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Quantum correlations with no causal order

Identifieur interne : 000005 ( Pmc/Corpus ); précédent : 000004; suivant : 000006

Quantum correlations with no causal order

Auteurs : Ognyan Oreshkov ; Fabio Costa ; Aslav Brukner

Source :

RBID : PMC:3493644

Abstract

The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.


Url:
DOI: 10.1038/ncomms2076
PubMed: 23033068
PubMed Central: 3493644

Links to Exploration step

PMC:3493644

Le document en format XML

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<label>2</label>
<institution>QuIC, Ecole Polytechnique, CP 165, Université Libre de Bruxelles</institution>
, Brussels 1050,
<country>Belgium</country>
.</aff>
<aff id="a3">
<label>3</label>
<institution>Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences</institution>
, Boltzmanngasse 3, Vienna A-1090,
<country>Austria</country>
.</aff>
</contrib-group>
<author-notes>
<corresp id="c1">
<label>a</label>
<email>oreshkov@ulb.ac.be</email>
</corresp>
</author-notes>
<pub-date pub-type="epub">
<day>02</day>
<month>10</month>
<year>2012</year>
</pub-date>
<volume>3</volume>
<fpage>1092</fpage>
<lpage></lpage>
<history>
<date date-type="received">
<day>29</day>
<month>05</month>
<year>2012</year>
</date>
<date date-type="accepted">
<day>17</day>
<month>08</month>
<year>2012</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright © 2012, Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-statement>
<copyright-year>2012</copyright-year>
<copyright-holder>Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.</copyright-holder>
<license license-type="open-access" xlink:href="http://creativecommons.org/licenses/by-nc-sa/3.0/">
<pmc-comment>author-paid</pmc-comment>
<license-p>This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit
<ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by-nc-sa/3.0/">http://creativecommons.org/licenses/by-nc-sa/3.0/</ext-link>
</license-p>
</license>
</permissions>
<abstract>
<p>The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition.</p>
</abstract>
<abstract abstract-type="web-summary">
<p>
<inline-graphic id="i1" xlink:href="ncomms2076-i1.jpg"></inline-graphic>
Causal order is a concept that is engrained in the standard understanding of time, both in classical and quantum mechanics. Oreshkov
<italic>et al.</italic>
generalize the standard formalism of quantum theory to a framework with no pre-existing causal order, and find a new class of correlations that have no analogue in the classical world.</p>
</abstract>
</article-meta>
</front>
<body>
<p>One of the striking features of quantum mechanics is that it challenges the view that physical properties are well defined before and independent of their measurement. This motivates an operational approach to the theory, where primitive laboratory procedures, such as measurements and preparations, are basic ingredients. Although significant progress has recently been made in this direction
<xref ref-type="bibr" rid="b1">1</xref>
<xref ref-type="bibr" rid="b2">2</xref>
<xref ref-type="bibr" rid="b3">3</xref>
<xref ref-type="bibr" rid="b4">4</xref>
<xref ref-type="bibr" rid="b5">5</xref>
<xref ref-type="bibr" rid="b6">6</xref>
<xref ref-type="bibr" rid="b7">7</xref>
<xref ref-type="bibr" rid="b8">8</xref>
, most approaches still retain a notion of space-time as a pre-existing 'stage' in which events take place. Even the most abstract constructions, in which no explicit reference to space-time is made, do assume a definite order of events: if a signal is sent from an event
<italic>A</italic>
to an event
<italic>B</italic>
in the run of an experiment, no signal can be sent in the opposite direction in that same run. But are space, time and causal order truly fundamental ingredients of nature? Is it possible that, in some circumstances, even causal relations would be 'uncertain', similarly to the way other physical properties of quantum systems are
<xref ref-type="bibr" rid="b9">9</xref>
?</p>
<p>Here, we show that quantum mechanics allows for such a possibility. We develop a framework that describes all correlations that can be observed by two experimenters under the assumption that in their local laboratories physics is described by the standard quantum formalism, but without assuming that the laboratories are embedded in any definite causal structure. These include non-signalling correlations arising from measurements on a bipartite state, as well as signalling ones, which can arise when a system is sent from one laboratory to another through a quantum channel. We find that, surprisingly, more general correlations are possible, which are not included in the standard quantum formalism. These correlations are incompatible with any underlying causal structure: they allow performing a task—the violation of a 'causal inequality'—that is impossible if events take place in a causal sequence. This is directly analogous to the famous violation of local realism: quantum systems allow performing a task—the violation of Bell's inequality
<xref ref-type="bibr" rid="b10">10</xref>
—that is impossible if the measured quantities have pre-defined local values. The inequality considered here, unlike Bell's, concerns signalling correlations: it is based on a task that involves communication between two parties. Nevertheless, it cannot be violated if this communication takes place in a causal space-time. Previous works about relativistic causality in quantum mechanics focused on non-signalling correlations between space-like separated experiments or on a finite speed of signalling
<xref ref-type="bibr" rid="b11">11</xref>
<xref ref-type="bibr" rid="b12">12</xref>
<xref ref-type="bibr" rid="b13">13</xref>
<xref ref-type="bibr" rid="b14">14</xref>
<xref ref-type="bibr" rid="b15">15</xref>
<xref ref-type="bibr" rid="b16">16</xref>
<xref ref-type="bibr" rid="b17">17</xref>
<xref ref-type="bibr" rid="b18">18</xref>
<xref ref-type="bibr" rid="b19">19</xref>
. In the present work, we go beyond such approaches as we do not assume the existence of a space-time (or more generally of a definite causal structure) on which the evolution of quantum systems and the constraints given by relativity are defined. One of the motivations for our approach comes from the problem of time in attempts to merge quantum theory and general relativity into a more fundamental theory
<xref ref-type="bibr" rid="b20">20</xref>
<xref ref-type="bibr" rid="b21">21</xref>
<xref ref-type="bibr" rid="b22">22</xref>
<xref ref-type="bibr" rid="b23">23</xref>
<xref ref-type="bibr" rid="b24">24</xref>
<xref ref-type="bibr" rid="b25">25</xref>
.</p>
<sec disp-level="1" sec-type="results">
<title>Results</title>
<sec disp-level="2">
<title>Causal inequality</title>
<p>The general setting that we consider involves a number of experimenters—Alice, Bob and others—who reside in separate laboratories. At a given run of the experiment, each of them receives a physical system (for instance, a spin-1/2 particle) and performs operations on it (for example, measurements or rotations of the spin), after which she/he sends the system out of the laboratory. We assume that during the operations of each experimenter, the respective laboratory is isolated from the rest of the world—it is only opened for the system to come in and to go out, but between these two events it is kept closed. It is easy to see that, under this assumption, causal order puts a restriction on the way in which the parties can communicate during a given run. For instance, imagine that Alice can send a signal to Bob. (Formally, sending a signal (or signalling) is the existence of statistical correlations between a random variable that can be chosen by the sender and another one observed by the receiver.) As Bob can only receive a signal through the system entering his laboratory, this means that Alice must act on her system before that. But this implies that Bob cannot send a signal to Alice as each party receives a system only once. Therefore, bidirectional signalling is forbidden.</p>
<p>Consider, in particular, the following communication task to be performed by two parties, Alice and Bob. After a given party receives the system in her/his laboratory, she/he will have to toss a coin (or use any other means) to obtain a random bit. Denote the bits generated by Alice and Bob in this way by
<italic>a</italic>
and
<italic>b</italic>
, respectively. In addition, Bob will have to generate another random bit
<italic>b</italic>
<sup>′</sup>
, whose value, 0 or 1, will specify their goal: if
