Original Article Journal Section ICP, inductively coupled plasma, DRIE, deep
reactive ion etching, PR, photoresist. [1]Equally contributing authors.
Michael Gerlt, Department of Mechanical and Process Engineering, Institute for
Mechanical Systems, Zurich, 8092, Switzerland<EMAIL_ADDRESS>This
work is supported by the ETH Zurich and, in part, an interdisciplinary grant
from the Swiss National Science Foundation (Grant Number CR22I2_166110).

# Reduced etch lag and high aspect ratios by deep reactive ion etching (DRIE)

Michael S. Gerlt  Nino F. Läubli  Michel Manser  Department of Mechanical and
Process Engineering, Institute for Mechanical Systems, Zurich, 8092,
Switzerland  Bradley J. Nelson  Department of Mechanical and Process
Engineering, Institute of Robotics and Intelligent Systems, Zurich, 8092,
Switzerland  Jürg Dual  Department of Mechanical and Process Engineering,
Institute for Mechanical Systems, Zurich, 8092, Switzerland

###### Abstract

Deep reactive ion etching (DRIE) with the Bosch process is one of the key
procedures used to manufacture micron-sized structures for MEMS and
microfluidic applications in silicon and, hence, of increasing importance for
miniaturization in biomedical research. While guaranteeing high aspect ratio
structures and providing high design flexibility, the etching procedure
suffers from reactive ion etching lag and often relies on complex oxide masks
to enable deep etching. In this work, we introduce an optimized Bosch process
that reduces the etch lag to below $1.5\text{\,}\mathrm{\char 37\relax}$.
Furthermore, we improved a three-step Bosch process, allowing the fabrication
of structures with $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness at
depths up to $180\text{\,}\mathrm{\SIUnitSymbolMicro m}$ while maintaining
their stability.

###### keywords:

fabrication, deep reactive ion etching, process optimization, reduced etch
lag, high aspect ratio, small structures

##  1 Introduction

With the rise of the semiconductor industry, cleanroom-based production
technologies in silicon gained a significant increase in popularity. The
ability to fabricate micron-sized structures has revolutionized numerous
industries and led to previously unseen products such as MEMS[1] and
microfluidic devices[2] that recently attained increasing relevance for
biotechnology and biomedical research.[3] The standard fabrication steps used
for the production of these microstructures consist of photolithography,
etching, and post-processing such as bonding, dicing, and packaging. Given the
high complexity of the etching step in current microfabrication processes,
various approaches have been developed, each having their advantages and
disadvantages. One of the most common processes for microstructure fabrication
is deep reactive ion etching (DRIE), where physical and chemical etching are
successfully combined.[4] The full potential of DRIE was revealed with the
invention of a time-multiplexed alternating process of passivation and etching
by Laemer and Schilp in 1996.[5] Named after their employer, the Bosch process
is nowadays one of the key processes in the silicon industry.[6] During the
passivation step, a thin polymer layer, typically consisting of
octafluorocyclobutan ($C_{4}F_{8}$), is deposited onto the substrate, while in
the subsequent etching step, the passivation layer and the underlying silicon
are etched. Due to the ions’ acceleration towards the target, significantly
higher vertical etch rates can be achieved in sidewall etching, leading to the
formation of high aspect ratio trenches (Fig. 1 (a)). The characteristic
scallops that result from the Bosch process are unavoidable. However, given
their size in the nanometre-range, the influence of the scallops is usually
negligible (Fig. 1 (b)).

Figure 1: Bosch process. (a) Sketch of the Bosch process. (1) Silicon wafer
with mask after one process cycle. (2) Deposition of a polymer layer during
passivation. (3) Physical and chemical etching. (4) Silicon wafer with mask
after two process cycles. (b) Scanning electron microscopy image of a
$10\text{\,}$x$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ silicon pillar. The
scallops that result from the sidewall etching are clearly visible. Scale bar
corresponding to $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

One of the Bosch process’s fundamental limitations is the reactive ion etching
(RIE) lag, which leads to narrow features being etched shallower than large
features (RIE lag) as the gas dynamics limit the gas transport to and from
structures with an aspect ratio exceeding 1:1.[7] Recently, several research
groups tried to tackle the RIE lag by adding process steps or advanced timing
controls, which often significantly increases process complexity and, thus,
requires expensive tools.[8, 9, 10, 11] In this work, we achieved a RIE lag
reduction to below 1.5% at an etch depth of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by solely adjusting parameters of a
two-step Bosch process while maintaining its simplicity and adaptability for a
large variety of fabrication machines.

Another limitation of the two-step Bosch process is its low selectivity, which
complicates the production of structures with high aspect ratios.[12]
Therefore, several approaches to optimize the selectivity, i.e., the ratio
between mask and silicon etch, were developed.[13, 14, 15] Nevertheless, for a
reliable etch depths of more than $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
silicon oxide masks are required, which demands an additional patterning step
in a Reactive Ion Etching (RIE) machine.[16, 17] To overcome this challenge,
we introduce a three-step Bosch process, in which the duration of the second
process step (anisotropic etching) is lowered to the time necessary to remove
the polymer layer on the bottom of the trench, i.e., breakthrough step.
Additionally, a third step is introduced, which does not rely on particle
acceleration towards the target, thus reduces the etch rate of the mask and
increases selectivity. Our improvement of the three-step Bosch process enabled
us to use a photoresist with $1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thickness to etch depths greater than $450\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ corresponding to a selectivity of more than 350. Furthermore, our results
illustrate etching angles near $89.7\text{\,}\mathrm{\SIUnitSymbolDegree}$,
which allows for the production of a silicon wall with
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness in between two
microchannels to a depth of more than $270\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ while maintaining its impermeability, crucial for subsequent use in
microfluidics.

In this work, we present an optimized two and three-step Bosch process that
tackles two important challenges of DRIE etching, i.e. the RIE lag and the
selectivity. We believe that our achievements concerning the well-described
and characterized processes simplify the adaptability and accessibility of
state-of-the-art technologies for a broader audience.

##  2 Results & Discussion

###  2.1 Bosch process optimization for reduced RIE lag

Figure 2: Reduced RIE lag. (a) Sketch of the Bosch process parameters for a
standard procedure (top) and the optimized process parameters (bottom). A gas
flow of 10 sscm is always maintained to enable faster switching times (violet
horizontal bar). (b) Optical microscopy images of diced silicon wafers to
illustrate the improvement in RIE lag. (top) Trenches etched with the standard
Bosch process, with $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide trenches reached a depths of
$64\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$47\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively, corresponding to an
RIE lag of $27\text{\,}\mathrm{\char 37\relax}$. (bottom) Trenches etched with
our optimized process, with $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width were etched
$$47.8\text{\,}$\pm$1.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ deep into the
silicon wafer, corresponding to a RIE lag of below $1.5\text{\,}\mathrm{\char
37\relax}$. Scale bars corresponding to $25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$. (c) Microscope pictures of a diced silicon wafer with a
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide trench. The maximal
achievable etch depth with a $1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ PR
layer (top) and a $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $SiO_{2}$ mask
(bottom) as the mask was $29\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$141\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively. Scale bar
corresponding to $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The Bosch process is one of the most common DRIE processes as it consists of
only two steps while not relying on fast response times only attainable with
high-end equipment, making it accessible for a broader audience and a large
variety of research fields. However, despite its simplistic approach, the
process is capable of providing etch angles close to
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ or even higher, enabling the
production of small structures while guaranteeing their stability on the
channel bottom, i.e., inside the trenches. Nevertheless, one major limitation
of the Bosch process is the RIE lag where channels with different widths are
not etched to the same depth due to the limited gas supply within the
constricted features, which can already be relevant for channels with widths
smaller than $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

We successfully overcame this limitation following a concept initially
introduced by Lai et al., focusing on adjusting the ratio between the duration
of the passivation process $t_{P}$ and etching process $t_{E}$.[10] While the
Bosch process only consists of two steps, one process cycle can be separated
into three individual tasks, i.e., polymer deposition, polymer etching, and
silicon etching (see Supporting Figure S1). It is essential to highlight that
the polymer deposition and the silicon etching are primarily chemical and, by
that, diffusion dependent, while the initial polymer etching at the bottom of
the trench is of physical nature and, therefore, mainly independent of the
aspect ratio.

To better understand why a substantial increase of the passivation time, in
comparison to a standard Bosch process (top of Fig 2 (a)), led to a
significant reduction in RIE lag, we have a brief look at the procedure for a
wide as well as a narrow trench. In narrow structures, the polymer deposition
is slower than in wide features, leading, for a given deposition duration, to
a thinner passivation layer at the confined space compared to open spaces. The
subsequent anisotropic physical polymer etching, however, is independent of
the aspect ratio. Therefore, the silicon in narrow trenches will be exposed
faster than for a wide area due to the thinner passivation layer. The last
step, i.e. the chemical silicon etching, is again slower in confined spaces.
However, as it starts earlier than in wide structures, due to the silicon at
the bottom being exposed faster, the process duration can be chosen such that,
after a single cycle of the Bosch process, the wide and the narrow trenches
end at the same depth. Following the assimilation of the two process durations
$t_{P}$ and $t_{E}$, slight adjustments have been performed through the
iterative increment of the chamber pressure during the etching step to improve
reliability and uniformity further. Please refer to Tables 1 and 2 in the
Supporting Information for a detailed overview of all processes’ parameters.

With these adjustments to the original process (bottom of Fig. 2 (a)), we were
able to etch channels with a width ranging from
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ down to a depth of
$48\text{\,}\mathrm{\SIUnitSymbolMicro m}$ while the smallest channel showed a
difference in height of less than $1.5\%$ (Fig. 2 (b)). Despite the etch angle
$\alpha=91\pm 0.5$\mathrm{\SIUnitSymbolDegree}$$ of our modified process, the
etch depth is limited. With a PR layer of
$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness, we achieved a maximal
etch depth of $29\text{\,}\mathrm{\SIUnitSymbolMicro m}$ corresponding to a
selectivity of 22. When using a $SiO_{2}$ mask with
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness, the maximal etch depth
increased to $141\text{\,}\mathrm{\SIUnitSymbolMicro m}$ corresponding to a
selectivity of 71 (Fig 2 (c)). Measurements of multiple structures from
different wafers revealed reproducibility of our process with etch angles of
$$91.14\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$0.69\text{\,}\mathrm{\SIUnitSymbolDegree}$$
(see Supporting Figure S4). In summary, by adjusting the etch to deposition
time ratio and reducing the pressure, we reduced the RIE lag to below
$1.5\text{\,}\mathrm{\char 37\relax}$ and produced structures as deep as
$141\text{\,}\mathrm{\SIUnitSymbolMicro m}$. It is important to highlight that
our approach allowed for the controlled etching of wafers with large
percentages of exposed area (as high as $90\text{\,}\mathrm{\char 37\relax}$),
a task which typically provides an additional challenge in MEMS fabrication.

