# Operational interpretation of the vacuum and process matrices for identical
particles

Ricardo Faleiro1<EMAIL_ADDRESS>Nikola Paunković1
<EMAIL_ADDRESS>Marko Vojinović2<EMAIL_ADDRESS>1Instituto
de Telecomunicac̃oes and Departamento de Matemática, Instituto Superior
Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001, Lisboa,
Portugal
2Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade,
Serbia

###### Abstract

This work overviews the single-particle two-way communication protocol
recently introduced by del Santo and Dakić (dSD), and analyses it using the
process matrix formalism. We give a detailed account of the importance and the
operational meaning of the interaction of an agent with the vacuum — in
particular its role in the process matrix description. This raises the issue
of counting such operations in an operational manner. Motivated by this
analysis, we extend the process matrix formalism to Fock space using the
framework of second quantisation, in order to characterise protocols with an
indefinite number of identical particles.

##  I Introduction

In recent years there have been advances in quantum information theory related
to new techniques for discussing quantum circuits and quantum computation. One
of those techniques is the recently developed process matrix formalism
ore:cos:bru:12 . This formalism is general enough to describe all known
quantum processes, in particular the superposed orders of operations in a
quantum circuit. Moreover, its most prominent feature is that it allows for a
description of more general situations of indefinite causal orders of
spacetime points. A formal example of such a process has been introduced and
discussed in ore:cos:bru:12 , leading to the violation of the so-called causal
inequalities. The latter represent device-independent conditions that need to
be satisfied in order for a given process to have a well-defined causal order.
It is still an open question whether such a process is physical and can be
realised in nature. Also, a lot of attention in the literature has been
devoted to the quantum switch operation, which has been discussed through both
theoretical descriptions chi:dar:per:val:13 ; ara:bra:cos:fei:gia:bru:15 ;
bad:ara:bru:que:19 ; gue:rub:bru:18  and experimental implementations
pro:etal:15 ; rub:roz:fei:ara:zeu:pro:bru:wal:17 ;
rub:roz:mas:ara:zyc:bru:wal:17 .

One of the interesting aspects of the quantum switch is that it gives rise to
a superposition of orders of quantum operations. In a recent work pau:voj:20 ,
the difference between the superposition of orders of quantum operations and
the superposition of causal orders in spacetime was discussed in detail, and
it was demonstrated that the latter can in principle be realised only in the
context of quantum gravity (see also vil:rio:ren:20 ; vil:17 ). The detailed
analysis of the causal structure of the quantum switch has revealed one
important qualitative aspect of the process matrix description — in order to
properly account for the causal structure of an arbitrary process, it is
necessary to introduce the notion of the quantum vacuum as a possible physical
state. Otherwise, the naive application of the process matrix formalism may
suggest a misleading conclusion that quantum switch implementations in flat
spacetime feature genuine superpositions of spacetime causal orders. This
demonstrates the importance of the concept of vacuum in quantum information
processing. Regarding the role of the vacuum as a control system in (causally
ordered) quantum circuits, see also abe:04  and the references therein.

Simultaneously with these developments, another interesting quantum process
has been recently proposed dak:san:18  by del Santo and Dakić (dSD protocol).
As it turns out, while this process enables Alice and Bob to guess each
other’s input bits with certainty by exchanging a single particle only once,
it cannot be correctly described within the process matrix formalism without
the introduction of the interaction between the vacuum and the apparatus as an
operation. Thus, it represents an additional motivation to introduce the
vacuum state into the process matrix formalism, independent of the reasons
related to the description of the quantum switch process.

Moreover, while the dSD protocol employs only one photon, it is also relevant
for multiphoton processes, which opens the question of the treatment of
identical particles within the process matrix formalism. Also, taking into
account the presence of the vacuum state, one is steered towards the
generalisation of the process matrix formalism to systems with variable number
of identical particles, to the second quantisation and ultimately quantum
field theory (QFT). See also a related work on the causal boxes framework
por:mat:mau:ren:tac .

In this work we give a detailed description and treatment of dSD protocol
within the process matrix formalism. We analyse in detail the role of the
vacuum in the protocol and the formalism, and its operational interpretation.
In particular, we compare the operation of interacting with the vacuum in the
dSD protocol with the same interaction in the optical switch protocol. This
comparison gives rise to a question of proper operational counting of general
operations. In addition, we make use of the dSD protocol as an illuminative
example to introduce the generalisation of the process matrix formalism to
multipartite systems of identical particles.

The paper is organised as follows. In Section II we give a short overview of
the process matrix formalism and the dSD protocol. Section III is devoted to
the process matrix formalism description of dSD protocol, and to the
discussion of the operational role and importance of the vacuum state for its
description. In Section IV we provide the basic rules for the generalisation
of the process matrix formalism to identical multiparticle systems. Section V
is devoted to the summary of our results, discussion and prospects for future
research. The Appendix contains various technical details of the calculations.

##  II State of the art

In this section, we present an overview of the relevant background results.
First, we give a short introduction to the process matrix formalism, and then
we present the dSD protocol. This overview is not intended to be complete or
self-contained, but merely of informative type. The reader should consult the
literature for more details.

###  II.1 The process matrix formalism

The process matrix formalism is based on an idea of a set of laboratories,
interacting with the outside world by exchanging quantum systems. Each
laboratory is assumed to be spatially local in the sense that one can consider
its size negligible for the problem under discussion. Inside the laboratory,
it is assumed that the ordinary laws of quantum theory hold. The laboratory
interacts with the outside world by receiving an input quantum system and by
sending an output quantum system. Inside the laboratory, the input and output
quantum systems are being manipulated using the notion of an instrument,
denoting the most general operation one can perform over quantum systems. Each
interaction is also assumed to be localised in time, such that each operation
of a given laboratory has a separate spacetime point assigned to it (see
Subsection III.4 for a discussion of time delocalised laboratories and
operations ore:19 ). Thus, we introduce the notion of a gate, which represents
the action of an instrument at a given spacetime point (see Section 2 of
pau:voj:20 ); for simplicity, both the gate, as well as its corresponding
spacetime point, we denote by the same symbol, $G$. By $G_{I}$ and $G_{O}$ we
denote the Hilbert spaces of the input and the output quantum systems,
respectively. These Hilbert spaces may be infinite-dimensional, finite-
dimensional or trivial. The action of the instrument is described by an
operator, ${\cal M}^{G}_{x,a}:G_{I}\otimes G_{I}^{*}\to G_{O}\otimes
G_{O}^{*}$, which may depend on some classical input information $a$ available
to the gate $G$, and some readouts $x$ of eventual measurement results that
may take place in $G$. Thus, the instrument maps a generic mixed input state
$\rho_{I}$ into the output state $\rho_{O}={\cal M}^{G}_{x,a}(\rho_{I})$.

