A no-go theorem for theories that decohere to quantum mechanics

0.17 MB·

by  Ciarán M. Lee

A no-go theorem for theories that decohere to quantum mechanics
Ciar´an M. Lee1,∗ and John H. Selby2,3,†
1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
2University of Oxford, Department of Computer Science,
Wolfson Building, Parks Road, Oxford OX1 3QD, UK.
3Imperial College London, London SW7 2AZ, UK.
Quantumtheoryisthemostexperimentallyverifiedphysicaltheoryinthehistoryofscience. Yet
it may be the case that quantum theory is only an effective description of the world, in the same
way that classical physics is an effective description of the quantum world. In this work we ask
whether there can exist an operationally-defined theory superseding quantum theory, but which
reducestoquantumtheoryviaadecoherence-likemechanism. Weprovethatnosuchpost-quantum
7
theory can exist if it is demanded that it satisfies two natural physical principles, causality and
1
0 purification. Here, causality formalises the statement that information propagates from present to
2 future,andpurificationthateachstateofincompleteinformationarisesinanessentiallyuniqueway
due to lack of information about an environment system. Hence, our result can either be viewed
n
as a justification of why thefundamental theory of Nature is quantum,or as showing in a rigorous
a
manner that any post-quantum theory must abandon the principle of causality, the principle of
J
purification, or both.
5
2
In 1903 Michelson wrote “The more important funda- hyperdecoherencemechanism. Here,causalityformalises
]
h mental laws and facts of physical science have all been the statement that information propagates from present
p discovered, and these are so firmly established that the to future, and purification that each state of incomplete
- possibilityoftheireverbeingsupplantedinconsequenceof information arises in an essentially unique way due to a
t
n new discoveries is exceedingly remote” [Mic03]. Within lack of information about some larger environment sys-
a two years Einstein had proposed the photoelectric effect tem. Inasense,purificationcanbethoughtofasastate-
u
[Ein05] and within thirty quantum theory was an estab- mentof“objectiveuncertainty”;anymissinginformation
q
lished field of scientific research. This new science rev- aboutthestateofagivensystemcanalwaysbeaccounted
[
olutionised our understanding of the physical world and forbyconsideringitaspartofalargersystem. Ourresult

1 brought with it a litany of classically counter-intuitive can either be viewed as a justification of why the funda-
v
featuressuchassuperposition,entanglement,andfunda- mental theory of Nature is quantum, or as highlighting
9
mental uncertainty. the ways in which any post-quantum theory must radi-
4
cally depart from a quantum description of the world.
4 Today, quantum theory is the most accurately tested
7 theory of Nature in the history of science. Yet, just as
0 I. DECOHERENCE
for Michelson, it may turn out to be the case that quan-
.
1 tum theory is only an effective description of our world.
0 There may be some more fundamental theory yet to be One of the standard descriptions of the quantum to
7 discovered that is as radical a departure from quantum classical transition is environment-induced decoherence
1
theory as quantum was from classical. If such a theory [Zur03]. In this description, a quantum system inter-
:
v exists, there should be some mechanism by which effects actsdeterministicallywithsomeenvironmentsystem,af-
Xi of this theory are suppressed, explaining why quantum ter which the environment is discarded, leading to a loss
theory is a good effective description of Nature. This of information. This procedure formalises the idea of a
r
a would be analogous to decoherence, which both sup- quantum system irretrievably losing information to an
presses quantum effects and gives rise to the classical environment, leading to an effective classical description
world [JZK+13, Zur03]. As such, we call such a mecha- of the decohered system. The decoherence process can
nism hyperdecoherence [48]. be viewed as inducing a completely positive trace pre-
We formalise such a hyperdecoherence mechanism serving map on the original quantum system, which is
within a broadframework of operationally-definedphys- termed the decoherence map.
