Non-classicality of temporal correlations

0.49 MB·English

by  Stephen Brierley

Non-classicality of temporal correlations
Stephen Brierley,1 Adrian Kosowski,2 Marcin Markiewicz,3,4,∗ Tomasz Paterek,5,6 and Anna Przysiężna4,7
1Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
2Inria and Université Paris Diderot, LIAFA, Case 7014, 75205 Paris Cedex 13, France
3Faculty of Physics, University of Warsaw, Pasteura 5, PL-02-093 Warszawa, Poland
4Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland
5School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
6Centre for Quantum Technologies, National University of Singapore, Singapore
7National Quantum Information Centre in Gdansk, Andersa 27, 81-824 Sopot
Theresultsofspace-likeseparatedmeasurementsareindependentofdistantmeasurementsettings,
apropertyonemightcalltwo-wayno-signalling. Incontrast,time-likeseparatedmeasurementsare
only one-way no-signalling since the past is independent of the future but not vice-versa. For this
5
reason some temporal correlations that are formally identical to non-classical spatial correlations
1
can still be modelled classically. We propose a new formulation of Bell’s theorem for temporal
0
correlations, namely we define non-classical temporal correlations as the ones which cannot be
2
simulated by propagating in time the classical information content of a quantum system given by
y the Holevo bound. We first show that temporal correlations between results of any projective
a
quantum measurements on a qubit can be simulated classically. Then we present a sequence of
M
POVMmeasurementsonasinglem-levelquantumsystemthatcannotbeexplainedbypropagating
in time an m-level classical system and using classical computers with unlimited memory.
6
2
Introduction. The violation of a Bell inequality [1–3] tem. Moreover the operational meaning of the assump-
]
h demonstrates that the outcomes of an experiment have tions themselves is still a subject of debate [7, 11, 15].
p contradicted a set of well defined classical intuitions. The fact that in the sequential scenario the evolving
- Quantum mechanics allows correlations between space- system caries information has significant consequences
t
n like separated parties that have no explanation in terms [10]. Let us first consider the simplest possible case: a
a of a hidden variable model, i.e. they cannot be repro- single two-level system which undergoes a sequence of
u
duced with the help of classical computers running pre- two black-box operations at time instances t and t .
q 1 2
[ agreed algorithms. However, when correlations are gen- Each black box has an input describing which settings
erated in a temporal scenario, by a sequence of time-like are chosen and an output describing the measurement
3 separated measurements, it is more difficult to demon- result. For a quantum implementation in which black
v strate their non-classical nature. The causal structure boxes perform projective measurements the inputs have
5
0 of physics implies only one-way no-signalling, namely the form of unit vectors (cid:126)a(t1) and (cid:126)a(t2), and outputs
5 the impossibility of sending communication backwards read α(t1) = ±1 and α(t2) = ±1, respectively. It can
3 in time. The only bound on forward signalling is the be verified [6], that the temporal correlation function
0 information capacity of the physical system. (cid:104)α(t )α(t )(cid:105) equals(cid:126)a(t )·(cid:126)a(t ). This is (up to the sign)
1 2 1 2
1. Hereweanalyseasinglequantumsystemmeasuredat the correlation function that would be generated by two
0 n points in time and consider to what extent one can separated parties that share the singlet state. Moreover,
5 prove that the temporal correlations between these mea- it leads to a maximal violation of the temporal CHSH
1 surement outcomescould not begenerated byaclassical inequality [6]. Can we conclude from these two facts
v: system. We assume an idealization, in which the m-level that our system gives rise to non-classical temporal cor-
i physical system carries no hidden degrees of freedom. In relations? The answer is negative — instead of a sin-
X
this case the classical information capacity of the system gle qubit one can communicate one classical bit which
ar is log2m; known as the Holevo bound [4]. together with black boxes equipped with classically cor-
Thepreviousapproachtodemonstratenon-classicality related real vectors (cid:126)λ implement the Toner-Bacon 1-bit
oftemporalquantumcorrelationsaresocalled“temporal protocolforsimulatingthesingletstate[16][17]. Further-
Bell inequalities” [5–14]. One of the problems with this more, beginning the evolution with an arbitrary qubit
approach,statedin[13],isthattheclassicalassumptions state any sequence of n projective measurements leads
behind the temporal inequalities, which are realism and to correlations that factor into pairs of dot products of
non-invasiveness,wereoriginallychosentotestthequan- the consecutive input vectors [6]. Thus the temporal
tumnessofatemporalevolutionofmacroscopic quantum correlations of n projective measurements on a qubit ad-
systems [5]. As such, they do not provide a convincing mit a classical simulation essentially using a sequence of
test of quantumness in the case of a single evolving sys- Toner-Bacon protocols.
