10.13125/CH/4492
application/pdf
2533-1825
4492
special II
en
urn:nbn:it:unica-26456
Giulio Chiribella
https://creativecommons.org/licenses/by/3.0/
Critical Hermeneutics
quantum information, quantum causality, indefinite causal order, quantum SWITCH, foundations of quantum mechanics
Quantum Superpositions of Causal Structures
Text.Serial.Journal
https://ojs.unica.it/index.php/ecch
4
Articles
2020-12-17
2020-12-17
2020-12-17
2020-12-17
1-24
1
Giulio Chiribella
2020/12/17
10.13125/CH/4492
1
2533-1825
special II
quantum information, quantum causality, indefinite causal order, quantum SWITCH, foundations of quantum mechanics
24
urn:nbn:it:unica-26456
Quantum Superpositions of Causal Structures
https://ojs.unica.it/index.php/ecch/article/download/4492/4575
4
Critical Hermeneutics, special 2 (2020)                   Received: 2/8/2020
Biannual International Journal of Philosophy               Accepted: 2/8/2020
http://ojs.unica.it/index.php/ecch/index                Published: 17/12/2020
ISSN 2533-1825 (on line); DOI 10.13125/CH/4492




      Quantum Superpositions of Causal Structures:
            From Foundations to New Technologies


                                  GIULIO CHIRIBELLA




Abstract
This presentation provides a non-technical overview of the notion of
quantum superposition of causal structures, of its applications, and of
its proposed physical realizations. The conceptual underpinning for
these investigations is a view of quantum theory as a new kind of
probability theory. At the axiomatic level, the principles of this new
kind of probability theory suggest new causal relations that have no
analogue in the classical world. These new causal relations are a
potential resource for new technologies, including computation and
communication technology.
Keywords: quantum information, quantum causality, indefinite causal
order, quantum SWITCH, foundations of quantum mechanics




1. Introduction
The subject of this presentation is causality in the quantum world. I
will mostly speak about how quantum theory can be viewed as a new
kind of probability theory, and how the principles of this new kind of
probability theory suggest new types of causal relation that have no
analogue in the classical world. I will show that these new types of
causal relations are a potential resource for new technologies, including
computation and communication technology. Finally, I will briefly
discuss the different physical situations in which such new causal
Giulio Chiribella, Quantum Superpositions of Causal Structures




relations could occur.


2. Quantum mechanics vs quantum probability theory
Let us start with a bit of background concerning the quantum theory of
probability. One of the important lessons of the last thirty years of
research in the foundations of quantum mechanics is that many of the
counterintuitive aspects of quantum physics have more to do with logic
and probability theory than they have to do with traditional physical
quantities such as position, velocity, mass, and energy (Chiribella and
Spekkens 2015).
      True that quantum mechanics was originally formulated in an
attempt to explain the physics of atoms and of the electromagnetic
radiation. Nevertheless, many of the aspects that make quantum
physics so strikingly special compared to classical physics have little to
do with “mechanics”, and much more to do with probability theory.
      For example, Bell’s celebrated Theorem (Bell 1964) shows that
quantum systems composed of multiple parts exhibit correlations that
are impossible in the classical world. Bell’s theorem holds universally,
independently of which specific quantum systems we consider. It is not
a consequence of the mechanics of the systems under consideration,
but rather of the way in which the quantum formalism assigns
probabilities to the outcomes of experiments.
      For this reason, it is convenient to distinguish between “quantum
mechanics” (viewed as the theory of time evolution of certain
microscopic systems) and “quantum probability theory” (viewed as a
set of abstract rules for assigning probabilities to the outcomes of
experiments). Quantum probability theory is the language in which the
contents of quantum mechanics are expressed, but in principle it can
be applied to more general situations.
      What I will discuss in the following refers primarily to quantum
probability theory. This approach, which transcends the details of



2
                                          Critical Hermeneutics, special 2 (2020)




specific physical systems, is broadly adopted in the field of Quantum
Information Theory.
    Quantum information theory is concerned with the information-
processing capabilities of abstract physical systems, characterized by
the maximum number of states that can be distinguished without error
through a single experiment. A quantum bit, or qubit, is an abstract
quantum system with two (and only two) perfectly distinguishable
states.
    For the purposes of information theory, it is irrelevant whether the
qubit is realized with the polarization of a single photon, the spin of a
nucleus, or the electronic state of an atom. In principle, all qubits are
created equal: they can all be used to perform the same logical
operations, the same computations, and the same communication
protocols. In this respect, quantum information theory is no different
from classical information theory, where the notion of bit is applied to
all classical systems with two (and only two) perfectly distinguishable
states.