ical theories by generalisingthe key features of quantum Aconcreteexampleservestoillustratethekeyfeatures
to classical decoherence. Using this we prove a no-go ofthismap. Considerthefollowingreversibleinteraction
result: there is no operationally-defined theory that sat- with an environment: U = Pi|iihi|⊗πi, where {|ii} is
isfies two natural physical principles, causality and pu- thecomputationalbasisandπi isaunitarywhichactson
rification, and which reduces to quantum theory via a the environment system as πi|0i = |ii, ∀ i,. Switching
to the density matrix formalism, the decoherence map
arising from the above interaction corresponds to
∗Electronicaddress: [email protected] D(ρ)=Tr Uρ⊗|0ih0| U† = hi|ρ|ii|iihi|,
†Electronicaddress: [email protected] E(cid:0) E (cid:1) X
i
2
where ρ is the input quantum state. Hence, in this con- some number of inputs and outputs, and one can intu-
crete setting, D is a de-phasing map. itivelythinkofsystems aspassingbetweentheinputand
It is clear that D(ρ) will always be diagonal in the outputs of a given process. Systems are labelled by dif-
{|ii} basis, regardless of the input. Hence, as they ferent types A,B,....
have no coherences between distinct elements of {|ii}, Processes can be connected together to form experi-
the states D(ρ) correspond to classical probability dis- ments, for example:
tributions. In fact, the entirety of classical probability
theory—corresponding to probability distributions over B g
classical outcomes, stochastic maps acting on said dis- C A
tributions, and measurements allowing one to read off h
f D
the probabilities of different possible outcomes—can be
D
seentoarisefromquantumtheorybyapplyingDtoden- A i
sity matrices ρ as D(ρ), completely positive trace pre-
serving maps E as D(E(D(·))), and POVM elements M This wiring together of processes must satisfy two con-
as Tr(MD(·)). In this manner, one can consider clas- ditions: firstly, system types must match, and secondly,
sical probability theory to be a sub-theory of quantum no cycles can be created. The relevant data for a partic-
theory—meaningthatapplyingstochasticmapstoprob- ular diagram is just the connectivity: which outputs are
abilitydistributionsresultsinprobabilitydistributions— connected to which inputs and the ordering of the free
where D is the map restricting quantum theory to the inputs and outputs. Any diagrams formed in this way
classical sub-theory. must also correspond to a valid process in the theory.
There are three key features of the decoherence map Onecanthink ofthediagramsformedbyconnectingdif-
that we will use to define our hyper-decoherence map in ferent processes as akin to circuits drawn in the field of
section III: quantum computation.
Processes with no inputs are known as states—
1. Itistracepreserving,correspondingtothefactthat
corresponding to density matrices in quantum theory—
it is a deterministic process.
those with no outputs effects—corresponding to POVM
elements in quantum theory, and those having both
2. It is idempotent, meaning
inputs and outputs transformations—corresponding to
D(D(ρ))=D(ρ), for all ρ. completely positive trace non-increasing maps in quan-
tum theory. When the diagramrepresentingthe connec-
Thiscorrespondsto the intuitive factthatclassical tions of processes in an experiment has no free inputs
systems have no more coherence ‘to lose’. or outputs, we associate it to the probability that all of
these processes occur when the experiment is run, for
3. IfD(ρ)isapureclassicalstate,i.e. D(ρ)=|iihi|for example:
somei,thenitisclearlyalsoapurequantumstate.
This is a consequence of the fact that decoherence g
arises from an irretrievable loss of information to A
C
an environment and if the state that results from D
h := Pr(f,g,h,i)
this procedure is a state of maximal information, f
D
then no information can have been lost to the en-
i
vironment.
