EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-291 LHCb-PAPER-2014-062

Search for long-lived particles decaying to jet pairs

The LHCb collaboration†††Authors are listed at the end of this paper.

A search is presented for long-lived particles with a mass between 25 and
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and a lifetime between 1 and
200${\rm\,ps}$ in a sample of proton-proton collisions at a centre-of-mass
energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an
integrated luminosity of 0.62$\mbox{\,fb}^{-1}$, collected by the LHCb
detector. The particles are assumed to be pair-produced by the decay of a
Standard Model-like Higgs boson. The experimental signature of the long-lived
particle is a displaced vertex with two associated jets. No excess above the
background is observed and limits are set on the production cross-section as a
function of the long-lived particle mass and lifetime.

Published as Eur. Phys. J. C75 (2015) 152

© CERN on behalf of the LHCb collaboration, license CC-BY-4.0.

##  1 Introduction

A variety of models for physics beyond the Standard Model (SM) feature the
existence of new massive particles whose coupling to lighter particles is
sufficiently small that they are long-lived. If these massive particles decay
to SM particles and have a lifetime between approximately 1${\rm\,ps}$ and
1${\rm\,ns}$, characteristic of weak decays, they can be identified by their
displaced decay vertex. Examples of such particles are the lightest
supersymmetric particle in SUSY models with baryon or lepton number violation
[1, *Butterworth:2009qa, *Kaplan:2007ap, *deCampos:2008re], the next-to-
lightest supersymmetric particle in gravity mediated SUSY [5] and the neutral
${\pi_{v}}$ particle in hidden valley (HV) models with a non-abelian gauge
symmetry [6, *Strassler:2006ri, *Han:2007ae]. The latter model is particularly
interesting as it predicts that experimental studies have sensitivity to the
production of long-lived particles in SM Higgs decays.

This paper reports on a search for ${\pi_{v}}$ particles, pair-produced in the
decay of a SM-like Higgs particle with a mass of
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, close to the mass of the scalar
boson discovered by the ATLAS and CMS experiments [9, 10].111The results are
equally valid for a Higgs particle with a mass up to
126${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ within a few percent. The
${\pi_{v}}$ candidates are identified by two hadronic jets originating from a
displaced vertex. The vertex is required to be displaced from the proton-
proton collision axis by more than 0.4 mm and less than 4.8 mm. The lower
bound is chosen to reject most of the background from heavy flavour decays.
The upper bound ensures that vertices are inside the LHCb beam pipe, which
generates a sizeable background of hadronic interaction vertices. The signal
is extracted from a fit to the di-jet invariant mass distribution. The
analysis is sensitive to a ${\pi_{v}}$ particle with a mass between 25 and
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and a lifetime between 1 and
200${\rm\,ps}$. The lower boundary on the mass range arises from the
requirement to identify two hadronic jets while the upper boundary is mostly
due to the geometric acceptance of the LHCb detector.

This analysis uses data collected in proton-proton ($pp$) collisions at a
centre-of-mass energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The data
correspond to an integrated luminosity of 0.62$\mbox{\,fb}^{-1}$, collected
during the second half of the year 2011 when an analysis-specific trigger
selection was implemented. Although similar searches have been reported by the
CDF [11], D0 [12], ATLAS [13] and CMS [14] experiments, LHCb has a unique
coverage for long-lived particles with relatively small mass and lifetime,
because its trigger makes only modest requirements on transverse momentum.

##  2 Detector description

The LHCb detector [15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region [16], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about 4 Tm, and three stations of
silicon-strip detectors and straw drift tubes [17] placed downstream of the
magnet. The tracking system provides a measurement of momentum, $p$, with a
relative uncertainty that varies from $0.4\text{\,}\mathrm{\char 37\relax}$ at
low momentum to $0.6\text{\,}\mathrm{\char 37\relax}$ at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex, the impact parameter, is measured with a resolution of
$(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the component
of $p$ transverse to the beam, in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Different types of charged hadrons are distinguished using information from
two ring-imaging Cherenkov detectors [18]. Photon, electron and hadron
candidates are identified by a calorimeter system consisting of scintillating-
pad and preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [19].

##  3 Event simulation

For the event simulation, $pp$ collisions are generated using Pythia 6.4 [20]
with a specific LHCb configuration [21] using CTEQ6L [22] parton density
functions. Decays of hadronic particles are described by EvtGen [23], in which
final-state radiation is generated using Photos [24]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [25, *Agostinelli:2002hh] as described in Ref. [27].

To simulate a signal event, a SM-like scalar Higgs boson with a mass of
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is generated with Pythia through
the gluon-gluon fusion mechanism, and is forced to decay into two spin-zero
${\pi_{v}}$ particles, each of which decays to ${b}{\overline{b}}$. Assuming
the decay occurs via a vector or axial-vector coupling, the
${b}{\overline{b}}$ final state is preferred to light quarks, due to helicity
conservation [6, *Strassler:2006ri, *Han:2007ae]. The average track
multiplicity of the ${\pi_{v}}$ decay, including tracks from secondary $b$ and
$c$ decays, varies from about 15 for a ${\pi_{v}}$ mass of
25${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to about 20 for larger masses.
Simulated events are retained if at least four charged tracks from the decay
of the generated ${\pi_{v}}$ particles are within the LHCb acceptance, which
corresponds to about $30\text{\,}\mathrm{\char 37\relax}$ of the cases. For
${\pi_{v}}$ particles within the acceptance on average about ten tracks can be
reconstructed.

Simulated samples with ${\pi_{v}}$ lifetimes of 10${\rm\,ps}$ and
100${\rm\,ps}$ and ${\pi_{v}}$ masses of 25, 35, 43 and
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are generated; other ${\pi_{v}}$
lifetimes are studied by reweighting these samples. Two additional samples are
generated in which ${\pi_{v}}$ particles with a lifetime of 10${\rm\,ps}$ and
a mass of 35${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ decay to either
${c}{\overline{c}}$ or ${s}{\overline{s}}$ quark pairs.

##  4 Event selection and signal extraction

The selection of candidates starts with the LHCb trigger [28], which consists
of a hardware stage, based on information from the calorimeter and muon
systems, followed by a software stage, which applies a full event
reconstruction. The hardware trigger (L0) requires a single high-$p_{\rm T}$
hadron, electron, muon or photon signature. The thresholds range from
$\mbox{$p_{\rm T}$}{}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for muons, to
transverse energy larger than $3.5\mathrm{\,Ge\kern-1.00006ptV}$ for hadrons.
The total L0 efficiency, dominated by the hadron trigger selection, depends on
the mass and final state of the ${\pi_{v}}$ particle and is typically
$20\text{\,}\mathrm{\char 37\relax}$, including the detector acceptance.

The software trigger is divided into two stages and consists of algorithms
that run a simplified version of the offline track reconstruction, which
allows identification of displaced tracks and vertices. For this analysis the
primary signature in the first software stage (HLT1) is a single high-quality
displaced track with high $p_{\rm T}$. The efficiency of HLT1 relative to L0
accepted events is typically $60\text{\,}\mathrm{\char 37\relax}$. However,
this efficiency reduces rapidly for vertices that are displaced by more than
about 5$\rm\,mm$ from the beamline due to limitations in the track
reconstruction in the vertex detector.

In the final trigger stage (HLT2) two different signatures are exploited. The
first of these relies on the generic reconstruction of a displaced vertex,
using an algorithm similar to that used for the primary vertex (PV)
reconstruction [29]. Secondary vertices are distinguished from PVs using the
distance to the interaction region in the transverse plane ($R_{xy}$). To
eliminate contributions from interactions with material, a so-called ‘material
veto’ removes vertices in a region defined as an envelope around the detector
material [30]. Events are selected when they have a displaced vertex with at
least four tracks, a sum of the scalar $p_{\rm T}$ of all tracks that is
larger than 3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a distance $R_{xy}$
larger than 0.4$\rm\,mm$ and an invariant mass of the particles associated
with this vertex $m_{\text{vtx}}$ above
4.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. To further refine the
selection, vertices are required to have either $R_{xy}>2\rm\,mm$ or
$m_{\text{vtx}}>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The second HLT2 signature is designed to identify two-, three- and four-body
exclusive $b$-hadron decays [31]. A multivariate algorithm is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron. The combined efficiency of the two HLT2 selections relative to events
accepted by L0 and HLT1 is about $60\text{\,}\mathrm{\char 37\relax}$.