3. Axiomatizations of quantum probability theory
The idea of the quantum formalism as a universal language,
independent of the details of the physical systems in question, is at the
basis of a large project of axiomatization of quantum theory, initiated
in 1936 by Garrett Birkhoff and John von Neumann (Birkhoff and von
Neumann 1936).
    This project led to the field of quantum logic, which produced
results of high technical value, although sometimes struggled to go
past its technicalities. A renewed interest in the axiomatization project
came with the advent of Quantum Information, and was strongly
advocated by Chris Fuchs (Fuchs 2003) and Gilles Brassard (Brassard
2005). The first result in this direction came in 2001 with the work of
Lucien Hardy (Hardy 2001), who proposed a list of simple axioms



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Giulio Chiribella, Quantum Superpositions of Causal Structures




concerning logical operations and the probabilistic structure of abstract
physical systems. Combining these axioms with a series of reasonable
assumptions on the formalism, Hardy showed that the mathematical
structure of quantum probability theory can be reconstructed from a
new starting point.
      Following Hardy, other authors sought to reconstruct the rules of
quantum probability theory without invoking ad hoc mathematical
axioms, and without invoking mechanical notions such as those of
position and momentum (Dakic and Brukner 2011, Masanes and Müller
2011, Chiribella, D’Ariano, and Perinotti 2011, Masanes, Müller,
Augusiak, and Peréz-García 2012, Barnum, Müller, Ududec 2014; see
also Chiribella and Spekkens 2015 and references therein).


4. The CDP axiomatization
One of the new axiomatizations was proposed by Mauro D’Ariano, Paolo
Perinotti, and myself in 2010. The original work was published in
Physical Review A (Chiribella, D’Ariano, and Perinotti 2011), and was
recently developed into a book, published by Cambridge University
Press (D’Ariano, Chiribella, and Perinotti 2017). Several non-technical
presentations of this work are also available (Chiribella, D’Ariano, and
Perinotti 2012; Chiribella and Yuan 2013; Chiribella and Scandolo
2015). Hereafter I will refer to this axiomatization as the CDP
axiomatization.
The key feature of the CDP axiomatization that it reconstructs the
quantum formalism from principles of informational nature. These
principles concern the ability to communicate without errors, to store
data with maximal efficiency, and to perform computations in a
reversible way.
      The CDP axiomatization is based on an abstract framework that
describes networks of events, connected with one another through the
transmission of physical systems. In general, every event is associated



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                                            Critical Hermeneutics, special 2 (2020)




to a set of physical systems in input, and to a set of physical systems
in output. For example, an event could be the collision of two particles:
in this case, the two particles are the physical systems in input, and
the result of the collision are the physical systems in output.
    Events can occur in experiments. In general, every experiment is
associated to a set of possible events, interpreted as alternative
outcomes. An example of experiment is the toss of a coin, in which the
possible events are “head” and “tails”. It is worth stressing that we use
the term “experiment” in a broad sense, without implying that an agent
should be present at every stage of the process.
    In   our   framework,    a   probabilistic   theory     consists     in   the
specification of all possible events and all possible experiments,
together with the specification of a rule that assigns probabilities to the
outcomes of such experiments.
    Note that the distinction between input systems and output
systems implies that our networks have a privileged direction, from the
input to the output. In the following, we will be interested in
applications of the formalism where the input-output direction is
identified with the arrow of time.


5. The Causality Principle
In the CDP axiomatization, the quantum theory of probability is
reconstructed from 6 principles. The first of them is the Causality
Principle, which forbids the transmission of information from the future
to the past. Informally, the Causality Principle stipulates that the
probability of an event in the present is independent of the choice of
experiments performed in the future.
    The Causality Principle is equivalent to the impossibility of
modifying the state of a system in the past. In other words, the
Causality Principle is the assumption of the “immodificability of the past”
which gives the title to this conference.



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Giulio Chiribella, Quantum Superpositions of Causal Structures




      The Causality Principle is essential in our derivation of the
quantum formalism. The reason for this is very simple: the standard
quantum formalism, which is found in most textbooks and is applicable
to most (if not all) of the physical systems we known, satisfies the
Causality Principle. For the purpose of reconstructing the standard
quantum framework it is therefore necessary to assume the Causality
Principle, or some combination of principles that implies it.
      On the other hand, one may ask whether the quantum formalism,
as we know it, could be extended to new situations in which the
Causality Principle has limited validity.
      A possible extension consists in having some physical systems that
obey the Causality Principle, alongside with some other physical
systems that violate it. Considering such an extension means
conceding (at least as a thought experiment) that certain system may
travel back in time.
      Another possible extension is to extend the whole framework,
going from a linear connection of events (through the input-output
distinction) to some new type of connection.
      In the following I will discuss both extensions. For this purpose, it
is useful to review two basic properties of the quantum formalism.