Thereare thentwo naturalassumptions thatwe make
II. GENERALISED THEORIES for theories within this framework. The first is tomog-
raphy: if two processes give the same probabilities in all
To begin to make progress on the question raised at experiments, then they are the same process. That is:
the start of this paper, we need to be able to describe
f =g ⇐⇒ ∀X, Pr(f,X)=Pr(g,X)
theories other than quantum and classical theory in a
consistent manner. This calls for a broad framework
whereX isanydiagramwhich,whencomposedwithf or
that candescribe anyconceivable and well-definedphys-
g, has no free inputs and outputs. The secondis convex-
ical theory. The framework we present here is based
ity: given a collection of processes with the same inputs
on [CK16, CDP10, Har11, LB15][49] and takes the view
andoutputs,itispossibletoobtainaprocesscorrespond-
that, ultimately, any physical theory is going to be ex-
ing to a convex combination of these, defined by
plored by experiments, and so must have an operational
description in terms of these experiments. A theory in
h= p f ⇐⇒ ∀X, Pr(h,X)= p Pr(f ,X)
this frameworkcanthereforebe describedas acollection X i i X i i
i i
of processes, each of which corresponds to a particular
outcome occurringina single use ofa piece of labequip- where p defines a probability distribution (i.e. p ∈ R+
i i
ment in some experiment. In general, each process has and p = 1). Convexity allows us to define purity of
Pi i
3
states. A state is pure if it is not a convex combination The purification principle, in conjunction with an-
of other distinct states. othernaturalprinciples,impliesmanyquantuminforma-
Inwhatfollows,wewillrequireourpost-quantumthe- tion processing [CDP10] and computational primitives
ory to satisfy two natural physical principles, causality [LS16b]. Examples include teleportation, no informa-
andpurification, whichwerefirstintroducedin[CDP10]. tion without disturbance, no-bit commitment [CDP10],
A process is deterministic if the piece of lab equipment and the existence of reversible controlled transforma-
it corresponds to only has one possible outcome. tions. Moreover,purification also leads to a well-defined
Causality [CDP10]: For each system of type A, notion of thermodynamics [CS15b, CS16a, CS16b].
there exists a unique deterministic effect, denoted as
A III. HYPERDECOHERENCE
In section I, the quantum to classical transition was
This may seem like a somewhat odd definition for
modelledbyadecoherencemaprestrictingquantumsys-
causality, however it can be shown to be equivalent to
temstoclassicalones. Wecananalogouslymodelapost-
the statement that future measurement choices do not
quantum to quantum transition with a hyperdecoherence
effect current experiments [CDP10]. It also implies the
map,representeddiagrammaticallyby ,whichrestricts
nosuperluminalsignallingprinciple[Coe14]andprovides
a unique definition of marginalisation for multi-system post-quantum systems—described by a generalised the-
states. A process f is said to be causal if oryfromsectionII—toquantumones. Wenowadaptthe
three key features of decoherence outlined at the end of
A section I to this general setting, ending this section with
f = B a formal definition of a post-quantum theory.
B As in the quantum to classical transition, we think of
thishyperdecoherencemapasarisingviasomedetermin-
Inquantumtheorytheuniquedeterministiceffectispro- isticinteractionwithanenvironmentsystem,afterwhich
videdbythe(partial)traceandsocausaltransformations the environment is discarded by marginalising with the
are precisely those that are trace preserving. It can be unique deterministic effect. Hence, as with standard de-
shown for general theories [CS15b] that both reversible coherence, hyperdecoherence can be thought of as an ir-
and deterministic transformations are causal. retrievable loss of information to an environment. As
Purification [CDP10]: For every state on a given deterministic processesare causal,the hyperdecoherence
system A, there exists a pure bipartite state on some map should be causal:
composite systemAB, suchthat the originalstate arises
as a marginalisation of this pure bipartite state:
=
A A
B This is the analogueof point 1 from the end of sectionI.
= .
ρ
ψ Moreover, hyperdecohering twice should be the same
as hyperdecohering once, as the hyperdecohered system
Here,ψissaidtopurify ρ. Moreover,anytwopurestates has no more ‘post-quantum-coherence’ to ‘lose’. Hence
ψ andψ′ onthesamecompositesystemwhichpurifythe this map should be idempotent:
same state are connected by a reversible transformation
=
A B
A B
R
B This is the analogueof point 2 from the end of sectionI.
=
ψ′ ψ AswasthecaseforclassicaltheoryinsectionI,onecan
constructtheentiretyofquantumtheoryasasub-theory
If one considers a pure state to be a state of maximal of the post-quantum theory by appropriately applying
information, then the purification principle formalises to states, transformations, and effects from the post-
the statement that each state of incomplete information quantum theory. That is, density matrices, completely
arisesin anessentially unique waydue to a lack ofinfor- positivetracenon-increasingmaps,andPOVMelements
mation about a larger environment system. In a sense, correspond to
purification can be thought of as a statement of “infor-
mationconservation”;anymissinginformationaboutthe B
A
stateofagivensystemcanalwaysbetracedbacktolack e
of information of some environment system. Or, more A , T , A
succinctly: information can only be discarded, not de- s
A
stroyed [CS15a]. A
4
Hence—as is idempotent—quantum states, transfor- left invariant as well—due to the fact that they can be
mations, and effects are those left invariant by the hy- steered to using a quantum state. Hence, for each sys-
perdecoherence map. tem, the hyperdecoherence map must be the identity, a
One can define a notion of purity relative to the sub- contradiction.