The offline candidate reconstruction starts from a generic secondary vertex
search, similar to that applied in the trigger, but using tracks from the
offline reconstruction as input. At this stage at least six tracks per vertex
are required and the sum of the scalar $p_{\rm T}$ of all tracks must be above
3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The vertex is required to have either
$R_{xy}>0.4\rm\,mm$ and
$m_{\text{vtx}}>9.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or
$R_{xy}>2.5\rm\,mm$ and
$m_{\text{vtx}}>8.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or
$R_{xy}>4\rm\,mm$ and
$m_{\text{vtx}}>6.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The vertex reconstruction is followed by a jet reconstruction procedure.
Inputs to the jet clustering are obtained using a particle flow approach [32]
that selects charged particles, neutral calorimeter deposits and a small
contribution from ${K}^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}$ decays. To reduce contamination from particles that do not
originate from the displaced vertex, only charged particles that have a
smaller distance of closest approach relative to the displaced vertex than to
any PV in the event are retained. Furthermore, the distance to the displaced
vertex is required to be less than 2$\rm\,mm$, which also allows tracks from
displaced $b$ and $c$ vertices in the
${\pi_{v}}\rightarrow{{b}{\overline{b}}}$ decay chain to be accepted.

The jet clustering uses the anti-$k_{\mathrm{T}}$ algorithm [33] with a cone
size of 0.7. Only jets with a $p_{\rm T}$ above
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are used. Additional requirements are
made to enhance the fraction of well-reconstructed hadronic jets: first, the
charged particle with the largest $p_{\rm T}$ in the jet must have a $p_{\rm
T}$ above 0.9${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, yet carry no more than
$70\text{\,}\mathrm{\char 37\relax}$ of the $p_{\rm T}$ of the jet. Second, to
remove jets whose energy is dominated by neutral particles, which cannot be
unambiguously associated with a vertex, at least $10\text{\,}\mathrm{\char
37\relax}$ of the $p_{\rm T}$ of the jet must be carried by charged particles.

The di-jet invariant mass is computed from the reconstructed four-momenta of
the two jets. Correction factors to the jet energy are determined from the
simulation and parameterised as a function of the number of reconstructed PVs
in the event, to account for effects due to multiple interactions and the
underlying event [32].

Two further requirements are made to enhance signal purity. First, a corrected
mass is computed as

$m_{\text{corr}}\;=\;\sqrt{m^{2}+\left(p\sin\theta\right)^{2}}\>+\>p\sin\theta\;,$
(1)

where $m$ is the di-jet invariant mass and $\theta$ is the pointing angle
between the di-jet momentum vector $\vec{p}$ and its displacement vector
$\vec{d}=\vec{x}_{\text{DV}}-\vec{x}_{\text{PV}}$, where $\vec{x}_{\text{DV}}$
is the position of the displaced vertex and $\vec{x}_{\text{PV}}$ the position
of the PV. To select candidates pointing back to a PV, only events with
$m/m_{\text{corr}}>0.7$ are retained. A requirement on this ratio is preferred
over a requirement on the pointing angle itself, since its efficiency depends
less strongly on the boost and the mass of the candidate.

Second, a requirement is made on the distance $\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$ between the two jets, where $\phi$ is
the azimuthal angle and $\eta$ the pseudorapidity. A background consisting of
back-to-back jet candidates, for example di-jet ${b}{\overline{b}}$-events,
appears mainly at large values of reconstructed mass, and is characterised by
a large difference between the jets in both $\phi$ and $\eta$. Only candidates
with $\Delta R<2.2$ are accepted.

Table 1: Average number of selected candidates per event (efficiency) in % for
the main stages of the offline selection for simulated
$H^{0}\rightarrow{\pi_{v}}{\pi_{v}}$ events with
${\pi_{v}}\rightarrow{{b}{\overline{b}}}$,
$m_{H^{0}}=120{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$,
$m_{{\pi_{v}}}=35{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and
$\tau_{{\pi_{v}}}=10{\rm\,ps}$. The pre-selection consists of the acceptance,
trigger and offline vertex reconstruction. It represents the first stage in
which the candidate yield on the total data sample, shown in the right column,
can be counted. The reported uncertainty on the efficiency is only the
statistical uncertainty from the finite sample size.

selection step signal efficiency yield in data pre-selection $2.125\pm 0.018$
2,555,377 jet reconstruction $1.207\pm 0.014$ 117,054 $m/m_{\text{corr}}$ and
$\Delta R$ $0.873\pm 0.012$ 58,163 trigger on candidate $0.778\pm 0.012$
29,921

Table 2: Average number of selected candidates per event (efficiency) in % for
different ${\pi_{v}}$ masses, lifetimes and decay modes. The reported
uncertainty is only the statistical uncertainty from the finite sample size.
No simulated samples were generated for the 100${\rm\,ps}$ decay to light
quarks.

signal efficiency decay $m_{{\pi_{v}}}$
[${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] $\tau_{{\pi_{v}}}=10{\rm\,ps}$
$\tau_{{\pi_{v}}}=100{\rm\,ps}$ ${\pi_{v}}\rightarrow{{b}{\overline{b}}}$ $25$
$0.373\pm 0.008$ $0.0805\pm 0.0019$ $35$ $0.778\pm 0.012$ $0.181\pm 0.005$
$43$ $0.743\pm 0.011$ $0.183\pm 0.003$ $50$ $0.573\pm 0.015$ $0.154\pm 0.004$
${\pi_{v}}\rightarrow{{c}{\overline{c}}}$ $35$ $2.18\pm 0.05$ –
${\pi_{v}}\rightarrow{{s}{\overline{s}}}$ $35$ $2.06\pm 0.04$ –

Finally, in order to facilitate a reliable estimate of the trigger efficiency,
only candidates triggered by particles belonging to one of the jets are kept.
Table 1 shows the efficiency to select a ${\pi_{v}}$ particle, for an
illustrative mass of 35${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and lifetime
of 10${\rm\,ps}$, together with the yield in the data after the most important
selection steps. The total efficiency for other masses and lifetimes, as well
as for the decays to light quark jets, is shown in Table 2. The efficiencies
listed in Tables 1 and 2 represent the number of selected candidates divided
by the number of generated events. As the selection efficiencies for the two
${\pi_{v}}$ particles in an event are practically independent, the fraction of
selected events with more than one candidate is less than a few percent in
simulated signal. In data no events with more than one ${\pi_{v}}$ candidate
are found.

(a)

(b)

Figure 1: Invariant mass (a) and $p_{\rm T}$ distribution (b) for di-jet
candidates in data and in hidden valley models with $25$, $35$ and
50${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ ${\pi_{v}}$ masses and
10${\rm\,ps}$ lifetime. For visibility, the simulated signal is scaled to
0.62$\mbox{\,fb}^{-1}$ assuming a Higgs cross-section of 10$\rm\,nb$ and
branching fractions of $100\text{\,}\mathrm{\char 37\relax}$ for ${\cal
B}(H\rightarrow{\pi_{v}}{\pi_{v}})$ and ${\cal
B}({\pi_{v}}\rightarrow{{b}{\overline{b}}})$.

Figure 1 shows the mass and $p_{\rm T}$ distributions for selected di-jet
candidates in data and in simulated signal events, assuming a ${\pi_{v}}$
particle with a mass of $25$, $35$ or
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The turn-on at low values in the
mass distribution of events observed in data (Fig. 1) is caused by the minimum
$p_{\rm T}$ requirement on the jets. The rest of the distribution falls off
exponentially. The $p_{\rm T}$ distribution shown in Fig. 1 illustrates that
long-lived particles with a higher mass have lower $p_{\rm T}$ as there is
less momentum available in the Higgs decay. This affects the selection
efficiency since for a given decay time the transverse decay length is
proportional to $p_{\rm T}$.

Studies on simulated events have shown that both the shape and the
normalisation of the mass distribution in data are compatible with the
expected background from ${b}{\overline{b}}$ production. It is not possible to
generate sufficiently large samples of ${b}{\overline{b}}$ events to use these
for a quantitative estimate of the background after the final selection.
Therefore, the signal yield is extracted by a fit to the invariant mass
distribution assuming a smooth shape for the background, as discussed in
Section 6.