6. Quantum superposition
In quantum information theory, the basic unit of information is the
quantum bit, or qubit.
      A qubit is a physical system with two perfectly distinguishable
states, |0> and |1>, plus an infinity of other states, often called
superposition states. Mathematically, the qubit is associated to a two-
dimensional complex vector space, and every vector of unit length
represents a valid state.
      An interesting example of qubit arises in the famous double slit
experiment, where a quantum particle can traverse one of two slits



6
                                           Critical Hermeneutics, special 2 (2020)




(Feynman, Leighton, and Sands 1965). In this example, the state |0>
describes the particle traversing the first slit, while the state |1>
describes the particle travelling the second slit.
     The superposition states are more difficult to visualize. First, they
are not states in which the particle traverses one slit with some
probability, and the other slit with the remaining probability. The
difference between a probabilistic mixture and a superposition state
can be experimentally detected by an interference experiment.
     With a bit of poetic license, we could describe the superposition
states as states in which a particle “traverses both slits at the same
time”, although it is important not to take this expression at face value.
     An interesting feature of the “double-slit qubit” is that it is not a
material system: the qubit is not the particle itself, but rather the
abstract system associated to the two alternative trajectories of the
particle.   At the logical level, we can define a qubit every time we
encounter two perfectly distinguishable alternatives. Considering these
two alternatives as the basic states |0> and |1>, we can in principle
conceive superposition states in which these two alternatives coexist.
For example, we can consider a particle in a superposition of two
alternative positions, or even a car in a superposition of turning left
and turning right.
     Whether the superposition states we defined correspond to an
experimentally accessible physical depends on the system under
consideration, and on the technology available to us. For particles, the
superposition states are experimentally accessible. For cars, no
superposition state has been observed so far.
     Regarding superposition states as a logical construction is useful
because it takes us to the heart of one of the most crucial features of
quantum theory: entanglement.




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Giulio Chiribella, Quantum Superpositions of Causal Structures




7. Entanglement
At the popular level, entanglement is associated to the so-called
“Schrödinger’s cat” (Schrödinger 1935). In this Gedankenexperiment,
a cat is in a room containing a vial of poison, and a machine designed
to open the vial if a certain radioactive particle decays. The particle can
be modelled as a qubit, with the state |0> bringing to no decay, and
the state |1> bringing to decay. The design of the machine implies that
when the particle is in the state |0> the cat survives, and when the
particle is in the state |1> the cat is killed by the poison.
      In short, we can represent the implication as:


      |0> → |non-decayed particle and alive cat >
      |1> → |decayed particle and dead cat >


The interesting situation arises when the particle is in a superposition
state. In this case, the particle and the cat are jointly in a superposition
state: precisely, a superposition of the states |non-decayed particle
and alive cat > and |non-decayed particle and alive cat >.
      The destinies of the particle and the cat are, as it were, joined in
a single superposition state.
      Please don’t get distracted by the rhetorical expedient of cat, and
by all the folklore that comes with it. The key point here is that the
superposition can propagate from a system to another: every time we
have an implication like


      |0> → |situation X>
      |1> → |situation Y>


a superposition of |0> and |1> can in principle lead to a superposition
of |situation X> and |situation Y>. This mechanism is general, in the
sense that it can be applied no matter what situations X and Y



8
                                          Critical Hermeneutics, special 2 (2020)




represent.
    In the following I will consider the case in which situations X and
Y correspond to two different ways to connect two events.


8. The quantum SWITCH
Regarding quantum theory as a new probability theory allows us to
apply the superposition principle to new situations. In particular, we
can consider the situation in which the two alternatives in superposition
are two different causal structures.
    The simplest case is the following. Consider two events, A and B.
For example, A could be the collision of two particles, and B could be
the passage of a photon through a crystal. Classically, we can imagine
two alternatives: in one alternative, the event A influences the event
B, in the other, the event B influences the event A. In our example, the
collision of two particles determine whether or not the photon passes
through the crystal, or vice-versa, the passage of the photon could
determine whether or not the two particles collide. Since we assumed
the Causality Principle, the first alternative requires A to occur before
B, and the second alternative requires B to occur before A.
    Now, the Schrödinger’s cat mechanism allows us consider a new
situation in which the causal influence takes place “simultaneously from
A to B and from B to A”. This idea is at the basis of the quantum SWITCH,
introduced in 2009 by Mauro D’Ariano, Paolo Perinotti, Benoit Valiron,
and myself, and published in Physical Review A (Chiribella, D’Ariano,
Perinotti, and Valiron 2013).
    The quantum SWITCH is a logical operation that connects two
events A and B either in the order AB or in the order BA depending on
the state of a control qubit, which plays the role of the radioactive
particle in Schrödinger’s cat.
    From an algorithmic point of view, the quantum SWITCH is a
higher-order function. The input of the function SWITCH are two