theoryconstructedviatheaboveprocedure. Astatefrom For ease of exposition, diagrammatic notation will be
the sub-theory is pure in the sub-theory if it cannot be usedthroughouttheproof. Notethatitcanalternatively
written as a convex combinationof other states from the be written using standard algebraic notation. For con-
sub-theory. Note that a state which is pure in the sub- venience we denote quantum states with a subscript q.
theory may not be pure in the full post-quantumtheory, Givena bipartite quantumstate ψq, it can be written as
asastatethatcannotbewrittenasaconvexcombination
opfossatabtleesafsroamcothnevesxubc-otmheboirnyatmioanyotfursntaotuestltyoinbge oduectsoimde- ψq = Pij rij φφiqiq χφiqjq rij ∈R.
the sub-theory. Ashyperdecoherencearisesfromanirre-
trievablelossofinformationtoanenvironment,ifastate
Idempotence of the hyperdecoherence map then gives
resulting from this process is a state of maximal infor-
mation, then no information can have been lost to the
estnavtiersonimnetnhte. sWube-ftohremoraylisaertehpisubreyidnemthaendpionsgt-tqhuaatnptuurme ψq = Pij rij φφiqiq χφiqjq = ψq (1)
theory. This is the analogue of point 3 from the end of
section I.
Next, consider the maximally mixed quantum state,
Finally, we say a hyperdecoherence map is trivial if it 1
µ := , of a d-dimensional system. For any pure quan-
is equal to the identity transformation: q d
tum state ψ , there is a state σ such that
q q
A = A µq = d1 ψq +(1− d1) σq (2)
To summarise all of the above, we now formally define a
Now, denote the Bell state 1 |iiihjj| for a d-
post-quantum theory. dPij
dimensional system diagrammatically as:
Post-quantum theory: A generalised theory is a
post-quantumtheoryif,foreachsystemtypeA,thereex-
A A
istsahyperdecoherencemap A satisfyingthefollowing
conditions:
A
A As the hyperdecoherence map is causal, marginalisation
1. A is causal: A =
in the post-quantum theory is the same as in quantum
theory. Hence, the marginals of the above Bell state are
A
2. A is idempotent: = A the maximally mixed quantum state
A
3. Pure states in the sub-theory are pure states.
= = (3)
µq
Moreover,thesub-theorydefinedbythecollectionn Ao
is quantum theory and at least one of the hyperdecoher- Eq. (3), in conjunction with the fact that re-
ence maps must be non-trivial. versible transformations are causal, implies that for any
reversible transformation G—including post-quantum
IV. RESULT
transformations—we have
Main Theorem. There is no post-quantum theory sat-
isfying both causality and purification. G
=
Proof. We prove that in any post-quantum theory sat-
isfying causality and purification, the hyperdecoherence
map must be trivial for all systems. The main idea As the marginalised systems are of the same type, the
of the proof is to show that by performing a suitable purification principle implies [50] the existence of a re-
post-quantum measurement on the quantum Bell state versible transformation T such that
andpost-selectingonasuitablepost-quantumeffect,any
post-quantumstatecanbesteeredto. Asquantumstates
G T
are left invariantby the hyperdecoherence map(even lo- =
cally, as we show below), all post-quantum states are
5
Hence, whereχ isapurequantumstate. Thepurificationprin-
q
ciple implies that these two purifications are connected
by a reversible transformation R :
φ
G G T
= = = (4)
µq µq R
φ
=
themaximallymixedquantumstateisinvariantunderall χq Sφ
reversible transformations in the post-quantum theory.
Usingpoint3above,itthenfollowsthatthereisaneffect
A standard result [CDP10] obtained from purification
e that steers the Bell state to φ
is transitivity: given any two pure states of the same φ
system there exists a reversible transformation between
them. This result, in conjunction with eq. (4) and (2), 0q
gives R
φ
eφ := χq = 1 (6)
G G G d φ
= = 1 +(1− 1) (5)
µq µq d ψq d ρq
This is true for any pure state φ in the theory.