<italic>b</italic>
<sup>′</sup>
=0, Bob will have to communicate the bit
<italic>b</italic>
to Alice, whereas if
<italic>b</italic>
<sup>′</sup>
=1, he will have to guess the bit
<italic>a</italic>
. Without loss of generality, we will assume that the parties always produce a guess, denoted by
<italic>x</italic>
and
<italic>y</italic>
for Alice and Bob, respectively, for the bit of the other (although the guess may not count depending on the value of
<italic>b</italic>
<sup>′</sup>
). Their goal is to maximize the probability of success</p>
<p>
<disp-formula id="eq1">
<label>1</label>
<inline-graphic id="d32e213" xlink:href="ncomms2076-m1.jpg"></inline-graphic>
</disp-formula>
</p>
<p>If all events obey causal order, no strategy can allow Alice and Bob to exceed the bound</p>
<p>
<disp-formula id="eq2">
<label>2</label>
<inline-graphic id="d32e218" xlink:href="ncomms2076-m2.jpg"></inline-graphic>
</disp-formula>
</p>
<p>Indeed, as argued above, in any particular order of events, there can be at most unidirectional signalling between the parties, which means that at least one of the following must be true: Alice cannot signal to Bob or Bob cannot signal to Alice. Consider, for example, a case where Bob cannot signal to Alice. Then, if
<italic>b</italic>
<sup>−</sup>
=1, they could in principle achieve up to
<italic>P</italic>
(
<italic>y</italic>
=
<italic>a</italic>
|
<italic>b′</italic>
<sup>−</sup>
=1)=1 (for instance, if Alice operates on her system before Bob, she could encode information about the bit
<italic>a</italic>
in the system and send it to him). However, if
<italic>b′</italic>
<sup>−</sup>
=0, the best guess that Alice can make is a random one, resulting in
<italic>P</italic>
(
<italic>x</italic>
=
<italic>b</italic>
|
<italic>b′</italic>
<sup>−</sup>
=0)=1/2 (see
<xref ref-type="fig" rid="f1">Fig. 1a</xref>
). Hence, the overall probability of success in this case will satisfy
<italic>p</italic>
<sub>succ</sub>
≤3/4. The same holds if Alice cannot signal to Bob. It is easy to see that no probabilistic strategy can increase the probability of success.</p>
<p>Formally, the assumptions behind the causal inequality (2) can be summarized as follows:</p>
<sec disp-level="3">
<title>Causal structure</title>
<p>The main events in the task (a system entering Alice's/Bob's laboratory, the parties obtaining the bits
<italic>a</italic>
,
<italic>b</italic>
and
<italic>b</italic>
′, and producing the guesses
<italic>x</italic>
and
<italic>y</italic>
) are localized in a causal structure. A causal structure (such as space-time) is a set of event locations equipped with a partial order
<inline-formula id="d32e295">
<inline-graphic id="d32e296" xlink:href="ncomms2076-m149.jpg"></inline-graphic>
</inline-formula>
that defines the possible directions of signalling. If
<italic>A</italic>
<inline-formula id="d32e300">
<inline-graphic id="d32e301" xlink:href="ncomms2076-m149.jpg"></inline-graphic>
</inline-formula>
<italic>B</italic>
, we say that
<italic>A</italic>
is in the causal past of
<italic>B</italic>
(or
<italic>B</italic>
is in the causal future of
<italic>A</italic>
). In this case, signalling from
<italic>A</italic>
to
<italic>B</italic>
is possible, but not from
<italic>B</italic>
to
<italic>A</italic>
. For more details on causal structures, see
<xref ref-type="supplementary-material" rid="S1">Supplementary Methods</xref>
.</p>
</sec>
<sec disp-level="3">
<title>Free choice</title>
<p>Each of the bits
<italic>a</italic>
,
<italic>b</italic>
and
<italic>b</italic>
can only be correlated with events in its causal future (this concerns only events relevant to the task). We assume also that each of them takes values 0 or 1 with probability 1/2.</p>
</sec>
<sec disp-level="3">
<title>Closed laboratories</title>
<p>Alice's guess
<italic>x</italic>
can be correlated with Bob's bit
<italic>b</italic>
only if the latter is generated in the causal past of the system entering Alice's laboratory. Analogously,
<italic>y</italic>
can be correlated with
<italic>a</italic>
only if
<italic>a</italic>
is generated in the causal past of the system entering Bob's laboratory.</p>
<p>In the
<xref ref-type="supplementary-material" rid="S1">Supplementary Methods</xref>
, we present a formal derivation of the inequality from these assumptions.Interestingly, we find that if the local laboratories are described by quantum mechanics, but no assumption about a global causal structure is made (
<xref ref-type="fig" rid="f1">Fig. 1b</xref>
), it is in principle possible to violate the causal inequality in physical situations in which one would have all the reasons to believe that the bits are chosen freely and the laboratories are closed. This would imply that the assumption
<italic>Causal structure</italic>
does not hold.</p>
</sec>
</sec>
<sec disp-level="2">
<title>Framework for local quantum mechanics</title>
<p>The most studied, almost epitomical, quantum correlations are the non-signalling ones, such as those obtained when Alice and Bob perform measurements on two entangled systems. Signalling quantum correlations exist as well, such as those arising when Alice operates on a system that is subsequently sent through a quantum channel to Bob who operates on it after that. The usual quantum formalism does not consider more general possibilities, as it does assume a global causal structure. Here, we want to drop the latter assumption while retaining the validity of quantum mechanics locally. For this purpose, we consider a multipartite setting of the type outlined earlier, where each party performs an operation on a system passing once through her/his laboratory, but we make no assumption about the spatio-temporal location of these experiments, not even that there exists a space-time or any causal structure in which they could be positioned (see
<xref ref-type="fig" rid="f2">Fig. 2</xref>
). Our framework is thus based on the central premise of local quantum mechanics, which is to say the local operations of each party are described by quantum mechanics.</p>
<p>More specifically, we assume that one party, say Alice, can perform all the operations she could perform in a closed laboratory, as described in the standard space-time formulation of quantum mechanics. These are defined as the set of quantum instruments
<xref ref-type="bibr" rid="b26">26</xref>
with an input Hilbert space
<inline-formula id="d32e391">
<inline-graphic id="d32e392" xlink:href="ncomms2076-m3.jpg"></inline-graphic>
</inline-formula>
(the system coming in) and an output Hilbert space
<inline-formula id="d32e394">
<inline-graphic id="d32e395" xlink:href="ncomms2076-m4.jpg"></inline-graphic>
</inline-formula>
(the system going out). (The set of allowed quantum operations can be used as a definition of 'closed quantum laboratory' with no reference to a global causal structure.) A quantum instrument can most generally be realized by applying a joint unitary transformation on the input system plus an ancilla, followed by a projective measurement on part of the resulting joint system, which leaves the other part as an output. (From the point of view of each party, the input/output systems most generally correspond to two subsystems of the Hilbert space associated with the local laboratory, each considered at a different instant—the time of entrance and the time of exit, respectively—where the subsystems and the respective instants are independent of the choice of operation that connects them.) When Alice uses a given instrument, she registers one out of a set of possible outcomes, labelled by
<italic>j</italic>
=1,...,
<italic>n</italic>
. Each outcome induces a specific transformation from the input to the output, which corresponds to a completely positive (CP) trace-non-increasing map
<xref ref-type="bibr" rid="b27">27</xref>
<inline-formula id="d32e406">
<inline-graphic id="d32e407" xlink:href="ncomms2076-m5.jpg"></inline-graphic>
</inline-formula>
, where
<inline-formula id="d32e409">
<inline-graphic id="d32e410" xlink:href="ncomms2076-m6.jpg"></inline-graphic>
</inline-formula>
, is the space of matrices over a Hilbert space
<inline-formula id="d32e412">
<inline-graphic id="d32e413" xlink:href="ncomms2076-m7.jpg"></inline-graphic>
</inline-formula>
of dimension
<italic>d</italic>
<sub>
<italic>X</italic>
</sub>
. The action of each
<inline-formula id="d32e422">
<inline-graphic id="d32e423" xlink:href="ncomms2076-m8.jpg"></inline-graphic>
</inline-formula>
on any matrix
<inline-formula id="d32e425">
<inline-graphic id="d32e426" xlink:href="ncomms2076-m9.jpg"></inline-graphic>
