistributionoverca|ndd; +)overa,c;and )over The preparation d|istiri∈buPtioHns3 µ(λ) for| siom6∈eBstate
| | |−
b,d. Fulldetailsofhowthesestatesbehaveandhowthey ψ ( ) form a set ∆ since different ways of pre-
|ψi
reproduce qubit phenomena is described in Ref. [9]. For |pairi∈ngPthHe same ψ may result in different distributions
thepurposeshere,itsufficestonoticethatallonticstates µ ∆ . If ∆ | iis a singleton for every ψ ( ),
|ψi |ψi
corresponding to the superpositions states +) and ) the∈ntheontologicalmodelispreparationnon|-cion∈tePxtuHal4
| |−
are also ontic states corresponding to either 0) or 1)—
for pure states (otherwise, it is preparation contextual).
| |
thistoytheoryhasnothingontheontologicallevelwhich def
Let Λ = λ Λ : µ(λ) > 0 be the support of the
canbeidentifiedasasuperpositionsothesuperpositions µ { ∈ }
distribution µ.
areepistemic. Such models,therefore,lend credibility to
AmeasurementM ofaquantumsystemcanberepres-
the idea that quantum superpositions themselves might,
ented as a set of outcomes: either vectors of some ONB
in a similar way, fail to have an ontologicalbasis.
′ (for an ONB measurement) or POVM elements (for
Previous single-system theorems that bound ontic B
a general POVM measurement). An ontological model
overlapstoarguefortheonticityofthequantumstate[2–
assignsasetΞ ofconditionalprobabilitydistributions,
6] share at least these shortcomings: (i) they prove that M
called response functions P Ξ , to M. A method
there exists some pair of quantum states (taken from a for performing measuremenMt M∈ seMlects some P Ξ
specificset)withboundedoverlap,ratherthanbounding M ∈ M
whichgivesthe probabilityofobtaining outcomeE M
overlapsbetweenarbitraryquantumstatesand(ii)when ∈
conditional on the ontic state of the system. A prepara-
the overlaps are proved to approach zero in some limit,
tion of ψ via µ ∆ followed by a measurement M
thequantumstatesinvolvedalsoapproachorthogonality via P | iΞ , th∈erefo|ψrei, returns outcome E M with
in that same limit [8]. M ∈ M ∈
probability
In this paper it is proved that, for a d > 3 dimen-
sional quantum system, almost all quantum superposi-
tions with respect to any given ONB must be ontic. A P (E µ)= dλµ(λ)P (E λ). (1)
M M
very similar argument can be used to obtain a general | ZΛ |
bound on ontic overlaps for d > 3, which addresses the
Transformations acting on a system must correspond
aboveshortcomings. Finally, the noise toleranceofthese
to stochastic maps on its space of ontic states Λ. An
results is discussed.
ontological model assigns a set Γ of stochastic maps γ
U
to each unitary transformation U over . A method for
H
performing U selects some γ Γ which, given that the
II. ONTOLOGICAL MODELS ∈ U
system is in ontic state λ′, describes a probability distri-
butionγ( λ′),so thatthe probablythatλ′ is mappedto
The appropriate framework for discussing epistemic λ under t·h|is operation is γ(λλ′). A preparation of ψ
realism is that of ontological models [8, 15, 16]. It is | | i
via µ ∆ followed by a transformation U via γ Γ
|ψi U
flexible enough for most realist approaches to quantum ∈ ∈
results in an ontic state distributed accordingto the dis-
ontology to be cast as ontological models [7] including,
but not limited to, Bohmian theories, spontaneous col-
1
lapse theories, and naïve wave-function-realisttheories.
An ontological model of a system has a set Λ of ontic 2Forsimplicity,taked<∞.
states λ Λ. The ontic state which the system occupies 3Infact,this treatment ofontological modelsisnotas general
∈
dictates the properties and behaviour of the system, re- as it should be. Reference [8] notes that, instead of probability
gardless of any other theory (such as quantum theory) distributions, one should consider general probability measures µ
over a measurable space (Λ,Σ) and ontological models can be re-
which may be used to describe it.
formulatedmeasure-theoretically. Thepresentationhereimplicitly,
An ontological model for a quantum system is con-
and problematically, assumes some canonical measure dλ over Λ
strained by the fact that it must reproduce the predic- with respect to which all of the probability distributions can be
tions of quantum theory (at least where they are em- defined. It is possible to derive the results presented here in the
morerigorousformulation,butdoingsowouldbeattheexpenseof
conceptualclarity. Inlightofthissimplificationsomeoftheproofs
presentedherewillalsolackinmathematicalrigouratcertainsteps,
thoughmorethoroughversionsofthesameresultscanbederived.
1Conversely, ontological models are irrelevant for any “anti- 4Preparation non-contextuality for pure states is often impli-
realist”,“instrumentalist”,“positivist”,or“Copenhagen-like” theor- citlyassumedbecauseitrarelyaffects arguments [8]. Rather, pre-
ies denying the existence of an underlying ontology. For example, parationcontextualityformixed quantumstatesismoreoftendis-
quantum-Bayesiantheoriesareexemptfromontologicalmodelana- cussed [17]. However, explicit preparation contextuality for pure
lysis. stateswillbeneededhere.