###  2.2 High rate process optimization for deep etch

Figure 3:  High aspect ratio etching of small structures. (a) Sketch of the
standard three-step Bosch process parameters as provided by the supplier (top)
and the optimized parameters (bottom). (b) Scanning electron microscopy (top)
and optical microscopy (bottom) images of micron-sized pillars with
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ height, produced with the standard
(top) and optimized (bottom) process parameters. Numbers on the side and the
bottom correspond to the length and width of the structures, respectively. The
same pillar has been marked with a red circle for demonstration purposes.
Scale bars corresponding to $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (c)
Optical microscopy pictures of a diced silicon wafer with two channels and a
thin wall in-between. We etched the tranches
$186\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (top) and
$277\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (bottom) deep into the silicon
wafer. The thin wall is $6.4\pm 0.2$\mathrm{\SIUnitSymbolMicro m}$$ wide (top)
and $8.4\pm 0.5$\mathrm{\SIUnitSymbolMicro m}$$ wide (bottom). The channels
have an etch angle of $89.83\degree$. Scale bar corresponding to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Due to the limitations in the etch depth of the two-step Bosch process, we
analyzed and optimized a three-step Bosch process for small high-aspect-ratio
structures. Small structures typically rely on thin photoresist layers to
ensure appropriate resolution during the preceding photolithography step as
vertical resist walls are yet unobtainable, limiting the etch depth to around
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Therefore, the combination of
deep etching with thin PR layers demands a process with high selectivity, such
as achievable through the means of a three-step Bosch process. Here, the
anisotropic etching step, where ions are accelerated towards the target with
platen power, solely focuses on etching away the thin polymer layer formed on
the bottom of the trench in the previous passivation step (Fig. 1 (a)).
Therefore, the anisotropic etching step has a much shorter duration in
comparison to the two-step Bosch process. A second etching step is introduced,
in which the platen power, which is responsible for the ion and radical
acceleration, is switched off, and the $SF_{6}$ gas-flow is increased by a
factor of four. In this second etching step, the etching happens predominately
chemical, leading to a significant increase in selectivity. One of the main
challenges of the three-step Bosch process is the fabrication of structures
with etch angles below $90\text{\,}\mathrm{\SIUnitSymbolDegree}$, which sets a
limit to the minimal producible feature size due to possible structural
instabilities at the bottom of the etched features. The etch angle of a
standard three-step Bosch process (Fig 3 (a)) is around
$87\text{\,}\mathrm{\SIUnitSymbolDegree}$ \-
$88\text{\,}\mathrm{\SIUnitSymbolDegree}$, depending on the etch depth.
Through precise adjustment of the anisotropic etching step and the chamber
pressure, we achieved an etch angle of
$$89.6\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$0.13\text{\,}\mathrm{\SIUnitSymbolDegree}$$
reproducible for various trench geometries (see Supporting Figure S5). To be
more specific, we lowered the pressure during the second step, i.e., the
breakthrough step, leading to a higher mean free path for the ions being
accelerated towards the sample and thus decreasing the sidewall etch.
Furthermore, we adjusted the passivation and etching times to our specific
design, leading to an overall thicker polymer layer while still ensuring
successful removal at the microchannels’ bottom. Depending on the etched
design, the passivation time might require individual adjustments to prevent
black silicon formation. Please refer to Tables 3 and 4 in the Supporting
Information for a detailed overview of all the processes’ parameters. It is
important to note that all parameters shown in Figure 3 (a) rely on a short
response time of the machine components and, therefore, high-end equipment
might be necessary. With the previously introduced adjustments, we were able
to achieve an etch angle of
$$89.6\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$0.13\text{\,}\mathrm{\SIUnitSymbolDegree}$$
enabling us to produce micrometer-sized pillars that maintained their
stability even at $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ etch depth.
Compared to a standard three-step Bosch process, where the smallest stable
structure was a $40\text{\,}$x$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$
pillar (top of Fig. 3 (b)), we were able to produce a
$40\text{\,}$x$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pillar with our
improved process (bottom of Fig. 3 (b)), lowering the critical feature size by
a factor of 4. Furthermore, our approach demonstrates suitability for
microfluidic applications. For this purpose, we etched two microfluidic
channels $186\text{\,}\mathrm{\SIUnitSymbolMicro m}$ deep into the silicon
with a distance of only $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ resulting in
a thin silicon wall (top of Figure 3 (c) ). To ensure the stability of the
produced structures, we bonded glass to the top of the silicon wafer and
applied a water flow of $7.5\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$
through one of the microfluidic channels to prove the wall’s impermeability
(see Supporting Figure S2). By increasing the distance of the two channels to
$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$, we were able to increase the etch
depth to $277\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (bottom of Figure 3 (c)
). Finally, we were able to etch more than
$450\text{\,}\mathrm{\SIUnitSymbolMicro m}$ deep into the silicon with a thin
PR layer of $1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (see Supporting Figure
S3), illustrating the high flexibility of our approach as required for complex
miniaturization in novel biomedical lab-on-a-chip devices.

##  3 Conclusion

In this work, we tackled two fundamental challenges of DRIE etching, namely
the RIE lag and high aspect ratio etching of small structures, and described
the optimized processes in detail. Through an improved two-step Bosch process,
we were able to etch trenches of various widths within a difference in etch
depth of less than $1.5\text{\,}\mathrm{\char 37\relax}$ while keeping the
simplicity of the Bosch process to ensure the broad and well-established
applicability of this procedure. Further, we optimized a three-step Bosch
process while focusing on the reproducibility of etch angles close to
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ and improving the overall
selectivity. With our refined process, we were able to etch a
$$40\text{\,}$x$5\text{\,}$x$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
pillar, lowering the critical feature size compared to the standard process by
a factor of 4. Additionally, we etched a $6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ thin wall in between two microchannels down to a depth of
$186\text{\,}\mathrm{\SIUnitSymbolMicro m}$, while maintaining impermeability
to water which is of substantial importance for subsequent lab-on-chip
applications. Our novel and improved procedures are of interest to a vast
number of research groups as they allow for the fabrication of novel
structures and devices as well as improving the performance of already
existing applications.

##  4 Experimental Section

All processes were carried out on standard single side polished silicon wafers
($500\pm$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness, Prolog Semicor,
Ukraine). To determine the limits in terms of etch depth for the Bosch-
process, we thermally evaporated $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$
$SiO_{2}$ on the polished side of some silicon wafers.

First, we processed the wafers in a yellow room with standard photolithography
processes. To enhance the adhesion of photoresist (PR), we deposited
hexamethyldisilazane (HMDS) on the wafers for $30\text{\,}\mathrm{s}$. Then,
$4\text{\,}\mathrm{ml}$ of PR was dropped on the polished side of the silicon
wafers. We used two different resists, namely S1813 (Shipley, United Kingdom)
and AZ nLOF 2070 (Microchemicals, Germany). We spin coated the resists at
$\mathrm{4}\mathrm{0}\mathrm{0}\mathrm{0}$ r.p.m. for $35\text{\,}\mathrm{s}$
onto the wafers resulting in a resist thickness of $1.4\pm
0.1$\mathrm{\SIUnitSymbolMicro m}$$ and $7.2\pm 0.2$\mathrm{\SIUnitSymbolMicro
m}$$ for the S1813 and the AZ nLOF 2070, respectively. Next, the samples were
soft-backed for $1\text{\,}\mathrm{min}$ per micrometre resist thickness at
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to evaporate solvents in the PR
layer. Afterwards, the wafers were re-hydrated within the cleanroom (40-50 %
humidity) for 10 minutes per micrometre.

After finishing the PR deposition, the samples were transferred to the Mask
Aligner (MABA6, Süss Microtec, Germany) and exposed to UV-light from a mercury
lamp. The exposure time depends on the exposure dose of the resists and the
measured intensity of the lamp. S1813 at
$1.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness required an exposure
dose of $230\text{\,}\mathrm{mJ}\text{\,}{\mathrm{cm}}^{-2}$ and thus, given
the UV lamps intensity of
$7.4\text{\,}\mathrm{mW}\text{\,}{\mathrm{cm}}^{-2}$, needed to be exposed for
$30\text{\,}\mathrm{s}$ in our machine. AZ nLOF 2070 at
$7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness required an exposure dose
of $170\text{\,}\mathrm{mJ}\text{\,}{\mathrm{cm}}^{-2}$ and thus, needed to be
exposed for $23\text{\,}\mathrm{s}$ in our machine. Further, we set the
contacting settings to hard vacuum.

Finally, we developed the wafers. AZ351B (Microchemicals, Germany) was diluted
with a ratio of 1:5 developer to distilled water and has been utilized for the
development of S1813. AZ 726 MIF (Microchemicals, Germany) has been used
undiluted for the development of AZ nLOF 2070. The development was assisted by
putting the Petri dish with developer solution and wafer into an ultrasonic
bath (ultrasonic power set to level 3 of 10). The wafers were removed from the
developer and put into a water bath to stop the development when the silicon
or the silicon dioxide gets visible. This typically happened after
$15\text{\,}\mathrm{s}$ for S1813 and after $60\text{\,}\mathrm{s}$ for AZ
nLOF 2070.

For the wafers covered with $SiO_{2}$, we transferred the samples into a
Reactive Ion Etching (RIE) machine (PlasmaPro NGP80, Oxford Instruments,
United Kingdom) and etched the complete $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ $SiO_{2}$ layer of the exposed areas. The process we used consisted of two
steps that were repeated twelve times. The first step (etch) consisted of 5
minutes $CHF_{3}$ flow at $40$ sscm, $CF_{4}$ flow at $40$ sscm, $O_{2}$ flow
at $5$ sscm and $130\text{\,}\mathrm{W}$ high frequency (HF,
$13.56\text{\,}\mathrm{MHz}$) power. Then, all flows and the HF power were
turned of for 5 minutes to let the sample cool down. All process steps were
carried out at $15\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and at 15 mTorr.

After photolithography or RIE etching, respectively, the samples were
transferred into the deep reactive ion etching (DRIE) machine (PlasmaPro
Estrelas100, Oxford Instruments, United Kingdom). The processes carried out
here are well described in the result section of this publication. Finally,
the wafers were diced into small pieces with a dicing SAW (DAD$3221$, Disco
corporation, Germany). We used a hub-blade (FTB R46 45130, Disco corporation,
Germany) and diced at 30000 rpm with
$0.5\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$ to avoid damaging the
small structures. However, as can be seen in Figure 2 (c), damage of the
wafers’ surface at the top of microfluidic channels could not be avoided
completely.

For evaluation of the results, the diced samples were inspected under a
microscope (Axioscope, Zeiss, Germany). We utilized a blue LED (505 nm) from
the top and no backlight to achieve the highest possible contrast. We
connected a camera with a one-inch sensor to the microscope and evaluated the
sample at two different magnifications, 20x and 40x respectively. Thin
structures such as pillars tend to rip off during dicing due to the high
momentum. These structures were inspected using a scanning electron microscope
(Nova NanoSEM 450, Thermo Fisher Scientific, USA). We tilted the samples to
examine the structures in all three spatial dimensions.

## acknowledgements

The authors would like to express their gratitude to Rasmus Pedersen from
Oxford Instruments for his continous support regarding etch process
optimisation. This work is supported by the ETH Zurich and, in part, by an
interdisciplinary grant from the Swiss National Science Foundation (Grant
Number CR22I2_166110) to B.J.N..