A similar situation can arise in the case of multi-point
correlations. For example, consider a sequence of three
black boxes with two-setting inputs φ(t ) = {0,π/2}
k
∗ [email protected] for k = 1,2,3 and binary outputs α(t ) = ±1 to-
k
2
gether with the promiss that the inputs fulfill the con- othernotionsofsimulability,whichcanbeusedtodefine
(cid:80)
straint φ(t ) = {0,π}. Let us assume, that the non-classicality (eg. in terms of computational complex-
k k
correlation function of outputs (cid:104)α(t )α(t )α(t )(cid:105) equals ity [32]). In this work we solely refer to simulability in
1 2 3
cos(φ(t )+φ(t )+φ(t )). Thisisthecorrelationfunction the context of communication complexity.
1 2 3
ofaGreenberger-Horne-Zeilinger(GHZ)state,andinthe Wedemonstratetheutilityofourdefinitionbypresent-
spatial scenario leads to the GHZ paradox [18]. In the ing a sequence of quantum measurements that give non-
temporal scenario such a function can be obtained by a classical temporal correlations provided the number of
sequenceoftwo-outcomePOVMmeasurementsonasin- measurements is sufficiently large. The quantum system
gle qubit [19]. However, as we prove later in the paper, and measurements we propose have the appealing prop-
the entire setup can be simulated by a classical protocol ertythattheyarewithinthereachofcurrentexperimen-
with exactly 1 bit of classical communication. Again, we taltechniques. Inthesimplestcase,wefindnon-classical
cannot verify that truly non-classical correlations have temporal correlations for a sequence of 16 POVM mea-
appeared. surements on a qubit system.
Definition of non-classical temporal correlations. The Non-classicality of temporal GHZ correlations. We
aboveexamplesshowthatidentifyingaviolationoftem- show that the temporal GHZ correlations, that is tem-
poral Bell inequalities with so called ’entanglement in poral correlations which have the same form as spatial
time’ [6] is not always accurate. Indeed, forward sig- correlations of an n-qumit GHZ state:
nalling of classical information in the sequential scenario
leaves a kind of communication loophole: correlation |GHZ(cid:105)= √1 (cid:88)m |i(cid:105)⊗n, (1)
functions which would be considered non-classical in the
m
spatial setting can be simulated by classical protocols i=1
that use a classical communication channel with capac-
are non-classical on condition that the number of mea-
itygivenbytheHolevoboundofthecorrespondingquan-
surement steps and the number of settings per observer
tum particle. The communication implicitly involved in
is sufficiently large. To prove this fact we utilize the so
a sequential process motivates adapting the concept of
called modulo-(m,d) games [33]. In the spatial scenario,
effective classical simulability with respect to the com-
thesegamesaredistributedcomputingtasksthatcanbe
munication cost (see Fig. 1):
solved with certainty with the help of shared GHZ-state
(1) but not with classical randomized algorithms. We
Definition 1. The temporal correlation function
translate these games into the sequential scenario and
E(Y ,...,Y |X ,...,X ) of the m level physical system
1 N 1 N
prove that they can be always solved exactly by a se-
is non-classical if all classical algorithms that simulate
quenceofPOVMmeasurementsonasinglequmitgiving
the function require more than log m bits of classical
2 rise to temporal GHZ correlations. We show that for
communication at some step of the simulation.
