Table Of ContentAppearance of causality in process matrices when performing fixed-basis
measurements for two parties
Veronika Baumann
Faculty of Informatics, Universita` della Svizzera italiana, Via G. Buffi 13, CH-6900 Lugano, Switzerland
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and
Institute for Quantum Optics and Quantum Information, Boltzmanngasse 3, 1090 Vienna, Austria
Cˇaslav Brukner
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and
Institute for Quantum Optics and Quantum Information, Boltzmanngasse 3, 1090 Vienna, Austria
(Dated: September21, 2016)
6
1
The recently developed framework for quantum theory with no global causal order allows for
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quantum processes in which operations in local laboratories are neither causally ordered nor in
2
a probabilistic mixture of definite causal orders. The causal relation between the laboratories is
p described by the process matrix. We show that, if the inputs of the laboratories are measured in
e afixedbasis, onecan introducean effectiveprocess matrix which is operationally indistinguishable
S fromtheoriginalone. ThiseffectiveprocessmatrixcanbeobtainedbyapplyingthevonNeumann-
Lu¨dersupdaterulefornonselectivemeasurementstotheoriginalprocessmatrixandinthebipartite
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case it is compatible with a definite causal order. The latter extends the original Oreshkov et al.
2
proof where one considers that both the measurement of the input and the re-preparation of the
] outputare performed in a fixed basis.
h
p
- Introduction: In all our theories, including quantum or can be decomposed in a classical (convex) mixture of
t
n field theory on curved space-time, causal structure is such correlations [10].
a fixed. However,this is unlikely to be present in a theory Inref. [4]itwasshownthatinthebipartitecase,ifthe
u
unifying quantum physics and general relativity, where inputs are measured in a fixed basis and the outputs are
q
[ the space-time geometry is expected to be both dynam- reprepared in a fixed basis, the causal relations between
ical, as in general relativity, as well as indefinite, due to the local operations are compatible with a global causal
2 quantum theory [1]. Recently, frameworks for quantum order. Hence, the effective process matrix the parties
v
theory have been developed that, unlike the standard can reconstruct from their data under this restricted set
0
formulations, do not presume a fixed background causal of operations is always causally separable. This is not
2
6 structure [1–5]. Moreover, the “process framework” [4] true for three or more parties where there exist classical
6 wasshowntoallowforprocessesinwhichtwooperations processmatrices (i.e. where bases arefixed for the input
0 areneithercausallyorderednorinaprobabilisticmixture measurement and the output preparation) that violate
1. of definite causal orders. causal inequalities [11].
0 The central object of the process framework is the Here we show that the causal relations between two
6 “process matrix”, which describes causal relations be- local laboratories are compatible with a global causal
1 tween local laboratories. Each such laboratory features order if only the inputs are measured in a fixed basis,
:
v a device with an input and an output. At every run of thus relaxing the assumption of the original proof [4].
i theexperiment,thelaboratoryisopenedfortheinputto The effective process matrix W can be obtained by
X eff
enter. Then a quantum operation (a complete-positive applying the von Neumann-Lu¨ders update rule [12, 13]
ar map) is performed while the laboratory is closed, after fornon-selectivemeasurementstotheoriginalW matrix.
which the output of the operation is sent away. For ex- W → P ⊗P WP ⊗P ≡ W , where P and
m,n n m n m eff n
ample,theoperationcanconsistofameasurementofthe P areprojectorsoftheinputsofA’sandB’slaboratory
m
input and then re-preparationof the output. onto thPe respective measurement basis. Moreover, W
eff
A bipartite process matrix W is called causally sepa- is causally separable. This is analogous to the situation
rable if it can be decomposed as a convex combination where anentangledstate is transformedinto a separable
of causally ordered ones: W =pWB(cid:22)A+(1−p)WA(cid:22)B, one, upon a non-selective measurement in a product ba-
where 0 ≤ p ≤ 1 and WA(cid:22)B is a process with which sis. The observed correlations are then compatible with
Bob cannot signal to Alice and WB(cid:22)A a process with the separable state ρsep that can be obtained from the
which Alice cannot signal to Bob. An explicit exam- originalρ byρ→ m,nPn⊗PmρPn⊗Pm ≡ρsep, where
ple of a causally non-separable process is the “quantum Pn andPm areorthogonalprojectorsonthesubspacesof
P
switch” where the causal structure is in quantum super- A and B.
position [6–9]. Moreover, certain causally non-separable Process formalism: We consider causal relations be-
processescanviolate“causalinequalities”whicharesat- tweentwolaboratories. Inthelaboratoriesquantumme-
isfied by all correlationsthat areunidirectional signaling chanics with a fixed (local) causal structure holds. The
(either from A to B, or from B to A) or non-signaling laboratories receive inputs ∈ L(HX1) and produce out-
2
puts ∈ L(HX2), X = A,B, but are otherwise isolated. matrix is then also diagonal W = weff |nihn|A1⊗
eff nrms
Each party applies a complete-positive map from a set |rihr|A2⊗|mihm|B1⊗|sihs|B2,with coefficientsweff =
nrms
{MX},whichisindexedbytheobservedoutcomek. The w . While in the bipartite cPase W has been
k nnrrmmss eff
joint probability for a pair of maps, (MA,MB), is given provento be causally separable[4], this is no longertrue
i j
by a generalized Born rule [4] for more than two parties [11].