3
tribution ν, given by From Eqs. (4,6) and Boole’s inequality, it is clear that
ν(λ)= dλ′µ(λ′)γ(λ λ′), λ Λ. (2) ̟(0 , φ ,... µ) ̟(0 µ)+̟(φ µ)+... (7)
| i | i | ≤ | i| | i|
Λ | ∀ ∈
Z
Quantum states are only perfectly distinguishable if
It is required that ν ∆ , since this an example of a
∈ U|ψi they are mutually orthogonal. Distinguishable states
procedure preparing the quantum state U ψ .
| i must be ontologically distinct (their preparation dis-
Fornow,assumethatmeasurementstatisticspredicted
tributions cannot overlap) in order to satisfy Eq. (3).
by quantum theory are exactly correct, so valid ontolo-
The opposite concept of anti-distinguishability is much
gicalmodels for quantumsystems mustreproduce them.
more useful in discussions of ontic overlaps [8]. A set
Therefore, for every ψ ( ), every unitary U over
| i ∈ P H ψ , φ ,... ( ) is anti-distinguishable if and only
, and every measurement M, any choices of prepara- {| i | i } ⊂ P H
tHfuionnctµion∈P∆|ψi,Ξsto,cmhausstticsamtisafpy γ ∈ ΓU, and response itfhathtere exists a measurement M = {E¬ψ,E¬φ,...} such
M M
∈
ψ E¬ψ ψ = φE¬φ φ =...=0, (8)
ψ U†EU ψ = dλ dλ′µ(λ′)γ(λλ′)P (E λ), E M. h | | i h | | i
M
h | | i Λ Λ | | ∀ ∈
Z Z i.e. the measurement can tell, with certainty, one state
(3)
from the set that was not prepared. It has been proven
Itshallbeusefultoconsiderthestabiliser subgroups of
2
def [4, 18] that if some inner products a = φψ , b =
unitaries S|ψi ={U : U|ψi= |ψi} for each |ψi∈P(H). 0ψ 2, c= 0φ 2 satisfy |h | i|
In particular, an ontological model is preparation non- |h | i| |h | i|
contextualwith respecttostabiliser unitaries of ψ ifand 2
| i a+b+c<1, (1 a b c) 4abc, (9)
only if for every µ ∆|ψi, U |ψi, and γ ΓU the − − − ≥
∈ ∈ S ∈
action of γ, according to Eq. (2), leaves the preparation
then the triple ψ , φ , 0 must be anti-
distribution µ unaffected (that is, ν in Eq. (2) would be {| i | i | i}
distinguishable. Anti-distinguishable triples
equal to µ).
ψ , φ , 0 are useful because Λ|ψi Λ|φi Λ|0i =
{| i | i | i} ∩ ∩ ∅
and therefore ̟(0 , φ µ) = ̟(0 µ)+̟(φ µ) for all
| i | i| | i| | i|
µ ∆ , as proved in appendix A.
III. MEASURING OVERLAPS ∈ |ψi
One way to quantify the overlap between preparation
IV. QUANTUM SUPERPOSITIONS ARE REAL
distributionsistheasymmetricoverlap ̟(φ µ)[2,3,7],
| i|
defined as the probability of obtaining an ontic state λ
accessible from some preparation of φ when sampling Define quantum superpositions with respect to some
| i
from µ. Formally, ONB and consider any superposition state ψ . If
B | i6∈B
every ontic state accessible by preparing any µ ∆ is
|ψi
∈
def alsoaccessiblebypreparingsome i ,then ψ hasno
̟(φ µ)= dλµ(λ) (4) | i∈B | i
| i| Λ ontologyindependentof intheontologicalmodel. Such
Z |φi B
a ψ is called an epistemic or statistical superposition
| i
where Λ|φi d=ef∪ν∈∆|φiΛν is the total support of all pos- and must satisfy
sible preparations of φ . By Eq. (3), the asymmetric
overlap must be upper| biounded by the Born rule meas- ̟(i µ) = 1, µ ∆|ψi, or equivalently,(10)
| i| ∀ ∈
urement probability (proof in appendix A) |Xii∈B
2
̟(φ µ) φψ 2, µ ∆|ψi. (5) ̟(|ii|µ) = |hi|ψi| , ∀|ii∈B,µ∈∆|ψi. (11)
| i| ≤|h | i| ∀ ∈
The alternative occurs when there exists some subset
Thatis,theprobabilityofobtainingaλcompatiblewith
of ontic states λ ΛB Λ for which µ(λ) > 0 for some
φ when preparing ψ cannot exceed the probability of ∈ ψ ⊂
|geitting the measure|mient outcome φ having prepared µ ∆|ψi, but ν(λ) = 0 for every ν ∆|ii∈B. That is,
ψ . | i the∈onticstatesinΛB areaccessibleby∈preparing ψ but
| i ψ | i
This quantifies overlaps between pairs of quantum notbypreparingany i ,making ψ anontic orreal
| i∈B | i
states, but what of multi-partite overlaps? The asym- superposition.