Since the background yield, the shape of the background invariant mass
distribution and the selection efficiency strongly depend on the radial
displacement $R_{xy}$, limits are extracted from a simultaneous maximum
likelihood fit to the di-jet invariant mass distribution in five bins of
$R_{xy}$. The intervals are chosen in the most sensitive region, between 0.4
and 4.8$\rm\,mm$. The events at larger radii are not used as they contribute
only marginally to the sensitivity. Figure 2 shows the distribution of
$R_{xy}$ of selected displaced vertices for data and simulated signal events,
together with the bin boundaries. The effect of the reduction in efficiency at
large radii due to the material veto and the HLT1 trigger is visible, as is
the effect of requirements on $R_{xy}$ in the trigger. The trigger effects are
more pronounced in data than in simulated signal, because signal events are
less affected by cuts on the vertex invariant mass.

Figure 2: Distribution of the distance of the displaced vertex to the
interaction region in the transverse plane for data and for a hidden valley
model with $m_{{\pi_{v}}}=35{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and
$\tau_{{\pi_{v}}}=10{\rm\,ps}$ after the full selection. For visibility, the
simulated signal is scaled to 0.62$\mbox{\,fb}^{-1}$ assuming a Higgs cross-
section of 10$\rm\,nb$ and branching fractions of $100\text{\,}\mathrm{\char
37\relax}$ for ${\cal B}(H\rightarrow{\pi_{v}}{\pi_{v}})$ and ${\cal
B}({\pi_{v}}\rightarrow{{b}{\overline{b}}})$. The boundaries of the intervals
used in the fit are indicated by the dotted lines. The generated $R_{xy}$
distribution is approximately exponential with an average of about 2$\rm\,mm$.

The background di-jet invariant mass distribution is characterised by an
exponential falloff, with a low-mass threshold determined mostly by the
minimum $p_{\rm T}$ requirement of the jets. It is modelled by a single-sided
exponential function convoluted with a bifurcated Gaussian function. The
parameters of the background model are fitted to data, independently in each
$R_{xy}$ bin. The signal is modelled by a bifurcated Gaussian function, whose
parameters are determined from simulated events in bins of $R_{xy}$. The
effect of the uncertainty on the jet-energy scale is included by a scale
parameter for the mass, which is common to all bins and constrained using a
sample of $Z+\text{jet}$ events, as explained in Section 5. Additional
nuisance parameters are added to account for the finite statistics of the
simulated samples and the systematic uncertainties on the signal efficiency
and the luminosity. The fit model is implemented using the RooFit [34]
package. Figure 3 shows the fit result in the five radial bins for a signal
model with $m_{{\pi_{v}}}=35{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$\tau_{{\pi_{v}}}=10{\rm\,ps}$.

Figure 3: Di-jet invariant mass distributions for each of the five $R_{xy}$
bins, superimposed with the fits for a hidden valley model with
$m_{{\pi_{v}}}=35{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and
$\tau_{{\pi_{v}}}=10{\rm\,ps}$. The blue line indicates the result of the
total fit to the data. The black short-dashed line is the background-only
contribution, and the red long-dashed line is the fitted signal contribution.
For illustration, the green dash-dotted line shows the signal scaled to a
cross-section of 17$\rm\,pb$, which corresponds to the SM Higgs production
cross-section at 7 TeV [35].

##  5 Systematic uncertainties

Several sources of systematic uncertainties have been considered. The
uncertainties depend on the ${\pi_{v}}$ mass and are summarised in Table 3.
The uncertainty on the vertex finding efficiency is assessed by comparing the
efficiency of the vertexing algorithm on a sample of
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{*0}$
with $K^{*0}\rightarrow{{K}^{+}}{{\pi}^{-}}$ events in data and simulation as
a function of $R_{xy}$. The efficiency difference is about
$7.5\text{\,}\mathrm{\char 37\relax}$ at large $R_{xy}$, which is taken as an
estimate of the uncertainty on the vertex finding algorithm efficiency. Since
the ${B}^{0}$ vertices have only four tracks, and the ${\pi_{v}}$ decays
studied in this paper have typically more tracks, this is considered a
conservative estimate. The uncertainty on the track finding efficiency for
prompt tracks in LHCb is $1.4\text{\,}\mathrm{\char 37\relax}$ per track, with
a small dependence on track kinematics [36]. The uncertainty for displaced
tracks was evaluated in the context of a recent LHCb measurement of $b$-hadron
lifetimes [37] and extrapolated to larger $R_{xy}$, leading to a per-track
uncertainty of $2\text{\,}\mathrm{\char 37\relax}$. Due to requirements on the
minimal number of tracks in the vertex, this translates into an uncertainty on
the vertex finding efficiency, which is estimated to be
$2\text{\,}\mathrm{\char 37\relax}$ for signal events. Adding in quadrature
the track efficiency and the vertex finding algorithm efficiency uncertainties
leads to a total uncertainty of $7.9\text{\,}\mathrm{\char 37\relax}$ on the
vertex reconstruction. The selection on the vertex sum-$p_{\rm T}$ and mass is
affected by the track finding efficiency as well. Propagating the per-track
uncertainty leads to an uncertainty on the vertex selection efficiency of up
to $2.9\text{\,}\mathrm{\char 37\relax}$, depending on the ${\pi_{v}}$ mass.

The uncertainties related to the jet selection are determined by comparing
jets in data and simulation on a sample of $Z+\text{jet}$ events, analogously
to a recent LHCb measurement of $Z+\text{jet}$ production [32]. The $Z$
candidate is reconstructed in the $\mu^{+}$ $\mu^{-}$ final state from two
oppositely charged tracks, identified as muons, that form a good vertex and
have an invariant mass in the range 60 – 120$\mathrm{\,Ge\kern-1.00006ptV}$.
Jets are reconstructed using the same selection of input particles as in the
reconstruction of jets for long-lived particles, except that the origin vertex
is in this case the PV consistent with the $Z$ vertex. The differences between
data and simulation in the $Z+\text{jet}$ sample are parameterised as function
of the jet $p_{\rm T}$ and subsequently propagated to the simulated hidden
valley signal samples.

The uncertainty on the jet energy scale is derived from the ratio of
transverse momenta of the jet and the $Z$, which are expected to have a back-
to-back topology, and correlated transverse momenta. Data and simulation agree
within about $2\text{\,}\mathrm{\char 37\relax}$, resulting in an uncertainty
on the di-jet invariant mass scale of $4\text{\,}\mathrm{\char 37\relax}$.
This uncertainty on the signal shape is taken into account in the fitting
procedure. The uncertainty on the jet-energy scale also affects the jet
reconstruction efficiency due to the requirement on the minimum jet $p_{\rm
T}$. It leads to an uncertainty on the efficiency between 0.3 and
$1.3\text{\,}\mathrm{\char 37\relax}$, depending on the assumed ${\pi_{v}}$
particle mass. The uncertainty on the hadronic jet identification requirements
are assessed using the $Z+\text{jet}$ sample as well and amount to about
$3\text{\,}\mathrm{\char 37\relax}$.

The resolutions on the pointing angle $\theta$ and on $\Delta R$ are dominated
by the resolution on the direction of the ${\pi_{v}}$ candidate, which in turn
is determined by the jet angular resolution. The latter is estimated from the
difference between data and simulation in the resolution of the azimuthal
angle between the jet and the $Z$. Due to the limited statistics in the
$Z+\text{jet}$ sample a relatively large uncertainty between 2.0 and
$4.6\text{\,}\mathrm{\char 37\relax}$ is obtained, depending on the
${\pi_{v}}$ mass.

The trigger selection efficiency on signal is determined from the simulation.
The trigger efficiencies in data and simulation are compared using a sample of
generic $B\rightarrow J/\psi X$ events that contain an offline reconstructed
displaced vertex, but are triggered independently of the displaced vertex
trigger lines. The integrated efficiency difference for the trigger stages L0,
HLT1 and HLT2 amounts to systematic uncertainties of at most 4.6, 4.3 and
$6.3\text{\,}\mathrm{\char 37\relax}$ respectively. This is a conservative
estimate since the trigger efficiencies for the sample of displaced $J/\psi$
vertices are smaller than the efficiencies for the signal, which consists of
heavier, more displaced objects with a larger number of tracks. Finally, the
uncertainty on the luminosity at the LHCb interaction point is
$1.7\text{\,}\mathrm{\char 37\relax}$ [38].