The idea of quantifying the degree of non-classicality
of a physical process distributed in space or time by
the amount of classical resources needed to simulate it
has already appeared in many contexts: communication
complexity of simulating spatial quantum correlations
[16, 20–27], memory complexity of simulating contextual
effects [28], and memory complexity of simulating uni-
tary evolution [13, 29–31]. The fact that simulation of
some sequential quantum procedures like contextuality
tests or unitary evolution demands resources exceeding
the Holevo bound has been already noticed in the litera-
ture[28,30]. Ourapproachgeneralizestheseideastothe
scenario of sequential measurements performed by black
boxes, with no restriction on their internal operations
or memory. Definition 1 provides a theory-independent
FIG. 1. Temporal correlation functions. a) A sequence of n
characterisation of non-classical temporal correlations:
consecutive measurements on a single quantum system with
one does not need to specify the physical implementa-
settings provided by inputs X and outcomes given by num-
tionwhichleadstogivencorrelations, butonlythenum- k
bers Y . We say that temporal correlations are non-classical
k
berofdegrees offreedomofthephysicalsystementering
ifthereisnoclassicalsimulationdepictedinpanelb). Atthe
the boxes. In contrast, in the most similar scenario of i-th step of the simulation, the i-th black box can perform
quantifying the memory cost of quantum contextuality local computations and send classical communication to the
[28] the counted resource is the total number of internal nextbox. Thecorrelationfunctionobtainedinthescenarioa)
states used by the simulating machines; which can be is non-classical if every classical simulation b) requires more
differentfromthesizeofthecommunicationbetweenthe communicationthantheclassicalinformationcapacityofthe
steps needed for simulation. We point out that there are quantum particle in at least one stage of the simulation.
3
some sets of parameters n,m,d the correlations cannot bit of communication at each stage (see Appendix A).
be simulated by any protocol whose communicates are Since the modulo-(2,2) problem for n = 3 is equivalent
lessthanthenumberofclassicalbitsgivenbytheHolevo to the scenario of the original GHZ paradox [37], dis-
bound. cussedintheintroduction,itfollowsthatthetwo-setting
GHZ qubit correlations, which in the spatial domain re-
Definition 2 (sequential n-point modulo-(m,d) prob-
vealstrongnon-classicality,donotfulfillthedefinitionof
lem). A sequential n-point modulo-(m,d) problem is a
non-classical temporal correlations.
communication complexity task, in which n separate or-
We now provide a general lower bound on the amount
dered parties are given (logd)-bit inputs X , with the
k of communication which is needed to classically solve
promise that (cid:80)n X modd = 0. The task of the par-
k=1 k these problems, which will imply the main result of our
ties is to output values Y ∈ {0,1,...,m−1} fulfilling
k work.
d(cid:80)n Y ≡ (cid:80)n X mod(md) in a sequential proto-
k=1 k k=1 k
col, which at k-th stage allows k-th party to produce her Theorem 1. Every classical protocol which solves the
local output Y and communicate a c -bit message M to sequential modulo-(m,d) problem with certainty uses at
k k k
the (k+1)-st party. least c = log(d/m) bits of communication in all stages
k
of the protocol except at most md − 1 (not necessarily
First we prove, that the above problem can be solved
consecutive) stages, when d is an integer power of 2 and
with certainty by a sequence of appropriate generalized
m is even.
quantum measurements on a single qumit. To solve the
modulo-(m,d) problem in a spatial domain [33], the n The proof comes by a minor modification of the ar-
parties share the state (1) and apply local unitary op- gument used in [30], adapted to the scenario of sequen-
erations Fm†(Sdm)Xk to their particles, where [Fm]αβ = tial measurements. We provide a stand-alone proof for
(cid:16) (cid:17) (cid:16) (cid:17) completeness, since the original proof applies an iterated
√1mexp 2iαmβπ and [Sdm]αβ = exp 2diβmπ δαβ, α,β = argument in a slightly informal way, leading to incorrect
0,1,...,m−1. Finally the states are locally measured in
constants in the analysis. To perform a proof by contra-
the standard basis. Let us now find a temporal counter-
diction, fix any classical protocol which claims to solve
part. First, note that the bond dimension of the matrix
the modulo-(m,d) problem with certainty, while using
product state representation [34] of the n-qumit GHZ
less than c bits of communication in some n = md
k 0
state is m, which implies that the state can be con-
stages. We will act as an adversary, constructing two
structed by a sequential cascade of two qumit gates U valid inputs of the modulo-(m,d) problem, X and X(cid:48)
[35, 36]. This property implies that one can map ar- suchthat(cid:80)n X (cid:54)≡(cid:80)n X(cid:48) modmd. Theinkputswilkl
bitrary local projective measurements in the spatial sce- be defined sok=t1hatkthe cokr=re1spoknding outputs Y and Y(cid:48)
nariointoasequenceofPOVMmeasurementsonasingle will be indistinguishable, i.e., Y =Y(cid:48), for all 1k≤k ≤nk.