Given such a setup, one defines a process, denoted ${\cal W}$, as a functional
over the instruments of all gates, as

$p(x,y,\dots|a,b,\dots)={\cal W}({\cal M}^{G^{(1)}}_{x,a}\otimes{\cal
M}^{G^{(2)}}_{y,b}\otimes\dots)\,,$

where $p(x,y,\dots|a,b,\dots)$ represents the probability of obtaining
measurement results $x,y,\dots$, given the inputs $a,b,\dots$. In order for
the right-hand side to be interpreted as a probability distribution, the
process ${\cal W}$ must satisfy three basic axioms,

${\cal W}\geq 0\,,\quad\mathop{\rm Tr}\nolimits{\cal W}=\prod_{i}\dim
G^{(i)}_{O}\,,\quad{\cal W}={\cal P}_{G}({\cal W})\,,$ (1)

where ${\cal P}_{G}$ is a certain projector onto a subspace of
$\bigotimes_{i}\left(G^{(i)}_{I}\otimes G^{(i)}_{O}\right)$ which, together
with the second requirement, ensures the normalisation of the probability
distribution (see ara:bra:cos:fei:gia:bru:15  for details).

In order to have a computationally manageable formalism, one often employs the
Choi-Jamiołkowski (CJ) map over the instrument operations, such that a given
operation ${\cal M}^{G}_{x,a}$ is being represented by a matrix,

$M^{G}_{x,a}=\\!\Big{[}\\!\left({\cal I}\otimes{\cal
M}^{G}_{x,a}\right)\left({|{\mathbbm{1}}\rangle\\!\rangle}{\langle\\!\langle{\mathbbm{1}}|}\right)\\!\Big{]}^{T}\\!\\!\\!\\!\in(G_{I}\otimes
G_{O})\otimes(G_{I}\otimes G_{O})^{*},$ (2)

where

${|{\mathbbm{1}}\rangle\\!\rangle}\equiv\sum_{i}{|{i}\rangle}\otimes{|{i}\rangle}\in
G_{I}\otimes G_{I}$ (3)

is the so-called transport vector, representing the non-normalised maximally
entangled state, and ${\cal I}$ is the identity operator. Then, one can
describe the process ${\cal W}$ using the process matrix $W$ to write

$p(x,y,\dots|a,b,\dots)=\mathop{\rm Tr}\nolimits\left[(M^{G_{1}}_{x,a}\otimes
M^{G_{2}}_{y,b}\otimes\dots)W\right]\,.$ (4)

Finally, if an instrument ${\cal M}^{G}_{x,a}$ is linear, one can also use a
corresponding “vector” notation (see Appendix A.1 in
ara:bra:cos:fei:gia:bru:15 ),

${|{({\cal M}^{G}_{x,a})^{*}}\rangle\\!\rangle}\equiv\left[{\cal
I}\otimes({\cal M}^{G}_{x,a})^{*}\right]{|{\mathbbm{1}}\rangle\\!\rangle}\in
G_{I}\otimes G_{O}\,,$ (5)

so that

$M^{G}_{x,a}={|{({\cal
M}^{G}_{x,a})^{*}}\rangle\\!\rangle}{\langle\\!\langle{({\cal
M}^{G}_{x,a})^{*}}|}\,.$

In cases where all instruments are linear, and in addition the process matrix
$W$ is a one-dimensional projector, one can introduce the the corresponding
process vector ${|{W}\rangle\\!\rangle}$, such that
$W={|{W}\rangle\\!\rangle}{\langle\\!\langle{W}|}$, and rewrite (4) in the
form:

$\begin{array}[]{c}p(x,y,\dots|a,b,\dots)=\vphantom{\displaystyle\int}\hphantom{mmmmmmmmmmmmm}\\\
\hphantom{mmmm}\left\|\left({\langle\\!\langle{{\cal
M}^{G^{(1)}*}_{x,a}}|}\otimes{\langle\\!\langle{{\cal
M}^{G^{(2)}*}_{y,b}}|}\otimes\dots\right){|{W}\rangle\\!\rangle}\right\|^{2}.\\\
\end{array}$ (6)

###  II.2 The dSD protocol

In a recent paper dak:san:18 , del Santo and Dakić have introduced a protocol
which allows two agents to guess each other’s input bits with certainty by
exchanging a single particle only once. The protocol goes as follows.
Initially, a single particle is prepared in a superposition state of being
sent to Alice and being sent to Bob. Upon receiving the particle, both Alice
and Bob perform unitary operations on it, encoding their bits of information,
$a$ and $b$, respectively, about the outcomes of their coin tosses. They do
this by changing the local phase of the particle by $(-1)^{a}$ and $(-1)^{b}$.
The particle is subsequently forwarded to a beam splitter, and after that
again to Alice and Bob, who now measure the presence or absence of the
particle.

This way, the state of the particle stays in coherent superposition of
different paths in a Mach-Zehnder interferometer. The interference of its
paths gives rise to deterministic outcome that depends on the relative phase
$e^{i\phi}=(-1)^{a\oplus b}$ between the two branches: in case $\phi=0$, the
particle will end up in Alice’s laboratory, while otherwise it will end in
Bob’s. Thus, knowing their own inputs and the outcomes of their local
measurements, both agents can determine each other’s inputs, allowing for two-
way communication using only one particle. This is clearly impossible in
classical physics, demonstrating yet another example of the advantage of
quantum over classical strategies.

The crucial aspect of the protocol lies in the fact that the absence of the
particle represents a useful piece of information for an agent. This gives
rise to the notion of the vacuum state as a carrier of information, playing
the central role in the protocol. Thus, in order to describe the protocol
using the process matrix formalism, one has to incorporate the notion of the
vacuum in the formalism itself. We show this in detail in the next section.

It is interesting to note that the crucial role of the vacuum plays an
important part not only in the dSD protocol, but also in a completely
different setup that has been discussed a lot in recent literature, namely the
quantum switch chi:dar:per:val:13 . As analysed in detail in pau:voj:20 , if
one takes care to distinguish the two temporal positions of a given laboratory
and introduces the notion of a vacuum explicitly, one can demonstrate that the
optical implementations of the quantum switch in flat spacetime do not feature
any superposition of causal orders induced by the spacetime metric. Instead,
it was argued that superpositions of spacetime causal orders can be present
only within the context of a theory of quantum gravity. As we shall see below,
the notion of the vacuum will prove essential to the process matrix
description of the dSD protocol as well.

##  III Process matrix description of the dSD protocol

###  III.1 The spacetime diagram

We begin by drawing the spacetime diagram of the process corresponding to the
dSD protocol (see Figure 1).

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{3}$$t_{f}$$0$AliceBob$A$$A^{\prime}$$B$$B^{\prime}$$S$$S^{\prime}$$L$$V$
Figure 1: The complete spacetime diagram of the process corresponding to the
dSD protocol.

At the initial time $t_{i}$ the laser $L$ creates a photon and shoots it
towards the beam splitter $S$, which at time $t_{1}$ performs the Hadamard
operation and entangles it with the incoming vacuum state (described by the
dotted arrow from the grey gate $V$). The entangled state of the photon and
the vacuum continues on towards Alice’s and Bob’s gates $A$ and $B$,
respectively. At time $t_{2}$, Alice and Bob generate their random bits $a$
and $b$, and encode them into the phase of the incoming photon-vacuum system.
The system then proceeds to the beam splitter $S^{\prime}$ which again
performs the Hadamard operation at time $t_{3}$. The photon-vacuum system then
proceeds to the gates $A^{\prime}$ and $B^{\prime}$, where it is measured at
time $t_{f}$ by Alice and Bob, respectively. Note that the spatial distance
$\Delta l$ between Alice and Bob is precisely equal to the time distance
between the generation of the random bits and the final measurements,

$\Delta l=c(t_{f}-t_{2})\,,$

so that a single photon has time to traverse the space between Alice and Bob
only once. Also, note that the gate $V$, which generates the vacuum state,
corresponds to a “trivial instrument”, since the vacuum does not require any
physical device to be generated. Nevertheless, the vacuum is still a
legitimate physical state of the EM field, so the appropriate gate $V$ has to
be formally introduced and accounted for in the process matrix formalism
calculations.