Table 3: Systematic uncertainties on the selection efficiency and luminosity
for simulated hidden valley events with a lifetime of 10${\rm\,ps}$ and
various ${\pi_{v}}$ masses.

source relative uncertainty (%) ${\pi_{v}}$ mass
[${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] 25 35 43 50 vertex
reconstruction 7.9 7.9 7.9 7.9 vertex scalar-$p_{\rm T}$ and mass 2.9 2.3 2.0
1.7 jet reconstruction 1.3 0.6 0.4 0.3 jet identification 2.9 3.0 3.2 3.2 jet
pointing 4.6 2.9 2.6 2.0 L0 trigger 4.6 4.5 4.5 4.4 HLT1 trigger 4.1 4.0 4.0
4.3 HLT2 trigger 5.9 5.9 6.1 6.3 luminosity 1.7 1.7 1.7 1.7 total 13.3 12.7
12.6 12.6

Several alternatives have been considered for the background mass model, in
particular with an additional exponential component, or a component that is
independent of the mass. With these models the estimated background yield at
higher mass is larger than with the nominal background model, leading to
tighter limits on the signal. As the nominal model gives the most conservative
limit, no additional systematic uncertainty is assigned.

##  6 Results

The fit procedure is performed for a ${\pi_{v}}$ mass of 25, 35, 43 and
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and for several values of the
lifetime in between 1 and 200${\rm\,ps}$. No significant signal is observed
for any combination of ${\pi_{v}}$ mass and lifetime. Upper limits are
extracted using the $\text{CL}_{\text{s}}$ method [39] with a frequentist
treatment of the nuisance parameters described above, as implemented in the
RooStats [40] package.

Limits are set on the Higgs production cross-section multiplied by the
branching fraction into long-lived particles $\sigma(H)\times{\cal
B}(H\rightarrow{\pi_{v}}{\pi_{v}})$. In the simulation it is assumed that both
${\pi_{v}}$ particles decay to the same final state. If the decay width of the
${\pi_{v}}$ particle is dominated by final states other than
${q}{\overline{q}}$, the limits scale as $1/({\cal
B}_{{{q}{\overline{q}}}}(2-{\cal B}_{{{q}{\overline{q}}}}))$ where ${\cal
B}_{{{q}{\overline{q}}}}$ is the ${\pi_{v}}\rightarrow{{q}{\overline{q}}}$
branching fraction. The obtained $95\text{\,}\mathrm{\char 37\relax}$ CL upper
limits on $\sigma(H)\times{\cal B}(H\rightarrow{\pi_{v}}{\pi_{v}})$, under the
assumption of a $100\text{\,}\mathrm{\char 37\relax}$ branching fraction to
${b}{\overline{b}}$, are shown in Table 4 and in Fig. 4. As the background
decreases with the observed di-jet invariant mass, the limits become stronger
with increasing ${\pi_{v}}$ mass. The sensitivity has an optimal value at a
lifetime of about 5${\rm\,ps}$.

Figure 4: Observed $95\text{\,}\mathrm{\char 37\relax}$ CL cross-section upper
limits on a hidden valley model [6, *Strassler:2006ri, *Han:2007ae] for
various ${\pi_{v}}$ masses, as a function of ${\pi_{v}}$ lifetime. Both
${\pi_{v}}$ particles are assumed to decay into ${b}{\overline{b}}$, unless
specified otherwise.

Additional limits are set on models with a ${\pi_{v}}$ particle decaying to
${c}{\overline{c}}$ and to ${s}{\overline{s}}$. The limits for ${\pi_{v}}$
decay to ${u}{\overline{u}}$ and ${d}{\overline{d}}$ are expected to be the
same as for ${s}{\overline{s}}$. The light quark decays result in a higher
displaced vertex track multiplicity, and lighter jets, leading to a higher
selection efficiency. Consequently, the limits for decays to light quark jets
are more stringent than those for decays to $b$-quark jets.

Table 4: Observed $95\text{\,}\mathrm{\char 37\relax}$ CL cross-section upper
limits on $\sigma(H)\times{\cal B}(H\rightarrow{\pi_{v}}{\pi_{v}})$ (in
$\rm\,pb$) on a hidden valley [6, *Strassler:2006ri, *Han:2007ae] model for
various ${\pi_{v}}$ masses and lifetimes. Both ${\pi_{v}}$ particles are
assumed to decay into ${b}{\overline{b}}$, unless specified otherwise.

${\pi_{v}}$ mass [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ ] ${\pi_{v}}$
lifetime [${\rm\,ps}$ ] 1 2 5 10 20 50 100 200 25 106.3 54.6 43.8 54.2 80.0
164.1 285.7 588.5 35 19.0 10.4 8.0 8.9 13.3 25.4 46.5 89.8 43 10.5 5.6 4.4 4.7
6.7 12.4 22.7 42.8 50 10.6 5.1 3.7 3.8 4.8 9.3 16.2 29.3 35
(${\pi_{v}}\rightarrow c\bar{c}$) 3.7 2.4 2.1 2.4 3.4 6.7 12.5 24.1 35
(${\pi_{v}}\rightarrow s\bar{s}$) 3.4 2.1 1.9 2.2 3.3 6.4 11.6 22.0

##  7 Conclusion

A search has been presented for massive, long-lived particles in a sample of
$pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, corresponding to
an integrated luminosity of 0.62$\mbox{\,fb}^{-1}$, collected by the LHCb
experiment. The long-lived spin-zero particles are assumed to be pair-produced
in the decay of a 120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ SM Higgs, and
to decay to two hadronic jets. They appear for instance as ${\pi_{v}}$
particles in hidden valley models. A single ${\pi_{v}}$ particle is identified
by a displaced vertex and two associated jets. No significant signal for
${\pi_{v}}$ particles with a mass between $25$ and
50${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and a lifetime between $1$ and
200${\rm\,ps}$ is observed. Assuming a $100\text{\,}\mathrm{\char 37\relax}$
branching fraction to $b$-quark jets, the $95\text{\,}\mathrm{\char 37\relax}$
CL upper limits on the production cross-section $\sigma(H)\,\times\,{\cal
B}(H\rightarrow{\pi_{v}}{\pi_{v}})$ are in the range 4 – 600$\rm\,pb$.

The results cover a region in mass and lifetime that so far has been
unexplored at the LHC. The obtained upper limits are more restrictive than
results from the Tevatron experiments in the same mass and lifetime region.
The best sensitivity is obtained for ${\pi_{v}}$ particles with a lifetime of
about 5${\rm\,ps}$ and a mass above approximately
40${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The SM Higgs cross-section at
7$\mathrm{\,Te\kern-1.00006ptV}$ is about 17$\rm\,pb$ [35]. The measurements
in the most sensitive region exclude branching fractions of greater than
$25\text{\,}\mathrm{\char 37\relax}$ for a SM Higgs boson to pair produce
${\pi_{v}}$ particles that decay to two hadronic jets.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); INFN (Italy);
FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). The Tier1 computing centres are
supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted
to the communities behind the multiple open source software packages on which
we depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia). Individual groups or
members have received support from EPLANET, Marie Skłodowska-Curie Actions and
ERC (European Union), Conseil général de Haute-Savoie, Labex ENIGMASS and
OCEVU, Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain),
Royal Society and Royal Commission for the Exhibition of 1851 (United
Kingdom).