                                                                               9
Giulio Chiribella, Quantum Superpositions of Causal Structures




physical processes, A and B, and a control qubit, which determines the
order in which A and B are connected. The output of the quantum
SWITCH is a new physical process, in which the two processes A and B
take place “simultaneously in the two orders AB and BA”.
      The quantum SWITCH lets the two processes A and B interact in a
new way, which was not possible in the classical world. We can think of
A and B as operations executed by two computers. For example,
computer A could take an integer number n as its input, and increments
it by 1, thus producing the number n+1 in output. Computer B could
take an integer number n as its input, and double it, thus producing
the number 2n in output.
      In this example, the order AB corresponds to the operation that
first increments by 1 and then doubles. The order BA corresponds to
the operation that first doubles and then increments by 1. Note that
the final result changes depending on the order in which the two
computers operate.
      Now, quantum theory allows us to conceive (at least in prinicple)
a new way to use our two computers, by connecting them in a
superposition of the two alternative configurations AB and BA.
      The mechanism is similar to that of Schrödinger’s cat: a quantum
particle, in a superposition of the two states |0> and |1>, controls the
choice between two alternatives, thus propagating the superposition to
all the systems involved in the interaction.
      At this point, it is natural to ask two questions. The first question
is which kind of operational consequences arise from this new way of
connecting physical processes. The second question is how to realize
the quantum SWITCH operation; indeed, quantum theory only tells us
that the quantum SWITCH operation is logically possible, but it does
not tell us how to realize it in practice.
      In the following I will briefly address both questions. I will start
from the easier one, about the operational consequences: if someone



10
                                           Critical Hermeneutics, special 2 (2020)




were to lend us a magical quantum SWITCH machine, what could we
do with it?


9. A game
A first indication that the quantum SWITCH enables us to achieve new
and potentially useful things came from a work published in Physical
Review A (Chiribella 2012). In this example, the quantum SWITCH
allows us to win in a game.
    The goal of the game is to classify two physical processes, by
distinguishing between two alternative hypotheses. The rules are the
following: the player is brought to a room, containing two devices, A
and B. The behaviour of the two devices is unknown to the player,
except for the following promises:
    -the two devices act on a given quantum system (for example, a
photon), known to the player
    -the action of the devices is described by two matrices, A and B,
respectively
    -one and only one of the following properties holds:


    (1) the matrices A and B commute, namely AB = BA
    (2) the matrices A and B anticommute, namely AB = - BA.


The goal of the player is to guess which alternative is the correct one.
A correct answer makes the player score one point, and a wrong
answer makes the player lose one point.
    Now, imagine that two players compete for a prize. The player who
scores more points wins the prize, while the other goes home empty
handed. If we wish, we could even imagine a crueller version of the
game, in which the prize is the player’s life.
    Imagine that the first player has a machine that connects the two
devices in the way described by the quantum SWITCH, while the



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Giulio Chiribella, Quantum Superpositions of Causal Structures




second player is constrained to use the two devices in a well-defined
order, say with A before B. For the first player, who has access to the
quantum SWITCH, the win is guaranteed. Using the interference
between the two orders AB and BA, the player can discover whether
the matrices A and B commute or anticommute. In principle, the player
is guaranteed to win every time the game is played: if the game is
played for 1000 times, the player will score 1000 points.
      For the second player, the situation is less promising. In the
original article, I showed that there is no way to win with certainty
when the two devices A and B are connected in a definite order. A later
work showed that the winning probability is limited by 92.98% (Araujo
et al. 2015). For 1000 repetitions of the game, this corresponds to an
expected score of approximately 929. The difference, of approximately
70 scores, is due to the ability to connect the two processes A and B in
a superposition of orders.
      The interest of this game is in the fact that it allows us to
demonstrate the advantage of the superposition of causal structures in
a simple and mathematically rigorous way. The game itself does not
have much practical relevance, at least for what we know at the
moment.
      Following up on these results, other games of similar nature have
been proposed (Araújo, Costa, and Brukner 2014). These games
employ a version of the quantum SWITCH with more than two
processes. For example, one can consider 3 processes A, B, and C, in
a superposition of the 6 possible orders ABC, BCA, CAB, ACB, CBA, and
BAC. More generally, N processes can be connected in N! orders, a
number that grows exponentially with N. The existing results suggest
that the advantage of the superposition increases with the number of
orders that are superposed. Still, a rigorous quantification of the
advantage as a function of N has not been provided so far.