</inline-formula>
can be written as
<xref ref-type="bibr" rid="b27">27</xref>
<inline-formula id="d32e430">
<inline-graphic id="d32e431" xlink:href="ncomms2076-m10.jpg"></inline-graphic>
</inline-formula>
, where the matrices
<inline-formula id="d32e433">
<inline-graphic id="d32e434" xlink:href="ncomms2076-m11.jpg"></inline-graphic>
</inline-formula>
satisfy
<inline-formula id="d32e436">
<inline-graphic id="d32e437" xlink:href="ncomms2076-m12.jpg"></inline-graphic>
</inline-formula>
, ∀
<italic>j</italic>
. If the operation is performed on a quantum state described by a density matrix ρ,
<inline-formula id="d32e443">
<inline-graphic id="d32e444" xlink:href="ncomms2076-m13.jpg"></inline-graphic>
</inline-formula>
describes the updated state after the outcome
<italic>j</italic>
up to normalization, whereas the probability to observe this outcome is given by
<inline-formula id="d32e449">
<inline-graphic id="d32e450" xlink:href="ncomms2076-m14.jpg"></inline-graphic>
</inline-formula>
. The set of CP maps
<inline-formula id="d32e452">
<inline-graphic id="d32e453" xlink:href="ncomms2076-m15.jpg"></inline-graphic>
</inline-formula>
corresponding to all the possible outcomes of a quantum instrument has the property that
<inline-formula id="d32e455">
<inline-graphic id="d32e456" xlink:href="ncomms2076-m16.jpg"></inline-graphic>
</inline-formula>
is CP and trace-preserving (CPTP) or equivalently
<inline-formula id="d32e459">
<inline-graphic id="d32e460" xlink:href="ncomms2076-m17.jpg"></inline-graphic>
</inline-formula>
, which reflects the fact that the probability to observe any of the possible outcomes is unity. A CPTP map itself corresponds to an instrument with a single outcome that occurs with certainty.</p>
<p>In the case of more than one party, the set of local outcomes corresponds to a set of CP maps
<inline-formula id="d32e464">
<inline-graphic id="d32e465" xlink:href="ncomms2076-m18.jpg"></inline-graphic>
</inline-formula>
. A complete list of probabilities
<inline-formula id="d32e467">
<inline-graphic id="d32e468" xlink:href="ncomms2076-m19.jpg"></inline-graphic>
</inline-formula>
for all possible local outcomes will be called process. (It is implicitly assumed that the joint probabilities are non-contextual, namely that they are independent of any variable concerning the concrete implementation of the local CP maps. For example, the probability for a pair of maps
<inline-formula id="d32e470">
<inline-graphic id="d32e471" xlink:href="ncomms2076-m20.jpg"></inline-graphic>
</inline-formula>
to be realized should not depend on the particular set
<inline-formula id="d32e473">
<inline-graphic id="d32e474" xlink:href="ncomms2076-m21.jpg"></inline-graphic>
</inline-formula>
of possible CP maps associated with Alice's operation.) A process can be seen as an extension of the notion of state as a list of probabilities for detection results
<xref ref-type="bibr" rid="b3">3</xref>
described by a positive operator-valued measure (POVM), which takes into account the transformation of the system after the measurement and can thus capture more general scenarios than just detection. Here, we will consider explicitly only the case of two parties (the generalization to arbitrarily many parties is straightforward). We want to characterize the most general probability distributions for a pair of outcomes
<italic>i</italic>
,
<italic>j</italic>
, corresponding to CP maps
<inline-formula id="d32e485">
<inline-graphic id="d32e486" xlink:href="ncomms2076-m22.jpg"></inline-graphic>
</inline-formula>
, to be observed, that is, to characterize all bipartite processes.</p>
<p>In quantum mechanics, operations obey a specific algebraic structure that reflects the operational relations between laboratory procedures
<xref ref-type="bibr" rid="b3">3</xref>
. For example, a probabilistic mixture of operations is expressed as a linear convex combination of CP maps. It can be shown (see Methods) that the only probabilities
<inline-formula id="d32e492">
<inline-graphic id="d32e493" xlink:href="ncomms2076-m23.jpg"></inline-graphic>
</inline-formula>
consistent with the algebraic structure of local quantum operations are bilinear functions of the CP maps
<inline-formula id="d32e495">
<inline-graphic id="d32e496" xlink:href="ncomms2076-m24.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e498">
<inline-graphic id="d32e499" xlink:href="ncomms2076-m25.jpg"></inline-graphic>
</inline-formula>
. Thus, the study of the most general bipartite quantum correlations reduces to the study of bilinear functions of CP maps.</p>
<p>It is convenient to represent CP maps by positive semi-definite matrices via the Choi-Jamiołkowsky (CJ) isomorphism
<xref ref-type="bibr" rid="b28">28</xref>
<xref ref-type="bibr" rid="b29">29</xref>
. The CJ matrix
<inline-formula id="d32e505">
<inline-graphic id="d32e506" xlink:href="ncomms2076-m26.jpg"></inline-graphic>
</inline-formula>
coresponding to a linear map
<inline-formula id="d32e508">
<inline-graphic id="d32e509" xlink:href="ncomms2076-m27.jpg"></inline-graphic>
</inline-formula>
is defined as
<inline-formula id="d32e511">
<inline-graphic id="d32e512" xlink:href="ncomms2076-m28.jpg"></inline-graphic>
</inline-formula>
, where
<inline-formula id="d32e514">
<inline-graphic id="d32e515" xlink:href="ncomms2076-m29.jpg"></inline-graphic>
</inline-formula>
is a (not normalized) maximally entangled state, the set of states
<inline-formula id="d32e518">
<inline-graphic id="d32e519" xlink:href="ncomms2076-m30.jpg"></inline-graphic>
</inline-formula>
is an orthonormal basis of
<inline-formula id="d32e521">
<inline-graphic id="d32e522" xlink:href="ncomms2076-m31.jpg"></inline-graphic>
</inline-formula>
is the identity map and T denotes matrix transposition (the transposition, absent in the original definition, is introduced for later convenience). Using this correspondence, the probability for two measurement outcomes can be expressed as a bilinear function of the corresponding CJ operators as follows:</p>
<p>
<disp-formula id="eq3">
<label>3</label>
<inline-graphic id="d32e526" xlink:href="ncomms2076-m32.jpg"></inline-graphic>
</disp-formula>
</p>
<p>where
<inline-formula id="d32e529">
<inline-graphic id="d32e530" xlink:href="ncomms2076-m33.jpg"></inline-graphic>
</inline-formula>
is a matrix in
<inline-formula id="d32e532">
<inline-graphic id="d32e533" xlink:href="ncomms2076-m34.jpg"></inline-graphic>
</inline-formula>
.</p>
<p>The matrix
<italic>W</italic>
should be such that probabilities are non-negative for any pair of CP maps
<inline-formula id="d32e540">
<inline-graphic id="d32e541" xlink:href="ncomms2076-m35.jpg"></inline-graphic>
</inline-formula>
. We require that this be true also for measurements in which the system interacts with any system in the local laboratory, including systems entangled with the other laboratory. This implies that
<inline-formula id="d32e543">
<inline-graphic id="d32e544" xlink:href="ncomms2076-m36.jpg"></inline-graphic>
</inline-formula>
must be positive semidefinite (see Methods). Furthermore, the probability for any pair of CPTP maps
<inline-formula id="d32e546">
<inline-graphic id="d32e547" xlink:href="ncomms2076-m37.jpg"></inline-graphic>
</inline-formula>
to be realized must be unity (they correspond to instruments with a single outcome). As a map
<inline-formula id="d32e549">
<inline-graphic id="d32e550" xlink:href="ncomms2076-m38.jpg"></inline-graphic>
</inline-formula>
is CPTP if and only if its CJ operator satisfies
<inline-formula id="d32e553">
<inline-graphic id="d32e554" xlink:href="ncomms2076-m39.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e556">
<inline-graphic id="d32e557" xlink:href="ncomms2076-m40.jpg"></inline-graphic>
</inline-formula>
(similarly for
<inline-formula id="d32e559">
<inline-graphic id="d32e560" xlink:href="ncomms2076-m41.jpg"></inline-graphic>
</inline-formula>
), we conclude that all bipartite probabilities compatible with local quantum mechanics are generated by matrices
<italic>W</italic>
that satisfy</p>
<p>
<disp-formula id="eq4">
<label>4</label>
<inline-graphic id="d32e567" xlink:href="ncomms2076-m42.jpg"></inline-graphic>
</disp-formula>
</p>
<p>
<disp-formula id="eq5">
<label>5</label>
<inline-graphic id="d32e570" xlink:href="ncomms2076-m43.jpg"></inline-graphic>
</disp-formula>
</p>
<p>We will refer to a matrix
<inline-formula id="d32e574">
<inline-graphic id="d32e575" xlink:href="ncomms2076-m44.jpg"></inline-graphic>
</inline-formula>
that satisfies these conditions as a process matrix. Conditions equivalent to equations (4) and (5) were first derived as part of the definition of a 'quantum comb'