metric multi-partite overlap ̟(0 , φ ,... µ) acts like From Eqs. (5,11), a superposition ψ can only
| i | i | | i 6∈ B
the union of the bipartite overlaps ̟(0 µ), ̟(φ µ), be epistemic if the asymmetric overlap ̟(i µ) is max-
| i| | i| | i|
etc. It is defined as the probability of obtaining a imal for every µ ∆ and all i . Therefore, the
|ψi
∈ | i ∈ B
λ Λ|0i Λ|φi ... when sampling from µ. Formally, statementthat“noteveryquantumsuperpositioncanbe
∈ ∪ ∪
epistemic” is rather weak. A more interesting question
def is whether an individual superposition state ψ can
̟(0 , φ ,... µ)= dλµ(λ). (6)
| i∈B
| i | i | ZΛ|0i∪Λ|φi∪... be epistemic.
4
Theorem 1. Consider a quantum system of dimension be lower bounded by the probability of obtaining a λ
∈
d>3anddefinesuperpositionswithrespecttosomeONB Λ|0i Λ|φi; formally,
∪
. Almost all quantum superposition states ψ are
Bontic. | i 6∈ B PM(0 1′ µ) ̟(0 , φ µ)=̟(0 µ)+̟(φ µ)
| i∨| i| ≥ | i | i| | i| | i|
2̟(0 µ) (16)
Proof. Let ψ be an arbitrary superposition state ψ ≥ | i|
| i | i 6∈
and assume only that ψ is not an exact 50:50 su- where the equality follows because 0 , ψ , φ is anti-
B | i {| i | i | i}
perposition of two states in . This is true for almost distinguishable and the final line follows from Eq. (15),
B
all superpositions and guarantees that there exists some which is found by assuming that ψ is an epistemic su-
0 such that 0ψ 2 (0,1). perposition. | i
| i∈B |h | i| ∈ 2
Define an ONB ′ = 0 i′ d−1 containing this In order to satisfy Eq. (3)
B {| i}∪{| i}i=1
0 such that
| i P (0 1′ µ)= 0ψ 2+ 1′ ψ 2 =α2+2α4. (17)
M
| i∨| i| |h | i| |h | i|
ψ =α0 +β 1′ +γ 2′ (12)
| i | i | i | i Combining Eqs. (16,17) it is found that
where α R, α (0, 1/√2), and β d=ef √2α2. Such 1
bases alwa∈ys exists∈since 0ψ 2 =α2 and α2+ β 2 = ̟(0 µ) α2 +α2 <α2. (18)
| i| ≤ 2
α2(1+2α2)<1. With re|shpe|cti|to the same| ′|, defi|n|e (cid:18) (cid:19)
B
But, this contradicts the assumption that ψ is an epi-
φ d=efδ 0 +η 1′ +κ3′ (13) stemic superposition which implies ̟(0 |µ)i = α2 by
| i | i | i | i | i|
Eq.(11). Therefore,ifthepredictionsofquantumtheory
where δd=ef1 2α2, ηd=ef√2α. This can always be norm- are to be exactly reproduced, any such ψ must be an
alised becaus−e δ 2+ η 2 =(1 2α2)2+2α2 <1. ontic, rather than epistemic, superpositio|n.i
| | | | −
Theaboveconstructionhasbeenchosensuchthat:
• 0ψ 2 = α2 = φψ 2 so there exists a unitary V. BOUNDS ON GENERAL OVERLAPS
|h | i| |h | i|
U for which U 0 = φ ; and
|ψi
∈S | i | i
Theorem 1 establishes the reality of almost all super-
2 2 2
• the inner products 0ψ , φψ , 0φ sat- positions in d > 3 by bounding an asymmetric overlap.
|h | i| |h | i| |h | i|
isfyEq.(9)andthereforethetriple ψ , φ , 0 is Thissuggeststhatasimilarmethodmaybeusedtoprove
{| i | i | i}
anti-distinguishable. a general bound on ontic overlaps.
Recall shortcomings (i) and (ii) of the previous single-
Proof. Forany preparationdistribution µ′ ∆|ψi of ψ , systemontologyargumentsasmentionedinSec.I.Short-
consider ̟(0 µ′). For any unitary V an∈d any cor|resi- coming (i) leaves open the possibility that many pairs
| i|
ponding γ ΓV, µ′ is evolved to some µ ∆V|ψi as in of quantum states could have significant ontic overlaps,
∈ ∈
Eq. (2). This operation cannot decrease the asymmetric while (ii) casts doubt on the significance of those zero-
overlap̟(V 0 µ) ̟(0 µ′) and, in particular,letting overlap limits (as orthogonal states are distinguishable
| i| ≥ | i|
V =U one finds and therefore must be trivially ontologically distinct).
The following theorem address these shortcomings.