## conflict of interest

The authors declare no conflict of interest

## supporting information

Supporting Information is available from the author

## References

  * GRAYSON et al. [2004] GRAYSON ACR, SHAWGO RS, JOHNSON AM, FLYNN NT, YAWEN LI, CIMA MJ, et al.  A BioMEMS review: MEMS technology for physiologically integrated devices.  Proceedings of the IEEE 2004;92(1):6–21.
  * Whitesides [2006] Whitesides GM.  The origins and the future of microfluidics.  Nature 2006;442(7101):368–373.  https://doi.org/10.1038/nature05058.
  * Dittrich et al. [2006] Dittrich PS, Tachikawa K, Manz A.  Micro Total Analysis Systems. Latest Advancements and Trends.  Analytical Chemistry 2006;78(12):3887–3908.  https://doi.org/10.1021/ac0605602, pMID: 16771530.
  * Wu et al. [2010] Wu B, Kumar A, Pamarthy S.  High aspect ratio silicon etch: A review.  Journal of Applied Physics 2010;108(5):051101.  https://doi.org/10.1063/1.3474652.
  * Franz Laermer, Andrea Schilp [granted on 26 March 1996] Franz Laermer, Andrea Schilp RBG, Method of anisotropically etching silicon; granted on 26 March 1996.  https://patents.google.com/patent/US5501893A/en.
  * Laermer and Urban [2019] Laermer F, Urban A.  MEMS at Bosch – Si plasma etch success story, history, applications, and products.  Plasma Processes and Polymers 2019 sep;16(9):1800207.
  * Laermer et al. [2020] Laermer F, Franssila S, Sainiemi L, Kolari K.  Deep reactive ion etching.  In: Handbook of Silicon Based MEMS Materials and Technologies Elsevier; 2020.p. 417–446.
  * Chen et al. [2007] Chen SC, Lin YC, Wu JC, Horng L, Cheng CH.  Parameter optimization for an ICP deep silicon etching system.  Microsystem Technologies 2007 mar;13(5-6):465–474.  https://link.springer.com/article/10.1007/s00542-006-0211-2.
  * Chung [2004] Chung CK.  Geometrical pattern effect on silicon deep etching by an inductively coupled plasma system.  Journal of Micromechanics and Microengineering 2004 apr;14(4):656–662.  https://iopscience.iop.org/article/10.1088/0960-1317/14/4/029.
  * Lai et al. [2006] Lai SL, Johnson D, Westerman R.  Aspect ratio dependent etching lag reduction in deep silicon etch processes.  Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 2006 Jul;24(4):1283–1288.  https://doi.org/10.1116/1.2172944.
  * Fu et al. [2001] Fu L, Miao JM, Li XX, Lin RM.  Study of deep silicon etching for micro-gyroscope fabrication.  Applied Surface Science 2001 jun;177(1-2):78–84.  https://linkinghub.elsevier.com/retrieve/pii/S0169433201001970.
  * Laermer and Urban [2003] Laermer F, Urban A.  Challenges, developments and applications of silicon deep reactive ion etching.  Microelectronic Engineering 2003 jun;67-68:349–355.  https://linkinghub.elsevier.com/retrieve/pii/S0167931703000893.
  * Abdolvand and Ayazi [2008] Abdolvand R, Ayazi F.  An advanced reactive ion etching process for very high aspect-ratio sub-micron wide trenches in silicon.  Sensors and Actuators A: Physical 2008 may;144(1):109–116.  https://linkinghub.elsevier.com/retrieve/pii/S0924424708000046.
  * Parasuraman et al. [2014] Parasuraman J, Summanwar A, Marty F, Basset P, Angelescu DE, Bourouina T.  Deep reactive ion etching of sub-micrometer trenches with ultra high aspect ratio.  Microelectronic Engineering 2014;113:35–39.  http://dx.doi.org/10.1016/j.mee.2013.06.010.
  * Morton et al. [2008] Morton KJ, Nieberg G, Bai S, Chou SY.  Wafer-scale patterning of sub-40 nm diameter and high aspect ratio (>50:1) silicon pillar arrays by nanoimprint and etching.  Nanotechnology 2008 aug;19(34):345301.  https://iopscience.iop.org/article/10.1088/0957-4484/19/34/345301.
  * Owen et al. [2012] Owen KJ, VanDerElzen B, Peterson RL, Najafi K.  High aspect ratio deep silicon etching.  In: 2012 IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS) No. February, IEEE; 2012. p. 251–254.  http://ieeexplore.ieee.org/document/6170138/.
  * Blauw et al. [2001] Blauw MA, Zijlstra T, van der Drift E.  Balancing the etching and passivation in time-multiplexed deep dry etching of silicon.  Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures 2001;19(6):2930.

## Table of Contents

Reactive ion etching with the Bosch process is one of the key procedures used
to manufacture micron-sized structures for MEMS and microfluidic applications.
We introduce an optimized Bosch process that reduces the etch lag to below
$1.5\text{\,}\mathrm{\char 37\relax}$. Furthermore, we improved a three-step
Bosch process, allowing the fabrication of
$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thin structures at depths up to
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ while maintaining their stability.


# A purification postulate for quantum mechanics with indefinite causal order

Mateus Araújo  Faculty of Physics, University of Vienna, Boltzmanngasse 5 1090
Vienna, Austria  Institute for Quantum Optics and Quantum Information (IQOQI),
Boltzmanngasse 3 1090 Vienna, Austria  Institute for Theoretical Physics,
University of Cologne, Germany  Adrien Feix  Faculty of Physics, University of
Vienna, Boltzmanngasse 5 1090 Vienna, Austria  Institute for Quantum Optics
and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria  Miguel
Navascués  Institute for Quantum Optics and Quantum Information (IQOQI),
Boltzmanngasse 3 1090 Vienna, Austria  Časlav Brukner  Faculty of Physics,
University of Vienna, Boltzmanngasse 5 1090 Vienna, Austria  Institute for
Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna,
Austria

(27th August 2024)

###### Abstract

To study which are the most general causal structures which are compatible
with local quantum mechanics, Oreshkov _et al._ [1] introduced the notion of a
process: a resource shared between some parties that allows for quantum
communication between them without a predetermined causal order. These
processes can be used to perform several tasks that are impossible in standard
quantum mechanics: they allow for the violation of causal inequalities, and
provide an advantage for computational and communication complexity.
Nonetheless, no process that can be used to violate a causal inequality is
known to be physically implementable. There is therefore considerable interest
in determining which processes are physical and which are just mathematical
artefacts of the framework. Here we make key progress in this direction by
proposing a purification postulate: processes are physical only if they are
purifiable. We derive necessary conditions for a process to be purifiable, and
show that several known processes do not satisfy them.

##  1 Introduction

It is widely believed that any future theory that unifies quantum mechanics
and gravity will feature quantum uncertainty in the metric tensor [2], which
should produce indefinite causal structures. Our understanding of the notion
of indefinite causal structures is, however, still lacking. To investigate
that, one approach is to consider simple, concrete models that are compatible
with the laws of quantum information processing. One such model – the process
matrix formalism – was introduced by Oreshkov _et al._ as the most general
causal structure that can connect local laboratories where standard quantum
mechanics is valid without creating paradoxical causal loops [1].

These processes have been shown to enable the realization of tasks that are
otherwise impossible: they allow for the violation of causal inequalities [1,
3, 4, 5, 6, 7, 8], can be detected by causal witnesses [9], provide an
advantage in quantum computation [10, 11, 12], and enable a reduction in
communication complexity [13, 14]. But even though one such process – the
quantum switch – has been implemented experimentally [15, 16], in general it
is not known if all process are physical or some of them are just mathematical
artefacts of the formalism. They were, after all, defined only from the
requirement of not generating logical contradictions, and processes realisable
in nature are likely to fulfil additional physical constraints.

One can, therefore, look for requirements beyond mere logical consistency in
order to shed light on the physicality of these processes. As seen from the
search for physical principles to determine the set of quantum correlations,
such meta-theoretical principles can provide nontrivial constraints on the
possible theories [17, 18, 19, 20, 21]. The principle we choose to investigate
here is reversibility of the transformations between quantum states [22]: it
is, after all, valid in all fundamental physical theories, and has been a
central ingredient in all of the reconstructions of quantum mechanics to date
[23, 24, 25, 26, 27, 28, 29].

To define what reversibility means for processes we first needed to generalise
their definition: we extend them with a global past and a global future, so
that they can be seen as inducing a transformation from quantum states in the
past to quantum states in the future, after passing through the causally
indefinite region of the local laboratories. We can then define pure processes
as those that preserve the reversibility of the underlying operations, i.e.,
those that induce a unitary transformation from the past to the future
whenever unitary transformations are also applied in the local laboratories.

With these definitions in hand, we can propose the purification postulate:
processes are physical only if they are purifiable, i.e., if they can be
expressed as a part of a pure process in a larger space. It turns out to be
rather difficult to determine if a given process is purifiable. We derive,
then, a necessary but not sufficient condition for purifiability, and show
that several known processes fail to satisfy this condition, among which is
the process from Ref. [1] that was the first one shown to violate a causal
inequality. In fact, we haven’t found a purification for any bipartite process
that was able to violate a causal inequality. There exists, however, a
tripartite process which is purifiable and able to violate causal inequalities
[6, 30]. If one takes the view that the violation of causal inequalities is
impossible in Nature, as done in Ref. [31], this implies that being purifiable
is not a sufficient condition for a process to be physical.

The paper is organized as follows: in Section 2 we generalise the definition
of a process and introduce the notion of pure processes. In Section 3 we
propose the purification postulate. In Section 4 we derive necessary and
sufficient conditions for a process to be purifiable, and in Section 5 we
derive a necessary but not sufficient condition for purifiability that can be
easily tested. In Section 6 we apply our condition to several known processes,
and in Section 7 we present a tripartite pure process that violates causal
inequalities.

##  2 Pure processes

In Ref. [1], the process matrix was introduced as the most general way to
allow two parties Alice and Bob to communicate – not necessarily in a causally
ordered way – but without creating paradoxes. These parties were assumed to
obey the laws of quantum mechanics locally, and the no-paradox condition means
that whatever probabilities Alice and Bob may extract from their local
experiments will be positive and normalised. In other words, the process
matrix was defined as a (multi)linear function that takes completely positive
(CP) maps to probabilities.

To be able to talk about purification, we need to extend this definition to
take CP maps not to probabilities, but to other CP maps. This view of a
process as a higher-order transformation is much in the spirit of quantum
combs [32], but with the crucial difference that combs are usually defined to
be causally ordered. Note also that it can be recovered from the multipartite
definition of process presented for example in Ref. [9]. We shall,
nevertheless, explicitly define bipartite processes as higher-order
transformations for clarity and to make this paper self-contained. The
multipartite case can be obtained by a straightforward generalisation of the
arguments presented here or from Ref. [9].

To this end, let $\mathcal{A}$ be a completely positive trace preserving
(CPTP) map that takes the Hilbert spaces111$A_{I}$ (and its analogues) is
defined as the set of $d_{A_{I}}\times d_{A_{I}}$ complex matrices.
$A_{I},A_{I}^{\prime}$ to $A_{O},A_{O}^{\prime}$, and $\mathcal{B}$ a CPTP map
that takes $B_{I},B_{I}^{\prime}$ to $B_{O},B_{O}^{\prime}$. A process is then
defined as the most general linear transformation that acts trivially on
$A_{I}^{\prime},B_{I}^{\prime},A_{O}^{\prime},B_{O}^{\prime}$ and takes
$\mathcal{A}$ and $\mathcal{B}$ to a CPTP map
$\mathcal{G}_{\mathcal{A},\mathcal{B}}$ from $A_{I}^{\prime},B_{I}^{\prime},P$
to $A_{O}^{\prime},B_{O}^{\prime},F$, as represented in Fig. 1. Note that the
Hilbert space $F$ cannot signal to any other Hilbert space, and can therefore
be interpreted as a global future. Conversely, none of the Hilbert spaces can
signal to $P$, which can then be interpreted as a global past.

$\mathcal{G}_{\mathcal{A},\mathcal{B}}$$A_{I}^{\prime}$$A_{O}^{\prime}$$B_{I}^{\prime}$$B_{O}^{\prime}$$F$$P$

$\quad=\quad$

$\mathcal{A}$$W$$\mathcal{B}$$A_{O}$$A_{O}^{\prime}$$A_{I}$$A_{I}^{\prime}$$B_{O}$$B_{O}^{\prime}$$B_{I}$$B_{I}^{\prime}$$F$$P$

Figure 1: A process $W$ is a bilinear function from the CPTP maps
$\mathcal{A},\mathcal{B}$ to a CPTP map
$\mathcal{G}_{\mathcal{A},\mathcal{B}}$. This map takes states from the global
past $P$ and auxiliary spaces $A_{I}^{\prime},B_{I}^{\prime}$ to the global
future $F$ and auxiliary spaces $A_{O}^{\prime},B_{O}^{\prime}$.