= 1 +(1− 1) Using Eq. (1) and Eq. (6), and noting that the Bell
d φ d σ
state for a composite system is the composite ofthe Bell
states for the single systems
where φ is an arbitrary pure state in the theory. Hence,
any pure state fromthe post-quantum theory arisesin a
A B
AB AB A B
decomposition of the quantum maximally mixed state.
Now, as every (non-trivial) quantum system A has at := (7)
leasttwoperfectlydistinguishablestates,{|0ih0|,|1ih1|},
giventhedecompositionofEq.(5), convexity impliesthe
following is a state in the post-quantum theory: we have, for all pure states ψ and all effects η, that
A A η
:= 1 +(1− 1) η
sφ d φ 0q d σ 1q eψ
A B = dAB A A B B
Consider a purification of this state, denoted Sφ, and ψ
note that it has the following properties:
A A P A A η e η
1. = ψ
Sφ sφ = dAB A A B = A B
B
ψ
A A P
2. =
Sφ µq
Thisresult,inconjunctionwithtomographyandconvex-
ity, implies that, for all systems A,
0q
A A P
3. S = d1 φ A = A
φ
Where the effect 0 is the quantum effect Tr(|0ih0|·)
q
which gives probability 1 for state 0 and probability 0
q
for 1 . As the product of two pure quantum states is a As we know that there exists a post-classical theory
q
pure quantum state, the definition of hyperdecoherence which satisfies causality and purification and decoheres
implies that the following is another purification of µ to classical theory, i.e. quantum theory, one might won-
q
with the same purifying system AP as Sφ deratwhatstageourproofbreaksdownwhenanalysing
this situation. The main reason is that the maximally
A A P correlated state in classical probability theory is mixed
and so Eq. 6 is no longer valid. Hence, the reason why
χq quantum theory cannot be extended in the manner pro-
posed here is the existence of pure entangled states.
6
V. DISCUSSION nite causal order between two processes has even been
showntobearesourcewhichcanbeexploitedtooutper-
form theories satisfying the causalityprinciple in certain
Fromthe famous theorems ofBell [Bel64] andKochen
information-theoretic tasks [ACB14, Chi12]. Moreover,
& Specker [KS67] to more recent results by Colbeck &
ithasbeensuggestedthatanytheoryofQuantumGrav-
Renner[CR11], andPusey,Barrett&Rudolph[PBR12],
itymustexhibitindefinitecausalorder[Har16a,Har16b].
no-go theorems have a long history in the foundations
Hence, as in the previous paragraph,our result provides
of quantum theory. Most previous no-go theorems have
further motivation for discarding the notion of definite
been concerned with ruling out certain classes of hidden
causalorder in the searchfor theories superseding quan-
variable models from some set of natural assumptions.
tum theory.
Hidden variables—or their contemporary incarnation as
ontologicalmodels[HS10]—aimtoprovidequantumthe- As purification seems to require a unique way
ory with an underlying classical description, where non- to marginalise multipartite states, one might wonder
classicalquantum features arise due to the fact that this whether one can define a notion of purification without
description is ‘hidden’ from us. the causality principle. Indeed, recent work [AFNB16]
has shown how one can formalise a purification princi-
Unlike these approaches, our result rules out certain
classes of operationally-defined physical theories which ple in the absence of causality, and forthcoming work of
one of the authors discusses a ‘time-symmetric’ notion
can supersede quantum theory, yet reduce to it via a
suitableprocess. Tothebestofourknowledge,ourno-go ofpurificationsatisfiedbyquantum, classicalandhybrid
quantum-classicalsystems [SSCng].