̟(φ µ) ̟(0 µ′). (14)
| i| ≥ | i| Theorem 2. Consider a d > 3 dimensional quantum
A proof of this is provided in appendix A. Therefore, system and any pair ψ , 0 ( ) such that 0ψ 2d=ef
there must exist preparation distributions µ,µ′ ∈ ∆|ψi α2 ∈ (0,41). Assum|eit|hait∈pPureHstate prepar|hati|onis| of
satisfying Eq. (14). ψ are non-contextual with respect to stabiliser unitar-
| i
Assume towards a contradiction that ψ is an epi- ies of ψ . For any preparation distribution µ ∆ ,
| i | i ∈ |ψi
stemic superposition so that Eq. (11) holds and, in par- the asymmetric overlap must satisfy
ticular, ̟(0 µ) = ̟(0 µ′) = α2. By Eq. (14) it is
| i| | i| 1+2α
therefore found that 2
̟(0 µ) α (19)
| i| ≤ d 2
(cid:18) − (cid:19)
̟(φ µ) ̟(0 µ). (15) lim ̟(0 µ) = 0 (20)
| i| ≥ | i| d→∞ | i|
Consider, then, a preparation of the state ψ via µ
followed by an ONB measurement M in the| i′ basis. andsobecomesarbitrarilysmallasdincreases,independ-
B ently of α.
Since ψ was prepared, λ Λ and the only possible
|ψi
measu|remi entoutcomesare∈0 , 1′ ,and 2′ . ByEq.(3), The proof, in appendix A, closely follows that of
| i | i | i
almost all λ Λ|0i must return the outcome 0 with Thm. 1. The assumption of pure state preparation non-
∈ | i
certainty. Similarly, almost all λ Λ can only return contextuality with respect to stabiliser unitaries is re-
|φi
0 , 1′ , or 3′ as the measuremen∈toutcome. Therefore, quired to replace the assumption used in Thm. 1 that
|thie |proibabi|litiy of obtaining outcomes 0 or 1′ must ψ is an epistemic superposition with respect to 0 .
| i | i | i | i
5
VI. NOISE TOLERANCE The proof is provided in appendix A. This theorem
makes Thm. 2 noise-tolerant at the expense of weaken-
ing the bound (and only applying for d>5). This is be-
Thus far Eq. (3) has been assumed, demanding that
cause the simple bound on symmetric overlap [Eq. (23)]
quantumstatistics areexactly reproducedby valid onto-
is lower than that for the asymmetric overlap [Eq. (5)]
logical models. However, it is impossible to verify this.
and therefore more difficult to improve upon.
At most, experiments demonstrate quantum probabilit-
Note that this theorem does not immediately imply
ies hold to within some finite additive error ǫ (0,1]. It
∈ that almost all superpositions are real. However, by
is therefore necessary to consider noise-tolerant versions
demonstratingthatThm.2’sargumentscanbemadero-
of the above theorems.
bust against error, it suggests that a noise-tolerant ver-
Unfortunately,theasymmetricoverlapisanoiseintol-
sionof Thm. 1 should also be possible. Even so, a noise-
erant quantity—there exist simple ontological models in
tolerantversionofThm.1wouldrequirethedefinitionof
which every pair of quantum states have unit asymmet-
“epistemic superposition” to be modified, since it is cur-
ric overlap and still reproduce quantum probabilities to
rentlydefinedintermsofthenoiseintolerantasymmetric
withinanygivenǫ (0,1]. However,analternativeover-
∈ overlapand is therefore noise intolerant.
lapmeasure,thesymmetricoverlap ω(ψ , φ )[2,4–6,8],
| i | i
is robust to small errors and Thm. 2 can be modified to
bound the symmetric overlapin a noise-tolerant way.
VII. DISCUSSION
Suppose you are given some λ Λ obtained by
∈
sampling from either µ or ν (each with equal a priori
probability). If you try to guess which of µ,ν was used, Assuming that quantum statistics are exactly correct,
thenω(µ,ν)/2 isdefinedto be the averageprobabilityof Thm. 1 proves that, for d> 3, almost all superpositions
error when using the optimal strategy. This is known to defined with respect to any given basis must be real.
B
correspond to [2, 4] Therefore,anyepistemicrealistaccountofquantumthe-
ory must include ontic features corresponding to super-
position states. The unfortunate cat cannot be put out
def
ω(µ,ν)= dλmin µ(λ),ν(λ) . (21)
Λ { } of its misery.
Z
A similar method and construction is used in Thm. 2
Extendingthistoquantumstatesthemselves,ratherthan to prove that, for arbitrary states satisfying φψ 2
1 |h | i| ∈
topreparationdistributions,givesthesymmetricoverlap (0, ), ontic overlap must approach zero as d increases
4
for fixed φψ 2. Theorem 3 makes this robust against
ω(ψ , φ )d=ef sup ω(µ,ν). (22) small err|ohrs| ini| quantum probabilities, at the expense
| i | i µ∈∆|ψi,ν∈∆|φi of weakening the bound. Both theorems require an ex-
traassumption: purestatepreparationnon-contextuality
Quantumtheoryprovidesanupperboundonthesym- with respect to stabiliser unitaries. Pure state prepar-
metric overlap, since any quantum procedure for dis- ation contextuality is often implicitly assumed whole-
tinguishing ψ , φ is also a method for distinguish- sale,sothisassumptionshouldnotbeverycontroversial.