system with dimension m whilst keeping the correlation k k
Hence, the modulo-(m,d) problem will be solved incor-
function fixed [19]. Following the construction presented rectly for at least one of the inputs X , X(cid:48), leading to a
in Ref. [19] one finds the measurement operators K k k
Yk contradiction.
corresponding to different outcomes Y by solving the
k The construction proceeds as follows. Let K =
0
system of equations:
{k ,k ,...,k } be the set of indices of the stages for
1 2 n0
which the protocol sends messages of size less than c .
m−1 k
S(U(|ψ(cid:105)⊗|0(cid:105)))= (cid:88) (KYk|ψ(cid:105))⊗Fm†(Sdm)Xj|Yk(cid:105), Wqueenptriaolcleye,dsowtihtahttthheefcoolnloswtriuncgtipornedoifcaintepsuatrseXfukl,fiXllek(cid:48)dsaet-
Yk=0
any step k:
whereS isaswapoperator,and|ψ(cid:105)isanarbitrarystate.
For odd dimensions m, measurement operators KYk ob- • For all j ≤k, Yj =Yj(cid:48),
tainedinthiswayarediagonalmatricesMwithjthdiag-
onal element equal to [M(d)] = √1 exp(cid:16)iXkπ(2j−2)(cid:17). • For all j ≤k, Mj =Mj(cid:48), where Mj is the message.
jj m d
The construction of inputs X ,X(cid:48) proceeds as follows:
For even m, measurement operators corresponding to k k
outputs Y are K = A M, where A is a diag-
k Yk Yk Yk • Foranystagek ∈/ K0, k (cid:54)=n, wesetXk arbitrarily,
onal matrix with elements ±1 and ±i. In particular and put X(cid:48) = X . Clearly, since M = M(cid:48)
for m = 2, A0 = diag(1,1), A1 = diag(1,−1); for by the indkuctive kassumption and X(cid:48)k−=1 X , kt−h1e
m = 4, A = diag(1,1,1,1), A = diag(1,−i,−1,i), k k
0 1 protocolwillactidenticallyinthek-thstepinboth
A2 = diag(1,−1,1,−1), A3 = diag(1,i,−1,−i); for cases, thus we have M =M(cid:48) and Y =Y(cid:48).
m = 6, A = A = A = diag(1,1,1,1,1,1), A = A = k k k k
0 2 4 1 3
A =diag(1,−1,1,−1,1,−1). • For any stage k ∈ K , given message M =
5 0 k−1
Wenowconsiderclassicalsimulationsofmodulo-(m,d) M(cid:48) we consider the set of outcome pairs p(x)=
k−1
games that use at most logm bits of classical communi- (M (x),Y (x)) of the execution k-th step of the
k k
cation at each step. Before presenting the main result, protocol, taken over all possible inputs x ∈
we first discuss the simplest case m = d = 2. It turns {0,1,...,d − 1}. Since |Mk| ≤ 2ck for k ∈ K0
out, that this problem can be simulated with a single and Y ∈ {0,1,...,m − 1}, the set of possible
k
4
output pairs p(x) has less than 2ckm elements, guarantees that the numbers m,n,d fulfill both of the
where we note that 2ckm = 2log(d/m)m = d. Con- above conditions.
sequently, there exists a pair of values x < x(cid:48),
x,x(cid:48) ∈{0,1,...,d−1},suchthatp(x)=p(x(cid:48)). We
denotex(cid:48) =x+∆ ,with∆ ∈{1,2,...,d−1}. We The above proposition shows that temporal GHZ cor-
k k
now put X =x, and choose X(cid:48) ∈{X ,X +∆ }, relations of any qumit reveal temporal non-classicality,
k k k k k
according to a rule which will be described later. if n is sufficiently large, which implies that any simulat-
Regardless of this choice, we have M = M(cid:48) and ing protocol uses more bits of communication than the
k k
Y =Y(cid:48). Holevo bound in at least one stage of the protocol. As
k k
a matter of fact, our theorem shows that this actually
• Finally,instagek =n,wesetXk sothattheinput happens in almost all stages of the protocol (for detailed
{Xk}nk=1 satisfies the modulo-d promise, and also analysis see Appendix C).