## References

  * [1] L. M. Carpenter, D. E. Kaplan, and E.-J. Rhee, _Six-quark decays of the Higgs boson in supersymmetry with $R$-parity violation_, Phys. Rev. Lett. 99 (2007) 211801, arXiv:hep-ph/0607204
  * [2] J. M. Butterworth, J. R. Ellis, A. R. Raklev, and G. P. Salam, _Discovering baryon-number violating neutralino decays at the LHC_ , Phys. Rev. Lett. 103 (2009) 241803, arXiv:0906.0728
  * [3] D. E. Kaplan and K. Rehermann, _Proposal for Higgs and superpartner searches at the LHCb experiment_ , JHEP 10 (2007) 056, arXiv:0705.3426
  * [4] F. de Campos, O. J. P. Eboli, M. B. Magro, and D. Restrepo, _Searching supersymmetry at the LHCb with displaced vertices_ , Phys. Rev. D79 (2009) 055008, arXiv:0809.0007
  * [5] F. de Campos and M. B. Magro, _Displaced vertices in GMSB models at LHC_ , arXiv:1306.5773
  * [6] M. J. Strassler and K. M. Zurek, _Echoes of a hidden valley at hadron colliders_ , Phys. Lett. B651 (2007) 374, arXiv:hep-ph/0604261
  * [7] M. J. Strassler and K. M. Zurek, _Discovering the Higgs through highly-displaced vertices_ , Phys. Lett. B661 (2008) 263, arXiv:hep-ph/0605193
  * [8] T. Han, Z. Si, K. M. Zurek, and M. J. Strassler, _Phenomenology of hidden valleys at hadron colliders_ , JHEP 07 (2008) 008, arXiv:0712.2041
  * [9] ATLAS collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , Phys. Lett. B716 (2012) 1, arXiv:1207.7214
  * [10] CMS collaboration, S. Chatrchyan et al., _Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC_ , Phys. Lett. B716 (2012) 30, arXiv:1207.7235
  * [11] CDF collaboration, T. Aaltonen et al., _Search for heavy metastable particles decaying to jet pairs in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV_, Phys. Rev. D85 (2012) 012007, arXiv:1109.3136
  * [12] D0 collaboration, V. M. Abazov et al., _Search for resonant pair production of neutral long-Lived particles decaying to $b\overline{b}$ in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$_, Phys. Rev. Lett. 103 (2009) 071801, arXiv:0906.1787
  * [13] ATLAS collaboration, G. Aad et al., _Search for a light Higgs boson decaying to long-lived weakly-interacting particles in proton-proton collisions at $\sqrt{s}=7$ TeV with the ATLAS detector_, Phys. Rev. Lett. 108 (2012) 251801, arXiv:1203.1303
  * [14] CMS, V. Khachatryan et al., _Search for long-lived neutral particles decaying to quark-antiquark pairs in proton-proton collisions at $\sqrt{s}=$ 8 TeV_, Phys. Rev. D91 (2015), no. 1 012007, arXiv:1411.6530
  * [15] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [16] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) 09007, arXiv:1405.7808
  * [17] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [18] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [19] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [20] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [21] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
  * [22] J. Pumplin et al., _New generation of parton distributions with uncertainties from global QCD analysis_ , JHEP 07 (2002) 012, arXiv:hep-ph/0201195
  * [23] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [24] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [25] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [26] Geant4 collaboration, S. Agostinelli et al., _Geant4: a simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [27] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [28] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [29] M. Kucharczyk, P. Morawski, and M. Witek, _Primary vertex reconstruction at LHCb_ , LHCb-PUB-2014-044
  * [30] LHCb collaboration, R. Aaij et al., _Search for the rare decay $K^{0}_{S}\rightarrow\mu^{+}\mu^{-}$_, JHEP 01 (2013) 090, arXiv:1209.4029
  * [31] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [32] LHCb collaboration, R. Aaij et al., _Study of forward Z + jet production in pp collisions at $\sqrt{s}=7$ TeV_, JHEP 01 (2014) 033, arXiv:1310.8197
  * [33] M. Cacciari, G. P. Salam, and G. Soyez, _The anti- $k_{T}$ jet clustering algorithm_, JHEP 04 (2008) 063, arXiv:0802.1189
  * [34] W. Verkerke and D. P. Kirkby, _The RooFit toolkit for data modeling_ , eConf C0303241 (2003) MOLT007, arXiv:physics/0306116
  * [35] LHC Higgs Cross Section Working Group, S. Heinemeyer et al., _Handbook of LHC Higgs Cross Sections: 3. Higgs Properties_ , arXiv:1307.1347
  * [36] LHCb collaboration, R. Aaij et al., _Measurement of the track reconstruction efficiency at LHCb_ , JINST 10 (2015) P02007, arXiv:1408.1251
  * [37] LHCb collaboration, R. Aaij et al., _Measurements of the $B^{+},B^{0},B^{0}_{s}$ meson and $\Lambda^{0}_{b}$ baryon lifetimes_, JHEP 04 (2014) 114, arXiv:1402.2554
  * [38] LHCb collaboration, R. Aaij et al., _Precision luminosity measurements at LHCb_ , JINST 9 (2014) P12005, arXiv:1410.0149
  * [39] A. L. Read, _Presentation of search results: The CL(s) technique_ , J. Phys. G28 (2002) 2693
  * [40] L. Moneta et al., _The RooStats Project_ , PoS ACAT2010 (2010) 057, arXiv:1009.1003

LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
S. Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, A. Bertolin22, M.-O.
Bettler47, M. van Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A.
Bizzeti17,i, P.M. Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59,
V. Bocci25, A. Bondar34, N. Bondar30,38, W. Bonivento15, S. Borghi54, A.
Borgia59, M. Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, D. Brett54,
M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46, A. Bursche40, J.
Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,p,
P. Campana18, D. Campora Perez38, L. Capriotti54, A. Carbone14,d, G.
Carboni24,l, R. Cardinale19,38,j, A. Cardini15, L. Carson50, K. Carvalho
Akiba2,38, RCM Casanova Mohr36, G. Casse52, L. Cassina20,k, L. Castillo
Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci23,t, M. Charles8, Ph.
Charpentier38, M. Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M.
Chrzaszcz40,26, X. Cid Vidal38, G. Ciezarek41, P.E.L. Clarke50, M.
Clemencic38, H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5,
V. Cogoni15,e, L. Cojocariu29, G. Collazuol22, P. Collins38, A. Comerma-
Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, A.C.
Crocombe48, M. Cruz Torres60, S. Cunliffe53, R. Currie53, C. D’Ambrosio38, J.
Dalseno46, P. David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, C.-T. Dean51, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N.
Déléage4, D. Derkach55, O. Deschamps5, F. Dettori38, B. Dey40, A. Di Canto38,
A Di Domenico25, H. Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, A.
Dosil Suárez37, D. Dossett48, A. Dovbnya43, K. Dreimanis52, G. Dujany54, F.
Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S.
Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S. Eisenhardt50, U.
Eitschberger9, R. Ekelhof9, L. Eklund51, I. El Rifai5, Ch. Elsasser40, S.
Ely59, S. Esen11, H.M. Evans47, T. Evans55, A. Falabella14, C. Färber11, C.
Farinelli41, N. Farley45, S. Farry52, R. Fay52, D. Ferguson50, V. Fernandez
Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M.
Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick39, T. Fiutowski27, P.
Fol53, M. Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38,
C. Frei38, M. Frosini17, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, S. Gallorini22,38, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. García Pardiñas37, J. Garofoli59, J. Garra Tico47, L. Garrido36, D.
Gascon36, C. Gaspar38, U. Gastaldi16, R. Gauld55, L. Gavardi9, G. Gazzoni5, A.
Geraci21,v, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Gianì39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C. Gotti20,k, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
E. Graverini40, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg63, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, J. He38, T. Head39, V. Heijne41, K.
Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E. van
Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54, W.
Hulsbergen41, N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P.
Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, A. Jawahery58,
F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, S.
Karodia51, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20,38,k, C.
Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O. Kochebina7, M. Kolpin11,
I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, L. Kravchuk33,
K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, W.
Kucewicz26,o, M. Kucharczyk20,26,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, G. Lanfranchi18, C. Langenbruch48, B. Langhans38,
T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R.
Lefèvre5, A. Leflat32, J. Lefrançois7, O. Leroy6, T. Lesiak26, B.
Leverington11, Y. Li3, T. Likhomanenko64, M. Liles52, R. Lindner38, C. Linn38,
F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51, J.H. Lopes2, P. Lowdon40,
D. Lucchesi22,r, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, A.
Malinin64, G. Manca15,e, G. Mancinelli6, A. Mapelli38, J. Maratas5, J.F.
Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,t, R. Märki39, J.
Marks11, G. Martellotti25, M. Martinelli39, D. Martinez Santos42, F. Martinez
Vidal65, D. Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C.
Matteuzzi20, A. Mazurov45, M. McCann53, J. McCarthy45, A. McNab54, R.
McNulty12, B. McSkelly52, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41,
D.A. Milanes62, M.-N. Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5,
M. Morandin22, P. Morawski27, A. Mordà6, M.J. Morello23,t, J. Moron27, A.-B.
Morris50, R. Mountain59, F. Muheim50, K. Müller40, M. Mussini14, B. Muster39,
P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21,
S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C.
Nguyen-Mau39,q, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A.
Novoselov35, D.P. O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S.
Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater66, M. Orlandea29,
B. Osorio Rodrigues1, J.M. Otalora Goicochea2, A. Otto38, P. Owen53, A.
Oyanguren65, B.K. Pal59, A. Palano13,c, F. Palombo21,u, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, L.L. Pappalardo16,f, C.
Parkes54, C.J. Parkinson9,45, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, A. Pearce54,49, A. Pellegrino41, G. Penso25,m, M. Pepe
Altarelli38, S. Perazzini14,d, P. Perret5, L. Pescatore45, E. Pesen67, K.
Petridis53, A. Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8,
A. Poluektov48,34, I. Polyakov31, E. Polycarpo2, A. Popov35, D. Popov10, B.
Popovici29, C. Potterat2, E. Price46, J.D. Price52, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,s, W.
Qian4, B. Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama23,
M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G. Raven42, F. Redi53, S.
Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, S. Richards46, M.
Rihl38, K. Rinnert52, V. Rives Molina36, P. Robbe7, A.B. Rodrigues1, E.
Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, H. Ruiz36, P. Ruiz Valls65,
J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino
Guimaraes2, C. Sanchez Mayordomo65, B. Sanmartin Sedes37, R. Santacesaria25,
C. Santamarina Rios37, E. Santovetti24,l, A. Sarti18,m, C. Satriano25,n, A.
Satta24, D.M. Saunders46, D. Savrina31,32, M. Schiller38, H. Schindler38, M.
Schlupp9, M. Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H.
Schune7, R. Schwemmer38, B. Sciascia18, A. Sciubba25,m, A. Semennikov31, I.
Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko64,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, I.
Skillicorn51, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, P. Spradlin51, S. Sridharan38, F.
Stagni38, M. Stahl11, S. Stahl11, O. Steinkamp40, O. Stenyakin35, F Sterpka59,
S. Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,t, M.
Straticiuc29, U. Straumann40, R. Stroili22, L. Sun57, W. Sutcliffe53, K.
Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28, P. Szczypka39,38,
T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G. Tellarini16,f, F. Teubert38, C.
Thomas55, E. Thomas38, J. van Tilburg41, V. Tisserand4, M. Tobin39, J. Todd57,
S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-Joergensen55, N. Torr55,
E. Tournefier4, S. Tourneur39, M.T. Tran39, M. Tresch40, A. Trisovic38, A.
Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38, A. Ukleja28,
A. Ustyuzhanin64, U. Uwer11, C. Vacca15,e, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11,
JVVB Viana Barbosa38, B. Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-
Cardona36,p, A. Vollhardt40, D. Volyanskyy10, D. Voong46, A. Vorobyev30, V.
Vorobyev34, C. Voß63, J.A. de Vries41, R. Waldi63, C. Wallace48, R. Wallace12,
J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D.
Websdale53, M. Whitehead48, D. Wiedner11, G. Wilkinson55,38, M. Wilkinson59,
M.P. Williams45, M. Williams56, H.W. Wilschut66, F.F. Wilson49, J.
Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, K. Wyllie38, Y. Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X.
Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, L. Zhang3, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
66Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41
67Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
vPolitecnico di Milano, Milano, Italy