###  III.2 Formulation of the process matrix

Based on the spacetime diagram, we formulate the process matrix description as
follows. All spacetime points, where interaction between the EM field and some
apparatus may happen, are assigned a gate and an operation which describes the
interaction. Each gate has an input and output Hilbert space, as follows:

$\begin{array}[]{rclcl}L:&&\mathbb{C}&\to&L_{O}\,,\\\
V:&&\mathbb{C}&\to&V_{O}\,,\\\ &&&&\\\ S:&&S_{I}&\to&S_{O}\,,\\\
S^{\prime}:&&S^{\prime}_{I}&\to&S^{\prime}_{O}\,,\\\
\end{array}\qquad\begin{array}[]{rclcl}A:&&A_{I}&\to&A_{O}\,,\\\
B:&&B_{I}&\to&B_{O}\,,\\\ &&&&\\\
A^{\prime}:&&A^{\prime}_{I}&\to&\mathbb{C}\,,\\\
B^{\prime}:&&B^{\prime}_{I}&\to&\mathbb{C}\,.\\\ \end{array}$

The initial gates, $L$ and $V$, have trivial input spaces and nontrivial
output spaces. The final gates, $A^{\prime}$ and $B^{\prime}$, have trivial
output spaces and nontrivial input spaces. The gates $A$ and $B$ have
nontrivial input and output spaces. Each nontrivial space is isomorphic to
${\cal H}_{1}\oplus{\cal H}_{0}\subset{\cal F}$, where ${\cal H}_{0}$ and
${\cal H}_{1}$ are the vacuum and single-excitation subspaces of the Fock
space ${\cal F}$ in perturbative QED. Namely, by design of the dSD protocol,
Alice and Bob may exchange at most one photon, which means that multiparticle
Hilbert subspaces of the Fock space can be omitted. Moreover, the resulting
probability distribution of the experiment outcomes does not in principle
depend on the frequency or the polarisation of the photon in use, so we can
approximate the single-excitation space as a one-dimensional Hilbert space.
Given that the vacuum Hilbert space ${\cal H}_{0}$ is by definition one-
dimensional, we can write

${\cal H}_{0}=\mathop{\rm
span}\nolimits\\{{|{0}\rangle}\\}\equiv\mathbb{C}\,,\qquad{\cal
H}_{1}\approx\mathop{\rm span}\nolimits\\{{|{1}\rangle}\\}\equiv\mathbb{C}\,,$

so that ${\cal H}_{0}\oplus{\cal H}_{1}\equiv\mathbb{C}\oplus\mathbb{C}$.
Here, ${|{0}\rangle}$ and ${|{1}\rangle}$ denote the states of the vacuum and
the photon in the occupation number basis of the Fock space. Therefore, we
have

$\begin{array}[]{c}L_{O}\cong V_{O}\cong A_{I}\cong
A_{O}\cong\hphantom{mmmmmmmmmm}\\\ B_{I}\cong B_{O}\cong A^{\prime}_{I}\cong
B^{\prime}_{I}\cong\mathbb{C}\oplus\mathbb{C}\,.\\\ \end{array}$

Finally, the input and output spaces of beam splitters $S$ and $S^{\prime}$
are “doubled”, since a beam splitter operates over two inputs to produce two
outputs. In particular,

$\begin{array}[]{lcl}S_{I}&=&S_{I}^{L}\otimes S_{I}^{V}\,,\\\
S_{O}&=&S_{O}^{A}\otimes S_{O}^{B}\,,\\\
\end{array}\qquad\begin{array}[]{lcl}S^{\prime}_{I}&=&S_{I}^{\prime A}\otimes
S_{I}^{\prime B}\,,\\\ S^{\prime}_{O}&=&S_{O}^{\prime A}\otimes S_{O}^{\prime
B}\,,\\\ \end{array}$

where again

$\begin{array}[]{c}S_{I}^{L}\cong S_{I}^{V}\cong S_{O}^{A}\cong
S_{O}^{B}\cong\hphantom{mmmmmmmmmm}\\\ S_{I}^{\prime A}\cong S_{I}^{\prime
B}\cong S_{O}^{\prime A}\cong S_{O}^{\prime
B}\cong\mathbb{C}\oplus\mathbb{C}\,.\\\ \end{array}$

With all relevant Hilbert spaces defined, we formulate the action of each
gate, using the CJ map in the form (5). The gates $L$ and $V$ simply generate
the photon and the vacuum,

${|{L^{*}}\rangle\\!\rangle}^{L_{O}}={|{1}\rangle}^{L_{O}}\,,\qquad{|{V^{*}}\rangle\\!\rangle}^{V_{O}}={|{0}\rangle}^{V_{O}}\,,$
(7)

where $*$ is the complex conjugation. The action of the beam splitters is

${|{S^{*}}\rangle\\!\rangle}^{S_{I}S_{O}}=\left[I^{S_{I}S_{I}}\otimes(H^{*})^{S_{O}S_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{I}S_{I}}\,,$
(8)

and

${|{S^{\prime*}}\rangle\\!\rangle}^{S^{\prime}_{I}S^{\prime}_{O}}=\left[I^{S^{\prime}_{I}S^{\prime}_{I}}\otimes(H^{*})^{S^{\prime}_{O}S^{\prime}_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{\prime}_{I}S^{\prime}_{I}}\,,$
(9)

where the Hadamard operator for $S$ is defined as

$\begin{array}[]{lcl}H^{S_{O}S_{I}}\\!\\!\\!&=&\\!\\!\displaystyle\frac{1}{\sqrt{2}}\left({|{1}\rangle}^{S_{O}^{A}}{|{0}\rangle}^{S_{O}^{B}}+{|{0}\rangle}^{S_{O}^{A}}{|{1}\rangle}^{S_{O}^{B}}\right){\langle{1}|}^{S_{I}^{L}}{\langle{0}|}^{S_{I}^{V}}\\\
\\!\\!\\!&&\\!\\!\displaystyle+\frac{1}{\sqrt{2}}\left({|{1}\rangle}^{S_{O}^{A}}{|{0}\rangle}^{S_{O}^{B}}-{|{0}\rangle}^{S_{O}^{A}}{|{1}\rangle}^{S_{O}^{B}}\right){\langle{0}|}^{S_{I}^{L}}{\langle{1}|}^{S_{I}^{V}},\\\
\end{array}$

and analogously for $H^{S^{\prime}_{O}S^{\prime}_{I}}$. The unit operator is
denoted as $I$. Next, in the gates $A$ and $B$, Alice and Bob generate their
random bits $a$ and $b$, and encode them into the phase of the photon. The
corresponding actions are defined as