<xref ref-type="bibr" rid="b30">30</xref>
, an object that formalizes quantum networks. Combs, however, are subject to additional conditions fixing a definite causal order, which are not assumed here.</p>
<p>A process matrix can be understood as a generalization of a density matrix and equation (3) can be seen as a generalization of Born's rule. In fact, when the output systems
<italic>A</italic>
<sub>2</sub>
,
<italic>B</italic>
<sub>2</sub>
are taken to be one-dimensional (that is, each party performs a measurement after which the system is discarded), the expression above reduces to
<inline-formula id="d32e591">
<inline-graphic id="d32e592" xlink:href="ncomms2076-m45.jpg"></inline-graphic>
</inline-formula>
, where now
<inline-formula id="d32e594">
<inline-graphic id="d32e595" xlink:href="ncomms2076-m46.jpg"></inline-graphic>
</inline-formula>
are elements of local POVMs and
<inline-formula id="d32e598">
<inline-graphic id="d32e599" xlink:href="ncomms2076-m47.jpg"></inline-graphic>
</inline-formula>
is a quantum state. This implies that a quantum state
<inline-formula id="d32e601">
<inline-graphic id="d32e602" xlink:href="ncomms2076-m48.jpg"></inline-graphic>
</inline-formula>
shared by Alice and Bob is generally represented by the process matrix
<inline-formula id="d32e604">
<inline-graphic id="d32e605" xlink:href="ncomms2076-m49.jpg"></inline-graphic>
</inline-formula>
. Signalling correlations can also be expressed in terms of process matrices. For instance, the situation where Bob is given a state
<inline-formula id="d32e607">
<inline-graphic id="d32e608" xlink:href="ncomms2076-m50.jpg"></inline-graphic>
</inline-formula>
and his output is sent to Alice through a quantum channel
<inline-formula id="d32e610">
<inline-graphic id="d32e611" xlink:href="ncomms2076-m51.jpg"></inline-graphic>
</inline-formula>
, which gives
<inline-formula id="d32e613">
<inline-graphic id="d32e614" xlink:href="ncomms2076-m52.jpg"></inline-graphic>
</inline-formula>
, is described by
<inline-formula id="d32e617">
<inline-graphic id="d32e618" xlink:href="ncomms2076-m53.jpg"></inline-graphic>
</inline-formula>
, where
<inline-formula id="d32e620">
<inline-graphic id="d32e621" xlink:href="ncomms2076-m54.jpg"></inline-graphic>
</inline-formula>
is the CJ matrix of the channel
<inline-formula id="d32e623">
<inline-graphic id="d32e624" xlink:href="ncomms2076-m55.jpg"></inline-graphic>
</inline-formula>
from
<italic>B</italic>
<sub>2</sub>
to
<italic>A</italic>
<sub>1</sub>
.</p>
<p>The most general bipartite situation typically encountered in quantum mechanics (that is, one that can be expressed in terms of a quantum circuit) is a quantum channel with memory, where, say, Bob operates on one part of an entangled state and his output plus the other part is transferred to Alice through a channel. This is described by a process matrix of the form
<inline-formula id="d32e638">
<inline-graphic id="d32e639" xlink:href="ncomms2076-m56.jpg"></inline-graphic>
</inline-formula>
. Conversely, all process matrices of this form represent channels with memory
<xref ref-type="bibr" rid="b30">30</xref>
. This is the most general situation in which signalling from Alice to Bob is not possible, a relation that we will denote by
<inline-formula id="d32e643">
<inline-graphic id="d32e644" xlink:href="ncomms2076-m57.jpg"></inline-graphic>
</inline-formula>
in accord with the causal notation introduced earlier. Process matrices of this kind will be denoted by
<inline-formula id="d32e646">
<inline-graphic id="d32e647" xlink:href="ncomms2076-m58.jpg"></inline-graphic>
</inline-formula>
(note that for non-signalling processes, both
<inline-formula id="d32e649">
<inline-graphic id="d32e650" xlink:href="ncomms2076-m59.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e653">
<inline-graphic id="d32e654" xlink:href="ncomms2076-m60.jpg"></inline-graphic>
</inline-formula>
are true). As argued earlier, if all events are localized in a causal structure, and Alice and Bob perform their experiments inside closed laboratories, at most unidirectional signalling between the laboratories is allowed. In a definite causal structure, it may still be the case that the location of each event, and thus the causal relation between events, is not known with certainty. A situation where
<inline-formula id="d32e656">
<inline-graphic id="d32e657" xlink:href="ncomms2076-m61.jpg"></inline-graphic>
</inline-formula>
with probability 0≤
<italic>q</italic>
≤1 and
<inline-formula id="d32e662">
<inline-graphic id="d32e663" xlink:href="ncomms2076-m62.jpg"></inline-graphic>
</inline-formula>
with probability 1−
<italic>q</italic>
is represented by a process matrix of the form</p>
<p>
<disp-formula id="eq6">
<label>6</label>
<inline-graphic id="d32e670" xlink:href="ncomms2076-m63.jpg"></inline-graphic>
</disp-formula>
</p>
<p>We will call the processes of this kind causally separable (note that the decomposition (6) need not be unique as non-signalling processes can be included either in
<inline-formula id="d32e673">
<inline-graphic id="d32e674" xlink:href="ncomms2076-m64.jpg"></inline-graphic>
</inline-formula>
or in
<inline-formula id="d32e676">
<inline-graphic id="d32e677" xlink:href="ncomms2076-m65.jpg"></inline-graphic>
</inline-formula>
). They represent the most general bipartite quantum processes for which the local experiments are performed in closed laboratories embedded in a definite causal structure. In particular, they generate the most general quantum correlations between measurements that take place at definite (though possibly unknown) instants of time. Clearly, according to the argument presented earlier, causally separable processes cannot be used by Alice an Bob to violate the causal inequality (2).</p>
<p>In the
<xref ref-type="supplementary-material" rid="S1">Supplementary Methods</xref>
, we provide a complete characterization of process matrices via the terms allowed in their expansion in a Hilbert–Schmidt basis, which we relate to the possible directions of signalling they allow (see
<xref ref-type="fig" rid="f3">Fig. 3</xref>
). We also provide possible interpretations of the terms that are not allowed in a process matrix (see
<xref ref-type="fig" rid="f4">Fig. 4</xref>
and
<xref ref-type="supplementary-material" rid="S1">Supplementary Fig. S1</xref>
).</p>
</sec>
<sec disp-level="2">
<title>A causally non-separable process</title>
<p>The question whether all local quantum experiments can be embedded in a global causal structure corresponds to the question whether all process matrices are causally separable. Note that this is not a question about entanglement: all possible entangled states, and more generally all quantum circuits, correspond to matrices of the form
<inline-formula id="d32e698">
<inline-graphic id="d32e699" xlink:href="ncomms2076-m66.jpg"></inline-graphic>
</inline-formula>
or
<inline-formula id="d32e701">
<inline-graphic id="d32e702" xlink:href="ncomms2076-m67.jpg"></inline-graphic>
</inline-formula>
, whereas the non-separable processes we are looking for cannot be written as quantum circuits or even as probabilistic mixtures of different circuits. Surprisingly, an example of such a kind exists. Consider the process matrix</p>
<p>
<disp-formula id="eq7">
<label>7</label>
<inline-graphic id="d32e706" xlink:href="ncomms2076-m68.jpg"></inline-graphic>
</disp-formula>
</p>
<p>where
<italic>A</italic>
<sub>1</sub>
,
<italic>A</italic>
<sub>2</sub>
,
<italic>B</italic>
<sub>1</sub>
and
<italic>B</italic>
<sub>2</sub>
are two-level systems (for example, the spin degrees of freedom of a spin-
<inline-formula id="d32e730">
<inline-graphic id="d32e731" xlink:href="ncomms2076-m69.jpg"></inline-graphic>
</inline-formula>
particle) and σ
<sub>
<italic>x</italic>
</sub>
and σ
<sub>
<italic>z</italic>
</sub>
are the Pauli spin matrices. It can be verified straightforwardly that conditions (4) and (5) are satisfied, hence (7) is a valid bipartite process. Having such a resource, Alice and Bob can play the game described above and exceed the bound on the probability of success (2) imposed by causal order. Indeed, if Bob measures in the
<italic>z</italic>
basis and detects one of the states |
<italic>z</italic>
<sub>±</sub>
〉, the corresponding CJ operator contains the factor
<inline-formula id="d32e750">
<inline-graphic id="d32e751" xlink:href="ncomms2076-m70.jpg"></inline-graphic>
</inline-formula>
. Inserting this, together with equation (7), into the expression (3) for the probabilities, the term containing
<inline-formula id="d32e753">