# Positive operator-valued kernels and non-commutative probability

Palle E.T. Jorgensen  (Palle E.T. Jorgensen) Department of Mathematics, The
University of Iowa, Iowa City, IA 52242-1419, U.S.A.  palle-
<EMAIL_ADDRESS>and  James Tian  (James F. Tian) Mathematical Reviews, 416
4th Street Ann Arbor, MI 48103-4816, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We prove new factorization and dilation results for general positive operator-
valued kernels, and we present their implications for associated Hilbert
space-valued Gaussian processes, and their covariance structure. Further
applications are to non-commutative probability theory, including a non-
commutative Radon–Nikodym theorem for systems of completely positive maps.

###### Key words and phrases:

Positive definite functions, kernels, Gaussian processes, covariance,
dilation, noncommutative Radon-Nikodym derivatives, completely positive maps,
measurement, quantum states, quantum gates.

###### 2020 Mathematics Subject Classification:

46E22, 46N50, 46L53, 47A20, 47N50, 60G15, 81P15, 81P47.

###### Contents

  1. 1 Introduction
  2. 2 $\mathcal{L}\left(H\right)$-valued kernels and $H$-valued Gaussian processes
  3. 3 A non-commutative Radon–Nikodym theorem
  4. 4 Completely positive maps

##  1\. Introduction

In this paper we present a result which offers a canonical link between a
general setting for operator valued completely positive maps on the one hand,
with an induced scalar valued counterpart. We further offer applications to a
variety of neighboring areas, including the following six closely interrelated
areas: (i) operator valued Gaussian processes, and their associated covariance
structure; (ii) universal factorizations; (iii) non-commutative operator
valued Radon-Nikodym derivatives and their applications to quantum gates and
to quantum states; (iv) partial orders on operator valued completely positive
maps via their associated reproducing kernel Hilbert spaces; (v) intertwining
operators for representations induced from completely positive maps; and (vi)
applications of (iii) to completely positive maps and associated quantum
gates.

The paper is organized as follows: In Section 2 we present the general
framework, and the main theorems, for operator valued completely positive
maps, as well as their associated structures, including operator valued
Gaussian processes. The applications to non-commutative operator valued Radon-
Nikodym derivatives are given in Section 3, while the focus in Section 4 is
that of completely positive maps.

For the reader’s benefit, we include the following citations: For the theory
of positive definite kernels [Gue22b, AA22, Dav21], Hilbert space valued
Gaussian processes [AJ22, Kre19, LM17], completely positive maps [Tho24,
Kod23, ZD22, SCC21], quantum gates [ZHL24, PY24, GLL24, LCH24], and operator
valued Radon-Nikodym derivatives [MPR20, Ahm13].

The literature on the theory of reproducing kernels is vast, encompassing both
theoretical foundations and recent applications. For a comprehensive overview,
including the latest developments, we refer to the following resources: [AS56,
Aro48, AJ21, PR16, AFMP94, LYA23, CY17, Alp91, AD93, AL95, AB97, AD06, AJ15].

Notation. Throughout the paper, we use the physics convention that inner
products are linear in the second variable. Let
$\left|a\left\rangle\right\langle b\right|$ denote Dirac’s rank-1 operator,
$c\mapsto a\left\langle b,c\right\rangle$. $\mathcal{L}\left(H\right)$ denotes
the algebra of all bounded linear operators in a Hilbert space $H$.

For a positive definite (p.d.) kernel $K:S\times S\rightarrow\mathbb{C}$, let
$H_{K}$ be the corresponding reproducing kernel Hilbert space (RKHS). $H_{K}$
is the Hilbert completion of

$span_{\mathbb{C}}\left\\{K_{y}\left(\cdot\right):=K\left(\cdot,y\right)\mid
y\in S\right\\}$ (1.1)

with respect to the inner product

$\left\langle\sum_{i}c_{i}K\left(\cdot,x_{i}\right),\sum_{i}d_{j}K\left(\cdot,y_{j}\right)\right\rangle_{H_{K}}:=\sum_{i}\sum_{j}\overline{c}_{i}d_{j}K\left(x_{i},x_{j}\right).$
(1.2)

The following reproducing property holds:

$\varphi\left(x\right)=\left\langle
K\left(\cdot,x\right),\varphi\right\rangle_{H_{K}},\forall x\in
S,\>\forall\varphi\in H_{K}.$ (1.3)

Any scalar-valued kernel $K$ as above is associated with a zero-mean Gaussian
process, where $K$ is the covariance:

$K\left(s,t\right)=\mathbb{E}\left[\overline{W_{s}}W_{t}\right],$ (1.4)

and $W_{s}\sim N\left(0,K\left(s,s\right)\right)$.