| i | i
ing µ ∆|ψi,ν ∆|φi in an ontological model. As Moreover, appendix B provides a heuristic argument to
∈ ∈
1 1 1 φψ 2 is the minimum average error the effect that this type of contextuality is a natural as-
2 − −|h | i| sumption in practice.
tphr(cid:16)eoobrayb5ipliittyfowllhowensdthisat(cid:17)tinωg(uµis,hνi)ng |1ψi,|φi1withφinψqu2anhtouldms These results aredamagingto any epistemic approach
toquantumtheorycompatiblewiththeontologicalmod-
≤ − −|h | i|
for every µ ∆ ,ν ∆ and so
∈ |ψi ∈ |φi p elsformalismthatreproducesquantumstatisticsexactly.
Such a programme can never hope to epistemically ex-
ω(ψ , φ ) 1 1 φψ 2. (23) plain superpositions, including macroscopic superposi-
| i | i ≤ − −|h | i|
tions. Moreover,foranymoderatelylargesystem,alarge
p
Theorem 3. Consider the assumptions of Thm. 2, but number of pairs of non-orthogonalstates cannot overlap
only assume that the probabilities predicted by quantum significantly, making it unlikely that such overlaps can
theoryareaccuratetowithin ǫ, forsomeǫ (0,1]. The satisfactorily explain quantum features.
± ∈
symmetric overlap must satisfy As a result tolerant to small errors, it is possible that
Thm.3couldbeexperimentallytested. Suchatestwould
2 1+2α (3d2 7d) require demonstration of small errorsin probabilities for
ω(0 , ψ ) α + − ǫ. (24)
| i | i ≤ d 2 2(d 2) a wide range of measurements on a d > 5 dimensional
(cid:18) − (cid:19) −
system.
This bound is tighter than Eq. (23) for d>5 for small ǫ. The methodology of Thms. 1,2 is tightly linked to
the asymmetric overlap as a probability, making noise-
tolerantversionsachallengetoextract. Iftheconclusion
fromThms.1,2couldbe obtainedthoughanoperational
5ByusingtheHelstrommeasurement [4,19]. methodology (closer to that of Bell’s theorem[10] or the
6
PBR theorem [1]) this would likely lead to better noise- sightful discussions as well as an anonymous referee for
tolerant extensions and better opportunities for experi- thorough and insightful comments. This work is sup-
mental investigation. Such an operational version may ported by: the Engineering and Physical Sciences Re-
also make it easier to discover any information theoretic searchCouncil(EPSRC);theEuropeanCoordinatedRe-
implications of these results. searchonLong-termChallengesinInformationandCom-
munication Sciences & Technologies (CHIST-ERA) pro-
ject on Device Independent Quantum Information Pro-
Acknowledgments cessing (DIQIP); and the Foundational Questions Insti-
tute (FQXi) Large Grant “Thermodynamic vs informa-
I would like to thank Jonathan Barrett, Owen Ma- tion theoretic entropies in probabilistic theories”.
roney, Dominic C. Horsman, and Matty Hoban for in-
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.
Appendix A: Proofs
1. Simple upper bound on asymmetric overlap
To proveEq. (5)fromEq. (3), firstconsider any φ ( ) andany ONB measurementM φ andno evolution
| i∈P H ∋| i
between preparation and measurement (U = and γ is trivial). Equation (3) then gives
1
1 = dλν(λ)P (φ λ) (A1)
M
Λ | i|
Z
= dλν(λ)P (φ λ) (A2)
M
Λ | i|
Z ν
7
for any ν ∆ . This can only be the case if P (φ λ) = 1 for almost all λ Λ and therefore6 (since ν ∆
|φi M ν |φi
is arbitrar∈y) P (φ λ) = 1 for almost all λ Λ |.iI|n other words, almost a∈ll ontic states in the support∈of any
M |φi
| i| ∈
preparation ν of φ must return the measurement result φ with certainty in any measurement M containing that
| i | i
result.
Now consider that ̟(φ µ) is the probability of obtaining some λ Λ when sampling µ. If µ ∆ for some
|φi |ψi
| i| ∈ ∈
ψ ( ), then
| i∈P H
def
̟(φ µ) = dλµ(λ) (A3)
| i| Λ
Z |φi
= dλµ(λ)P (φ λ) (A4)
M
Λ | i|
Z |φi
dλµ(λ)P (φ λ)= φψ 2 (A5)
M
≤ Λ | i| |h | i|
Z
by Eqs. (3,4), thus proving Eq. (5).
2. Anti-distinguishability and multi-partite asymmetric overlaps
The main text states that if ψ , φ , 0 is an anti-distinguishable triple, then Λ|ψi Λ|φi Λ|0i = which
{| i | i | i} ∩ ∩ ∅
further implies that ̟(0 , φ µ) = ̟(0 µ)+̟(φ µ) µ ∆ . Here, a more general statement, necessary for
|ψi
| i | i| | i| | i| ∀ ∈
the proofs of Thms. 2,3, is proved. Define the set = 0 , φ ,... and let µ ∆ be a preparation distribution
|ψi
A {| i | i } ∈
for a state ψ . The statement is that if each triple 0 , ψ , φ , where 0 , φ are unequal states from , is
| i 6∈ A {| i | i | i} | i | i A
anti-distinguishable, then Eq. (7) holds with equality.