${|{A^{*}}\rangle\\!\rangle}^{A_{I}A_{O}}=\left[I^{A_{I}A_{I}}\otimes(A^{*})^{A_{O}A_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{I}A_{I}}\,,$
(10)

and

${|{B^{*}}\rangle\\!\rangle}^{B_{I}B_{O}}=\left[I^{B_{I}B_{I}}\otimes(B^{*})^{B_{O}B_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{I}B_{I}}\,,$
(11)

where

$A^{A_{O}A_{I}}=(-1)^{a}{|{1}\rangle}^{A_{O}}{\langle{1}|}^{A_{I}}\oplus{|{0}\rangle}^{A_{O}}{\langle{0}|}^{A_{I}}\,,$

and

$B^{B_{O}B_{I}}=(-1)^{b}{|{1}\rangle}^{B_{O}}{\langle{1}|}^{B_{I}}\oplus{|{0}\rangle}^{B_{O}}{\langle{0}|}^{B_{I}}\,.$

Finally, the gates $A^{\prime}$ and $B^{\prime}$ describe Alice’s and Bob’s
measurement of the incoming state in the occupation number basis,

${|{A^{\prime*}}\rangle\\!\rangle}^{A^{\prime}_{I}}={|{a^{\prime}}\rangle}\,,\qquad{|{B^{\prime*}}\rangle\\!\rangle}^{B^{\prime}_{I}}={|{b^{\prime}}\rangle}\,,$
(12)

where their respective measurement outcomes $a^{\prime}$ and $b^{\prime}$ take
values from the set $\\{0,1\\}$, depending on whether the vacuum or the photon
has been measured, respectively.

After specifying the actions of the gates, the last step is the construction
of the process vector ${|{W_{dSD}}\rangle\\!\rangle}$ itself. The dSD protocol
assumes that the state of the photon remains unchanged during its travel
between the gates. Therefore, the process vector will be a tensor product of
transport vectors (3), one for each line connecting two gates in the spacetime
diagram. The input and output spaces of the gates connected by the line
determine the spaces of the corresponding transport vector. Thus, the total
process vector is:

$\begin{array}[]{lcl}{|{W_{dSD}}\rangle\\!\rangle}&=&{|{\mathbbm{1}}\rangle\\!\rangle}^{L_{O}S_{I}^{L}}{|{\mathbbm{1}}\rangle\\!\rangle}^{V_{O}S_{I}^{V}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{A}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{B}B_{I}}\\\
&&{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}S_{I}^{\prime
A}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}S_{I}^{\prime
B}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
A}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
B}B^{\prime}_{I}}\,.\\\ \end{array}$ (13)

###  III.3 Evaluation of the probability distribution

Now that the process vector and the operations of all gates have been
specified in detail, we can evaluate the probability distribution

$p(a^{\prime},b^{\prime}|a,b)=\|{\cal M}\|^{2}\,,$ (14)

where the probability amplitude ${\cal M}$ is obtained by taking the scalar
product of ${|{W_{dSD}}\rangle\\!\rangle}$ with the tensor product of all
gates, see (6). It is most instructive to perform the computation iteratively,
taking the partial scalar product of ${|{W_{dSD}}\rangle\\!\rangle}$ with each
gate, one by one. The explicit calculation of each step is based on two lemmas
from Appendix A.

We begin by taking the partial scalar product of (13) and the preparation
gates (7). Using Lemma 1 from Appendix A, we obtain:

$\begin{array}[]{l}\left({\langle\\!\langle{L^{*}}|}^{L_{O}}\otimes{\langle\\!\langle{V^{*}}|}^{V_{O}}\right){|{W_{dSD}}\rangle\\!\rangle}=\\\
\hphantom{mmmmm}{|{1}\rangle}^{S_{I}^{L}}{|{0}\rangle}^{S_{I}^{V}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{A}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{B}B_{I}}\\\
\hphantom{mmmmm}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}S_{I}^{\prime
A}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}S_{I}^{\prime
B}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
A}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
B}B^{\prime}_{I}}\,.\\\ \end{array}$

Next we take the partial scalar product with the beam splitter S gate
operation (8). Using Lemma 2 from Appendix A, we obtain:

$\begin{array}[]{l}\left({\langle\\!\langle{S^{*}}|}^{S_{I}S_{O}}\otimes{\langle\\!\langle{L^{*}}|}^{L_{O}}\otimes{\langle\\!\langle{V^{*}}|}^{V_{O}}\right){|{W_{dSD}}\rangle\\!\rangle}=\\\
\hphantom{mmmmm}\displaystyle\frac{1}{\sqrt{2}}\left({|{1}\rangle}^{A_{I}}{|{0}\rangle}^{B_{I}}+{|{0}\rangle}^{A_{I}}{|{1}\rangle}^{B_{I}}\right)\\\
\hphantom{mmmmm}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}S_{I}^{\prime
A}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}S_{I}^{\prime
B}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
A}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
B}B^{\prime}_{I}}\,.\\\ \end{array}$

Now we apply the Alice’s gate operation (10) to obtain:

$\begin{array}[]{l}\left({\langle\\!\langle{A^{*}}|}^{A_{I}A_{O}}\otimes{\langle\\!\langle{S^{*}}|}^{S_{I}S_{O}}\otimes\right.\\\
\hphantom{mm}\left.{\langle\\!\langle{L^{*}}|}^{L_{O}}\otimes{\langle\\!\langle{V^{*}}|}^{V_{O}}\right){|{W_{dSD}}\rangle\\!\rangle}=\\\
\hphantom{mmmmm}\displaystyle\frac{1}{\sqrt{2}}\left((-1)^{a}{|{1}\rangle}^{S_{I}^{\prime
A}}{|{0}\rangle}^{B_{I}}+{|{0}\rangle}^{S_{I}^{\prime
A}}{|{1}\rangle}^{B_{I}}\right)\\\
\hphantom{mmmmm}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}S_{I}^{\prime
B}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
A}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
B}B^{\prime}_{I}}\,.\\\ \end{array}$

Similarly, applying Bob’s gate (11) we get:

$\begin{array}[]{l}\left({\langle\\!\langle{B^{*}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{A^{*}}|}^{A_{I}A_{O}}\otimes{\langle\\!\langle{S^{*}}|}^{S_{I}S_{O}}\otimes\right.\\\
\hphantom{mmmm}\left.{\langle\\!\langle{L^{*}}|}^{L_{O}}\otimes{\langle\\!\langle{V^{*}}|}^{V_{O}}\right){|{W_{dSD}}\rangle\\!\rangle}=\\\
\hphantom{mm}\displaystyle\frac{1}{\sqrt{2}}\left((-1)^{a}{|{1}\rangle}^{S_{I}^{\prime
A}}{|{0}\rangle}^{S_{I}^{\prime B}}+(-1)^{b}{|{0}\rangle}^{S_{I}^{\prime
A}}{|{1}\rangle}^{S_{I}^{\prime B}}\right)\\\
\hphantom{mm}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
A}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{O}^{\prime
B}B^{\prime}_{I}}\,.\\\ \end{array}$

The next step is the application of the second beam splitter gate (9). After a
little bit of algebra, the result is:

$\begin{array}[]{l}\left({\langle\\!\langle{S^{\prime*}}|}^{S^{\prime}_{I}S^{\prime}_{O}}\otimes{\langle\\!\langle{B^{*}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{A^{*}}|}^{A_{I}A_{O}}\otimes\right.\\\
\hphantom{mmmm}\left.{\langle\\!\langle{S^{*}}|}^{S_{I}S_{O}}\otimes{\langle\\!\langle{L^{*}}|}^{L_{O}}\otimes{\langle\\!\langle{V^{*}}|}^{V_{O}}\right){|{W_{dSD}}\rangle\\!\rangle}=\\\
\displaystyle\frac{(-1)^{a}+(-1)^{b}}{2}{|{1}\rangle}^{A^{\prime}_{I}}{|{0}\rangle}^{B^{\prime}_{I}}+\frac{(-1)^{a}-(-1)^{b}}{2}{|{0}\rangle}^{A^{\prime}_{I}}{|{1}\rangle}^{B^{\prime}_{I}}\,.\\\
\end{array}$