12
                                          Critical Hermeneutics, special 2 (2020)




10. Applications to the theory of communication
An application of the superposition of order that is closer to practical
application concerns the theory of communication. This is a more
recent development, published in Physical Review Letters (Ebler, Salek,
and Chiribella 2018).
    Consider the following situation. A sender wants to transmit a
message to a receiver. On the way from the sender to the receiver, the
message is forced to traverse two noisy transmission lines, A and B, in
which its content is altered. For example, the message could be a
sequence of zeros and ones, and the transmission lines could turn some
zeros into ones and vice-versa. The problem is to find the best way to
communicate in the presence of this kind of errors.
    In this scenario, the quantum SWITCH offers a rather spectacular
advantage. Suppose that the two transmission lines are to completely
noisy channels, which erase the message, replacing it with a random
sequence of zeros and ones. When the message traverses A and B in a
definite order, communication is impossible: as soon as the first
transmission line is traversed, the content of the message is obliterated.
The situation is radically different when the two transmission lines A
and B are combined through the quantum SWITCH. In this case, the
message sent by the sender ends up in a superposition of two
trajectories, one that traverses channel A before channel B, and one
that traverses channel B before channel A. Surprisingly, communication
becomes possible, despite the fact that each transmission line is
disastrously noisy. Paradoxically, the superposition of two useless
channels yields a useful channel.
    The secret lies in the correlations between the message reaching
the receiver and the qubit that controls the order of the two channels
A and B. The net result of the superposition of orders AB and BA is to
transfer information from the original message to the correlations
between the message and the control qubit. Now, if the control qubit



                                                                              13
Giulio Chiribella, Quantum Superpositions of Causal Structures




is accessible to the receiver, then the receiver can exploit these
correlations to decode the message. The efficiency of the transmission
is not too high, but the interesting thing is that, by using a suitable
code, the sender and receiver can now communicate.
      Another      example       of    this   kind     concerns        the   security   of
communication (Salek, Ebler, and Chiribella 2018; Chiribella et al.
2018). We can imagine a scenario in which a message is forced to
traverse two regions controlled by spies, corresponding to two insecure
communication channels A and B. When the channels are used in a
definite order, AB or BA, the resulting channel is insecure. In contrast,
when A and B are combined in a superposition of orders, it becomes
possible to transmit messages securely.
      Examples of this type are abundant. In general, the composition
of two noisy communication channels allows us to mitigate, and
sometime even cancel the noise.
      From the technological point of view, the reduction of noise
through the superposition of orders is quite interesting. Several
prototype experiments inspired by the quantum SWITCH have been
realized at the University of Queensland (Goswami, Romero, and White
2018), the University of Science and Technology of China (Guo et al.
2020), and the University of Vienna (Rubino et al. 2020). Moreover, the
communication advantages of the quantum SWITCH stimulated the
interest in new communication protocols where the configuration of the
devices is in a quantum superposition (Abbott et al. 2018; Chiribella
and    Kristjánsson        2019;      Guerín,     Rubino,        and    Brukner    2019;
Kristjánsson, Chiribella, Ebler, Salek, and Wilson 2020).
      For the moment, it is worth summarizing what has been discussed
so far. Quantum theory allows us to imagine a machine, the quantum
SWITCH, that combines two events A and B in two alternative orders
AB and BA. This machine offers several advantages. First, it increases
our chances to win in certain games, where the player has to find out



14
                                                Critical Hermeneutics, special 2 (2020)




a property of two unknown processes. Second, the quantum SWITCH
increases our ability to communicate through noisy channels.


11. Impossibility of realizing the quantum SWITCH in a
standard causal network
It is now time to ask ourselves how the quantum SWITCH could be
realized in nature. Remind that the quantum SWITCH was defined
abstractly as a function that transforms two physical processes in input
into a physical process in output. In general, there may be many
physically inequivalent ways to realize the same function.
    Let us see first how the quantum SWITCH cannot be realized.
    For sure, the quantum SWITCH cannot be realized by inserting the
two processes A and B in standard causal network, that is, a circuit
where time is well-defined and every process is localized in time. The
advantages we have seen are already a proof that such realization is
impossible. It   is   also   interesting   to     consider      another      line   of
demonstration, presented in the original paper on the quantum
SWITCH (Chiribella, D’Ariano, Perinotti, and Valiron 2013).
    The proof is by contradiction. Assume that it is possible to realize
the quantum SWITCH by inserting the processes A and B in a sequence
of processes, in which each process happens with certainty. Based on
this premise, one can show that such a sequence of processes, should
contain a “time travel”, that is, a process that takes a message in the
future and sends it back to the past.
    In other words, the realization of the quantum SWITCH in a
standard causal network requires the ability to modify the state of a
system in the past, in open violation of the Causality Principle. Since
standard quantum theory satisfies the Causality Principle, it follows
that such a realization is impossible.
    On the other hand, if we entertain the idea that the Causality
Principle may not hold in certain situations, the quantum SWITCH could