Recall that ̟(0 , φ ,...µ) is the probability of obtaining a λ Λ by sampling from µ, while, for every
|ai∈A |ai
| i | i | ∈ ∪
φ , ̟(φ µ) is the probability of obtaining a λ Λ from µ. The event corresponding to the probability
|φi
| i ∈ A | i| ∈
̟(0 , φ ,...µ) must therefore be the disjunction of the events corresponding to each probability ̟(φ µ).
| i | i | | i ∈ A|
Applying Boole’s inequality therefore gives Eq. (7).
Now suppose that each triple 0 , ψ , φ , where 0 , φ are unequal states from , is anti-distinguishable. Can
{| i | i | i} | i | i A
the events corresponding to ̟(0 µ) and ̟(φ µ) occur simultaneously (or are they mutually exclusive)? This is
| i| | i|
only possible if there exists a finite-measure set of ontic states λ Λµ Λ|0i Λ|φi. It shall now be shown that
∈ ∩ ∩
anti-distinguishability and Eq. (3) prevent this.
Let χ ∆|0i,ν ∆|φi be any relevant pair of preparation distributions and let M = E¬0,E¬ψ,E¬φ be the
∈ ∈ { }
anti-distinguishing measurement for 0 , ψ , φ . Equations (3,8) imply that
{| i | i | i}
dλχ(λ)PM(E¬0 λ)= dλµ(λ)PM(E¬ψ λ)= dλν(λ)PM(E¬φ λ)=0. (A6)
Λ | Λ | Λ |
Z χ Z µ Z ν
Theserespectivelyimply the following: foralmostallλ Λχ,PM(E¬0 λ)=0;foralmostallλ Λµ,PM(E¬ψ λ)=0;
and for almost all λ Λν, PM(E¬φ λ) = 0. Since this∈holds for arb|itrary χ and ν, it follow∈s that7 for alm| ost all
λ Λ|0i, PM(E¬0 λ)=∈0, and for alm|ost all λ Λ|φi, PM(E¬φ λ)=0.
∈ | ∈ |
Therefore,for almostallλ Λµ Λ|0i Λ|φi it followsthatPM(E¬0 λ)=PM(E¬ψ λ)=PM(E¬φ λ)=0. However,
thisisimpossiblesincesomeo∈utcom∩emus∩toccurinanymeasurement,r|equiringPM(E|¬0 λ)+PM(E¬|ψ λ)+P(E¬φ λ)=
| | |
1. So Λµ Λ|0i Λ|φi must be of measure zero and the events corresponding to ̟(0 µ) and ̟(φ µ) cannot occur
∩ ∩ | i| | i|
simultaneously—they are mutually exclusive.
SinceBoole’sinequalityholdswithequalityformutuallyexclusiveevents,itfollowsthatEq.(7)holdswithequality
whenever every such triple 0 , ψ , φ is anti-distinguishable.
{| i | i | i}
6Amoremathematically rigorous treatment wouldfullyconsiderthisstepinthelightof∆|φi beinguncountable inthegeneral case.
Such a discussion is omitted for the sake of conceptual clarity and since a more rigorous treatment would also have to account for the
issuesraisedinfootnote3.
7Similarly to the previous footnote, a fully rigorous treatment would include a proof of this step, which is omitted for conceptual
clarityandsincemathematicalrigourhasalreadybeensacrificedforconceptual clarityearlierinthepaper.
8
3. Unitary transformations never decrease ontic overlaps
Consider quantum states ψ , φ ( ), a unitary transformation γ Γ , and preparation distribution ν ∆
U |φi
| i | i∈P H ∈ ∈
so that under γ, ψ transforms to U ψ and ν to ν′ ∆ . By Eqs. (2,4) one finds
U|φi
| i | i ∈
̟(U ψ ν′) = dλ′ν′(λ′) (A7)
| i| Λ
Z U|ψi
= dλ′ dλν(λ)γ(λ′ λ) (A8)
Λ Λ |
Z U|ψi Z
dλν(λ) dλ′γ(λ′ λ). (A9)
≥ Λ Λ |
Z |ψi Z U|ψi
Consider the transition probability dλ′γ(λ′ λ), where λ Λ . Suppose towards a contradiction that this
ΛU|ψi | ∈ |ψi
probability is less than unity dλ′γ(λ′ λ) < 1 for some finite measure of λ Λ . This implies that8 there is
ΛU|ψi R | ∈ |ψi
some preparation µ ∆ of ψ such that
∈ |ψi R| i
1 > dλµ(λ) dλ′γ(λ′ λ) (A10)
Λ Λ |
Z Z U|ψi
= dλ′µ′(λ′) (A11)
Λ
Z U|ψi
= dλ′µ′(λ′) (A12)
Λ
Z
byEq.(2)whereµ′ ∆ isobtainedfromµviaγ,whichisacontradictionsincepreparationsmustalwaysproduce
U|ψi
∈
some ontic state dλ′µ′(λ′)=1. Therefore, dλ′γ(λ′ λ)=1 and so
Λ ΛU|ψi |
R R
̟(U ψ ν′) dλν(λ)=̟(ψ ν), (A13)
| i| ≥ Λ | i|
Z |ψi
thus proving Eq. (14).