To find a matrix representation for a process, we make use of the Choi-
Jamiołkowski (CJ) isomorphism, which we recap in Appendix A. Let then
$A=\mathfrak{C}(\mathcal{A})$, $B=\mathfrak{C}(\mathcal{B})$, and
$G_{A,B}=\mathfrak{C}\mathopen{}\mathclose{{}\left(\mathcal{G}_{\mathcal{A},\mathcal{B}}}\right)$
be the CJ representations of the CPTP maps $\mathcal{A}$, $\mathcal{B}$, and
$\mathcal{G}_{\mathcal{A},\mathcal{B}}$. By linearity we can represent the
mapping from $A,B$ to $G_{A,B}$ as

$G_{A,B}=\operatorname{tr}_{A_{I}A_{O}B_{I}B_{O}}[W^{T_{A_{I}A_{O}B_{I}B_{O}}}(A\otimes
B)],$ (1)

where $W\in P\otimes A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes F$ is
the process matrix we are defining, $\cdot^{T_{A_{I}A_{O}B_{I}B_{O}}}$ denotes
partial transposition on the subsystems $A_{I},A_{O},B_{I},$ and $B_{O}$, and
identity matrices on the subsystems $P$, $F$, $A_{I}^{\prime}$,
$A_{O}^{\prime}$, $B_{I}^{\prime}$, and $B_{O}^{\prime}$ are left implicit.
This formula can be rewritten in a more convenient way using the link product,
which was designed to conveniently express the CJ representation of a
composition of quantum operations [32], and we recap in Appendix B:

$G_{A,B}=W*(A\otimes B).$ (2)

We need $G_{A,B}$ to be a valid CPTP map for all valid CPTP maps $A$ and $B$.
This imposes the following restrictions on $W$, which we derive in Appendix C:

$\displaystyle W\geq 0,$ (3)
$\displaystyle\operatorname{tr}W=d_{A_{O}}d_{B_{O}}d_{P},$ (4) $\displaystyle
W=L_{V}(W),$ (5)

where $L_{V}$ is a projector on the linear subspace of valid process matrices
defined in equation (64).

As an explicit example of a process defined as a higher-order transformation
consider a situation where a state in $P=P_{1}\otimes P_{2}$ is sent to Alice
and Bob, and the resulting state from Alice and Bob is sent to $F=F_{1}\otimes
F_{2}$. The process matrix is given by
$\mathopen{}\mathclose{{}\left|w_{\text{state}}\middle\rangle\middle\langle
w_{\text{state}}}\right|$, where

$|w_{\text{state}}\rangle={|{\mathds{1}}\rangle\\!\rangle}^{P_{1}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{P_{2}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}F_{1}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}F_{2}},$
(6)

and ${|{\mathds{1}}\rangle\\!\rangle}^{P_{1}A_{I}}$ is the “pure” CJ
representation of the identity map between $P_{1}$ and $A_{1}$, as defined in
Appendix A. Here and throughout the paper we shall use the superscript of a
vector to indicate the Hilbert space to where its projector belongs, e.g.,
${|{\mathds{1}}\rangle\\!\rangle}^{P_{1}A_{I}}$ means that
${{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}}\in
P_{1}\otimes A_{I}$.

A less trivial example is the process where a state in $P$ goes first to
Alice, then to Bob, and finally to $F$. Using again the vector representation
the process is

$|w_{\text{channel}}\rangle={|{\mathds{1}}\rangle\\!\rangle}^{PA_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}F}.$
(7)

Finally, an example of a process that encodes an indefinite causal order is
the quantum switch [10, 9], which in this representation is:

$|w_{\text{switch}}\rangle=|0\rangle^{P_{1}}{|{\mathds{1}}\rangle\\!\rangle}^{P_{2}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}F_{2}}|0\rangle^{F_{1}}+|1\rangle^{P_{1}}{|{\mathds{1}}\rangle\\!\rangle}^{P_{2}B_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathds{1}}\rangle\\!\rangle}^{A_{O}F_{2}}|1\rangle^{F_{1}}.$
(8)

To connect this version of the quantum switch with the one in Refs. [10, 9],
one should prepare the control qubit and send it to $P_{1}$, and prepare the
target qubit and send it to $P_{2}$. After the process is done the control
qubit will be found in $F_{1}$ (unchanged), and the target qubit in $F_{2}$.

This definition of processes allows a natural definition of what it means for
a process to be “pure”. Just as we can define unitaries as the most general
linear transformations that map pure states to pure states of the same
dimension222More precisely, a linear map $\mathcal{E}$ is a unitary iff for
all pure states $|\psi\rangle$ the transformed state
$\mathcal{I}\otimes\mathcal{E}(\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|)$
is a pure state of the same dimension., we can define pure processes to be the
most general linear transformations that map unitaries to unitaries. More
precisely:

######  Definition 1.

A process $W$ is pure if for all unitaries $\mathcal{A}$, $\mathcal{B}$ the
resulting transformation $\mathcal{G}_{\mathcal{A},\mathcal{B}}$ is also a
unitary.

It turns out that pure processes are unitary transformations from
$A_{O},B_{O},P$ to $A_{I},B_{I},F$ and can be conveniently represented as
vectors, as shown in the following theorem:

######  Theorem 2.

A process $W$ is pure if and only if
$W={{|{U_{w}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{w}}|}}$ for some
unitary $U_{w}$.

###### Proof.

If $W$ is pure, then $\mathcal{G}_{\mathcal{A},\mathcal{B}}$ must be a unitary
in particular when $\mathcal{A}$ and $\mathcal{B}$ are SWAPs that map
$A_{I}^{\prime},A_{I}$ to $A_{O},A_{O}^{\prime}$ and $B_{I}^{\prime},B_{I}$ to
$B_{O},B_{O}^{\prime}$. Then

$G_{A,B}=W*\mathopen{}\mathclose{{}\left({{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}}^{A_{I}^{\prime}A_{O}}{{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}}^{A_{I}A_{O}^{\prime}}\otimes{{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}}^{B_{I}^{\prime}B_{O}}{{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}}^{B_{I}B_{O}^{\prime}}}\right)=W,$
(9)

so the resulting transformation
$\mathcal{G}_{\mathcal{A},\mathcal{B}}=\mathfrak{C}^{-1}\mathopen{}\mathclose{{}\left(G_{A,B}}\right)=\mathfrak{C}^{-1}(W)=:\mathcal{W}$
is just the process itself, with the relabelling $A_{I}\mapsto
A_{O}^{\prime}$, $A_{O}\mapsto A_{I}^{\prime}$, $B_{I}\mapsto B_{O}^{\prime}$,
and $B_{O}\mapsto B_{I}^{\prime}$, so $\mathcal{W}$ must be a unitary
transformation from $A_{O},B_{O},P$ to $A_{I},B_{I},F$. Writing
$\mathcal{W}(\rho)=U_{w}\rho U_{w}^{\dagger}$, we have that its CJ
representation is
$W=\mathfrak{C}(\mathcal{W})={{|{U_{w}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{w}}|}}$.

Conversely, if
$W={{|{U_{w}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{w}}|}}$,
$A={{|{U_{a}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{a}}|}}$, and
$B={{|{U_{b}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{b}}|}}$, then

$G_{A,B}={{|{U_{w}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{w}}|}}*\mathopen{}\mathclose{{}\left({{|{U_{a}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{a}}|}}\otimes{{|{U_{b}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{b}}|}}}\right)={{|{U_{g}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{g}}|}},$
(10)

where

$U_{g}^{A_{I}^{\prime}B_{I}^{\prime}P}=\operatorname{tr}_{A_{I}B_{I}}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(U_{w}^{PA_{I}B_{I}}\otimes\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}}}\right)\mathopen{}\mathclose{{}\left(\mathds{1}^{P}\otimes
U_{a}^{A_{I}A_{I}^{\prime}}\otimes
U_{b}^{B_{I}B_{I}^{\prime}}}\right)}\right].$ (11)

Since by assumption $\mathcal{G}_{\mathcal{A},\mathcal{B}}$ is trace
preserving, we have that
$\operatorname{tr}_{A_{O}^{\prime}B_{O}^{\prime}F}G_{A,B}=\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}P}$.
Substituting
$G_{A,B}={{|{U_{g}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{g}}|}}$ we get

$\operatorname{tr}_{A_{O}^{\prime}B_{O}^{\prime}F}{{|{U_{g}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{g}}|}}=U_{g}^{\dagger}U_{g}=\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}P}.$
(12)

Since, furthermore, $U_{g}$ is a square finite-dimensional matrix, it has a
right inverse which is equal to its left inverse, so $U_{g}$ is a unitary, and
so is $\mathcal{G}_{\mathcal{A},\mathcal{B}}$. ∎

A different definition of purity for process matrices was used in the Appendix
A of Ref. [9]: there they defined pure processes simply as those that can be
written as vectors, i.e., that are proportional to rank-one projectors. We are
not going to use this definition because rank-one processes do not necessarily
induce reversible transformations between quantum states, and therefore fail
to capture the essential feature we want from “purity”. This is because for
some isometries $V_{w}$ the matrix
${{|{V_{w}}\rangle\\!\rangle}\\!{\langle\\!\langle{V_{w}}|}}$ is a valid rank-
one process, that however maps unitaries $\mathcal{A}$ and $\mathcal{B}$ to a
non-unitary isometry $\mathcal{G}_{\mathcal{A},\mathcal{B}}$, which increases
the dimension of the Hilbert space and is therefore not reversible. Moreover,
this definition is trivial in the sense that it would make every process
purifiable, as can be seen from the discussion in Section 4.

##  3 A purification postulate

With the definition of a pure process in hand, we can now define what it means
to purify a process:

######  Definition 3.

A process $W\in P\otimes A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes
F$ is purifiable if one can recover it from a pure process $S\in P\otimes
P^{\prime}\otimes A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes F\otimes
F^{\prime}$ by inputting the state333We can fix the state to be $|0\rangle$
instead of allowing an arbitrary state without loss of generality. $|0\rangle$
in $P^{\prime}$ and tracing out $F^{\prime}$, i.e., if

$W=S*(\mathopen{}\mathclose{{}\left|0\middle\rangle\middle\langle
0}\right|^{P^{\prime}}\otimes\mathds{1}^{F^{\prime}}).$ (13)

Note that there are no restrictions on the dimensions of $P,P^{\prime},F$, and
$F^{\prime}$, so for instance a pure process $W$ is trivially purifiable by
setting $S=W$ and $d_{P^{\prime}}=d_{F^{\prime}}=1$. One can also ask whether
a process $W$ with trivial $P$ and $F$ is purifiable by setting
$d_{P}=d_{F}=1$, as shown in Fig. 2. This case is in fact the focus of this
paper, as most processes considered in the literature have trivial $P$ and $F$
and those are the ones we shall test for purifiability.

$W$$A_{O}$$A_{I}$$B_{O}$$B_{I}$

$\quad=\quad$

$S$$A_{O}$$A_{I}$$B_{O}$$B_{I}$$F^{\prime}$$P^{\prime}$$|0\rangle$

Figure 2: A process $W$ with trivial $P$ and $F$ is purifiable if it can be
recovered from a pure process $S$ by inputting the state $|0\rangle$ in
$P^{\prime}$ and tracing out $F^{\prime}$.

We propose then the purification postulate: a process is physical only if it
is purifiable. This postulate is motivated by the fact that a non-purifiable
process would fundamentally map unitaries onto isometries or non-unitary CPTP
maps, destroying the reversibility of the theory. Reversibility, in its turn,
is a cherished principle [22]: all fundamental physical theories are time
reversible444Note that to treat the collapse of the wave function during a
measurement as an irreversible process one needs to use objective collapse
models [33, 34, 35] instead of standard quantum mechanics. We are taking the
view that collapse is not a physical process. and the hint that the formation
and evaporation of a black hole might be an irreversible evolution is one of
the major problems in modern physics [36, 37]. Moreover, in all
reconstructions of quantum mechanics to date [23, 24, 25, 26, 27, 28, 29]
reversibility has been used as a central ingredient. This supports the idea
that it is indeed a fundamental part of quantum mechanics, and it should not
be done away with lightly. Furthermore, although there is speculation that in
a full theory of quantum gravity new degrees of freedom are created by the
expansion of the Universe [38, 39, 40], in concrete models these degrees of
freedom are born in the vacuum state [41], making the time evolution actually
reversible. In any case, current inflationary models using quantum field
theory on a curved expanding spacetime do not feature creation of new degrees
of freedom [42].

As a sanity check, note that all processes known to be physical – all
processes where the order between the parties is fixed, e.g.
$|w_{\text{channel}}\rangle$ (equation (7)), controlled by a quantum system,
e.g. $|w_{\text{switch}}\rangle$ (equation (8)), or incoherently mixed – are
purifiable.