                                                                                    15
Giulio Chiribella, Quantum Superpositions of Causal Structures




be realized in a circuit in which the processes A and B are localized in
time, provided that the circuit contains some process that sends the
state of a system backward in time.


12. Realization of the quantum SWITCH through a closed time-
like curve
A realization of the quantum SWITCH using a time-travelling system
was provided in the original article (Chiribella, D’Ariano, Perinotti, and
Valiron 2013). In that realization, the processes A and B take place in
parallel, and one of the physical systems travels back in time.
       In principle, a circuit of this kind could be realized in a spacetime
that contains closed time-like curves. Spacetimes of this kind are
known in general relativity: a famous example is due to Gödel, who
derived a solution of Einstein’s equations where time flows in a cyclic
way.
       An entirely different matter is whether the universe in which we
live corresponds to a solution of this type. In this sense, neither the
quantum theory of probability nor the relativistic theory of spacetime
give us a direct answer: both theories provide us a spectrum of logical
possibilities, but leave our experiments the burden to decide whether
or not these logical possibilities are realized in nature.
       So far, no experiment has reported any violation of the Causality
Principle, or any indication that closed time-like curves exist in some
parts of the universe. What we can conclude is that, in the physical
regime accessible to our experiments, the past is not modifiable, and
the quantum SWITCH cannot be realized by placing processes A and B
in a sequence of processes.


13. Experiments with photons
The impossibility of realizing the quantum SWITCH with known physics
seems to be in contradiction with numerous experiments performed



16
                                             Critical Hermeneutics, special 2 (2020)




with photons (Procopio et al. 2015; Rubino et al., Goswami et al. 2018,
Wei et al. 2019). The working principle of these is based on ordinary
physics, in which time flows in a linear way. How can they realize the
quantum SWITCH?
    In fact, the experiments are not in contradiction with the
impossibility of realizing the quantum SWITCH in a standard causal
network. The key point is that the experiments do not utilize two
processes A and B that are localized in time, but rather two new
processes A’ and B’ that are distributed over time. What the
experiments demonstrate is the possibility to simulate the quantum
SWITCH with alternative resources that are accessible in ordinary
physics.
    From the technological point of view, these experiments are
important, because they suggest new applications. For example, they
show that certain correlations in time allow us to realize the advantages
of the quantum SWITCH for communication theory. In the longer term,
these experiments could provide the foundation of a new technology of
quantum communication networks in which the messages travel in a
superposition    of   trajectories,   thus   improving       the     quality     of
communication. A discussion of this perspective is presented in a
recent work published in Proceedings of the Royal Society A (Chiribella
and Krisjánsson 2019).


14. Realization of the quantum SWITCH in a quantum spacetime
So far we have seen a radical realization of the quantum SWITCH in a
scenario in which time is a classical, well-defined variable, but certain
physical systems can travel back in time. A different kind of realization
arises when the time order is not a classical variable. Suppose that two
processes A and B take place in two spacetime events P and Q,
respectively,   defined   by   some    suitable    operational       procedure.
Classically, we can imagine a spacetime configuration in which P



                                                                                 17
Giulio Chiribella, Quantum Superpositions of Causal Structures




precedes Q, and another configuration in which Q precedes P. Now if
spacetime itself is a quantum system, we can imagine it in a
superposition of these two configurations. In this scenario, the qubit
that controls the order of processes A and B is the configuration of the
spacetime in which the processes take place.
      This situation is conceivable in a quantum theory of gravity.
Examplea of gravitational realizations of the quantum SWITCH was
provided in terms of superposition of massive objects (Zych, Costa,
Pikovski, and Brukner 2019; Paunković and Vojinović 2020). In fact,
the possibility of having quantum superpositions of spacetime
configurations is considered one of the distinctive traits of quantum
gravity. In this sense, the search for a physical realization of the
quantum SWITCH is intertwined with another fascinating project,
namely the search for experimental evidence that spacetime can be in
a quantum superposition. Recent proposals by Bose et al. at University
College London (Bose et al. 2019) and by Marletto and Vedral at Oxford
(Marletto and Vedral 2019) suggest that experiments of this kind could
be realized in the not too far future.