An $\mathcal{L}\left(H\right)$-valued kernel $K:S\times
S\rightarrow\mathcal{L}\left(H\right)$ is p.d. if

$\sum_{i=1}^{n}\left\langle
a_{i},K\left(s_{i},s_{j}\right)a_{j}\right\rangle_{H}\geq 0$ (1.5)

for all $\left(a_{i}\right)_{1}^{n}$ in $H$, and all $n\in\mathbb{N}$.

##  2\. $\mathcal{L}\left(H\right)$-valued kernels and $H$-valued Gaussian
processes

In this section, we begin by revisiting a universal construction for operator-
valued positive definite kernels. This construction will be adapted for
various applications, which will be explored in the following sections.

######  Theorem 2.1 (Universal Factorization).

Let $K:S\times S\rightarrow\mathcal{L}\left(H\right)$ be a p.d. kernel. Then
there exists a RKHS $H_{\tilde{K}}$, and a family of operators
$V\left(s\right):H\rightarrow H_{\tilde{K}}$, $s\in S$, such that

$H_{\tilde{K}}=\overline{span}\left\\{V\left(s\right)a:a\in H,s\in S\right\\}$
(2.1)

and

$K\left(s,t\right)=V\left(s\right)^{*}V\left(t\right).$ (2.2)

Conversely, if there is a Hilbert space $L$ and operators
$V\left(s\right):H\rightarrow L$, $s\in S$, such that

$L=\overline{span}\left\\{V\left(s\right)a:a\in H,s\in S\right\\}$ (2.3)

and (2.2) holds, then $L\simeq H_{\tilde{K}}$.

###### Proof.

For a detailed proof, we refer the reader to [JT24b, JT24a]. We provide a
brief outline of the main steps below:

Given a p.d. kernel $K:S\times S\rightarrow\mathcal{L}\left(H\right)$, define
$\tilde{K}:\left(S\times H\right)\times\left(S\times
H\right)\rightarrow\mathbb{C}$ as

$\tilde{K}\left(\left(s,a\right),\left(t,b\right)\right):=\left\langle
a,K\left(s,t\right)b\right\rangle_{H}.$ (2.4)

Then $\tilde{K}$ is a scalar-valued p.d. kernel. Let $H_{\tilde{K}}$ be the
associated RKHS.

Set $V\left(s\right):H\rightarrow H_{\tilde{K}}$ by

$V\left(s\right)a=\tilde{K}_{\left(s,a\right)}=\tilde{K}\left(\cdot,\left(s,a\right)\right):\left(S\times
H\right)\rightarrow\mathbb{C},\quad\forall a\in H.$ (2.5)

One verifies that

$V\left(s\right)^{*}\tilde{K}\left(\cdot,\left(t,b\right)\right)=K\left(s,t\right)b.$

and the factorization $K\left(s,t\right)=V\left(s\right)^{*}V\left(t\right)$
holds. ∎

######  Remark 2.2.

The setting in Theorem 2.1 includes and extends well known constructions in
classical dilation theory. Notably, transitioning from an operator-valued
kernel $K$ to the scalar-valued positive definite kernel $\tilde{K}$ enables a
function-based approach to dilation theory, contrasting with the traditional
abstract spaces of equivalence classes. The literature is vast, and here we
call attention to [Arv10], and the papers cited there.

Building upon Ito’s seminal work, Hilbert spaces and their corresponding
operators have become essential tools in the stochastic analysis of Gaussian
processes. For more detailed insights, see e.g., [Sch52, AJ15, AJ21], as well
as the earlier works cited therein. We explore two principal types of Gaussian
processes in this field: (i) scalar-valued processes that are indexed by a
Hilbert space, typically through an Ito-isometry; and (ii) processes where the
Gaussian values are embedded directly within a Hilbert space (see e.g.,
[Lop23, Gue22a, KM20, FS20, Kat19, JT23]). This paper concentrates on the
second type, which provides enhanced adaptability in modeling covariance
structures—a key element in processing large datasets. We now proceed to
discuss the details of this approach.

######  Theorem 2.3.

Every operator-valued p.d. kernel $K:S\times
S\rightarrow\mathcal{L}\left(H\right)$ is associated with an $H$-valued
Gaussian process $\left\\{W\left(s\right)\right\\}_{s\in S}$ with zero-mean,
realized on some probability space $\left(\Omega,\mathbb{P}\right)$, such that

$K\left(s,t\right)=\int_{\Omega}\left|W\left(s\right)\left\rangle\right\langle
W\left(t\right)\right|d\mathbb{P}.$ (2.6)

Conversely, every $H$-valued Gaussian process is obtained from such a
$\mathcal{L}\left(H\right)$-valued kernel.

######  Remark 2.4.

More precisely, the identity (2.6) holds in the sense that

$\mathbb{E}\left[\left\langle a,W\left(s\right)\right\rangle_{H}\left\langle
W\left(t\right),b\right\rangle_{H}\right]=\left\langle
a,K\left(s,t\right)b\right\rangle_{H}.$ (2.7)

for all $s,t\in S$, and all $a,b\in H$. For a detailed proof, see [JT24a].

######  Definition 2.5.

Consider the $H$-valued Gaussian process $W_{s}:\Omega\rightarrow H$,

$W\left(t\right)=\sum_{i}\left(V\left(t\right)^{*}\varphi_{i}\right)Z_{i},$
(2.8)

where $\left(\varphi_{i}\right)$ is an ONB in $H_{\tilde{K}}$. Following
standard conventions, here $\left\\{Z_{i}\right\\}$ refers to a choice of an
independent identically distributed (i.i.d.) system of standard scalar
Gaussians $N(0,1)$ random variables, and with an index matching the choice of
ONB.

######  Theorem 2.6.

We have

$\mathbb{E}\left[\left\langle a,W\left(s\right)\right\rangle_{H}\left\langle
W\left(t\right),b\right\rangle_{H}\right]=\left\langle
a,K\left(s,t\right)b\right\rangle_{H}.$ (2.9)

###### Proof.

Note that $\mathbb{E}\left[Z_{i}Z_{j}\right]=\delta_{i,j}$. From this, we get

$\displaystyle\text{LHS}_{\left(\ref{eq:d9}\right)}$
$\displaystyle=\mathbb{E}\left[\left\langle
a,W\left(s\right)\right\rangle_{H}\left\langle
W\left(t\right),b\right\rangle_{H}\right]$
$\displaystyle=\sum_{i}\sum_{j}\left\langle
a,V\left(s\right)^{*}\varphi_{i}\right\rangle\left\langle
V\left(t\right)^{*}\varphi_{j},b\right\rangle\mathbb{E}\left[Z_{i}Z_{j}\right]$
$\displaystyle=\sum_{i}\left\langle
V\left(s\right)a,\varphi_{i}\right\rangle_{H_{\tilde{K}}}\left\langle\varphi_{i},V\left(t\right)b\right\rangle_{H_{\tilde{K}}}$
$\displaystyle=\left\langle
V\left(s\right)a,V\left(t\right)b\right\rangle_{H_{\tilde{K}}}=\left\langle
a,V\left(s\right)^{*}V\left(t\right)b\right\rangle_{H}=\left\langle
a,K\left(s,t\right)b\right\rangle_{H}.$

∎

######  Remark 2.7.

The Gaussian process $\left\\{W\left(t\right)\right\\}_{t\in S}$ in (2.8) is
well defined and possesses the stated properties. This is an application of
the central limit theorem to the choice $\left\\{Z_{i}\right\\}$ of i.i.d.
$N\left(0,1\right)$ Gaussians on the right-side of (2.8).

Varying the choices of ONBs $\left(\varphi_{i}\right)$ and i.i.d.
$N\left(0,1\right)$ Gaussian random variables $\left(Z_{i}\right)$ will result
in different Gaussian processes $\left\\{W\left(t\right)\right\\}_{t\in S}$,
but all will adhere to the covariance condition specified in (2.9).
Importantly, once the ONB $\left(\varphi_{i}\right)$ and the random variables
$\left(Z_{i}\right)$ are fixed, the resulting Gaussian process
$\left\\{W\left(t\right)\right\\}_{t\in S}$ is uniquely determined by its
first two moments (mean and covariance).