Finally, applying the measurement gates (12), we obtain the complete
probability amplitude,

${\cal
M}=\frac{(-1)^{a}+(-1)^{b}}{2}\delta_{a^{\prime}1}\delta_{b^{\prime}0}+\frac{(-1)^{a}-(-1)^{b}}{2}\delta_{a^{\prime}0}\delta_{b^{\prime}1}\,,$

and substituting this into (14), we obtain the desired probability
distribution of the dSD process:

$\begin{array}[]{lcl}p(a^{\prime},b^{\prime}|a,b)&=&\displaystyle\frac{1+(-1)^{a+b}}{2}\delta_{a^{\prime}1}\delta_{b^{\prime}0}\\\
&&\displaystyle+\frac{1-(-1)^{a+b}}{2}\delta_{a^{\prime}0}\delta_{b^{\prime}1}\,.\\\
\end{array}$

From the probability distribution we can now conclude that there are two
distinct possibilities: either Alice detects the photon and Bob does not,
$a^{\prime}=1,b^{\prime}=0$, or vice versa, $a^{\prime}=0,b^{\prime}=1$. In
the first case, because total probability must be equal to one, we have

$\frac{1+(-1)^{a+b}}{2}=1\,,\qquad\frac{1-(-1)^{a+b}}{2}=0\,.$

The only solution to these equations is $a=b$, which means that Alice and Bob
have initially generated equal bits. Since both know the probability
distribution and their own bit, they both know each other’s bit as well, with
certainty. In the second case, when Bob detects the photon, we instead have

$\frac{1+(-1)^{a+b}}{2}=0\,,\qquad\frac{1-(-1)^{a+b}}{2}=1\,,$

and the only solution is $a\neq b$, meaning that Alice and Bob have initially
generated opposite bits. Again, both parties know the probability distribution
and their own bit, and therefore each other’s bit as well, with certainty.

In order to formalise this result, one can also introduce the parity
$\pi\equiv a\oplus b$ and rewrite the probability distribution in the form

$\\!\\!p(a^{\prime},b^{\prime}|\pi)=\frac{1+(-1)^{\pi}}{2}\delta_{a^{\prime}1}\delta_{b^{\prime}0}+\frac{1-(-1)^{\pi}}{2}\delta_{a^{\prime}0}\delta_{b^{\prime}1}\,.$
(15)

Thus, if Alice detects the photon, then $\pi$ is even, while if Bob detects
the photon, $\pi$ must be odd. In both cases, they can “guess” each other’s
bits with certainty by calculating

$x=\pi\oplus a\,,\qquad y=\pi\oplus b\,,$

where $x$ is Alice’s prediction of the value of Bob’s bit, and $y$ is Bob’s
prediction of the value of Alice’s bit. Therefore, the probability of guessing
each other’s input bit is

$p_{\text{success}}\equiv p(x=b\wedge y=a)=1\,.$ (16)

###  III.4 Analysis of the process matrix description — operational
interpretation of the vacuum

After we have given the detailed process matrix description of the dSD
protocol and derived the main result (16), we analyse in more detail the role
of the vacuum in the formalism, giving its operational interpretation.

In the dSD protocol four operations (gates), $A,A^{\prime},B$ and $B^{\prime}$
are performed (see Figure 1). Note that for each choice of input bits $a$ and
$b$ one of the two operations performed, $A^{\prime}$ and $B^{\prime}$, is of
a special form: it represents the absence of the particle. This gives rise to
an operational interpretation of the vacuum state as a carrier of information,
playing the central role in the protocol — the very interaction between the
apparatus and the vacuum (the absence of a particle) plays exactly the same
role in this protocol as any other operation, i.e., not detecting a particle
(“seeing the vacuum”) is an operation on its own. From the mathematical point
of view, supported by the structure of the process vector (13) that explicitly
features the vacuum state, it is perfectly natural to consider the interaction
between the apparatus and the vacuum state on equal footing with the
interaction between the apparatus and the field excitation (i.e., the
particle). Both interactions equally represent operations.

Let us now consider the optical quantum switch, a similar protocol in which
the notion of the vacuum also plays a role. Current optical implementations of
the quantum switch feature four spacetime points, the same as the dSD protocol
pau:voj:20 ; vil:rio:ren:20 ; vil:17 , thus having the similar type of the
spacetime schematic description, see Figure 2. However, by introducing the
notion of time delocalised operations it was argued that the optical switch
implements only two operations, $U$ in spacetime points $A$ or $A^{\prime}$,
and $V$ in spacetime points $B$ or $B^{\prime}$ ore:19 . Nevertheless, the
optical switch features the same apparatus-vacuum interaction as the one from
the dSD protocol: whenever the particle is in, say, the blue branch, and the
operations $U$ and $V$ are applied at spacetime points $A$ and $B^{\prime}$,
respectively, Alice’s and Bob’s labs experience the interaction with the
vacuum at spacetime points $B$ and $A^{\prime}$ (and analogously for the red
branch). Therefore, the treatment of the vacuum in the optical quantum switch
is mutually incoherent with the treatment of the vacuum in the dSD protocol.

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{3}$$t_{4}$$t_{f}$$0$AliceBob$D_{1}$$D_{2}$$A$$A^{\prime}$$B$$B^{\prime}$$S$$S^{\prime}$$L$$V$
Figure 2: The complete spacetime diagram of the process corresponding to the
optical quantum switch. Upon receiving the photon, Alice rotates its
polarisation by the unitary $U$. Analogously, Bob performs rotation $V$ on the
photon entering his lab.

Our analysis can thus serve as motivation for a search towards a more coherent
treatment of the vacuum within the operational approach, since the same
physical situation — interaction between the apparatus and the vacuum — is
currently treated differently in the descriptions of the two protocols.

One might consider the following possible chain of inference. From the
examples of both the quantum switch and the dSD protocol, we have that unitary
operations (be it “genuine rotations” $U$ and $V$, as well as phase flips $\pm
I$) are considered to be operations. From the example of the dSD protocol, we
see that the interaction with the vacuum is an operation as well. Further, in
reference ore:19  it was argued that the optical switch features two “time-
delocalised operations”, $U$ and $V$. Thus, by the same token, it follows that
within this operational approach the optical switch should feature two
additional “time-delocalised operations”: interactions with the vacuum, one
performed by Alice, and the other by Bob (see Appendix B). Note that this is a
possible treatment of the vacuum, which still features superposition of orders
of operations $U$ and $V$ in the optical switch.

It is obvious that the interaction with the vacuum plays a prominent role in
achieving the goal of the dSD protocol — communication between Alice and Bob.
But interactions with the vacuum are also crucial in the optical switch.
Indeed, without those operations, it would be impossible to achieve
superposition of orders of operations $U$ and $V$ in flat spacetime with fixed
causal order of spacetime points pau:voj:20 .