##  4 Necessary and sufficient conditions for purification

We shall now derive necessary and sufficient conditions for a process $W$ with
trivial $P$ and $F$ to be purifiable, i.e., to be recoverable via equation
(13) from a pure process $S$. From Theorem 2, we know that $S$ is pure iff
$S={{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}$
for some unitary $U_{s}$. Defining

$|w_{\psi}\rangle^{A_{I}A_{O}B_{I}B_{O}F^{\prime}}:=\langle\psi^{*}|^{P^{\prime}}{|{U_{s}}\rangle\\!\rangle}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}$
(14)

and noting that

${{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}*(\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|^{P^{\prime}}\otimes\mathds{1}^{F^{\prime}})=\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|^{A_{I}A_{O}B_{I}B_{O}F^{\prime}}*\mathds{1}^{F^{\prime}}=\operatorname{tr}_{F^{\prime}}\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|^{A_{I}A_{O}B_{I}B_{O}F^{\prime}},$ (15)

we can rewrite equation (13) as

$W^{A_{I}A_{O}B_{I}B_{O}}=\operatorname{tr}_{F^{\prime}}\mathopen{}\mathclose{{}\left|w_{0}\middle\rangle\middle\langle
w_{0}}\right|^{A_{I}A_{O}B_{I}B_{O}F^{\prime}},$ (16)

where $|w_{0}\rangle^{A_{I}A_{O}B_{I}B_{O}F^{\prime}}=\langle
0|^{P^{\prime}}{|{U_{s}}\rangle\\!\rangle}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}$.

We can also state the condition that ${|{U_{s}}\rangle\\!\rangle}$ is a valid
process purely in terms of the vectors555From now on we shall write
${|{U_{s}}\rangle\\!\rangle}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}$ and
$|w_{\psi}\rangle^{A_{I}A_{O}B_{I}B_{O}F^{\prime}}$ without their superscripts
for brevity. $|w_{\psi}\rangle$. To do this, first note that, as discussed in
Appendix C, we only need to consider the ancillary spaces
$A_{I}^{\prime},A_{O}^{\prime},B_{I}^{\prime}$, and $B_{O}^{\prime}$ in the
definition of processes to conclude that a process matrix must be positive
semidefinite. Since
${{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}$ must already be
positive semidefinite from its form, the definition of process reduces to
saying that ${{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}$ is
valid iff
${{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*(A\otimes B)$ is
a CPTP map for all CPTP maps $A\in A_{I}\otimes A_{O}$ and $B\in B_{I}\otimes
B_{O}$.

In its turn,
${{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*(A\otimes B)$ is
a CPTP map iff for all input states $|\psi\rangle^{P^{\prime}}$ its output
$\Big{[}{{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*(A\otimes
B)\Big{]}*\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|^{P^{\prime}}$
is a valid quantum state. Note, however, that

$\Big{[}{{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*(A\otimes
B)\Big{]}*\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|^{P^{\prime}}=\Big{(}{{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|^{P^{\prime}}\Big{)}*(A\otimes
B),$ (17)

and the condition that the right hand side is a valid quantum state for all
$|\psi\rangle$, $A$, and $B$ is precisely the condition that the process with
trivial $P^{\prime}$ given by
$({{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}*\mathopen{}\mathclose{{}\left|\psi\middle\rangle\middle\langle\psi}\right|^{P^{\prime}})=\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|$ produces a valid CPTP map (with trivial $P^{\prime}$) when
linked with the CPTP maps $A$ and $B$, which means that
$\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|$ is a valid process. Writing out the validity conditions
explicitly, we have

$\forall|\psi\rangle\quad\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|=L_{V}(\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|)\quad\text{and}\quad\operatorname{tr}\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|=d_{A_{O}}d_{B_{O}},$ (18)

where $L_{V}$ is the projector onto the valid subspace defined in equation
(64) particularized for $d_{P^{\prime}}=1$.

With this, we reduce the problem of purifying a process $W$ to that of finding
a set of vectors $|w_{\psi}\rangle$ such that equations (16) and (18) are
satisfied. We can simplify it further by noting that equation (16) is just the
purification of a positive matrix, which admits a simple solution. Let then
the eigendecomposition of $W$ be

$W=\sum_{i=0}^{r-1}\lambda_{i}\mathopen{}\mathclose{{}\left|\lambda_{i}\middle\rangle\middle\langle\lambda_{i}}\right|^{A_{I}A_{O}B_{I}B_{O}},$
(19)

where $r$ is the rank of $W$, and $\lambda_{i},|\lambda_{i}\rangle$ its
nonzero eigenvalues and corresponding eigenvectors. Then a valid purification
for it is

$|w_{0}\rangle=\sum_{i=0}^{r-1}\sqrt{\lambda_{i}}|\lambda_{i}\rangle^{A_{I}A_{O}B_{I}B_{O}}|i\rangle^{F^{\prime}},$
(20)

which is unique modulo some isometry $V$ on the purifying system. But this
isometry has no effect on the validity of
$\mathopen{}\mathclose{{}\left|w_{\psi}\middle\rangle\middle\langle
w_{\psi}}\right|$, as it only affects the state output by the process, and an
isometry maps valid quantum states to valid quantum states. This implies that
we can choose without loss of generality the isometry to be the identity, and
the dimension of the purifying system $d_{F^{\prime}}=d_{P^{\prime}}$ to be
the rank of $W$. This allows us to use an $r^{2}$-dimensional basis for
$P^{\prime}$ to simplify condition (18). Choosing $\\{|i\rangle\langle
j|\\}_{i,j=0}^{r-1}$ as this basis, and rewriting the equations explicitly in
terms of the vectors $|w_{i}\rangle=\langle
i|^{P^{\prime}}{|{U_{s}}\rangle\\!\rangle}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}$,
the condition translates to

$\forall i,j\quad L_{V}^{\perp}(|w_{i}\rangle\langle
w_{j}|)=0\quad\text{and}\quad\mathopen{}\mathclose{{}\left\langle
w_{i}\middle|w_{j}}\right\rangle=d_{A_{O}}d_{B_{O}}\delta_{ij},$ (21)

where for brevity we are using the projector
$L_{V}^{\perp}:=\mathds{1}-L_{V}$.

We summarize the results of this Section in the following theorem:

######  Theorem 4.

A process matrix $W$ of rank $r$ and eigendecomposition

$W=\sum_{i=0}^{r-1}\lambda_{i}\mathopen{}\mathclose{{}\left|\lambda_{i}\middle\rangle\middle\langle\lambda_{i}}\right|^{A_{I}A_{O}B_{I}B_{O}}$
(22)

is purifiable if and only if there exists a set of vectors
$\\{|w_{i}\rangle\\}_{i=0}^{r-1}$ such that

$|w_{0}\rangle=\sum_{i=0}^{r-1}\sqrt{\lambda_{i}}|\lambda_{i}\rangle^{A_{I}A_{O}B_{I}B_{O}}|i\rangle^{F^{\prime}}$
(23)

and

$\forall i,j\quad L_{V}^{\perp}(|w_{i}\rangle\langle
w_{j}|)=0\quad\text{and}\quad\mathopen{}\mathclose{{}\left\langle
w_{i}\middle|w_{j}}\right\rangle=d_{A_{O}}d_{B_{O}}\delta_{ij}.$ (24)

If they exist, a process
$S={{|{U_{s}}\rangle\\!\rangle}\\!{\langle\\!\langle{U_{s}}|}}$ that purifies
$W$ is given by

$\forall i\quad\langle
i|^{P^{\prime}}{|{U_{s}}\rangle\\!\rangle}^{P^{\prime}A_{I}A_{O}B_{I}B_{O}F^{\prime}}:=|w_{i}\rangle$
(25)

##  5 Necessary condition for purification

It is not easy to find a set of vectors $\\{|w_{i}\rangle\\}$ that satisfies
the conditions of Theorem 4, as the constraint
$L_{V}^{\perp}(|w_{i}\rangle\langle w_{j}|)=0$ is quadratic on them. We shall
not solve this problem in full, but rather prove an upper bound on the number
of vectors that can satisfy those conditions for a given process. If this
upper bound is smaller than the rank of the process, this is a proof that this
process is not purifiable.

We shall start by characterising the vector space $V_{W}$ formed by the
vectors $|v\rangle$ such that $L_{V}^{\perp}(|v\rangle\langle w_{0}|)=0$.
Since this condition is linear on $|v\rangle$ it is straightforward to do it.
Furthermore, since this is one of the conditions in Theorem 4, it should be
clear that the set of vectors $\\{|w_{i}\rangle\\}$, if it exists, must belong
to $V_{W}$. Restricting our attention to this (hopefully small) subspace makes
it easier to consider the other, non-linear, conditions.

To construct an orthonormal basis for $V_{W}$, first we transform the
projector $L_{V}^{\perp}$ to act on the “double-kets” of process matrices,
i.e., we define the matrix $\Pi_{L_{V}^{\perp}}$ such
that666$\Pi_{L_{V}^{\perp}}$ can be explicitly written as
$\Pi_{L_{V}^{\perp}}=\mathfrak{C}(L_{V}^{\perp})^{R}$, where $R$ is the
reshuffling operation [43].

$\forall M\in A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes
F^{\prime}\quad\Pi_{L_{V}^{\perp}}{|{M}\rangle\\!\rangle}={\Big{|}{L_{V}^{\perp}(M)}\Big{\rangle}\\!\Big{\rangle}}.$
(26)

We have then

$L_{V}^{\perp}(|v\rangle\langle
w_{0}|)=0\iff\Pi_{L_{V}^{\perp}}|w_{0}^{*}\rangle|v\rangle=0,$ (27)

so if we define

$O_{W}:=\big{(}\langle
w_{0}^{*}|\otimes\mathds{1}\big{)}\Pi_{L_{V}^{\perp}}\big{(}|w_{0}^{*}\rangle\otimes\mathds{1}\big{)}$
(28)

we have that

$O_{W}|v\rangle=0\iff\Pi_{L_{V}^{\perp}}|w_{0}^{*}\rangle|v\rangle=0,$ (29)

so the null eigenvectors of $O_{W}$ span $V_{W}$, and we can use them to
restrict the projector $\Pi_{L_{V}^{\perp}}$ to the subspace $V_{W}^{*}\otimes
V_{W}$.