15. Conclusion
Our brief tour of quantum causality has reached its end. Our journey
can be summarized as follows. The conceptual core of quantum theory
is an extension of probability theory. This extension includes new states,
called superposition states, in which alternative classical configurations
coexist. When a quantum system is in a superposition state, the
interaction      with     other     physical      systems        can   propagate   the
superposition to them, as highlighted by the Schrödinger’s cat
mechanism. This mechanism suggests the in-principle possibility of a
machine, called the quantum SWITCH, which connects two physical
processes A and B in a superposition of two alternative orders AB and
BA, where the order is correlated with the state of a quantum bit.



18
                                            Critical Hermeneutics, special 2 (2020)




     The quantum SWITCH offers a new resource for information
technology. It increases our chances to win in certain games, and it
enhances our ability to communicate through noisy transmission lines.
     The quantum SWITCH is an abstract operation, and admits
radically different physical realization. One of them requires a violation
of the Causality Principle, and might be possible if our universe were
to contain closed time-like curves. Another, less radical, possibility is
that the processes A and B are not localized in time. This realization is
compatible with the known physics, in which time flows in a linear
fashion and the Causality Principle is satisfied. This realization is at the
basis of many recent experiments, in which the superposition of orders
is generated by sending a photon along two alternative trajectories, as
in the double-slit experiment. Finally, a third realization is possible in a
quantum spacetime, in which in principle the processes A and B can be
instantaneous, but the relation between the corresponding instants is
not well defined. In this scenario, the realization of the quantum
SWITCH is intertwined with the search for experimental evidences of
the quantum nature of spacetime.
     As you might have noticed, the analysis provided in this
presentation does not offer answers to the most radical questions “is it
possible to modify the past” or “is it possible to observe situations in
which the order of two spacetime events is in a superposition?”. In this
respect, the abstract theory of quantum probabilities and its extensions
only indicate the space of what is logically conceivable, suggesting
possible effects that would arise in that scenario. In this context, the
research on foundations invites us to explore new frontiers of
experimental physics and engineering, with the scope of determining
in which of the many logically possible universes we actually live.
     I hope that this presentation may have offered an interesting
perspective on the topic of this conference, and few stimuli for future
discussions. Thank you for your attention.



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Giulio Chiribella, Quantum Superpositions of Causal Structures




References
Abbott A. A., Wechs J., Horsman D., Mhalla M., and Branciard C.
(2018).     Communication           through      coherent        control   of   quantum
channels, arXiv:1810.09826.
Araújo M, Costa F, Brukner Č. (2014). Computational advantage from
quantum-controlled ordering of gates. Physical Review Letters 113,
250402.
Araújo M,, Branciard C., Costa F., Feix A., Giarmatzi C., and Brukner Č
(2015). Witnessing causal nonseparability. New Journal of Physics 17,
102001.
Barnum, Müller, Ududec (2014). Higher-order interference and single-
system postulates characterizing quantum theory. New Journal of
Physics 16, 123029.
Bell J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics
Physique Fizika 1, 195.
Birkhoff G. and von Neumann J. (1936). The logic of quantum
mechanics. Annals of Mathematics 37, 823.
Bose S., Mazumdar A., Morley G. W., Ulbricht H., Toroš M., Paternostro
M., Geraci A. A., Barker P. F., Kim M. S., Milburn G. (2017). Spin
entanglement witness for quantum gravity. Physical Review Letters
119, 240401.
Brassard G. (2005). Is information the key? Nature Physics 1, 2.
Chiribella G., D’Ariano G. M., and Perinotti P. (2011). Informational
derivation of quantum theory. Physical Review A 84, 012311.
Chiribella G. (2012). Perfect discrimination of no-signalling channels
via quantum superposition of causal structures. Physical Review A(R)
86, 040301.