In ara:cos:bru:14  the so-called “flag” systems were introduced to count the
number of operations performed in a lab without destroying the superposition,
which count only one operation per each lab of the optical switch. Note though
that using this method, which effectively counts the number of times a
particle enters the lab, one would count three rather than four operations in
the dSD protocol. This means that either the method is not appropriate, or in
fact the dSD protocol features three, instead of four operations. In the case
of the former, it would be useful to introduce a formal operational definition
of a general method of counting operations, given that the above “flag” method
cannot count interactions with the vacuum.

On the other hand, if the interaction between the vacuum and the apparatus
would not be considered an operation, an issue with formulating the process
vector for the dSD protocol would arise. The one we formulated in (13)
contains input and output Hilbert spaces associated with the interaction
between the vacuum and the detectors. It is not possible to formulate a
process matrix for the dSD protocol that would feature three operations,
without the mentioned interaction with the vacuum. Namely, depending on the
choice of input bits $a$ and $b$, the photon will end up either in Alice’s or
Bob’s lab, rendering it impossible to know in advance which of the two agents
is supposed to perform the final operation. Thus, it is not possible to
formulate a process matrix which features only one operation at the final
moment $t_{f}$. Note that the process matrices themselves were introduced as
the main tool for describing quantum processes in the operational approach. In
other words, the impossibility of formulating the main operational tool for
the dSD protocol without introducing the interaction with the vacuum as an
operation, suggests that the latter should be considered as an operation in
that protocol.

Note that, if the dSD and the optical switch protocols featured incoherent
mixtures of the two possible paths instead of coherent superpositions, then
one could formulate the corresponding process matrices without treating the
interaction with the vacuum as an operation, indeed without even mentioning
the vacuum at all. These would be purely classical processes, which would not
feature any interference effects. In general, ommiting the vacuum is a natural
point of view in classical physics. However, if one wants to describe quantum
physics, the notions of the vacuum and its interaction with the apparatus are
unavoidable.

##  IV Identical particles

The above analysis shows that the vacuum state plays a physically relevant
role in transmitting information, and cannot be ignored. From the point of
view of QFT this is a perfectly natural state of affairs, but from the point
of view of quantum mechanics (QM) it is not, since the notion of vacuum as a
physical state does not exist in QM. This advantage of QFT over QM is the main
motivation for the generalisation of the process matrix formalism to QFT.

In this section, we give basic elements of the process matrix formalism, when
applied to systems of identical particles. In order to avoid working with
(anti/̄)symmetrised vectors of multi-particle states that contain non-physical
entanglement whenever two or more identical particles are fully
distinguishable (say, one photon is in Alice’s, and another in Bob’s lab), we
will use the representation of the second quantisation in which the effects of
particle statistics are governed by the creation and annihilation
(anti-)commutation rules. First, we need to move from the single-particle
Hilbert spaces associated to the gates and the process matrix to the
corresponding Fock spaces.

To each gate $G$, we assign the input/output Fock spaces, ${\cal G}_{I/O}$,
given in terms of the vacuum state ${|{0}\rangle}$ and the single-particle
Hilbert spaces $G_{I/O}$. The single-particle input Hilbert space is given as

$G_{I}=\mbox{span}\\{{|{i}\rangle}=a_{i}^{\dagger}{|{0}\rangle}\ |\
i=1,2,\dots d_{I}\\},$

such that its creation and annihilation operators satisfy the standard
(anti-)commutation relations,

$[a_{i}^{\dagger},a_{j}^{\dagger}]_{\pm}=[a_{i},a_{j}]_{\pm}=0\,,\qquad[a_{i},a_{j}^{\dagger}]_{\pm}=\delta_{ij}\,,$
(17)

where $[\\_\,,\\_]_{+}$ stands for anti-commutator, and $[\\_\,,\\_]_{-}$ for
commutator. The overall bosonic input Fock space is then

${\cal G}_{I}=\bigoplus_{\ell=0}^{\infty}G_{I}(\ell),$ (18)

where $G_{I}(0)=\mbox{span}\\{{|{0}\rangle}\\}$ is the zero-particle,
$G_{I}(1)=G_{I}$ the single-particle, and

$G_{I}(\ell)=\\{[(a_{1}^{\dagger})^{s_{1}}...(a_{d_{I}}^{\dagger})^{s_{d_{I}}}]{|{0}\rangle}\
|\ s_{1}+...+s_{d_{I}}=\ell\\}$

are the $\ell$-particle orthogonal subspaces of the input Fock space. For
fermions, each $s_{i}\in\\{0,1\\}$, and the orthogonal sum in Equation (18)
goes until $d_{I}$, instead of $\infty$. For a given gate, the output Fock
space ${\cal G}_{O}$ is defined analogously, and we denote its creation and
annihilation operators as $\tilde{a}_{i}^{\dagger}$ and $\tilde{a}_{i}$,
respectively, in order to distinguish them from the corresponding operators in
${\cal G}_{I}$.

Our formalism is constructed for quantum circuits which consist of finite
number of gates. This means that we work in the approximation of a finite
number of spacetime points, as opposed to the standard QFT where one works
with an uncountably infinitely many spacetime points. Thus, given the algebra
(17) for the creation and annihilation operators at a single gate, the full
algebra across all gates is normalised to a Kronecker delta, instead of the
standard Dirac delta function. Moreover, the operators in (17) are operators
in coordinate space, as opposed to the momentum space operators which are
standard in QFT, since they create and annihilate modes at a given gate (i.e.,
a given spacetime point), instead of modes with a given momentum. Since the
gates are distinguishable, the modes assigned to different gates always
(anti/̄)commute.

We restrict ourselves to the Minkowski spacetime, so that the global Poincaré
symmetry implies that the vacuum state ${|{0}\rangle}$ is identical across
different gates, as well as between input and output Fock spaces for a given
gate. In this sense, each gate is assumed to be stationary in some inertial
reference frame, since the Fock spaces of non-inertial gates would be subject
to the Unruh effect. We leave the discussion of non-inertial gates and
spacetimes with more general geometries for future work.

Once the Fock spaces have been defined, we pass on to the process matrix
description of gate operations. A gate operation is represented via a CJ
isomorphism of the corresponding operator between the input and the output
Fock spaces, defined as a generalisation of (2),

$M=\sum_{k=0}^{\infty}\Big{[}\left({\cal I}\otimes{\cal
M}_{k}\right)\left({|{\mathbbm{1}_{k}}\rangle\\!\rangle}{\langle\\!\langle{\mathbbm{1}_{k}}|}\right)\Big{]}^{T}\,,$

where the $k$-transport vectors are defined as

${|{\mathbbm{1}_{k}}\rangle\\!\rangle}=\sum_{s_{1}+...+s_{d}=k}\Big{[}\prod_{i=1}^{d}\frac{(a_{i}^{\dagger})^{s_{i}}}{\sqrt{s_{i}!}}\Big{]}\otimes\Big{[}\prod_{i=1}^{d}\frac{(a_{i}^{\dagger})^{s_{i}}}{\sqrt{s_{i}!}}\Big{]}{|{0}\rangle}\,,$
(19)

and ${\cal M}_{k}$ represents the $k$-particle operator for the gate.