To do that, let $\\{|r_{i}\rangle\\}_{i=0}^{\dim(V_{W})-1}$ be the
(orthonormal) null eigenvectors of $O_{W}$, and let
$\\{|i\rangle\\}_{i=0}^{\dim(V_{W})-1}$ be a generic computational basis of
dimension $\dim(V_{W})$. The restriction is then done via the isometry
$R=\sum_{i=0}^{\dim(V_{W})-1}|i\rangle\langle r_{i}|$, which maps a vector
space of dimension
$d_{A_{I}}d_{A_{O}}d_{B_{I}}d_{B_{O}}\operatorname{rank}(W)$ to a vector space
of dimension $\dim(V_{W})$, usually drastically smaller. The restricted
operator $\Pi_{L_{V}^{\perp}|V_{W}}$ is then given by

$\Pi_{L_{V}^{\perp}|V_{W}}=(R^{*}\otimes R)\
\Pi_{L_{V}^{\perp}}\,(R^{T}\otimes R^{\dagger}).$ (30)

Using $\Pi_{L_{V}^{\perp}|V_{W}}$ we can now particularize the condition that
$L_{V}^{\perp}(|a\rangle\langle b|)=0$ to vectors $|a\rangle$ and $|b\rangle$
inside $V_{W}$. Let $|t\rangle=R|a\rangle$ and $|u\rangle=R|b\rangle$. Then
$L_{V}^{\perp}(|a\rangle\langle b|)=0$ iff
$\Pi_{L_{V}^{\perp}|V_{W}}|u^{*}\rangle|t\rangle=0$. Let then
$\\{|m_{k}\rangle\\}_{k=0}^{d_{m}-1}$ be the set of eigenvectors of
$\Pi_{L_{V}^{\perp}|V_{W}}$ with nonzero eigenvalue777It is possible that this
set is empty. In this case all vectors in $V_{W}$ respect the conditions of
Theorem 4, and the process is purifiable iff
$\dim(V_{W})\geq\operatorname{rank}(W)$.. Then

$\Pi_{L_{V}^{\perp}|V_{W}}|u^{*}\rangle|t\rangle=0\iff\forall k\quad\langle
u^{*}|\langle t||m_{k}\rangle=0.$ (31)

We want to rewrite the inner product $\langle u^{*}|\langle t||m_{k}\rangle$
in a more convenient way. For that, note that
$\langle\psi|A|\phi\rangle=\langle\phi^{*}|\langle\psi|{|{A^{T}}\rangle\\!\rangle}$,
so $\langle u^{*}|\langle t||m_{k}\rangle=\langle t|M_{k}|u\rangle$ for
matrices $M_{k}$ such that ${|{M_{k}^{T}}\rangle\\!\rangle}=|m_{k}\rangle$.
These matrices are in general not Hermitian, so for convenience we define
$C_{k}=M_{k}+M_{k}^{\dagger}$ and $C_{k+d_{m}}=i(M_{k}-M_{k}^{\dagger})$. Then

$\Pi_{L_{V}^{\perp}|V_{W}}|u^{*}\rangle|t\rangle=0\iff\forall k\quad\langle
t|C_{k}|u\rangle=0.$ (32)

Let now $d_{C_{k}}$ be the dimension of the largest subspace such that for all
vectors $|t\rangle,|u\rangle$ in this subspace $\langle t|C_{k}|u\rangle=0$.
We can easily calculate $d_{C_{k}}$ using the the null-square lemma proved in
Appendix D: let $n_{k+},n_{k-}$, and $n_{k0}$ be the number of positive,
negative, and null eigenvalues of $C_{k}$. Then

$d_{C_{k}}=n_{k0}+\min\\{n_{k+},n_{k-}\\},$ (33)

and a simple upper bound on the dimension of the largest subspace of vectors
that respect the conditions of Theorem 4 is

$d_{\text{max}}(W):=\min_{i}d_{C_{i}}.$ (34)

Therefore if $d_{\text{max}}(W)<\operatorname{rank}(W)$ the process $W$ is not
purifiable.

##  6 Examples

We shall now apply the necessary condition derived in the previous Section to
several process matrices from the literature. The calculation of
$d_{\text{max}}$ for all the matrices in this Section was done numerically
with the MATLAB code available as the ancillary file purification.m on the
arXiv.

In this Section we shall omit superscripts that identify subsystems and tensor
products to avoid clutter. The expression (e.g.) $Z\mathds{1}XZ$ should be
understood as $Z^{A_{I}}\otimes\mathds{1}^{A_{O}}\otimes X^{B_{I}}\otimes
Z^{B_{O}}$.

Let

$W_{\text{OCB}}=\frac{1}{4}\mathopen{}\mathclose{{}\left[\mathds{1}\mathds{1}\mathds{1}\mathds{1}+\frac{\mathds{1}ZZ\mathds{1}+Z\mathds{1}XZ}{\sqrt{2}}}\right]$
(35)

be the process introduced in Ref. [1]. It was proven, under a restriction on
the allowed operations, to produce the maximal violation of the original
causal inequality for any dimension [44]. Since
$\operatorname{rank}(W_{\text{OCB}})=8$ and $d_{\text{max}}(W_{\text{OCB}})=5$
it is not purifiable.

Let

$W_{\text{max}}=\frac{1}{4}\Big{[}\mathds{1}\mathds{1}\mathds{1}\mathds{1}+a_{0}\,Z\mathds{1}Z\mathds{1}-a_{1}\big{(}Z\mathds{1}\mathds{1}\mathds{1}+\mathds{1}\mathds{1}Z\mathds{1}\big{)}-a_{2}\big{(}Z\mathds{1}\mathds{1}Z+\mathds{1}ZZ\mathds{1}\big{)}+a_{3}\big{(}Z\mathds{1}ZZ+ZZZ\mathds{1}\big{)}\\\
+a_{4}\big{(}Z\mathds{1}XX-Z\mathds{1}YY+XXZ\mathds{1}-YYZ\mathds{1}\big{)}\Big{]},$
(36)

where $a_{0}\approx 0.2744$, $a_{1}\approx 0.2178$, $a_{2}\approx 0.3628$,
$a_{3}\approx 0.3114$, and $a_{4}\approx 0.2097$. This process was introduced
in Appendix C of Ref. [5] and conjectured to produce the maximal violation of
GYNI and LGYNI inequalities for its dimension. Since
$\operatorname{rank}(W_{\text{max}})=11$ and
$d_{\text{max}}(W_{\text{max}})=10$ it is not purifiable.

Let

$W_{\text{opt}}=\frac{1}{4}\mathopen{}\mathclose{{}\left[\mathds{1}\mathds{1}\mathds{1}\mathds{1}+\frac{1}{\sqrt{3}}\,Z\mathds{1}XZ+\frac{\sqrt{3}-1}{3}\big{(}\mathds{1}XX\mathds{1}+\mathds{1}YY\mathds{1}+\mathds{1}ZZ\mathds{1}\big{)}}\right]$
(37)

be the process introduced in Ref. [31]. By itself it cannot violate any causal
inequalities, but it becomes able to do so when it is extended with an
entangled ancilla. However, when admixed with a small amount of noise, the
known violations disappear, so this noisy version was conjectured to be unable
to violate any causal inequalities [31], that is, to be “extensibly causal”
[7]. Since extensibly causal processes seem physically reasonable, but we have
no physical interpretation for $W_{\text{opt}}$ or its noisy version, it would
very interesting to test its purifiability to gather evidence about its
physicality. Unfortunately, since $\operatorname{rank}(W_{\text{opt}})=12$ and
$d_{\text{max}}(W_{\text{opt}})=17$ our test is inconclusive. This is also the
case for the noisy version, which has rank $16$ and $d_{\text{max}}$ equal to
$41$.

##  7 Counter-example

Since the processes examined in the previous Section either are not purifiable
or have unknown status, it might be tempting to conjecture that all processes
that can violate causal inequalities are not purifiable. While this might be
true for the bipartite case, there exists a counter-example for the general
case: a tripartite process which can violate causal inequalities and is
purifiable. Note that although Definition 1, Theorem 2, and Definition 3 were
stated explicitly only for the bipartite case, their multipartite
generalisation is straightforward, and we shall use it in this Section.

This process, which was first published in Ref. [6], can be concisely
described as a function from $A_{O},B_{O},C_{O}$ to $A_{I},B_{I},C_{I}$ that
incoherently maps the basis states $|a,b,c\rangle$ to the basis states $|\lnot
b\land c,\lnot c\land a,\lnot a\land b\rangle$, where $\lnot$ and $\land$
represent logical negation and logical conjunction. It can be represented in a
more cumbersome way as the process matrix

$W_{\text{det}}=\sum_{abc}\mathopen{}\mathclose{{}\left|a,b,c\middle\rangle\middle\langle
a,b,c}\right|\otimes\mathopen{}\mathclose{{}\left|\lnot b\land c,\lnot c\land
a,\lnot a\land b\middle\rangle\middle\langle\lnot b\land c,\lnot c\land
a,\lnot a\land b}\right|,$ (38)

where unlike in the other processes, we wrote the subsystems in the order
$A_{O}$,$B_{O}$,$C_{O}$,$A_{I}$,$B_{I}$,$C_{I}$, to match the function
described above. To purify this process [30] we use the standard trick to turn
a irreversible function into a reversible one, that is, changing the function
from $|x\rangle\mapsto|f(x)\rangle$ to
$|x\rangle|y\rangle\mapsto|x\rangle|y\oplus f(x)\rangle$. In terms of our
function this means mapping $|a,b,c\rangle|i,j,k\rangle$ to
$|a,b,c\rangle|i\oplus\lnot b\land c,j\oplus\lnot c\land a,k\oplus\lnot a\land
b\rangle$. This reversible function is then the purification of
$W_{\text{det}}$, and can be written as the process vector

$|w_{\text{det}}\rangle=\sum_{\begin{subarray}{c}abc\\\
ijk\end{subarray}}|a,b,c\rangle|i,j,k\rangle\otimes|a,b,c\rangle|i\oplus\lnot
b\land c,j\oplus\lnot c\land a,k\oplus\lnot a\land b\rangle,$ (39)

where the subsystems are written in the order
$A_{O}$,$B_{O}$,$C_{O}$,$P$,$F$,$A_{I}$,$B_{I}$,$C_{I}$. Note that unlike the
other subsystems, $P$ and $F$ consist of three qubits each.

##  8 Conclusion

We have proposed purifiability as a necessary condition for the physicality of
a process, motivated by considerations that are independent of the process
matrix formalism. This allows us for the first time to declare a large class
of processes to be unphysical. This can be made part of an axiomatic, top-down
approach to determining the set of physical processes. Ideally, one should
marry it with a bottom-up approach, that shows processes to be physical by
actually constructing laboratory implementations for them. As of yet, however,
there are still processes which are neither excluded by the purification
principle nor known to be implementable, showing that more work must be done
in order to consummate this marriage.

In this grey zone we find, for example, $|w_{\text{det}}\rangle$ (equation
(39)), showing that the set of purifiable processes is rather rich, and that
the purification principle does not rule out the violation of causal
inequalities in Nature. It might still be the case, however, that processes
that cannot violate causal inequalities, i.e., extensibly causal processes,
are always purifiable. In order to decide this, it would be interesting to
know whether the noisy version of $W_{\text{opt}}$ (equation (37)) is
purifiable.

##  9 Acknowledgements

We thank Ämin Baumeler, Fabio Costa, Paolo Perinotti, and Stefan Wolf for
useful discussions. We acknowledge support from the Austrian Science Fund
(FWF) through the Special Research Programme FoQuS, the Doctoral Programme
CoQuS and the projects No. P-24621 and I-2526. This work has been supported by
the Excellence Initiative of the German Federal and State Governments (Grant
ZUK 81). This publication was made possible through the support of a grant
from the John Templeton Foundation. The opinions expressed in this publication
are those of the authors and do not necessarily reflect the views of the John
Templeton Foundation.