Thesameresultalsoholdsforthesymmetricoverlap[Eq.(22)]betweenanypairofquantumstates ψ , φ . Consider
| i | i
any pair µ ∆ ,ν ∆ , then ω(µ,ν) is simply twice the optimal average probability of error when attempting
|ψi |φi
to guess wh∈ich of µ o∈r ν a given λ Λ was sampled from. For any stochastic map γ that transforms µ to µ′ and ν
∈
to ν′, a strategy for distinguishing µ′,ν′ is also a strategy for distinguishing µ,ν. Therefore, the optimal strategy for
distinguishingµ′,ν′ cannot,bydefinition,havealowerprobabilityoferrorthantheoptimalstrategyfordistinguishing
µ,ν. It immediately follows that
ω(µ′,ν′) ω(µ,ν) (A14)
≥
and by Eq. (23) that ω(U ψ ,U φ ) ω(ψ , φ ) for any unitary U.
| i | i ≥ | i | i
4. Theorem 2: Bounding general state overlaps
The proof strategy is almost identical to that of Thm. 1, but modified to make use of higher dimensional systems.
thiAsncyasseucαh2|ψ+i cβan2b=e wαr2it+ten2iαn3th<e f1orsmo tohfeEcqo.n(s1tr2u)cftoironsormemeaOinNsBpBos′s=ibl{e|.0iS}im∪i{la|ir′liy},di=a−11seatnodfwsthaetreesβd=φef√2d−α132.caInn
| | | | | | | | {| ii}i=3
be defined with respect to the same basis by
φ d=efδ 0 +η 1′ +κi′ (A15)
i
| i | i | i | i
8Onceagain,suchamorerigorous formulationoftheproblemwouldrequireafulljustificationofthisstep.
9
with δ d=ef 1 2α2 and η d=ef √2α32. Again, this is possible since δ 2 + η 2 = (1 2α2)2 +2α3 < 1. Note that the
− | | | | −
definitions of β and η have changed from those used in Thm. 1.
It may be verified by Eq. (9) that both 0 , ψ , φ and ψ , φ , φ are anti-distinguishable triples for all
i i j
{| i | i | i} {| i | i | i}
i=j.
6 Notethat φ ψ 2 =α2 = 0ψ 2 foralli,sothereexiststabiliserunitaries U d−1 forwhichU 0 = φ .
|h i| i| |h | i| { i}i=3 ⊂S|ψi i| i | ii
Consider preparing ψ via some µ ∆ then transforming with U via any γ Γ . By assumption, preparations
| i ∈ |ψi i i ∈ Ui
of ψ are non-contextual with respect to such stabiliser unitaries, so µ simply transforms to itself. Therefore, by
| i
Eq. (14), it is found that
̟(φ µ) ̟(0 µ) i. (A16)
i
| i| ≥ | i| ∀
So, prepare the state ψ via µ and then perform a measurement M in the ′ basis. Since 0 and 1′ are the only
mmueastsulroewmerenbtoouuntdcotmheesp|rcoobimapbailtiitbyleofwoibthtaλin∈ingΛ|eψitih∩er(Λ0|0io∪rdi=−131′ Λ. |Oφini)e,tthheernefothrBeefiansydms mtheattric o|veirlap w|ithi these states
| i | i
PM(0 1′ µ) ̟(0 , φ3 ,..., φd−1 µ)
| i∨| i| ≥ | i | i | i|
d−1
= ̟(0 µ)+ ̟(φ µ)
i
| i| | i|
i=3
X
(d 2)̟(0 µ) (A17)
≥ − | i|
where the second line follows because each of the sets 0 , ψ , φ and ψ , φ , φ are anti-distinguishable. So
i i j
if quantum predictions are exactly reproduced, one find{s| tih|atiP| (i}0 1{′|µi)=| αi2|+i2}α3 and
M
| i∨| i|
1+2α
2
̟(0 µ) α . (A18)
| i| ≤ d 2
(cid:18) − (cid:19)
Which completes the proof.