theoremisthefirstofitskind. Thismayseemsurprising
given that it is an obvious question to ask. However, to Another assumption in our theorem was the man-
even begin posing such questions in a rigorous manner nerinwhichourhyperdecoherencemap—themechanism
requires a consistent way to define operational theories by which the post-quantum theory reduces to quantum
beyondquantumandclassicaltheory. Themathematical theory—wasformalised. However,itmaynotbethecase
underpinnings of such a framework have only recently thatpost-quantumphysicsgivesrisetoquantumphysics
been developed and investigated in the field of quantum via such a mechanism. Indeed, alternate proposals for
foundations. how some hypothetical post-quantum theory reduces to
Aswithallno-gotheorems,ourresultisonlyasstrong quantumtheoryhavebeenproposed[KOSW13]. Despite
as the assumptions which underlie it. We now critically this,ourunderstandingofthe quantumtoclassicaltran-
examine each of our assumptions, outlining for each one sitionintermsofdecoherencesuggestshyperdecoherence
thesenseinwhichitcanbeconsidered‘natural’,yetalso as the natural mechanism by which this should occur.
suggesting ways in which a hypothetical post-quantum Hence we see this as the least objectionable of the as-
theory could violate it and hence escape the conclusion sumptions made in this work.
of our theorem. The last assumption underlying our no-go theorem
Our first assumption is purification. As noted in sec- is the generalised framework itself, introduced in sec-
tion II, the purification principle provides a way of for- tion II. While the operational methodology underlying
malising the natural idea that information can only be thisframeworkispartandparcelofthescientificmethod,
discarded [CS15a], and any lack of information about it may not be the case that the correct way to formalise
thestateofagivensystemarisesinanessentiallyunique this methodology is by asserting that pieces of labora-
way due to a lack of information about some larger en- tory equipment can be composed together to result in
vironment system. However, proposals for construct- experiments, as described in section II. Indeed, it may
ing theories in which information can be fundamen- be the case that the standard manner in which elements
tally destroyed have been suggested and investigated of a theory are composed together—resulting in other
[OR09, BSP84, UW95]. Such proposals take their in- elements—needs to be revised in order to go beyond the
spiration from the Black Hole Information loss paradox. quantum formalism. Work in this direction has already
Our result can therefore be thought of as providing an- begun [Har13].
other manner in which the fundamental status of infor- Ourresultcaneither be viewedas demonstratingthat
mation conservation can be challenged. the fundamental theory of Nature is quantum mechani-
Our second assumption is causality. This principle al- cal, or as showing in a rigorous manner that any post-
lows one to uniquely define a notion of “past” and “fu- quantum theory must radically depart from a quantum
ture” for a given process in a diagram, and is equiva- description of the world by abandoning the principle of
lent to the statement that future measurement choices causality, the principle of purification, or both.
do not affect current experimental outcomes. As such, Acknowledgements— The authors thank H. Barnum
this principle appears to be fundamental to the scien- and B. Coecke for useful discussions and D. Browne, M.
tific method. Despite this, recent work has shown how Hoban,&J.Richensforproofreadingadraftofthe cur-
one can relax this principle to arrive at a principle of rent paper. They also acknowledge encouragement from
indefinite causality [OCB12, OC14, CDP16, Har07]. In J. J. Barry. This work was supported by the EPSRC
this case, there may be no matter of fact about whether throughtheUCLEPSRCDoctoralPrize,theControlled
a given process causally precedes another. The indefi- Quantum Dynamics Centre for Doctoral Training, and
7
the OxfordDepartmentofComputer Science. This work is supported by the Government of Canada through the
began while the authors were attending the “Formulat- Departmentof Innovation,Science andEconomic Devel-
ing and Finding Higher-order Interference” workshop at opment Canada and by the Province of Ontario through
the Perimeter Institute. Research at Perimeter Institute the Ministry of Research, Innovation and Science.
[ACB14] Mateus Arau´jo, Fabio Costa, and Cˇaslav Brukner. arXiv:0911.0695, 2009.
Computational advantage from quantum-controlled or- [Ein05] AlbertEinstein. Thephotoelectriceffect. Ann.Phys,
dering of gates. Physical review letters, 113(25):250402, 17(132):4, 1905.
2014. [Har07] Lucien Hardy. Towards quantum gravity: a frame-
[AFNB16] Mateus Araujo, Adrien Feix, Miguel Navas- work for probabilistic theories with non-fixed causal
cus, and Caslav Brukner. A purification postulate structure. Journal of Physics A: Mathematical and The-
for quantum mechanics with indefinite causal order. oretical, 40(12):3081, 2007.
arXiv:1611.08535, 2016. [Har11] Lucien Hardy. Reformulating and reconstructing
[Bel64] JohnSBell. Ontheeinsteinpodolskyrosenparadox, quantum theory. arXiv preprint arXiv:1104.2066, 2011.