## References

  * [1] O. Oreshkov, F. Costa and Č. Brukner  “Quantum correlations with no causal order”  In _Nat. Commun._ 3, 2012, pp. 1092  DOI: 10.1038/ncomms2076
  * [2] A. Ashtekar  “Large Quantum Gravity Effects: Unforeseen Limitations of the Classical Theory”  In _Phys. Rev. Lett._ 77, 1996, pp. 4864–4867  DOI: 10.1103/PhysRevLett.77.4864
  * [3] Ä. Baumeler and S. Wolf  “Perfect signaling among three parties violating predefined causal order”  In _Information Theory (ISIT), 2014 IEEE International Symposium on_ , 2014, pp. 526–530  DOI: 10.1109/ISIT.2014.6874888
  * [4] Ämin Baumeler, Adrien Feix and Stefan Wolf  “Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios”  In _Phys. Rev. A_ 90 American Physical Society, 2014, pp. 042106  DOI: 10.1103/PhysRevA.90.042106
  * [5] Cyril Branciard et al.  “The simplest causal inequalities and their violation”  In _New J. Phys._ 18.1, 2015, pp. 013008  DOI: 10.1088/1367-2630/18/1/013008
  * [6] Ä. Baumeler and S. Wolf  “The space of logically consistent classical processes without causal order”  In _New J. Phys._ 18.1, 2016, pp. 013036  DOI: 10.1088/1367-2630/18/1/013036
  * [7] O. Oreshkov and C. Giarmatzi  “Causal and causally separable processes”  In _New J. Phys._ 18, 2015, pp. 093020  DOI: 10.1088/1367-2630/18/9/093020
  * [8] A. A. Abbott, C. Giarmatzi, F. Costa and C. Branciard  “Multipartite Causal Correlations: Polytopes and Inequalities”  In _Phys. Rev. A_ 94.3, 2016, pp. 032131  DOI: 10.1103/PhysRevA.94.032131
  * [9] Mateus Araújo et al.  “Witnessing causal nonseparability”  In _New J. Phys._ 17.10, 2015, pp. 102001  DOI: 10.1088/1367-2630/17/10/102001
  * [10] G. Chiribella, G. M. D’Ariano, P. Perinotti and B. Valiron  “Quantum computations without definite causal structure”  In _Phys. Rev. A_ 88.2, 2013, pp. 022318  DOI: 10.1103/PhysRevA.88.022318
  * [11] G. Chiribella  “Perfect discrimination of no-signalling channels via quantum superposition of causal structures”  In _Phys. Rev. A_ 86.4, 2012, pp. 040301  DOI: 10.1103/PhysRevA.86.040301
  * [12] M. Araújo, F. Costa and Č. Brukner  “Computational Advantage from Quantum-Controlled Ordering of Gates”  In _Phys. Rev. Lett._ 113.25, 2014, pp. 250402  DOI: 10.1103/PhysRevLett.113.250402
  * [13] A. Feix, M. Araújo and Č. Brukner  “Quantum superposition of the order of parties as a communication resource”  In _Phys. Rev. A_ 92 American Physical Society, 2015, pp. 052326  DOI: 10.1103/PhysRevA.92.052326
  * [14] P. Allard Guérin, A. Feix, M. Araújo and Č. Brukner  “Exponential communication complexity advantage from quantum superposition of the direction of communication”  In _Phys. Rev. Lett._ 117 American Physical Society, 2016, pp. 100502  DOI: 10.1103/PhysRevLett.117.100502
  * [15] Lorenzo M Procopio et al.  “Experimental superposition of orders of quantum gates”  In _Nat. Commun._ 6, 2015, pp. 7913  DOI: 10.1038/ncomms8913
  * [16] G. Rubino et al.  “Experimental verification of an indefinite causal order”  In _Sci. Adv._ 3.3 American Association for the Advancement of Science, 2017  DOI: 10.1126/sciadv.1602589
  * [17] G. Brassard et al.  “Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial”  In _Phys. Rev. Lett._ 96.25, 2006, pp. 250401  DOI: 10.1103/PhysRevLett.96.250401
  * [18] Noah Linden, Sandu Popescu, Anthony J. Short and Andreas Winter  “Quantum Nonlocality and Beyond: Limits from Nonlocal Computation”  In _Phys. Rev. Lett._ 99 American Physical Society, 2007, pp. 180502  DOI: 10.1103/PhysRevLett.99.180502
  * [19] M. Pawłowski et al.  “Information causality as a physical principle”  In _Nature_ 461, 2009, pp. 1101–1104  DOI: 10.1038/nature08400
  * [20] M. Navascués and H. Wunderlich  “A glance beyond the quantum model”  In _Proc. Royal Soc. A_ 466, 2009, pp. 881–890  DOI: 10.1098/rspa.2009.0453
  * [21] T. Fritz et al.  “Local orthogonality as a multipartite principle for quantum correlations”  In _Nat. Commun._ 4, 2013, pp. 2263  DOI: 10.1038/ncomms3263
  * [22] Giulio Chiribella, Giacomo Mauro D’Ariano and Paolo Perinotti  “Probabilistic theories with purification”  In _Phys. Rev. A_ 81 American Physical Society, 2010, pp. 062348  DOI: 10.1103/PhysRevA.81.062348
  * [23] L. Hardy  “Quantum Theory From Five Reasonable Axioms”, 2001  arXiv:quant-ph/0101012
  * [24] B. Dakić and Č. Brukner  “Deep Beauty: Understanding the Quantum World through Mathematical Innovation”  Cambridge University Press, 2011, pp. 365–392  arXiv:0911.0695 [quant-ph]
  * [25] L. Masanes and M. P. Müller  “A derivation of quantum theory from physical requirements”  In _New J. Phys._ 13.6, 2011, pp. 063001  DOI: 10.1088/1367-2630/13/6/063001
  * [26] Giulio Chiribella, Giacomo Mauro D’Ariano and Paolo Perinotti  “Informational derivation of quantum theory”  In _Phys. Rev. A_ 84 American Physical Society, 2011, pp. 012311  DOI: 10.1103/PhysRevA.84.012311
  * [27] H. Barnum, M. P. Müller and C. Ududec  “Higher-order interference and single-system postulates characterizing quantum theory”  In _New J. Phys._ 16.12, 2014, pp. 123029  DOI: 10.1088/1367-2630/16/12/123029
  * [28] P. A Höhn  “Toolbox for reconstructing quantum theory from rules on information acquisition”, 2014  arXiv:1412.8323 [quant-ph]
  * [29] P. A Höhn and C. Wever  “Quantum theory from questions”  In _Phys. Rev. A_ 95 American Physical Society, 2017, pp. 012102  DOI: 10.1103/PhysRevA.95.012102
  * [30] Ä. Baumeler and S. Wolf Private communication (2015).
  * [31] A. Feix, M. Araújo and Č. Brukner  “Causally nonseparable processes admitting a causal model”  In _New J. Phys._ 18.8, 2016, pp. 083040  DOI: 10.1088/1367-2630/18/8/083040
  * [32] G. Chiribella, G. M. D’Ariano and P. Perinotti  “Theoretical framework for quantum networks”  In _Phys. Rev. A_ 80.2, 2009, pp. 022339  DOI: 10.1103/PhysRevA.80.022339
  * [33] G. C. Ghirardi, A. Rimini and T. Weber  “Unified dynamics for microscopic and macroscopic systems”  In _Phys. Rev. D_ 34 American Physical Society, 1986, pp. 470–491  DOI: 10.1103/PhysRevD.34.470
  * [34] Gian Carlo Ghirardi, Philip Pearle and Alberto Rimini  “Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles”  In _Phys. Rev. A_ 42 American Physical Society, 1990, pp. 78–89  DOI: 10.1103/PhysRevA.42.78
  * [35] A. Bassi et al.  “Models of wave-function collapse, underlying theories, and experimental tests”  In _Rev. Mod. Phys._ 85, 2013, pp. 471–527  DOI: 10.1103/RevModPhys.85.471
  * [36] S. W. Hawking  “Breakdown of predictability in gravitational collapse”  In _Phys. Rev. D_ 14 American Physical Society, 1976, pp. 2460–2473  DOI: 10.1103/PhysRevD.14.2460
  * [37] D. Harlow  “Jerusalem Lectures on Black Holes and Quantum Information”  In _Rev. Mod. Phys._ 88 American Physical Society, 2016, pp. 015002  DOI: 10.1103/RevModPhys.88.015002
  * [38] T. Jacobson  “Trans-Planckian Redshifts andthe Substance of the Space-Time River”  In _Prog. Theor. Phys._ 136, 1999, pp. 1–17  DOI: 10.1143/PTPS.136.1
  * [39] M. Bojowald, D. Cartin and G. Khanna  “Lattice refining loop quantum cosmology, anisotropic models, and stability”  In _Phys. Rev. D_ 76.6, 2007, pp. 064018  DOI: 10.1103/PhysRevD.76.064018
  * [40] S. Gielen and L. Sindoni  “Quantum Cosmology from Group Field Theory Condensates: a Review”  In _SIGMA_ 12, 2016, pp. 082  DOI: 10.3842/SIGMA.2016.082
  * [41] P. A. Höhn  “Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces”  In _J. Math. Phys._ 55.8, 2014, pp. 083508  DOI: 10.1063/1.4890558
  * [42] V.F. Mukhanov  “Physical Foundations of Cosmology”  Cambridge University Press, 2005
  * [43] Karol Życzkowski and Ingmar Bengtsson  “Geometry of Quantum States”  Cambridge University Press, 2006
  * [44] Č. Brukner  “Bounding quantum correlations with indefinite causal order”  In _New J. Phys._ 17.8, 2015, pp. 083034  DOI: 10.1088/1367-2630/17/8/083034
  * [45] Antoine Royer  “Wigner function in Liouville space: A canonical formalism”  In _Phys. Rev. A_ 43.1, 1991, pp. 44–56  DOI: 10.1103/PhysRevA.43.44
  * [46] Samuel L. Braunstein, Giacomo M. D’Ariano, G. J. Milburn and Massimiliano F. Sacchi  “Universal Teleportation with a Twist”  In _Phys. Rev. Lett._ 84.15, 2000, pp. 3486–3489  DOI: 10.1103/PhysRevLett.84.3486

##  Appendix A The Choi-Jamiołkowski isomorphism

We use two levels of the Choi-Jamiołkowski isomorphism. The first level is the
“pure” CJ isomorphism, used to represent matrices as vectors. The CJ operator
of a matrix $A:\mathcal{H}_{I}\to\mathcal{H}_{O}$ is its “double-ket” [45,
46]:

${|{A}\rangle\\!\rangle}:=\mathds{1}\otimes
A{|{\mathds{1}}\rangle\\!\rangle}=\sum_{i=0}^{d_{\mathcal{H}_{I}}-1}|i\rangle
A|i\rangle.$ (40)

The second level is the “mixed” CJ isomorphism, used to represent linear
operators that act on matrices as matrices themselves. The CJ operator of a
map $\mathcal{M}:\mathcal{L}(\mathcal{H}_{I})\to\mathcal{L}(\mathcal{H}_{O})$
is

$\mathfrak{C}(\mathcal{M}):=\mathcal{I}\otimes\mathcal{M}({{|{\mathds{1}}\rangle\\!\rangle}\\!{\langle\\!\langle{\mathds{1}}|}})=\sum_{i,j=0}^{d_{\mathcal{H}_{I}}-1}|i\rangle\langle
j|\otimes\mathcal{M}(|i\rangle\langle j|).$ (41)

Note that this convention differs from the one in [1, 9] by a transposition.
This is necessary to avoid representing completely positive maps with a non-
positive matrix, as would happen with the convention from [1, 9] when, for
example, we take in equation (1) the process as being identity from $P$ to
$A_{I}$ and identity from $A_{O}$ to $F$, and the map $\mathcal{A}$ to be a
SWAP.

##  Appendix B The link product

The motivation for defining the link product is to express the CJ operator of
a composition of two linear maps $\mathcal{M}$ and $\mathcal{N}$ directly as a
function of the CJ operators of the individual maps [32], as

$\mathfrak{C}(\mathcal{N}\circ\mathcal{M})=\mathfrak{C}(\mathcal{N})*\mathfrak{C}(\mathcal{M}).$
(42)

We do not need, however, to explicitly use the CJ isomorphism, as the link
product can be more simply defined as an abstract operation on two matrices.
Namely, the link product between two operators $M\in\bigotimes_{i\in I}A^{i}$
and $N\in\bigotimes_{i\in J}A^{i}$ is

$N*M:=\operatorname{tr}_{I\cap J}[(\mathds{1}^{J\setminus I}\otimes
M^{T_{I\cap J}})(N\otimes\mathds{1}^{I\setminus J})],$ (43)

where $\cdot^{T_{I\cap J}}$ denotes partial transposition on the subsystems
$I\cap J$.

Note that when $I\cap J=\emptyset$ the link product reduces to $N*M=N\otimes
M$ and, conversely, when $I\cap J=I\cup J$, it reduces to
$N*M=\operatorname{tr}M^{T}N$.