<inline-graphic id="d32e754" xlink:href="ncomms2076-m71.jpg"></inline-graphic>
</inline-formula>
in the process matrix is annihilated and what remains corresponds to a noisy channel from Alice to Bob. If Alice encodes her bit in the
<italic>z</italic>
basis with the CJ operator
<inline-formula id="d32e759">
<inline-graphic id="d32e760" xlink:href="ncomms2076-m72.jpg"></inline-graphic>
</inline-formula>
, this channel allows Bob to guess Alice's bit with probability
<inline-formula id="d32e762">
<inline-graphic id="d32e763" xlink:href="ncomms2076-m73.jpg"></inline-graphic>
</inline-formula>
. If, on the other hand, Bob measures in the
<italic>x</italic>
basis, equation (7) is reduced to a similar noisy channel from Bob to Alice. Bob is thus able to activate a channel in the desired direction by choosing the measurement basis (see Methods for a detailed calculation and analysis of the protocol). In this way they can achieve</p>
<p>
<disp-formula id="eq8">
<label>8</label>
<inline-graphic id="d32e771" xlink:href="ncomms2076-m74.jpg"></inline-graphic>
</disp-formula>
</p>
<p>which proves that (7) is not causally separable. We see that, depending on his choice, Bob can effectively end up 'before' or 'after' Alice, each possibility with a probability
<inline-formula id="d32e774">
<inline-graphic id="d32e775" xlink:href="ncomms2076-m75.jpg"></inline-graphic>
</inline-formula>
. This is remarkable, because if Alice and Bob perform their experiments inside laboratories that they believe are isolated from the outside world for the duration of their operations (for example, by walls made of impenetrable material), and if they believe that they are able to freely choose the bits
<italic>a</italic>
,
<italic>b</italic>
and
<italic>b</italic>
' (for example, by tossing a coin), they will have to conclude that the events in their experiment do not take place in a causal sequence. Indeed, the framework only assumes that the local operations from the input to the output system of each party are correctly described by quantum mechanics, and it is compatible with any physical situation in which one would have all the reasons to believe that each party's operations are freely chosen in a closed laboratory.</p>
<p>Interestingly, both the classical bound (2) and the quantum violation (8) match the corresponding numbers in the CHSH-Bell inequality
<xref ref-type="bibr" rid="b31">31</xref>
, which strongly resembles inequality (2). However, the physical situations to which these inequalities correspond is very different: Bell inequalities can be violated in space-like separated laboratories, while (8) cannot be achieved neither with space-like nor with time-like separated laboratories. It is an open question whether (8) is the maximal possible violation allowed by quantum mechanics.</p>
</sec>
<sec disp-level="2">
<title>Classical processes are causally separable</title>
<p>It is not difficult to see that if the operations of the local parties are classical, they can always be understood as taking place in a global causal structure. Classical operations can be described by transition matrices
<inline-formula id="d32e795">
<inline-graphic id="d32e796" xlink:href="ncomms2076-m76.jpg"></inline-graphic>
</inline-formula>
, where
<italic>P</italic>
(λ
<sub>2</sub>
,
<italic>j</italic>
|λ
<sub>1</sub>
) is the conditional probability that the measurement outcome
<italic>j</italic>
is observed and the classical output state λ
<sub>2</sub>
is prepared given that the input state is λ
<sub>1</sub>
. They can be expressed in the quantum formalism as CP maps diagonal in a fixed ('pointer') basis, and the corresponding CJ operators are
<inline-formula id="d32e820">
<inline-graphic id="d32e821" xlink:href="ncomms2076-m77.jpg"></inline-graphic>
</inline-formula>
. Thus, to express arbitrary bipartite probabilities of classical maps, it is sufficient to consider process matrices that are diagonal in the pointer basis. In the Supplementary Methods, we provide a detailed proof that all such processes are causally separable.</p>
</sec>
</sec>
<sec disp-level="1" sec-type="discussion">
<title>Discussion</title>
<p>We have seen that by relaxing the assumption of definite global causal order and requiring that the standard quantum formalism holds only locally, we obtain the possibility for global causal relations that are not included in the usual formulation of quantum mechanics. The latter is reminiscent of the situation in general relativity, where by requiring that locally the geometry is that of flat Minkowski space-time, one obtains the possibility of having more general, curved space-times.</p>
<p>The natural question is whether 'non-causal' quantum correlations of the kind described by our formalism can be found in nature. One can speculate that they may exist in unprobed physical regimes, such as, for example, those in which quantum mechanics and general relativity become relevant. Indeed, our result that classical theories can always be understood in terms of a global causal structure suggests the possibility that the observed causal order of space-time might not be a fundamental property of nature but rather emerge from a more fundamental theory
<xref ref-type="bibr" rid="b32">32</xref>
<xref ref-type="bibr" rid="b33">33</xref>
<xref ref-type="bibr" rid="b34">34</xref>
in a quantum-to-classical transition due to, for example, decoherence
<xref ref-type="bibr" rid="b35">35</xref>
or coarse-grained measurements
<xref ref-type="bibr" rid="b36">36</xref>
. Once a causal structure is present, it is possible to derive relativistic space-time from it under appropriate conditions
<xref ref-type="bibr" rid="b37">37</xref>
<xref ref-type="bibr" rid="b38">38</xref>
. Furthermore, as the conformal space-time metric is a description of the causal relation between space-time points
<xref ref-type="bibr" rid="b39">39</xref>
<xref ref-type="bibr" rid="b40">40</xref>
, one can expect that an extension of general relativity to the quantum domain would involve situations where different causal orders could coexist 'in superposition'. The formalism we presented may offer a natural route in this direction: based only on the assumption that quantum mechanics is valid locally, it yields causal relations that cannot be understood as arising from a definite, underlying order.</p>
<p>It is also worth noting that exotic causal structures already appear in the classical theory of general relativity. For example, there exist solutions to the Einstein equation containing closed time-like curves (CTCs)
<xref ref-type="bibr" rid="b41">41</xref>
. In this context, it should be noted that any process matrix
<italic>W</italic>
in our framework can be interpreted as a CPTP map from the outputs,
<italic>A</italic>
<sub>2</sub>
,
<italic>B</italic>
<sub>2</sub>
, of the parties, to their inputs,
<italic>A</italic>
<sub>1</sub>
,
<italic>B</italic>
<sub>1</sub>
. In other words, any process can be thought of as having the form of a CTC, where information is sent back in time through a noisy channel (see also
<xref ref-type="fig" rid="f1">Fig. 1b</xref>
). The existence of processes that do not describe definite causal order is therefore not incompatible with general relativity in principle. It is sometimes argued that CTCs should not exist as they generate logical paradoxes, such as an agent going back in time and killing his grandfather. The possible solutions that have been proposed
<xref ref-type="bibr" rid="b42">42</xref>
<xref ref-type="bibr" rid="b43">43</xref>
<xref ref-type="bibr" rid="b44">44</xref>
<xref ref-type="bibr" rid="b45">45</xref>
<xref ref-type="bibr" rid="b46">46</xref>
<xref ref-type="bibr" rid="b47">47</xref>
, in which quantum mechanics and CTCs might coexist, involve non-linear extensions of quantum theory that deviate from quantum mechanics already at the level of local experiments. Our framework, on the other hand, is by construction linear and in agreement with local quantum mechanics, and yet paradoxes are avoided, in accordance with the Novikov principle
<xref ref-type="bibr" rid="b48">48</xref>
, due to the noise in the evolution 'backward in time'.</p>
<p>Finally, we remark that instances of indefinite causal orders may also emerge in situations closer to possible laboratory implementations. As already noted, our formalism describes more general correlations than those that can be realized with a quantum circuit, that is, as a sequence of quantum gates. Recently, a new model of quantum computation that goes beyond the causal paradigm of quantum circuits by using superpositions of the 'wires' connecting different gates was proposed