We will now show that for the bipartite case the effec-
P(MA,MB)=Tr W(MA1A2 ⊗MB1B2) , (1) tive process matrices that are diagonal only in the input
i j i j
spaces of the two parties are still causally separable. We
h i
where MX1X2 are the Choi-Jamiolkowski (CJ) matrices consider effective process matrices of the form
k
[14, 15] of the operations MX andW ∈L(HA1⊗HA2⊗
HB1 ⊗ HB2) is the processkmatrix. Summing over all Weff = wnefiif′mjj′
(4)
outcomes of any party should give a complete-positive, |nihn|A1X⊗|iihi′|A2 ⊗|mihm|B1 ⊗|jihj′|B2,
trace preserving map, which means the CJ matrices sat-
isfy TrX2( kMkX1X2) = 1X1. Requiring that probabili- where {|ii} and {|ji} are arbitraryorthonormalbases of
tiesarepoPsitiveandsumuptoone(wherethepartiesare HA2 and HB2 and wnefiif′mjj′ = wnnii′mmjj′ analogous to
allowed to share entangled states) implies W ≥ 0 and a
the classical case. As any two-party process matrix, (4)
certain form of W, which automatically excludes causal
can be written as
loops,
1
W = ((1+λ )1+κ +κ ), (5)
1 eff d 0 1 2
W = 1A1A2B1B2 + c WA1A2B1B2 . (2)
d i i
!
Xi where λ0 ∈ [−1,0] is the minimal eigenvalue of
c WA1A2B1B2, κ acts trivially on HB2, κ acts triv-
Here d=dim(HA1)·dim(HB1), ci ∈R and WiA1A2B1B2 ialilyioniHA2 and κ1+κ ≥0. In the original2proof (see
1 2
are traceless and have trivial entry in at least one of P
supplementary information of ref. [4]), one makes use of
the output spaces and nontrivial entry in the respec- terms P =|nihn|A1 ⊗1A2⊗|mihm|B1 ⊗1B2, where
tive input space, i.e they are either of wA1A2B1 ⊗ 1B2 (n,m)
i {|nihn|}and{|mihm|}formorthogonalsets. Theseterms
or wB1B2A1 ⊗1A2 form. For example, a channel from
i are added to and subtracted from κ1 and κ2 for ev-
AlicetoBobisdescribedbyaprocessmatrixoftheform
ery pair (n,m) in a systematic way, such that the sum
W = wA1A2B1 ⊗1B2, while one from Bob to Alice by
κ +κ (and hence W ) remains unchanged. In that
W =wB1B2A1 ⊗1A2. Equations (1) and (2) can readily 1 2 eff
way, one can produce positive eigenvalues and arrive at
be generalized to more parties. matrices κA1A2B1 ≥ 0 and κA1B1B2 ≥ 0 (the super-
Effective process matrix: The process matrix encodes
script denotes spaces with non trivial entries) such that
causal relations between local laboratories. If local op- (1+λ )1+κ +κ =κA1A2B1+κA1B1B2. HenceW =
0 1 2 eff
erations are restricted to a set, one can find an effective 1(κA1A2B1+κA1B1B2)=pWA(cid:22)B+(1−p)WB(cid:22)A where
process matrix Weff that returns the same joint proba- pd = Tr(κA1A2B1)/d′ with d′ = d·dim(HA2)·dim(HB2)
bilities as the original process matrix W: and 1−p=Tr(κA1B1B2)/d′.
Examining the proof in detail shows that the system-
Tr W(MA1A2 ⊗MB1B2) =!