##  Appendix C Conditions for validity

In this Appendix we derive the constraints on $W$ such that the map

$G_{A,B}=W*(A\otimes B)$ (44)

is a valid CPTP map for all valid CPTP maps $A$ and $B$. This derivation
follows closely the one presented in Appendix B of Ref. [9]. Recall that a
linear map $\mathcal{A}:A_{I}\to A_{O}$ is completely positive and trace
preserving if and only if its CJ operator $A=\mathfrak{C}(\mathcal{A})$ is
positive and $\operatorname{tr}_{A_{O}}A=\mathds{1}^{A_{I}}$. Therefore we
need that

$\displaystyle\operatorname{tr}_{A_{O}^{\prime}B_{O}^{\prime}F}W*(A\otimes
B)=\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}P}\quad\text{and}\quad W*(A\otimes
B)\geq 0$ (45) $\displaystyle\forall A,B\quad\text{s.t.}\quad A\geq
0,\quad\operatorname{tr}_{A_{O}A_{O}^{\prime}}A=\mathds{1}^{A_{I}A_{I}^{\prime}}$
(46) $\displaystyle B\geq
0,\quad\operatorname{tr}_{B_{O}B_{O}^{\prime}}B=\mathds{1}^{B_{I}B_{I}^{\prime}}.$
(47)

First we show that $W*(A\otimes B)\geq 0$ iff $W\geq 0$. To see that, note
that when $\mathcal{A}$ and $\mathcal{B}$ are swaps we have $W*(A\otimes
B)=W$, so $W\geq 0$ is a necessary condition for $G_{A,B}\geq 0$. To see that
it is also sufficient, just note that the link product of two positive
operators is always positive.

To get the constraints on $W$ such that
$\operatorname{tr}_{A_{O}^{\prime}B_{O}^{\prime}F}W*(A\otimes
B)=\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}P}$, first note that the positivity
of $A,B$, and $W$ is not relevant for that, since the set of positive
operators is a full-dimensional subset of the space of Hermitian operators.
Therefore we only need the normalisation condition

$\displaystyle\operatorname{tr}_{A_{O}^{\prime}B_{O}^{\prime}F}W*(A\otimes
B)=\mathds{1}^{A_{I}^{\prime}B_{I}^{\prime}P}$ (48) $\displaystyle\forall
A,B\quad\text{s.t.}\quad\operatorname{tr}_{A_{O}A_{O}^{\prime}}A=\mathds{1}^{A_{I}A_{I}^{\prime}},\quad\operatorname{tr}_{B_{O}B_{O}^{\prime}}B=\mathds{1}^{B_{I}B_{I}^{\prime}}.$
(49)

Note also that the set of operators of the form
$\operatorname{tr}_{A_{I}^{\prime}A_{O}^{\prime}}A$ for CPTP $A$ does not
depend888It would if the CPTP maps $A$ were restricted to be unitaries. on the
dimension of $A_{I}^{\prime}$ and $A_{O}^{\prime}$, so we can without loss of
generality set
$d_{A_{I}^{\prime}}=d_{A_{O}^{\prime}}=d_{B_{I}^{\prime}}=d_{B_{O}^{\prime}}=1$,
simplifying this condition to

$\displaystyle\operatorname{tr}_{F}W*(A\otimes B)=\mathds{1}^{P}$ (50)
$\displaystyle\forall
A,B\quad\text{s.t.}\quad\operatorname{tr}_{A_{O}}A=\mathds{1}^{A_{I}},\quad\operatorname{tr}_{B_{O}}B=\mathds{1}^{B_{I}}.$
(51)

Note also that

$\operatorname{tr}_{A_{O}}A=\mathds{1}^{A_{I}}\iff
A=a-{}_{A_{O}}a+\frac{\mathds{1}^{A_{I}A_{O}}}{d_{A_{O}}}$ (52)

for some Hermitian operator $a$, where we are using the shorthand notation

${}_{X}W=\frac{\mathds{1}^{X}}{d_{X}}\otimes\operatorname{tr}_{X}W.$ (53)

Using this in Alice and Bob’s sides allows us to reduce the normalization
condition to

$\forall
a,b\quad(\operatorname{tr}_{F}W)*\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(a-{}_{A_{O}}a+\frac{\mathds{1}^{A_{I}A_{O}}}{d_{A_{O}}}}\right)\otimes\mathopen{}\mathclose{{}\left(b-{}_{B_{O}}b+\frac{\mathds{1}^{B_{I}B_{O}}}{d_{B_{O}}}}\right)}\right]=\mathds{1}^{P}.$
(54)

For $a=b=0$ this yields the condition

$(\operatorname{tr}_{F}W)*\mathopen{}\mathclose{{}\left(\frac{\mathds{1}^{A_{I}A_{O}}}{d_{A_{O}}}\otimes\frac{\mathds{1}^{B_{I}B_{O}}}{d_{B_{O}}}}\right)=\mathds{1}^{P}.$
(55)

For $b=0$ and $a=0$, respectively, equation (55) together with (54) imply that

$\displaystyle\forall
a\quad(\operatorname{tr}_{F}W)*\mathopen{}\mathclose{{}\left[(a-{}_{A_{O}}a)\otimes\frac{\mathds{1}^{B_{I}B_{O}}}{d_{B_{O}}}}\right]=0$
(56) $\displaystyle\forall
b\quad(\operatorname{tr}_{F}W)*\mathopen{}\mathclose{{}\left[\frac{\mathds{1}^{A_{I}A_{O}}}{d_{A_{O}}}\otimes(b-{}_{B_{O}}b)}\right]=0$
(57)

which then imply that

$\forall
a,b\quad(\operatorname{tr}_{F}W)*[(a-{}_{A_{O}}a)\otimes(b-{}_{B_{O}}b)]=0.$
(58)

Using the fact that the maps ${}_{A_{O}}$ and ${}_{B_{O}}$ are self-adjoint
under the Hilbert-Schmidt inner product, we see that conditions (56)-(58) are
equivalent to

$\displaystyle\operatorname{tr}_{B_{I}B_{O}F}W={}_{A_{O}}\operatorname{tr}_{B_{I}B_{O}F}W,$
(59)
$\displaystyle\operatorname{tr}_{A_{I}A_{O}F}W={}_{B_{O}}\operatorname{tr}_{A_{I}A_{O}F}W,$
(60)
$\displaystyle\operatorname{tr}_{F}W={}_{A_{O}}\operatorname{tr}_{F}W+{}_{B_{O}}\operatorname{tr}_{F}W-{}_{A_{O}B_{O}}\operatorname{tr}_{F}W.$
(61)

These conditions, together with (55), completely characterize the affine
subspace to which valid process matrices belong. It is, however, convenient to
rewrite them so that we characterize this affine subspace as a linear subspace
together with a single constraint on the trace of $W$. To do that, note that
conditions (59)-(61) define linear subspaces, so we only need to rewrite
condition (55). It is easy to see that it is equivalent to:

$\displaystyle\operatorname{tr}_{A_{I}A_{O}B_{I}B_{O}F}W=\operatorname{tr}_{PA_{I}A_{O}B_{I}B_{O}F}W,$
(62) $\displaystyle\operatorname{tr}W=d_{P}d_{A_{O}}d_{B_{O}}.$ (63)

So the linear subspace of valid process matrices is the intersection of the
linear subspaces defined in the equations (59)-(62). Since the projectors onto
these subspaces commute, we can obtain the projector onto the subspace of
valid process matrices $L_{V}$ simply by composing them. Elementary
manipulations give us then

$L_{V}(W)=W-{}_{F}W+{}_{A_{O}F}W+{}_{B_{O}F}W-{}_{A_{O}B_{O}F}W-{}_{A_{I}A_{O}F}W+{}_{A_{I}A_{O}B_{O}F}W\\\
-{}_{B_{I}B_{O}F}W+{}_{A_{O}B_{I}B_{O}F}W-{}_{A_{I}A_{O}B_{I}B_{O}F}W+{}_{PA_{I}A_{O}B_{I}B_{O}F}W.$
(64)

To summarize the results of this Appendix, $W$ is a valid process matrix iff

$\displaystyle W\geq 0,$ (65)
$\displaystyle\operatorname{tr}W=d_{P}d_{A_{O}}d_{B_{O}},$ (66) $\displaystyle
W=L_{V}(W).$ (67)

##  Appendix D The null-square lemma

The purpose of this Appendix is to prove that, for any Hermitian matrix $C$,
the greatest null square submatrix of $C$ achievable via a change of basis has
size $d_{C}\equiv\min(n_{+},n_{-})+n_{0}$, where $n_{+},n_{-},n_{0}$ denote,
respectively, the dimensionality of the subspaces spanned by the eigenvectors
of $C$ with positive, negative and zero eigenvalues.

We will first establish that the above quantity is an upper bound for the
greatest zero-square submatrix. Let then $S$ be a subspace with the property
that $\langle v|C|w\rangle=0$, for $|v\rangle,|w\rangle\in S$, and denote by
$\Pi_{0}$ the projector onto the null space of $C$.

Note that $\mbox{dim}(S)\leq\mbox{dim}(S_{0})+\mbox{dim}(S_{+-})$, where
$S_{0}\equiv\Pi_{0}S$ and $S_{+-}\equiv(\mathds{1}-\Pi_{0})S$. Notice as well
that $\langle v|C|w\rangle=0$, for $|v\rangle,|w\rangle\in S_{+-}$. Clearly,
the dimension of $S_{0}$ is upper bounded by $n_{0}$, hence all we need to do
is show that the dimension of $S_{+-}$ is upper bounded by
$\min(n_{+},n_{-})$.

Let $\\{|u^{(i)}\rangle\\}_{i}$ be a basis for $S_{+-}$, and let
$\\{\lambda_{k}\\}_{k=1}^{n_{+}}$ and $\\{-\mu_{k}\\}_{k=1}^{n_{-}}$ be the
set of positive and negative eigenvalues of $C$ (where $\mu_{k}\geq 0$), with
corresponding eigenvectors $\\{|\psi_{k}\rangle\\}_{k=1}^{n_{+}}$ and
$\\{|\phi_{k}\rangle\\}_{k=1}^{n_{-}}$. Then,

$|u^{(i)}\rangle=\sum_{k=1}^{n_{+}}v^{(i)}_{k}|\psi_{k}\rangle+\sum_{k=1}^{n_{-}}w^{(i)}_{k}|\phi_{k}\rangle.$
(68)

From the condition $\langle u^{(i)}|C|u^{(j)}\rangle=0$ we have that

$\sum_{k=1}^{n_{+}}(v^{(i)}_{k})^{*}v^{(j)}_{k}\lambda_{k}-\sum_{k=1}^{n_{-}}(w^{(i)}_{k})^{*}w^{(j)}_{k}\mu_{k}=0,$
(69)

or, in other words,
$\mathopen{}\mathclose{{}\left\langle\tilde{v}^{(i)}\middle|\tilde{v}^{(j)}}\right\rangle=\mathopen{}\mathclose{{}\left\langle\tilde{w}^{(i)}\middle|\tilde{w}^{(j)}}\right\rangle$,
with $\tilde{v}^{(i)}_{k}\equiv\sqrt{\lambda_{k}}v^{(i)}_{k}$,
$\tilde{w}^{(i)}_{k}\equiv\sqrt{\mu_{k}}w^{(i)}_{k}$. This implies that there
exists an isometry $U$ such that
$|\tilde{v}^{(i)}\rangle=U|\tilde{w}^{(i)}\rangle$. The vectors
$|\tilde{v}^{(i)}\rangle\oplus|\tilde{w}^{(i)}\rangle$ (and thus
$|u^{(i)}\rangle$) are generated by applying a linear transformation on
$|\tilde{w}^{(i)}\rangle$ and so the space they span cannot be greater than
$n_{-}$. An identical argument shows that the space spanned by
$\\{|u^{(i)}\rangle\\}_{i=1}^{m}$ cannot be greater than $n_{+}$ either. We
have just proven that $d_{C}$ is an upper bound for the size of the null
square.

To prove that it is also a lower bound, note that the $\min(n_{+},n_{-})$
linearly independent vectors
$\\{|u_{i}\rangle\equiv\lambda_{k}^{-1/2}|\psi_{k}\rangle+\mu_{k}^{-1/2}|\phi_{k}\rangle:i=1,...,\min(n_{+},n_{-})\\}$
satisfy $\langle u_{i}|C|u_{j}\rangle=0$ for all $i,j$. Adding all $n_{0}$
vectors from the null space of $C$, we obtain a basis
$\\{|u_{i}\rangle\\}_{i=1}^{d_{C}}$ with the property that $\langle
u_{i}|A|u_{j}\rangle=0$.