20
                                           Critical Hermeneutics, special 2 (2020)




Chiribella G., D’Ariano G. M., and Perinotti P. (2012). Quantum theory,
namely the pure and reversible theory of information. Entropy 14,
1877.
Chiribella G, D’Ariano G. M., Perinotti P, and Valiron B (2013). Quantum
computations without definite causal structure. Physical Review A 88,
022318.
Chiribella G, and Yuan X (2013). Quantum theory from quantum
information: the purification route. Canadian Journal of Physics, 91,
475.
Chiribella G, and Scandolo C. M. (2015). Conservation of information
and the foundations of quantum mechanics. EPJ Web of Conferences
95, 03003.
Chiribella G. and Spekkens R. W., eds. (2015). Quantum Theory:
Informational Foundations and Foils. Dordrecht: Springer.
Chiribella G., Banik M., Bhattacharya S. S., Guha T., Alimuddin M., Roy
A., Saha S., Agrawal S., Kar G. (2018). Indefinite causal order enables
perfect   quantum    communication      with   zero     capacity      channel.
arXiv:1810.10457.
Chiribella G. and Kristjánsson H. (2019). Quantum Shannon theory
with superpositions of trajectories. Proceedings of the Royal Society A
475, 20180903.
Coecke B., Moore D., and Wilce A., eds. (2000). Current research in
operational quantum logic. Dordrecht: Springer.
Dakic B. and Brukner Č. (2011). Quantum Theory and Beyond: Is
Entanglement    Special?,   in   H.   Halvorson    (Ed.),    Deep      Beauty:
Understanding the Quantum World through Mathematical Innovation.
Cambridge: Cambridge University Press.
D’ Ariano G. M., Chiribella G., and Perinotti P. (2017). Quantum Theory
from First Principles. Cambridge: Cambridge University Press.
Feynman R. P., Leighton R. B., and Sands M. (1965). The Feynman
Lectures on Physics, Vol. 3. Reading (Mass): Addison-Wesley.



                                                                               21
Giulio Chiribella, Quantum Superpositions of Causal Structures




Ebler D., Salek S., and Chiribella G. (2018). Enhanced communication
with the assistance of indefinite causal order. Physical Review Letters
120, 120502.
Fuchs C. A. (2003). Quantum mechanics as quantum information,
mostly. Journal of Modern Optics 50, 987.
Goswami K., Giarmatzi C., Kewming M., Costa F., Branciard C., Romero
J., White A. G. (2018). Indefinite causal order in a quantum switch.
Physical Review Letters 121, 090503.
Goswami K., Romero J., and White A. G. (2018). Communicating via
ignorance. arXiv:1807.07383.
Guérin P. A., Rubino G., Brukner Č (2019). Communication through
quantum-controlled noise. Physical Review A 99, 062317.
Guo Y., Hu X. M., Hou Z. B., Cao H., Cui J. M., Liu B. H., Huang Y. F.,
Li C. F., Guo G. C., and Chiribella G. (2020). Experimental transmission
of quantum information using a superposition of causal orders. Physical
Review Letters 124, 030502.
Hardy L. (2001). Quantum theory from five reasonable axioms. quant-
ph/0101012.
Kristjánsson H., Chiribella G., Salek S., Ebler D., and Wilson M. (2020).
Resource theories of communication. New Journal of Physics 22
073014.
Masanes L. and Müller M. P. (2011). A derivation of quantum theory
from physical requirements. New Journal of Physics 13, 063001.
Masanes L., Müller M. P., Augusiak R., and Pérez-García D. (2013).
Existence of an information unit as a postulate of quantum theory.
Proceedings of the National Academy of Sciences 110, 16373.
Salek S., Ebler D., and Chiribella G. (2018). Quantum communication
in a superposition of causal orders. arXiv:1809.06655.
Paunković N. and Vojinović M. (2020). Causal orders, quantum circuits
and spacetime: distinguishing between definite and superposed causal
orders. Quantum 4, 275.



22
                                              Critical Hermeneutics, special 2 (2020)




Procopio L. M., Moqanaki A., Araújo M., Costa F., Calafell I. A., Dowd
E. G., Hamel D. R., Rozema L. A. , Brukner Č., and Walther P. (2015).
Experimental superposition of orders of quantum gates. Nature
communications 6, 1.
Rubino G., Rozema L. A., Feix A., Araújo M., Zeuner J. M., Procopio L.
M., Brukner Č., and Walther P. (2017). Experimental verification of an
indefinite causal order. Science Advances 3, 1602589.
Schrödinger,      E.   (1935).   Die     gegenwärtige      Situation      in    der
Quantenmechanik. Naturwissenschaften 23, 807.
Wei K., Tischler N., Zhao S. R., Li Y. H., Arrazola J. M., Liu Y., Zhang
W., Li H., You L., Wang Z., Chen Y.A. (2019). Experimental quantum
switching   for    exponentially       superior   quantum       communication
complexity. Physical Review Letters 122, 120504.
Zych M., Costa F., Pikovski I., and Brukner Č. (2019). Bell’s theorem
for temporal order. Nature communications 10, 1.




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Giulio Chiribella, Quantum Superpositions of Causal Structures




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