1964. [Har13] Lucien Hardy. On the theory of composition in
[BMU+14] Howard Barnum, PM Markus, Cozmin Ududec, physics. In Computation, Logic, Games, and Quan-
et al. Higher-order interference and single-system pos- tum Foundations. The Many Facets of Samson Abram-
tulates characterizing quantum theory. New Journal of sky, pages 83–106. Springer, 2013.
Physics, 16(12):123029, 2014. [Har16a] Lucien Hardy. Operational general relativity: Pos-
[Bol16] Arkady Bolotin. On the ongoing experiments look- sibilistic, probabilistic, and quantum. arXiv preprint
ing for higher-order interference: What are they really arXiv:1608.06940, 2016.
testing? arXiv preprint arXiv:1611.06461, 2016. [Har16b] Lucien Hardy. Reconstructing quantum theory. In
[BSP84] Thomas Banks, Leonard Susskind, and Michael E Quantum Theory: Informational Foundations and Foils,
Peskin. Difficulties for the evolution of pure states into pages 223–248. Springer, 2016.
mixed states. Nuclear Physics B, 244(1):125–134, 1984. [Hen15] Joe Henson. Bounding quantum contextuality with
[CDP10] Giulio Chiribella, Giacomo Mauro DAriano, and lack of third-order interference. Physical review letters,
Paolo Perinotti. Probabilistic theories with purification. 114(22):220403, 2015.
Physical Review A, 81(6):062348, 2010. [HS10] Nicholas Harrigan and Robert W Spekkens. Ein-
[CDP16] Giulio Chiribella, Giacomo Mauro DAriano, and stein,incompleteness,andtheepistemicviewofquantum
Paolo Perinotti. Quantum from principles. In Quan- states. Foundations of Physics, 40(2):125–157, 2010.
tumTheory: InformationalFoundationsandFoils,pages [JZK+13] Erich Joos, H Dieter Zeh, Claus Kiefer,
171–221. Springer, 2016. Domenico JW Giulini, Joachim Kupsch, and Ion-
[Chi12] Giulio Chiribella. Perfect discrimination of no- OlimpiuStamatescu. Decoherence and the appearance of
signalling channels via quantum superposition of causal a classical world in quantum theory. Springer Science &
structures. Physical Review A, 86(4):040301, 2012. Business Media, 2013.
[CK16] B Coecke and A Kissinger. Picturing quantum pro- [KOSW13] Matthias Kleinmann, Tobias J Osborne,
cesses. a first course in quantum theory and diagram- Volkher B Scholz, and Albert H Werner. Typical
matic reasoning, 2016. local measurements in generalized probabilistic theories:
[Coe14] Bob Coecke. Terminality implies non-signalling. Emergence of quantum bipartite correlations. Physical
arXiv preprint arXiv:1405.3681, 2014. review letters, 110(4):040403, 2013.
[CR11] Roger Colbeck and Renato Renner. No extension of [KS67] Simon Kochen and Ernst P Specker. The problem of
quantumtheorycanhaveimprovedpredictivepower.Na- hidden variables in quantum mechanics. 1967.
ture communications, 2:411, 2011. [LB15] Ciar´an M Lee and Jonathan Barrett. Computation
[CS15a] Giulio Chiribella and Carlo Maria Scandolo. Con- in generalised probabilisitic theories. New Journal of
servationofinformationandthefoundationsofquantum Physics, 17(8):083001, 2015.
mechanics. InEPJWebofConferences,volume95,page [LS16a] Ciar´an M Lee and John H Selby. Deriving grover’s
03003. EDPSciences, 2015. lower bound from simple physical principles. New Jour-
[CS15b] Giulio Chiribella and Carlo Maria Scandolo. En- nal of Physics, 18(9):093047, 2016.