put X(cid:48) =X .
n n We point out that the above proposition holds also
for a single qubit case and its possibility follows from
It remains to show that it is possible to fix X(cid:48) from
k allowing POVM measurements. Therefore we provide a
among each pair of considered values {x,x+∆} for k ∈
K , so that (cid:80) X (cid:54)≡ (cid:80) X(cid:48) mod(md). This is first demonstration, that simulation of a temporal corre-
0 k∈K0 k k∈K0 k lation function on qubit demands resources that exceed
possiblebythefollowinglemma,whoseproofispresented
theHolevobound(notethatasimilareffectforaunitary
in Appendix B.
evolution of a single qubit was shown before [30]).
Lemma 1. Let {∆ } be any sequence of integers,
with∆ ∈{1,2,...,mk dk∈−K10},wheremisevenandd=2s Conclusions. In general terms, correlations between
k
for some integer s > 0. Then, there exists a subset of physical systems can be considered in two distinct sce-
indices K(cid:48) ⊆ K such that (cid:80) ∆ ≡ 0modd and narios: spatial and temporal. In the spatial scenario lo-
0 0 k∈K(cid:48) k
(cid:80) 0 cal measurements are performed by space-like separated
∆ (cid:54)≡0mod(md).
k∈K0(cid:48) k parties who may share a source of joint randomness but
Using Lemma 1, we then pick X(cid:48) = X +∆ for all
k k k who are unable to communicate. Temporal correlations,
stepsk ∈K(cid:48),andputX(cid:48) =X forallstepsk ∈K \K(cid:48).
0 k k 0 0 ontheotherhandarisefromasequenceofmeasurements
This completes our construction.
on a single physical system at different time instances.
In the above proof, we restricted our considerations to
Communication is now allowed from one time instance
deterministic protocols. For randomized protocols, the
to the next but is limited by the information capacity of
claimofthepropositionalsoholdsinthefollowingsense:
the system.
any randomized protocol which does not satisfy the as-
Weshowedthatinthetemporalmeasurementscenario
sumptions of the proposition will lead to an incorrect
one can define a notion of non-classical n-point correla-
output for some instances of the modulo-md problem,
tions with a clear operational interpretation. Namely,
with strictly positive probability.
such correlations cannot be simulated by any classical
The above theorem can be treated as a temporal ver-
protocol whose communication is limited by the Holevo
sion of Bell inequalities with auxiliary communication in
capacity of the evolving quantum system. In addition,
the spatial scenario [38, 39], and directly leads to the
we demonstrated that the temporal analogue of gener-
main result of our work:
alisedGHZcorrelationsarisingfromsequentialmeasure-
Proposition 1. The temporal GHZ correlations arising ments on a single qumit, reveal non-classicality in the
from a sequential measurements on a single qumit, where temporalscenarioprovidedthenumberofmeasurements
m is even, are non-classical for n≥2m3. is large enough. Apart from these foundational issues,
weprovidedthefirstgenerallowerboundonthecommu-
Proof. It suffices to show that there exists a modulo- nication complexity of simulating multi-point quantum
(m,d) game for some d and n, for which classical sim- correlations in a sequential measurement scenario (see
ulation uses in at least one stage of the protocol more Ref. [27] for results on the classical communication cost
than logm bits of communication. Using Theorem 1, of simulating spatial GHZ correlations).
we need to choose parameters so that the following two Acknowledgements.—We would like to thank Marcin
conditions are fulfilled: Pawłowski, Rafał Demkowicz-Dobrzański and Marek
Żukowski for helpful discussions. MM and AP grate-
• log(d/m) > logm, which means that the classical
fully acknowledge the financial support by Polish Min-
communication needed is greater than the Holevo
istryofScienceandHigherEducationGrantno. IdP2011
bound for the system,
000361. This work is supported by the National
Research Foundation, Ministry of Education of Sin-
• n ≥ md, which guarantees that the amount of
gapore Grant No. RG98/13, start-up grant of the
communication equal to log(d/m) > logm bits is
Nanyang Technological University, the NCN Grant No.
needed in at least one stage.