In case of a linear gate operation, one can analogously use the “vector”
formalism, and the generalisation of the CJ vector (5), where now we can write

${\cal M}=\sum_{k=0}^{\infty}{\cal M}_{k}\,,$

and

${|{\mathbbm{1}}\rangle\\!\rangle}=\sum_{k=0}^{\infty}{|{\mathbbm{1}_{k}}\rangle\\!\rangle}\,.$
(20)

Here, it is assumed that by definition

${\cal M}_{k}{|{\mathbbm{1}_{k^{\prime}}}\rangle\\!\rangle}\equiv 0\,,\qquad
k\neq k^{\prime}\,.$

For example, one can consider a single-particle unitary operator

$U=\sum_{i,j}u_{ij}\tilde{a}_{i}^{\dagger}a_{j}\,,$

and then its Fock-space generalisation is given as

${\cal
M}={|{0}\rangle}{\langle{0}|}+\sum_{k=1}^{\infty}\frac{1}{k!}:U^{\otimes
k}:\,,$

where $:U^{\otimes k}:$ is the normal ordering of $U^{\otimes k}$.

Given the Fock spaces and actions of instruments in all gates, a process
matrix is defined in the same way as in Section II, according to Eq. (4). In
the field theory case, the process matrix maps the tensor product of output
Fock spaces for all gates into the tensor product of input Fock spaces for all
gates. For example, if the process under consideration is a quantum circuit
(see Section 2 of pau:voj:20 ), the corresponding process matrix can be
represented as a tensor product of transport vectors, each corresponding to a
wire connecting two gates. Transport vectors are defined in the same way as
(20), where in (19) the first set of creation operators corresponds to the
input Fock space of the wire, while the second set corresponds to its output
Fock space. Given that a wire is connecting two gates, its input and output
Fock spaces correspond to the output and input Fock subspaces of the two
gates, respectively. A gate can in general have multiple incoming or outgoing
wires attached to it. Therefore, its input (output) Fock space is a tensor
product of all output (input) Fock spaces of the corresponding wires.

The final step in generalising the process matrix formalism to the second
quantisation framework consists of formulating the axioms that a process
matrix is required to satisfy, namely the generalisations of (1). The first
and third of these axioms have the same form as in the case of finite number
of dimensions. The second axiom also has the same form in cases when the
number of particles is known to be bounded from above for a given process
(such as the discussed case of dSD protocol), since one can truncate the full
Fock spaces to a subspace of finite number of degrees of freedom. However, in
the general case (for example when discussing coherent states), the second
axiom requires a much more careful formulation, since the issue of
normalisation and renormalisation in QFT is highly nontrivial. As such, this
topic is outside the scope of the current paper, and is left for future work.

##  V Conclusions

###  V.1 Summary of the results

In this work we have presented a detailed account of the dSD protocol,
formulating it within the process matrix formalism. Analysing the role of the
vacuum state in the dSD protocol and its process matrix description, we gave
the operational interpretation of the vacuum. Furthermore, we generalised the
process matrix formalism to Fock space using the second quantisation
framework, allowing the description of systems of identical particles.

###  V.2 Discussion

The first important point of this work is the necessity of explicitly
introducing the interaction with the vacuum as a legitimate operation in the
dSD protocol, on equal footing with any other operation. Indeed, the very lack
of detection of the particle in the protocol provides an equal amount of
information as its detection (explicit interaction). As a consequence, instead
of interpreting the absence of particle as noninteraction, one should instead
interpret it as the interaction between the vacuum and the apparatus, and thus
as an operation. Including the interaction with the vacuum as an operation
poses a question of the method of counting operations in a given protocol,
given that the operations corresponding to the interaction with the vacuum
cannot be counted.

The introduction of the vacuum into the process matrix formalism gives a
natural motivation to extend the latter to the case of identical particles,
both bosons and fermions, which is the second important point of this work.
However, note that while employing the formalism of second quantisation, our
construction still features only a discrete number of gates. This discreteness
means that we still work in particle ontology (i.e., mechanics). Nevertheless,
our construction is an important first step towards defining the process
matrix formalism in field ontology, i.e., fully fledged QFT.

###  V.3 Future lines of investigation

As mentioned in the discussion, a natural next line of investigation would be
a generalisation of the process matrix formalism to full QFT. This would
include an analysis of non-inertial gates and the corresponding Unruh effect.
In addition, a mathematically rigorous formulation of the axioms for the
process matrix description in Fock spaces, as discussed at the end of Section
IV, is also an important topic to be addressed. Finally, addressing in more
detail the interaction between the agent and the vacuum within the operational
approach is an interesting topic of future research.

Acknowledgments

The authors wish to thank Borivoje Dakić and Flavio del Santo for useful
discussions.

RF acknowledges support from DP-PMI and Fundação para a Ciência e Tecnologia
(FCT) through Grant PD/BD/128636/2017. NP’s work was partially supported by
Security and Quantum Information Group of Instituto de Telecomunicações, by
Programme (COMPETE 2020) of the Portugal 2020 framework [Project Q.DOT with
Nr. 039728 (POCI-01-0247-FEDER-039728)] and the Fundação para a Ciência e a
Tecnologia (FCT) through national funds, by FEDER, COMPETE 2020, and by
Regional Operational Program of Lisbon, under UIDB/50008/2020 (actions
QuRUNNER, QUESTS), Project QuantumMining POCI-01-0145-FEDER-031826, Project
PREDICT PTDC/CCI-CIF/29877/2017, Project CERN/FIS-PAR/0023/2019 and Project
QuantumPrime PTDC/EEI-TEL/8017/2020. The bilateral scientific cooperation
between Portugal and Serbia through the project “Noise and measurement errors
in multi-party quantum security protocols”, no. 451-03-01765/2014-09/04
supported by the FCT, Portugal, and the Ministry of Education, Science and
Technological Development of the Republic of Serbia is also acknowledged. MV
was supported by the Ministry of Education, Science and Technological
Development of the Republic of Serbia, and the bilateral scientific
cooperation between Austria and Serbia through the project “Causality in
Quantum Mechanics and Quantum Gravity — 2018-2019”, no.
451-03-02141/2017-09/02 supported by the Austrian Academy of Sciences (ÖAW),
Austria, and the Ministry of Education, Science and Technological Development
of the Republic of Serbia.

##  Appendix A Two lemmas for the process matrix evaluation

Lemma 1. Let
${|{\Psi^{*}}\rangle\\!\rangle}^{X_{O}}={|{\Psi^{*}}\rangle}^{X_{O}}$
represent a gate which has no input, while it prepares the state
${|{\Psi}\rangle}\in X_{O}$ as its output. Then, the scalar product of that
vector and the transport vector
${|{\mathbbm{1}}\rangle\\!\rangle}^{X_{O}Y_{I}}$ is given as:

${}^{X_{O}}{\langle\\!\langle{\Psi^{*}}|{\mathbbm{1}}\rangle\\!\rangle}^{X_{O}Y_{I}}={|{\Psi}\rangle}^{Y_{I}}\,.$

Proof. Using the fact that the transport vector is an unnormalized maximally
entangled state, the explicit calculation goes as follows:

$\begin{array}[]{lcl}{}^{X_{O}}{\langle\\!\langle{\Psi^{*}}|{\mathbbm{1}}\rangle\\!\rangle}^{X_{O}Y_{I}}&=&\displaystyle{\langle{\Psi^{*}}|}^{X_{O}}\sum_{k}{|{k}\rangle}^{X_{O}}{|{k}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{k}\langle\Psi^{*}|k\rangle\;{|{k}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{k}\langle k|\Psi\rangle\;{|{k}\rangle}^{Y_{I}}\\\
&=&\displaystyle{|{\Psi}\rangle}^{Y_{I}}\,,\\\ \end{array}$

where we have used the unit decomposition $\displaystyle
I=\sum_{k}{|{k}\rangle}{\langle{k}|}$ and the fact that
$\langle\Psi^{*}|k\rangle=\langle\Psi|k\rangle^{*}=\langle k|\Psi\rangle$.