<xref ref-type="bibr" rid="b49">49</xref>
. This possibility may allow breaking the assumption that events are localized in a causal structure. As the instant when a system enters a device depends on how the device is wired with the rest of the computer's architecture, superpositions of wires may allow creating situations in which events are not localized in time (similarly to the way in which a quantum particle may not be localized in space). Although it is an open question whether violating the causal inequality (2) can be achieved by similar means, the present work suggests that new quantum resources for information processing might be available—beyond entanglement, quantum memories and even 'superpositions of wires'—and the formalism introduced provides a natural framework for exploring them.</p>
</sec>
<sec disp-level="1" sec-type="methods">
<title>Methods</title>
<sec disp-level="2">
<title>Definition of process matrices</title>
<p>In this section, we will derive the linear representation (3), as well as the conditions (4) and (5) that a process matrix has to satisfy.</p>
<sec disp-level="3">
<title>Linearity of probabilities</title>
<p>A quantum instrument
<xref ref-type="bibr" rid="b26">26</xref>
is defined as a set
<inline-formula id="d32e894">
<inline-graphic id="d32e895" xlink:href="ncomms2076-m78.jpg"></inline-graphic>
</inline-formula>
of CP maps such that
<inline-formula id="d32e897">
<inline-graphic id="d32e898" xlink:href="ncomms2076-m79.jpg"></inline-graphic>
</inline-formula>
is a CPTP map. Our main assumption is that the description of the operations in the individual laboratories is in agreement with quantum mechanics. In particular, we derive linearity from the quantum mechanical representation of probabilistic mixtures and of coarse-graining of operations. Consider first an instrument
<inline-formula id="d32e900">
<inline-graphic id="d32e901" xlink:href="ncomms2076-m80.jpg"></inline-graphic>
</inline-formula>
defined as the randomization of two different instruments
<inline-formula id="d32e903">
<inline-graphic id="d32e904" xlink:href="ncomms2076-m81.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e907">
<inline-graphic id="d32e908" xlink:href="ncomms2076-m82.jpg"></inline-graphic>
</inline-formula>
, where the first is performed with probability
<italic>p</italic>
and the second with probability (1−
<italic>p</italic>
). The probability to observe the outcome
<italic>j</italic>
is, by definition,
<inline-formula id="d32e919">
<inline-graphic id="d32e920" xlink:href="ncomms2076-m83.jpg"></inline-graphic>
</inline-formula>
. In quantum mechanics, randomization is described as a convex linear combination,
<inline-formula id="d32e922">
<inline-graphic id="d32e923" xlink:href="ncomms2076-m84.jpg"></inline-graphic>
</inline-formula>
. We can then conclude that the probability must respect linear convex combinations:
<inline-formula id="d32e926">
<inline-graphic id="d32e927" xlink:href="ncomms2076-m85.jpg"></inline-graphic>
</inline-formula>
. Consider then the coarse-graining of an instrument
<inline-formula id="d32e929">
<inline-graphic id="d32e930" xlink:href="ncomms2076-m86.jpg"></inline-graphic>
</inline-formula>
. This is realized when two or more outcomes, for example, those corresponding to the labels
<italic>j</italic>
=
<italic>n</italic>
−1 and
<italic>j</italic>
=
<italic>n</italic>
, are treated as a single one. In the resulting instrument
<inline-formula id="d32e945">
<inline-graphic id="d32e946" xlink:href="ncomms2076-m87.jpg"></inline-graphic>
</inline-formula>
, all non-coarse-grained outcomes correspond to the original CP maps
<inline-formula id="d32e948">
<inline-graphic id="d32e949" xlink:href="ncomms2076-m88.jpg"></inline-graphic>
</inline-formula>
, whereas the probability of the coarse-grained outcome is given by
<inline-formula id="d32e951">
<inline-graphic id="d32e952" xlink:href="ncomms2076-m89.jpg"></inline-graphic>
</inline-formula>
. In quantum mechanics, the CP map corresponding to the coarse graining of two outcomes is represented by the sum of the respective CP maps,
<inline-formula id="d32e954">
<inline-graphic id="d32e955" xlink:href="ncomms2076-m90.jpg"></inline-graphic>
</inline-formula>
, from which it follows that
<inline-formula id="d32e957">
<inline-graphic id="d32e958" xlink:href="ncomms2076-m91.jpg"></inline-graphic>
</inline-formula>
. Randomization and coarse graining together impose linearity. The argument can be repeated for two (or more) parties, yielding the conclusion that all bipartite probabilities compatible with a local quantum mechanical description are bilinear functions,
<inline-formula id="d32e960">
<inline-graphic id="d32e961" xlink:href="ncomms2076-m92.jpg"></inline-graphic>
</inline-formula>
, of the local CP and trace-non-increasing maps
<inline-formula id="d32e964">
<inline-graphic id="d32e965" xlink:href="ncomms2076-m93.jpg"></inline-graphic>
</inline-formula>
.</p>
<p>Thanks to the CJ isomorphism, it is possible to represent bilinear functions of CP maps as bilinear functions of matrices:
<inline-formula id="d32e969">
<inline-graphic id="d32e970" xlink:href="ncomms2076-m94.jpg"></inline-graphic>
</inline-formula>
. In general, multilinear functions on a set of vector spaces
<italic>V</italic>
<sup>1</sup>
×
<italic>V</italic>
<sup>2</sup>
×... are isomorphic to linear functions on
<inline-formula id="d32e982">
<inline-graphic id="d32e983" xlink:href="ncomms2076-m95.jpg"></inline-graphic>
</inline-formula>
, hence the probabilities can be written as linear functions on
<inline-formula id="d32e986">
<inline-graphic id="d32e987" xlink:href="ncomms2076-m96.jpg"></inline-graphic>
</inline-formula>
. Using the Hilbert–Schmidt scalar product, we can identify each real linear function with an element of the same space,
<inline-formula id="d32e989">
<inline-graphic id="d32e990" xlink:href="ncomms2076-m97.jpg"></inline-graphic>
</inline-formula>
, arriving at the representation (3).</p>
</sec>
<sec disp-level="3">
<title>Non-negativity and normalization of probabilities</title>
<p>The requirement that the probabilities are non-negative for any pair of CP maps
<inline-formula id="d32e997">
<inline-graphic id="d32e998" xlink:href="ncomms2076-m98.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e1000">
<inline-graphic id="d32e1001" xlink:href="ncomms2076-m99.jpg"></inline-graphic>
</inline-formula>
imposes the restriction that
<italic>W</italic>
is positive on pure tensors
<xref ref-type="bibr" rid="b50">50</xref>
with respect to the partition
<italic>A</italic>
<sub>1</sub>
<italic>A</italic>
<sub>2</sub>
−
<italic>B</italic>
<sub>1</sub>
<italic>B</italic>
<sub>2</sub>
. These are matrices such that</p>
<p>
<disp-formula id="eq9">
<label>9</label>
<inline-graphic id="d32e1029" xlink:href="ncomms2076-m100.jpg"></inline-graphic>
</disp-formula>
</p>
<p>The condition has to be imposed for arbitrary positive semidefinite matrices
<inline-formula id="d32e1032">
<inline-graphic id="d32e1033" xlink:href="ncomms2076-m101.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e1035">
<inline-graphic id="d32e1036" xlink:href="ncomms2076-m102.jpg"></inline-graphic>
</inline-formula>
because these are the CJ matrices of CP maps.</p>
<p>We additionally assume that the parties can share arbitrary (possibly entangled) ancillary states independent of the process, and use them in their local operations. The latter means that each party can extend the input space of her/his operations to the ancillas, which we denote by
<inline-formula id="d32e1040">
<inline-graphic id="d32e1041" xlink:href="ncomms2076-m103.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e1043">
<inline-graphic id="d32e1044" xlink:href="ncomms2076-m104.jpg"></inline-graphic>
</inline-formula>
for Alice and Bob, respectively, and apply arbitrary quantum operations with CP maps
<inline-formula id="d32e1046">
<inline-graphic id="d32e1047" xlink:href="ncomms2076-m105.jpg"></inline-graphic>
</inline-formula>
. (One can similarly extend the output systems, but this is not necessary for our argument.) The assumption that the ancillary systems contain a joint quantum state independent of the process means that if separate operations are applied on the ancillas and the original systems, the joint probability distribution for the outcomes is a product of two distributions—one for the outcomes on the ancillas, which is the same as one arising from a measurement on a quantum state
<inline-formula id="d32e1049">
<inline-graphic id="d32e1050" xlink:href="ncomms2076-m106.jpg"></inline-graphic>