tanglementandthermodynamicsingeneralprobabilistic [LS16b] Ciar´an M Lee and John H Selby. Generalised
theories. New Journal of Physics, 17(10):103027, 2015. phase kick-back: the structure of computational algo-
[CS16a] Giulio Chiribella and Carlo Maria Scandolo. Entan- rithmsfromphysicalprinciples. New Journal of Physics,
glement as an axiomatic foundation for statistical me- 18(3):033023, 2016.
chanics. arXiv preprint arXiv:1608.04459, 2016. [LS17] Ciar´an M.LeeandJohnH.Selby. Higher-orderinter-
[CS16b] Giulio Chiribella and Carlo Maria Scandolo. Purity ference in extensions of quantumtheory. Foundations of
in microcanonical thermodynamics: a tale of three re- Physics, Volume 47, Issue 1, pp 89–112, 2017.
source theories. arXiv preprint arXiv:1608.04460, 2016. [Mic03] Albert Abraham Michelson. Light waves and their
[CSTng] Bob Coecke, John Selby,and Sean Tull. Two roads uses. University of Chicago Press, 1903.
to classicality. forthcoming. [Nie13] GerdNiestegge. Three-slitexperimentsandquantum
[DB09] BorivojeDakicandCaslavBrukner.Quantumtheory nonlocality.FoundationsofPhysics,43(6):805–812,2013.
and beyond: is entanglement special? arXiv preprint [OC14] Ognyan Oreshkov and Nicolas J Cerf. Operational
8
quantumtheorywithoutpredefinedtime. arXivpreprint theory. Physical Review D, 52(4):2176, 1995.
arXiv:1406.3829, 2014. [Zur03] Wojciech Hubert Zurek. Decoherence, einselection,
[OCB12] Ognyan Oreshkov, Fabio Costa, and Cˇaslav andthequantumoriginsoftheclassical. Reviewsofmod-
Brukner. Quantum correlations with no causal order. ern physics, 75(3):715, 2003.
Nature communications, 3:1092, 2012. [Z˙yc08] Karol Z˙yczkowski. Quartic quantum theory: an ex-
[OR09] Jonathan OppenheimandBenniReznik. Fundamen- tension of the standard quantum mechanics. Journal of
tal destruction of information and conservation laws. PhysicsA:MathematicalandTheoretical,41(35):355302,
arXiv preprint arXiv:0902.2361, 2009. 2008.
[PBR12] Matthew F Pusey, Jonathan Barrett, and Terry [48] To the best of the authors knowledge, the notion of hy-
Rudolph. On the reality of the quantum state. Nature perdecoherence was first discussed in [Z˙yc08] and has
Physics, 8(6):475–478, 2012. commonly been considered as a mechanism to explain
[SCng] John Selbyand Bob Coecke. Leaks: quantum,classi- why we do not observed post-quantum effects, such as
cal, intermediate and more. forthcoming. in [DB09], and, in particular, in the context of higher-
[Sor94] Rafael D Sorkin. Quantum mechanics as quantum orderinterference[Sor94,LS17,LS16a,LS16b,BMU+14,
measure theory. Modern Physics Letters A, 9(33):3119– Bol16,Nie13,Hen15,UBE11,SVS15].Theformalisation
3127, 1994. of the idea of hyperdecoherence presented here is built
[SSCng] JohnSelby,Carlo MariaScandolo, andBobCoecke. on work presented in [SCng, CSTng] where quantum to
A classical purification postulate: what separates quan- classical decoherence is discussed in terms of ‘leaks’ in
tum and classical theory? forthcoming. generalised process theories.
[SVS15] Aninda Sinha, Aravind H Vijay, and Urbasi Sinha. [49] Tobeprecise,onecanviewatheoryinthisframeworkas
On the superposition principle in interference experi- ageneralised probabilistic theory [CDP10,Har11]where
ments. Scientific reports, 5, 2015. the standard assumptions of finite dimensionality and
[UBE11] Cozmin Ududec, Howard Barnum, and Joseph closure of theset of states are not made, or as a process
Emerson. Three slit experiments and the structure of theory [CK16] with the added assumptions of tomogra-
quantumtheory. Foundations of Physics, 41(3):396–405, phy and convexity.
2011. [50] Notethatapplyingareversibletransformation toapure
[UW95] William G Unruh and Robert M Wald. Evolution state results in a pure state [CS15b].
laws takingpurestates tomixed states in quantumfield