2012/05/E/ST2/02352 and by the EC under the FP7 IP
Now, let d be the smallest integer power of 2 larger than project SIQS co-financed by the Polish Ministry of Sci-
m2, we have d ≤ 2m2. Taking any n ≥ 2m3 ≥ md, ence and Higher Education.
5
Appendix A: Communication complexity of Appendix B: Proof of Lemma 1
sequential modulo-(2,2) problem
W.l.o.g. let K = {1,...,n} within the proof of this
claim. Define S be the set of all modulo-md remainders
Corollary 1. The sequential modulo-(2,2) problem can i
which can be obtained using subset sums of the first i
be solved with certainty with one bit of (classical) com-
elements,
munication at each stage of the protocol (that is c = 1
k
for all k =1,...,N −1).  
(cid:88)
Si ={ ∆jmod(md):I ⊆{1,...,i}}.
Proof. Recall that in the sequential modulo-(2,2) prob- j∈I
lem, the parties are given bit inputs X ∈ {0,1} with
k
(cid:80) Consider the sequence of sets, S ,S ,...S . Since S ⊆
the promise X ≡ 0mod2 and output Y ∈ {0,1} so 1 2 n i
i i (cid:80) (cid:80) k S ⊆1,...,md−1fori=1...nandn≥md,wemust
that the outputs fulfill 2Y − X ≡ 0mod4. Let i+1
i i i i havethatS =S forsomea. Theelement∆ ∈S
us consider the following deterministic protocol: a a+1 a+1 a
and the set S is invariant with respect to a modulo-md
a
shift by ∆ ,
a+1
• The message is initialised to M =0,
0
S =∆ +S modmd. (B1)
a a+1 a
Let p be the unique odd integer such that ∆ =p·2r,
a+1
• if X =M =1, then the i-th party returns Y =
i i−1 i for some integer r ≥ 0. Equation (B1) implies that all
1, and otherwise it returns Y =0,
i multiples of ∆ are also in S (modmd) and in partic-
a+1 a
ular,2s−r∆ =pdmodmd∈S . Sincepisoddandm
a+1 a
even, 2s−r∆ (cid:54)= 0modmd and 2s−r∆ = 0modd,
a+1 a+1
• the i-th party sends Mi = (Xi+Mi−1) mod 2 to as required.
the (i+1)-st party.
Appendix C: Communication complexity properties
(cid:80)
The protocol works due to the property: i≤jXi = of sequential protocols simulating GHZ correlations
(cid:80)
M + 2Y , which can be shown by induction on
j i≤j i
(cid:80)
j. Since Mj ≡ i≤jXimod2, by the promise on the Corollary 2. Any classical protocol simulating temporal
(cid:80)
input for j = N we have M = 0. Hence, X = GHZ correlations of a single qumit on n parties must:
N i≤N i
(cid:80)
2Y , and the claim follows.
i≤N i 1. Send Ω(εlogn) bits of communication to the next
party for each of at least n−O(nε) parties, for any
ε>0.
2. Contain a sequence of Ω(n1−ε) consecutive parties,
The protocol is valid because the expectation value of
(cid:80)n Y for settings that satisfy the promise (assum- each of which needs to send Ω(εlogn) bits of com-
k=1 k munication to the next party, for any ε>0.
ing X corresponds to xˆ and yˆ directions on the Bloch
k
sphere) is either equal to 0 (for even number of pairs of Proof. It suffices to take d=Θ(nε) in Theorem 1.
settings equal to 1) or 1 (for odd number of the pairs).
It is therefore sufficient to keep track of the parity of Note that in order to obtain a violation of the Holevo
the number of settings equal to 1. This is exactly acom- boundforalmostallparties,weneedtochooseanappro-
plished by the protocol. Note that the "local" (single- priate value of n=mΩ(1/ε), so that the required amount
point) expectation values are not modelled by this pro- of communication of Ω(εlogn), which follows from the
tocol. above proposition, exceeds logm.
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