</inline-formula>
, and another one for the outcomes on the original systems, which is given by equation (3) with the original
<inline-formula id="d32e1052">
<inline-graphic id="d32e1053" xlink:href="ncomms2076-m107.jpg"></inline-graphic>
</inline-formula>
. These requirements imply that the extended process matrix is given by
<inline-formula id="d32e1056">
<inline-graphic id="d32e1057" xlink:href="ncomms2076-m108.jpg"></inline-graphic>
</inline-formula>
. If we then require that the probabilities for extended operations are non-negative, one has</p>
<p>
<disp-formula id="eq10">
<label>10</label>
<inline-graphic id="d32e1061" xlink:href="ncomms2076-m109.jpg"></inline-graphic>
</disp-formula>
</p>
<p>It was shown
<xref ref-type="bibr" rid="b50">50</xref>
that condition (10) is satisfied if and only if
<inline-formula id="d32e1066">
<inline-graphic id="d32e1067" xlink:href="ncomms2076-m110.jpg"></inline-graphic>
</inline-formula>
is positive semidefinite (a class strictly smaller than positive on pure tensors), which is condition (4).</p>
<p>Additionally, probabilities must be normalized:
<inline-formula id="d32e1071">
<inline-graphic id="d32e1072" xlink:href="ncomms2076-m111.jpg"></inline-graphic>
</inline-formula>
, which means</p>
<p>
<disp-formula id="eq11">
<label>11</label>
<inline-graphic id="d32e1076" xlink:href="ncomms2076-m112.jpg"></inline-graphic>
</disp-formula>
</p>
<p>Condition (5) can be deduced from equation (11) simply by noticing that for a CPTP map
<italic>M</italic>
the corresponding CJ matrix satisfies the condition
<inline-formula id="d32e1082">
<inline-graphic id="d32e1083" xlink:href="ncomms2076-m113.jpg"></inline-graphic>
</inline-formula>
. To see that this is also a sufficient condition for a map to be trace-preserving, it is enough to consider the inverse direction of the CJ isomorphism, </p>
<p>
<disp-formula id="eq12">
<label>12</label>
<inline-graphic id="d32e1087" xlink:href="ncomms2076-m114.jpg"></inline-graphic>
</disp-formula>
</p>
</sec>
</sec>
<sec disp-level="2">
<title>Violation of the causal inequality</title>
<p>The process described by equation (7) can be exploited for the task described above in the following way. Alice always measures the incoming qubit in the
<italic>z</italic>
basis, assigning the value
<italic>x</italic>
=0 to the outcome |
<italic>z</italic>
<sub>+</sub>
〉 and
<italic>x</italic>
=1 to |
<italic>z</italic>
<sub>
<italic>−</italic>
</sub>
〉. She then reprepares the qubit, encoding
<italic>a</italic>
in the same basis, and sends it away. It is easy to see that the CP map corresponding to the detection of a state |ψ〉 and repreparation of another state |φ〉 has CJ matrix
<inline-formula id="d32e1117">
<inline-graphic id="d32e1118" xlink:href="ncomms2076-m115.jpg"></inline-graphic>
</inline-formula>
. Accordingly, the possible operations performed by Alice can be represented compactly by the CJ matrix</p>
<p>
<disp-formula id="eq13">
<label>13</label>
<inline-graphic id="d32e1122" xlink:href="ncomms2076-m116.jpg"></inline-graphic>
</disp-formula>
</p>
<p>Bob adopts the following protocol. If he wants to read Alice's bit (
<italic>b</italic>
<sup>′</sup>
=1), he measures the incoming qubit in the
<italic>z</italic>
basis and assigns
<italic>y</italic>
=0,
<italic>y</italic>
=1 to the outcomes |
<italic>z</italic>
<sub>+</sub>
〉, |
<italic>z</italic>
<sub>
<italic>−</italic>
</sub>
〉, respectively (the repreparation is unimportant in this case). If he wants to send his bit (
<italic>b</italic>
′=0), he measures in the
<italic>x</italic>
basis, and if the outcome is |
<italic>x</italic>
<sub>+</sub>
〉, he encodes
<italic>b</italic>
in the
<italic>z</italic>
basis of the outgoing qubit as 0→|
<italic>z</italic>
<sub>+</sub>
〉, 1→|
<italic>z</italic>
<sub>
<italic>−</italic>
</sub>
〉, whereas if the outcome is |
<italic>x</italic>
<sub>
<italic>−</italic>
</sub>
〉, he encodes it as 0→|
<italic>z</italic>
<sub>
<italic>−</italic>
</sub>
〉, 1→|
<italic>z</italic>
<sub>+</sub>
〉. The CJ matrix representing Bob's CP map is</p>
<p>
<disp-formula id="eq14">
<label>14</label>
<inline-graphic id="d32e1200" xlink:href="ncomms2076-m117.jpg"></inline-graphic>
</disp-formula>
</p>
<p>
<disp-formula id="eq15">
<label>15</label>
<inline-graphic id="d32e1203" xlink:href="ncomms2076-m118.jpg"></inline-graphic>
</disp-formula>
</p>
<p>
<disp-formula id="eq16">
<label>16</label>
<inline-graphic id="d32e1206" xlink:href="ncomms2076-m119.jpg"></inline-graphic>
</disp-formula>
</p>
<p>where
<inline-formula id="d32e1209">
<inline-graphic id="d32e1210" xlink:href="ncomms2076-m120.jpg"></inline-graphic>
</inline-formula>
is the arbitrary state prepared when
<italic>b</italic>
′=1 (with
<inline-formula id="d32e1215">
<inline-graphic id="d32e1216" xlink:href="ncomms2076-m121.jpg"></inline-graphic>
</inline-formula>
) and ⊕ denotes the sum modulo 2. Note that in equation (16), Bob's assignment |
<italic>x</italic>
<sub>+</sub>
〉→
<italic>y</italic>
=0, |
<italic>x</italic>
<sub>
<italic>−</italic>
</sub>
〉→
<italic>y</italic>
=1 for the outcome of his measurement is arbitrary, because for
<italic>b</italic>
′=0 he is not trying to correlate
<italic>y</italic>
with
<italic>a</italic>
.</p>
<p>The probabilities for different possible outcomes, when the described protocol is applied to the process (7), are given, according to (3), by
<inline-formula id="d32e1247">
<inline-graphic id="d32e1248" xlink:href="ncomms2076-m122.jpg"></inline-graphic>
</inline-formula>
. To calculate the success probability, we need as intermediate steps
<inline-formula id="d32e1250">
<inline-graphic id="d32e1251" xlink:href="ncomms2076-m123.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e1253">
<inline-graphic id="d32e1254" xlink:href="ncomms2076-m124.jpg"></inline-graphic>
</inline-formula>
. Notice that when the outcome of one party is ignored, it is always possible to identify a specific state in which the other party receives the qubit. For example, to average out Alice's outcomes one has to calculate</p>
<p>
<disp-formula id="eq17">
<inline-graphic id="d32e1258" xlink:href="ncomms2076-m125.jpg"></inline-graphic>
</disp-formula>
</p>
<p>The process observed by Bob is therefore described by the reduced matrix</p>
<p>
<disp-formula id="eq18">
<label>17</label>
<inline-graphic id="d32e1264" xlink:href="ncomms2076-m126.jpg"></inline-graphic>
</disp-formula>
</p>
<p>The matrix
<inline-formula id="d32e1267">
<inline-graphic id="d32e1268" xlink:href="ncomms2076-m127.jpg"></inline-graphic>
</inline-formula>
represents the CPTP map performed by Alice when the outcomes of her measurement are ignored (the explicit dependence on
<italic>a</italic>
accounts for the possibility of signalling). Using (13), we find
<inline-formula id="d32e1273">
<inline-graphic id="d32e1274" xlink:href="ncomms2076-m128.jpg"></inline-graphic>
</inline-formula>
, which, plugged into equation (17) together with equation (7), gives</p>
<p>
<disp-formula id="eq19">
<label>18</label>
<inline-graphic id="d32e1278" xlink:href="ncomms2076-m129.jpg"></inline-graphic>
</disp-formula>
</p>
<p>When this is measured with the map (15), we find</p>
<p>
<disp-formula id="eq20">
<label>19</label>
<inline-graphic id="d32e1283" xlink:href="ncomms2076-m130.jpg"></inline-graphic>
</disp-formula>
</p>
<p>from which we obtain
<inline-formula id="d32e1286">
<inline-graphic id="d32e1287" xlink:href="ncomms2076-m131.jpg"></inline-graphic>
</inline-formula>
.</p>
<p>Consider now the case when
<italic>b</italic>
<sup>′</sup>
=0. When Bob's outcomes are ignored, he performs the CPTP map described by
<inline-formula id="d32e1296">
<inline-graphic id="d32e1297" xlink:href="ncomms2076-m132.jpg"></inline-graphic>
</inline-formula>
. From this we can calculate, as in the previous case, the effective state received by Alice, which is</p>
<p>
<disp-formula id="eq21">
<label>20</label>
<inline-graphic id="d32e1301" xlink:href="ncomms2076-m133.jpg"></inline-graphic>
</disp-formula>
</p>
<p>from which we find
<inline-formula id="d32e1304">
<inline-graphic id="d32e1305" xlink:href="ncomms2076-m134.jpg"></inline-graphic>
</inline-formula>
. In conclusion, the protocol described yields the probability of success (8), which proves that the process in equation (7) is not causally separable.</p>
</sec>
</sec>
<sec disp-level="1">
<title>Author contributions</title>
<p>All authors contributed extensively to the work presented in this paper.</p>
</sec>
<sec disp-level="1">
<title>Additional information</title>
<p>
<bold>How to cite this article:</bold>
Oreshkov, O.