i j atic modification of the eigenvalues requires
(3)
h i
Tr W (MA1A2 ⊗MB1B2) ,
eff i j [κ ,κ ]=[κ ,P ]=[P ,κ ]=0 (6)
1 2 1 (n,m) (n,m) 2
h i
for all i,j fromthe set. One might think ofWeff as rep- andthejointeigenvectorsofκ1,κ2andP(n,m)tobeprod-
resentingthe causalstructureobserversinclosedlabora- uct vectors
torieswoulddeduce fromthe correlationsbetweenthem,
when their operations are limited. The idea of an effec- |ψ(n,m)i=|niA1 ⊗|a(n,m)iA2 ⊗|miB1 ⊗|b(n,m)iB2 (7)
tive process matrix can straightforwardly be generalized
to many local laboratories. with {|ni} and {|mi} being orthonormal bases of the
An example of restricted operations are classical ones input spaces HA1 and HB1. For every pair (n,m) the
for which all CJ matrices are diagonal both in the input {|a(n,m)i}, {|b(n,m)i} are orthogonal bases of HA2 and
andthe output space, MA1A2 = p(i,r|n)|nihn|A1⊗ HB2 respectively. Together (6) and (7) ensure that
i n,r
|rihr|A2 and MB1B2 = p(j,s|m)|mihm|B1 ⊗ m(n,a,m,b) = m1(n,a,m)+m2(n,m,b), where m(·) is
j mP,s
the eigenvalue of κ +κ andm (·), m (·) are the eigen-
|sihs|B2, where p(k,t|l) is the probability of outcome k 1 2 1 2
P values of κ , κ respectively. For process matrices (4)
and re-preparation of state t given input l. (In what 1 2
the commutation relations are trivially fulfilled. More-
follows sums in expressions of process matrices are al-
over,as (7) areeigenvectorsofP onehas|ψ(n,m)i=
ways taken over all appearing indices). Given a gen- (n,m)
eral process matrix W = wnn′rr′mm′ss′|nihn′|A1 ⊗ jlcjl|niA1 ⊗|jiA2 ⊗|miB1 ⊗|liB2. For a given (n,m)
|rihr′|A2 ⊗ |mihm′|B1 ⊗ |s′ihs′|B2, the effective process aPnyκ1 from(4)actson|ψ(n,m)ias1A1⊗AA(n2,m)⊗1B1B2
P
3
and analogously κ as 1A1A2B1 ⊗ BB2 , with hermi- Considertheprojectionofageneralstateuponamea-
2 (n,m)
tian matrices A and B . Therefore, |ψ(n,m)i = surement onto a product basis states. While the orig-
(n,m) (n,m)
inal bipartite state ρ might be entangled and contain
|niA1 ⊗|a(n,m)iA2 ⊗|miB1 ⊗|b(n,m)iB2, where |a(n,m)i
quantum correlations, the state after the measurement
and |b(n,m)i are eigenvectorsof AA2 and BB2 . Note
(n,m) (n,m) n,mPn⊗PmρPn⊗Pm is separable and contains only
that conditions (6) and (7) are sufficient for causal sep- classical correlations. Analogously, while the original W
arability but not necessary. To see this, simply consider mPight encode correlations with no definite causal order,
W0 = pd(1−σzA1 ⊗σzA2 ⊗σxB1) +1−dp(1+ 12σzA1 ⊗σxB2 + after the measurements are performed in a fixed basis,
1σA1 ⊗ σB1 ⊗ σB2). The two nontrivial terms do not the process matrix is causally separable and the correla-
2 x x z
commute. Moreover no nontrivial P(n,m) exists, which tions are compatible with a global causal order.
commutes with both terms, since it would have to com-
Note that there is no analogy to the update rule for
mute with both σ and σ . Hence neither (6) nor (7) is
x z selectivemeasurements. Ingeneral,P WP isnot
(n,m) (n,m)
fulfilled,althoughW iscausallyseparableperdefinition.
0 a valid process matrix. It contains terms that correlate
Interpretation: We considered bipartite local opera-
the output spaces of Alice and Bob, or the input and
tions that consist of a measurement of the input and
output space of a single party andis therefore not of the
a re-preparation of the output. The CJ representation
form (2).
of the complete-positive map has the form MX1X2 =
φ1φ2 In processes violating causal inequalities the causal
|φ ihφ |X1 ⊗|φ ihφ |T X2, where |φ ihφ | describes the
1 1 2 2 1 1 order cannot be assumed to be fixed prior to and
measuredinputstateand|φ ihφ |isthepreparedoutput
2 2 independent of local operations. However, by choosing
state. Following Holevo’s classification [16] of entangle-
the measurement basis for the inputs, local observers
ment breaking quantum channels, one can denote maps,
are able to (non-selectively) “project” the initial process
where both the measurement and the re-preparationare
matrix to a causally separable one – a desired (noisy)
performed in a fixed basis, as classical-classicalmaps. A
channelbetween them. In general,this channelwill be a
classical-quantummapthen correspondsto anoperation
convex mixture of a channel from A to B and a channel
in which the input is measured in a fixed basis, but the
from B to A. This provides a way to interpret violation
output is arbitrary. The CJ representation of such map
of causal inequalities by processes.
is MX1X2 = p(i|n)|nihn|X1 ⊗ ρX2, where p(i|n) is
i n i
the probability to prepare state ρX2 given the input is
P i
measured to be in state |ni.
If the local operations are restricted to classical-
quantum maps, the effective process matrix (4) is ACKNOWLEDGMENTS
causally separable. Moreover, it is related to the orig-
inal process matrix through the update rule
This work has been supported by the European Com-
W →W = P WP , (8) missionProjectRAQUELandtheAustrianScienceFund
eff (n,m) (n,m)
(FWF) throughSFB FoQuS, and the Individual Project
n,m
X 2462. We also acknowledgethe support of the Swiss Na-
where P = P ⊗P , which is analogous to the von tionalScienceFoundation(SNF)andtheNationalCentre
(n,m) n m
Neumann-Lu¨der rule for the state update under a non- of Competence in Research Quantum Science and Tech-
selective measurement. nology (QSIT).
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