Lemma 2. Let

${|{U^{*}}\rangle\\!\rangle}^{X_{I}X_{O}}=\left[I^{X_{I}X_{I}}\otimes(U^{*})^{X_{O}X_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{X_{I}X_{I}}$

represent a gate which performs the operation $U:X_{I}\to X_{O}$, and let
${|{W}\rangle\\!\rangle}={|{\Psi}\rangle}^{X_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{X_{O}Y_{I}}$.
Then the scalar product of the two is

${}^{X_{I}X_{O}}{\langle\\!\langle{U^{*}}|{W}\rangle\\!\rangle}=\Big{(}U{|{\Psi}\rangle}\Big{)}^{Y_{I}}\,.$

Proof. Again using the expansion of the transport vectors as unnormalized
maximally entangled states, the explicit calculation goes as follows:

$\begin{array}[]{lcl}{}^{X_{I}X_{O}}{\langle\\!\langle{U^{*}}|{W}\rangle\\!\rangle}&=&\displaystyle{\langle\\!\langle{\mathbbm{1}}|}^{X_{I}X_{I}}\left[I^{X_{I}X_{I}}\otimes(U^{T})^{X_{I}X_{O}}\right]{|{\Psi}\rangle}^{X_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{X_{O}Y_{I}}\vphantom{\displaystyle\sum_{k}}\\\
&=&\displaystyle\sum_{k}{\langle{k}|}^{X_{I}}{\langle{k}|}^{X_{I}}\left[I^{X_{I}X_{I}}\otimes(U^{T})^{X_{I}X_{O}}\right]{|{\Psi}\rangle}^{X_{I}}\sum_{m}{|{m}\rangle}^{X_{O}}{|{m}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{k,m}\Big{(}{\langle{k}|}^{X_{I}}I^{X_{I}X_{I}}{|{\Psi}\rangle}^{X_{I}}\Big{)}\Big{(}{\langle{k}|}^{X_{I}}(U^{T})^{X_{I}X_{O}}{|{m}\rangle}^{X_{O}}\Big{)}{|{m}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{k,m}\Big{(}\langle
k|\Psi\rangle\Big{)}\Big{(}{\langle{m}|}U{|{k}\rangle}\Big{)}{|{m}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{m}{\langle{m}|}U\left(\sum_{k}{|{k}\rangle}{\langle{k}|}\right){|{\Psi}\rangle}\;{|{m}\rangle}^{Y_{I}}\\\
&=&\displaystyle\sum_{m}{\langle{m}|}U{|{\Psi}\rangle}\;{|{m}\rangle}^{Y_{I}}\\\
&=&\displaystyle\Big{(}U{|{\Psi}\rangle}\Big{)}^{Y_{I}}\,,\\\ \end{array}$

where we have again used the unit decomposition and the fact that
${\langle{k}|}U^{T}{|{m}\rangle}={\langle{m}|}U{|{k}\rangle}$.

##  Appendix B Time-delocalised operations in the optical switch

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{3}$$t_{4}$$t_{f}$$0$AliceBob$D_{1}$$D_{2}$$A$$A^{\prime}$$B$$B^{\prime}$$S$$S^{\prime}$$L$$V$ |  | spacetime$t_{i}$$t_{1}$$t_{2}$$t_{3}$$t_{4}$$t_{f}$$0$AliceBob$D_{1}$$D_{2}$$A$$A^{\prime}$$B$$B^{\prime}$$S$$S^{\prime}$$L$$V$
---|---|---
Figure 3: Two branches of the coherent superposition of the optical switch
protocol.

Figure 3 depicts two branches coherently superposed in the optical switch. The
left diagram represents the branch in which the photon first enters Alice’s
lab, and then Bob’s. On the right, the photon first visits Bob’s lab, and then
Alice’s. Whenever the photon enters Alice’s lab, she applies unitary $U$ (in
$A$, left diagram, or $A^{\prime}$, right diagram), while Bob interacts with
the vacuum (in $B$, left diagram, or $B^{\prime}$, right diagram).
Analogously, whenever the photon enters Bob’s lab, he applies unitary $V$ (in
$B$, right diagram, or $B^{\prime}$, left diagram), while Alice interacts with
the vacuum (in $A$, right diagram, or $A^{\prime}$, left diagram).

Since applying the unitaries in a quantum protocol are operations, and since
in the optical switch they are applied by Alice ($U$) and Bob ($V$) at two
different times, we say that the optical switch features two time-delocalised
operations $U$ (at $A$ and $A^{\prime}$) and $V$ (at $B$ and $B^{\prime}$).

Since the interaction with the vacuum in the dSD protocol is an operation, and
since in the optical switch it is applied by Alice and Bob at two different
times, one can say that the optical switch features two time-delocalised
operations of the interaction with the vacuum (at $A$ and $A^{\prime}$, as
well as at $B$ and $B^{\prime}$).

## References

  * (1) O. Oreshkov, F. Costa and Č. Brukner, Nature Communications 3, 1092 (2012).
  * (2) G. Chiribella, G. M. D’Ariano, P. Perinotti and B. Valiron, Physical Review A 88, 022318 (2013).
  * (3) M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi and Č. Brukner, New Journal of Physics 17, 102001 (2015).
  * (4) J. Bavaresco, M. Araújo, Č. Brukner and M. T. Quintino (2019), eprint arXiv:1903.10526.
  * (5) P. Allard Guérin, G. Rubino and Č. Brukner (2018), eprint arXiv:1812.06848.
  * (6) L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. A. Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner and P. Walther, Nature Communications 6, 7913 (2015).
  * (7) G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner and P. Walther, Science Advances 3, e1602589 (2017).
  * (8) G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner and P. Walther (2017), eprint arXiv:1712.06884.
  * (9) N. Paunković and M. Vojinović, Quantum 4, 275 (2020), eprint arXiv:1905.09682.
  * (10) V. Vilasini, L. del Rio and R. Renner, “Causality in definite and indefinite spacetimes”, https://wdi.centralesupelec.fr/users/valiron/qplmfps/papers/qs01t3.pdf (2020), extended abstract for QPL 2020. Talk: https://www.youtube.com/watch?v=iBCntE0zxg8.
  * (11) V. Vilasini, An introduction to causality in quantum theory (and beyond), Master’s thesis, ETH, Zürich, Switzerland (2017).
  * (12) J. Åberg, Physical Review A 70, 012103 (2004).
  * (13) F. Del Santo and B. Dakić, Physical Review Letters 120, 060503 (2018).
  * (14) C. Portmann, C. Matt, U. Maurer, R. Renner and B. Tackmann, IEEE Transactions on Information Theory 63, 3277 (2017).
  * (15) O. Oreshkov, Quantum 3, 206 (2019).
  * (16) M. Araújo, F. Costa and Č. Brukner, Physical Review Letters 113, 250402 (2014).