5. Theorem 3: A noise-tolerant bound on the symmetric overlap
This proof uses the the assumptions, notation, and constructions from Thm. 2, except this time it is only assumed
that the ontological model reproduces quantum probabilities to within some additive error ǫ (0,1]. It will also be
∈
necessary to define the tri-partite symmetric overlapbetween three probability distributions µ,ν,χ [8]
def
ω(µ,ν,χ)= dλ min µ(λ),ν(λ),χ(λ) . (A19)
Λ { }
Z
Consider any pair of preparation distributions µ ∆|ψi, ν ∆|0i. From Thm. 2 it is known that there exist
∈ ∈
U such that U 0 = φ . For each U consider any corresponding stochastic map γ Γ . By assumption,
i ∈ S|ψi i| i | ii i i ∈ Ui
preparationsof ψ arenon-contextualwithrespecttostabiliserunitariessoeachγ mapsµ toitself. Leteachγ map
i i
| i
ν to some χi ∈∆|φii. For notationalconvenience,let |φ0id=ef|0i andχ0d=efν then define the sets I˜d=ef{3,...,d−1}and
Id=ef 0 I˜. By Eq. (A14) it is therefore seen that
{ }∪
ω(µ,χ ) ω(µ,ν), i I˜. (A20)
i
≥ ∀ ∈
Consider a preparationof ψ via µ, followedby a measurementM in the basis ′. Similarly to Thm. 2, the aim is
to bound ω(µ,ν) by consider|ingi the probability of obtaining either of the measuremBentoutcomes 0 or 1′ , given by
| i | i
P (0 1′ µ) α2+β2+ǫ. (A21)
M
| i∨| i| ≤
The trick is to do this in such a way that all possible errors are accounted for.
In order to link this quantum probability to symmetric overlaps, consider the following subsets of Λ.
def
• For each i I consider Ω = λ Λ : 0 < µ(λ) χ (λ) . Roughly, Ω is the region of the overlap between µ
i i i
∈ { ∈ ≤ }
and χ for which µ is smaller than χ .
i i
def
• For each i I consider Θ = λ Λ : 0 < χ (λ) < µ(λ); j < i, χ (λ) χ (λ); j > i, 0 < χ (λ) < χ (λ) .
i i j i j i
∈ { ∈ ∀ ≤ ∀ }
Roughly, this is the region of the overlap between µ and χi for which χi is greater than all other χj6=i, but
smaller than µ.
10
• For each i<j I consider Θj d=ef λ Λ : 0<χ (λ) χ (λ); χ (λ)<µ(λ) . Roughly, this is the regionof the
∈ i { ∈ i ≤ j i }
tri-partite overlap of µ,χ ,χ in which χ is the minimum of the three.
i j i
• Similarly, for each i>j I consider Θj d=ef λ Λ : 0<χ (λ)<χ (λ); χ (λ)<µ(λ) .
∈ i { ∈ i j i }
• For every unequal pair i,j I, let Ω =Ω Ω .
ij i j
∈ ∩
Note that these sets are defined to be disjoint, for i=j: Θ Ω =Θ Θ =Θ Θj = .
6 i∩ j i∩ j i∩ i ∅
Thepointofthesesubsetsisthewayinwhichtheyrelatetosymmetricoverlaps. Fromthedefinitionsofsymmetric
overlaps it is not difficult to verify that
ω(µ,χ ) = dλµ(λ)+ dλχ (λ) (A22)
i i
ZΩi ZΘi∪[∪j6=iΘji]
ω(µ,χi,χj6=i) = dλµ(λ)+ dλχi(λ)+ dλχj(λ). (A23)
Ω Θj Θi
Z ij Z i Z j
Proceed by separating the probability Eq. (A21) according to subsets in which λ may obtain:
P (0 1′ µ) P (0 1′ ,λ Ω µ)
M M i
| i∨| i| ≥ | i∨| i ∈ |
i∈I
X
+ P (0 1′ ,λ Θ µ)
M i
| i∨| i ∈ |
i∈I
X
P (0 1′ ,λ Ω µ), (A24)
M ij
− | i∨| i ∈ |
i,j<i
X
P (0 1′ ,λ Ω µ)
M i
≥ | i∨| i ∈ |
i∈I
X
+ P (0 1′ ,λ Θ µ)
M i
| i∨| i ∈ |
i∈I
X
dλµ(λ). (A25)
− Ω
i,j<iZ ij
X
The final line follows simply because P (0 1′ ,λ Ω µ) P (λ Ω µ)= dλµ(λ).
M | i∨| i ∈ ij| ≤ M ∈ ij| Ωij
For the i=0 term in the first line of Eq. (A25), define the function ξ(λ)d=ef1 PRM(0 λ) so that
− | i|
PM(0 1′ ,λ Ω0 µ) = dλµ(λ) PM(0 λ)+PM(1′ λ) (A26)
| i∨| i ∈ | ZΩ0 { | i| | i| }
dλµ(λ)P (0 λ) (A27)
M
≥ ZΩ0 | i|
= dλµ(λ) dλµ(λ)ξ(λ) (A28)
ZΩ0 −ZΩ0
dλµ(λ) dλν(λ)ξ(λ). (A29)
≥ ZΩ0 −ZΩ0
This can be simplified by noting that, for any Ω Λ,
⊆
dν(λ)ξ(λ) = dλν(λ) dλν(λ)P (0 λ)+ dλν(λ)P (0 λ) (A30)
M M
ZΩ ZΩ −ZΛ | i| ZΛ\Ω | i|
dλν(λ) 1+ǫ+ dλν(λ) (A31)
≤ ZΩ − ZΛ\Ω
= ǫ (A32)
so that
PM(0 1′ ,λ Ω0 µ) dλµ(λ) ǫ. (A33)
| i∨| i ∈ | ≥ZΩ0 −