<italic>et al</italic>
. Quantum correlations with no causal order.
<italic>Nat. Commun.</italic>
3:1092 doi: 10.1038/ncomms2076 (2012).</p>
</sec>
<sec sec-type="supplementary-material" id="S1">
<title>Supplementary Material</title>
<supplementary-material id="d32e23" content-type="local-data">
<caption>
<title>Supplementary Information</title>
<p>Supplementary Figure S1, Supplementary Methods and Supplementary References</p>
</caption>
<media xlink:href="ncomms2076-s1.pdf" mimetype="application" mime-subtype="pdf"></media>
</supplementary-material>
</sec>
</body>
<back>
<ack>
<p>We thank G. Chiribella for discussions. This work was supported by FWF projects P19570-N16 and SFB-FOQUS, FQXi, the European Commission Project Q-ESSENCE (No. 248095) and the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P6-10 〈〈
<email>photonics@be 〉</email>
〉. O.O. acknowledges the support of the European Commission under the Marie Curie Intra-European Fellowship Programme (PIEF-GA-2010-273119). F.C. is a member of the FWF Doctoral Program CoQuS (W 1210).</p>
</ack>
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<floats-group>
<fig id="f1">
<label>Figure 1</label>
<caption>
<title>Strategy for accomplishing communication task by using processes with definite and indefinite causal order.</title>
<p>(
<bold>a</bold>
) There exists a global background time according to which Alice's actions are strictly before Bob's. She sends her input
<italic>a</italic>
to Bob, who can read it out at some later time and give his estimate
<italic>y</italic>
=
<italic>a</italic>
. However, Bob cannot send his bit
<italic>b</italic>
to Alice as the system passes through her laboratory at some earlier time. Consequently, she can only make a random guess of Bob's bit. This results in a probability of success of 3/4. (
<bold>b</bold>
) If the assumption of a definite order is dropped, it is possible to devise a resource (that is, a process matrix
<italic>W</italic>
) and a strategy that enables a probability of success
<inline-formula id="d32e1355">
<inline-graphic id="d32e1356" xlink:href="ncomms2076-m135.jpg"></inline-graphic>
</inline-formula>
(see text).</p>
</caption>
<graphic xlink:href="ncomms2076-f1"></graphic>
</fig>
<fig id="f2">
<label>Figure 2</label>
<caption>
<title>Local quantum experiments with no assumption of a pre-existing background time or global causal structure.</title>
<p>Although the global causal order of events in the two laboratories is not fixed in advance and in general not even definite (here illustrated by the 'shifted' relative orientation of the two laboratories), the two agents, Alice and Bob, are each certain about the causal order of events in their respective laboratories.</p>
</caption>
<graphic xlink:href="ncomms2076-f2"></graphic>
</fig>
<fig id="f3">
<label>Figure 3</label>
<caption>
<title>Terms appearing in a process matrix.</title>
<p>A matrix satisfying condition (4) can be expanded as
<inline-formula id="d32e1370">
<inline-graphic id="d32e1371" xlink:href="ncomms2076-m136.jpg"></inline-graphic>
</inline-formula>
, where the set of matrices
<inline-formula id="d32e1373">
<inline-graphic id="d32e1374" xlink:href="ncomms2076-m137.jpg"></inline-graphic>
</inline-formula>
, with
<inline-formula id="d32e1376">
<inline-graphic id="d32e1377" xlink:href="ncomms2076-m138.jpg"></inline-graphic>
</inline-formula>
and
<inline-formula id="d32e1379">
<inline-graphic id="d32e1380" xlink:href="ncomms2076-m139.jpg"></inline-graphic>
</inline-formula>
for
<inline-formula id="d32e1382">
<inline-graphic id="d32e1383" xlink:href="ncomms2076-m140.jpg"></inline-graphic>
</inline-formula>
, provide a basis of
<inline-formula id="d32e1386">
<inline-graphic id="d32e1387" xlink:href="ncomms2076-m141.jpg"></inline-graphic>
</inline-formula>
. We refer to terms of the form
<inline-formula id="d32e1389">
<inline-graphic id="d32e1390" xlink:href="ncomms2076-m142.jpg"></inline-graphic>
</inline-formula>
as of the type
<italic>A</italic>
<sub>1</sub>
, terms of the form
<inline-formula id="d32e1397">
<inline-graphic id="d32e1398" xlink:href="ncomms2076-m143.jpg"></inline-graphic>
</inline-formula>
as of the type
<italic>A</italic>
<sub>1</sub>
<italic>A</italic>
<sub>2</sub>
and so on. In the
<xref ref-type="supplementary-material" rid="S1">Supplementary Information</xref>
, we prove that a matrix satisfies condition (5) if it contains the terms listed in this table. Each of the terms can allow signalling in at most one direction and can be realized in a situation in which either Bob's actions are not in the causal past of Alice's
<inline-formula id="d32e1413">
<inline-graphic id="d32e1414" xlink:href="ncomms2076-m144.jpg"></inline-graphic>
</inline-formula>
or vice versa
<inline-formula id="d32e1416">
<inline-graphic id="d32e1417" xlink:href="ncomms2076-m145.jpg"></inline-graphic>
</inline-formula>
. The most general unidirectional process is a quantum channel with memory. Measurements of bipartite states that lead to non-signalling probabilities can be realized in both situations. The most general process matrix can contain terms from both rows and may not be decomposable into a mixture of quantum channels from Alice to Bob and from Bob to Alice.</p>
</caption>
<graphic xlink:href="ncomms2076-f3"></graphic>
</fig>
<fig id="f4">
<label>Figure 4</label>
<caption>
<title>Terms not appearing in a process matrix.</title>
<p>These terms are not compatible with local quantum mechanics because they yield non-unit probabilities for some CPTP maps. A possible interpretation of these terms within our framework is that they correspond to statistical sub-ensembles of possible processes. For example, terms of the type
<italic>A</italic>
<sub>2</sub>
can be understood as postselection. One specific case is when a system enters a laboratory in a maximally mixed state, is subject to the map
<italic>M</italic>
and, after going out of the laboratory, is measured to be in some state |ψ . The corresponding probability is given by
<inline-formula id="d32e1433">
<inline-graphic id="d32e1434" xlink:href="ncomms2076-m146.jpg"></inline-graphic>
</inline-formula>
, generated in our formalism by
<inline-formula id="d32e1436">
<inline-graphic id="d32e1437" xlink:href="ncomms2076-m147.jpg"></inline-graphic>
</inline-formula>
. Notably, correlations of the type
<italic>A</italic>
<sub>1</sub>
<italic>A</italic>
<sub>2</sub>
have been exploited in models for describing CTCs
<xref ref-type="bibr" rid="b43">43</xref>
<xref ref-type="bibr" rid="b45">45</xref>
. The pictures are only suggestive of the possible interpretations.</p>
</caption>
<graphic xlink:href="ncomms2076-f4"></graphic>
</fig>
</floats-group>
</pmc>
</record>

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