##  3\. A non-commutative Radon–Nikodym theorem

As is known, the theory of von Neumann algebras offers a framework for non-
commutative measure theory, see e.g., [Sak65]. In this interpretation, the
projections in the algebra are the characteristic functions of the (non-
commuting) “measurable sets.” From the elements of the Hilbert space $H$ we
then build the bounded measurable functions; and the (normal) states are the
probability measures on the underlying (non-commutative) measure space. Here
we shall supplement this theory with Aronszajn’s notion of systems of ordered
p.d. kernels and contractive containment of the corresponding reproducing
kernel Hilbert spaces (RKHSs.) From this we then build a natural (but
different) notion of non-commutative Radon–Nikodym derivatives.

For the sake of completeness, we include a proof of Aronszajn’s inclusion
theorem below. It states that, for two (scalar-valued) p.d. kernels $K$ and
$L$ on $S\times S$, $K\leq L$ (i.e., $L-K$ is p.d.) if and only if $H_{K}$ is
contractively contained in $H_{L}$ (see e.g., [Aro50]).

######  Theorem 3.1 (Aronszajn).

If $K\leq L$, then $H_{K}\subset H_{L}$ and
$\left\|f\right\|_{H_{L}}\leq\left\|f\right\|_{H_{K}}$ for all $f\in H_{K}$.

###### Proof.

Let $f\in H_{K}$. To verify that $f\in H_{L}$, it is equivalent to show that

$\left|\sum
c_{i}f\left(s_{i}\right)\right|^{2}\leq\text{const}\cdot\left\|\sum
c_{i}L_{s_{i}}\right\|_{H_{L}}^{2}$

for all $\left(c_{i}\right)_{i=1}^{n}$ in $\mathbb{C}$,
$\left(s_{i}\right)_{i=1}^{n}$ in $S$, and $n\in\mathbb{N}$. This follows from
the reproducing property, and the assumption $K\leq L$:

$\displaystyle\left|\sum c_{i}f\left(s_{i}\right)\right|^{2}$
$\displaystyle=\left|\left\langle\sum\overline{c_{i}}K_{s_{i}},f\right\rangle_{H_{K}}\right|^{2}\leq\left\|f\right\|_{H_{K}}^{2}\left\|\sum\overline{c_{i}}K_{s_{i}}\right\|_{H_{K}}^{2}$
$\displaystyle\leq\left\|f\right\|_{H_{K}}^{2}\sum\overline{c_{i}}c_{j}K\left(s_{i},s_{j}\right)\leq\left\|f\right\|_{H_{K}}^{2}\sum\overline{c_{i}}c_{j}L\left(s_{i},s_{j}\right)$
$\displaystyle=\left\|f\right\|_{H_{K}}^{2}\left\|\sum
c_{i}L_{s_{i}}\right\|_{H_{L}}^{2}.$

∎

######  Proposition 3.2.

Suppose $K,L$ are p.d. on $S\times S$, and $K\leq L$. There exists a unique
positive selfadjoint operator $T$ on $H_{L}$, such that $0\leq T\leq
I_{H_{K}}$, and

$K\left(s,t\right)=\left\langle
T^{1/2}L_{s},T^{1/2}L_{t}\right\rangle_{H_{L}},\quad s,t\in S.$ (3.1)

###### Proof.

Define a map $\Phi$ as

$\Phi\left(L_{s},L_{t}\right)=K\left(s,t\right)$

and extend it by linearity:

$\Phi\left(\sum_{i=1}^{m}c_{i}L_{s_{i}},\sum_{j=1}^{n}d_{j}L_{t_{j}}\right)=\sum_{i=1}^{m}\sum_{j=1}^{n}\overline{c_{i}}d_{j}K\left(s_{i},t_{j}\right).$

Then $\Phi$ extends by density to a unique bounded sesquilinear form on
$H_{L}$.

The remaining assertions follow from the general theory of quadratic forms. In
particular, if $j:H_{K}\rightarrow H_{L}$ is the inclusion map, then
$T=jj^{*}:H_{L}\rightarrow H_{L}$ is the desired operator. ∎

We now consider $\mathcal{L}\left(H\right)$-valued p.d. kernels.

######  Definition 3.3.

Suppose $K,L$ are $\mathcal{L}\left(H\right)$-valued p.d. kernels defined on
$S\times S$. We say that $K\leq L$ if

$\sum\left\langle
a_{i},K\left(s_{i},s_{j}\right)a_{j}\right\rangle_{H}\leq\sum\left\langle
a_{i},L\left(s_{i},s_{j}\right)a_{j}\right\rangle_{H}$ (3.2)

for all $\left(s_{i}\right)_{i=1}^{n}$ in $S$, $\left(a_{i}\right)_{i=1}^{n}$
in $H$, and $n\in\mathbb{N}$.

Let $\tilde{K}$, $\tilde{L}$ be the associated scalar-valued p.d. kernels on
$S\times S$, and $H_{\tilde{K}},H_{\tilde{L}}$ be the respective RKHSs (see
(2.4)). Definition 3.3 means that

$K\leq L\Longleftrightarrow\tilde{K}\leq\tilde{L}.$ (3.3)

######  Theorem 3.4.

Let $K,L:S\times S\rightarrow\mathcal{L}\left(H\right)$ be operator-valued
p.d. kernels on $S\times S$. Let $\tilde{K}$, $\tilde{L}$ be the associated
scalar-valued p.d. kernels, and $H_{\tilde{K}}$, $H_{\tilde{L}}$ be the
respective RKHSs.

The following are equivalent:

  1. (1)

$K\leq L$ in the sense of (3.2)–(3.3).

  2. (2)

There exists a positive selfadjoint operator $T:H_{\tilde{L}}\rightarrow
H_{\tilde{L}}$, $0\leq T\leq I_{H_{\tilde{L}}}$, such that

$K\left(s,t\right)=V_{L}\left(s\right)^{*}TV_{L}\left(t\right).$ (3.4)

Equivalently,

$\displaystyle V_{K}\left(t\right)$ $\displaystyle=$ $\displaystyle
T^{1/2}V_{L}\left(t\right).$ (3.5)

Here, $L\left(s,t\right)=V_{L}\left(s\right)^{*}V_{L}\left(t\right)$ and
$K\left(s,t\right)=V_{K}\left(s\right)^{*}V_{K}\left(t\right)$ are the
respective canonical factorizations of $L$ and $K$ (Theorem 2.1).

###### Proof.

(1) $\Rightarrow$ (2) Assume $K\leq L$. By Theorem 3.1, $H_{\tilde{K}}\subset
H_{\tilde{L}}$ and
$\left\|f\right\|_{H_{\tilde{L}}}\leq\left\|f\right\|_{H_{\tilde{K}}}$ for all
$f\in H_{\tilde{K}}$.

Recall that $L$ factors as

$L\left(s,t\right)=V_{L}\left(s\right)^{*}V_{L}\left(t\right)$

where $V_{L}\left(t\right):H\rightarrow H_{\tilde{L}}$ is given by
$V_{L}\left(t\right)a=\tilde{L}_{\left(t,a\right)}$, for all $t\in S$ and
$a\in H$ (see (2.5)).

Apply Proposition 3.2 to the scalar-valued kernels $\tilde{K}$ and
$\tilde{L}$: There exists a unique selfadjoint operator $T$ in
$H_{\tilde{L}}$, with $0\leq T\leq I_{H_{\tilde{L}}}$, so that

$\displaystyle\left\langle a,K\left(s,t\right)b\right\rangle_{H}$
$\displaystyle=\left\langle\tilde{K}_{\left(s,a\right)},\tilde{K}_{\left(t,b\right)}\right\rangle_{H_{\tilde{K}}}$
$\displaystyle=\left\langle
T^{1/2}\tilde{L}_{\left(s,a\right)},T^{1/2}\tilde{L}_{\left(t,b\right)}\right\rangle_{H_{\tilde{L}}}$
$\displaystyle=\left\langle
T^{1/2}V_{L}\left(s\right)a,T^{1/2}V_{L}\left(t\right)b\right\rangle_{H_{\tilde{L}}}$
$\displaystyle=\left\langle
a,V_{L}\left(s\right)^{*}TV_{L}\left(t\right)b\right\rangle_{H}$

for all $a,b\in H$, and all $s,t\in S$. Therefore, (3.4) holds. The identity
(3.5) follows from the above argument and the proof of Proposition 3.2.

(2) $\Rightarrow$ (1) Conversely, from (3.4) and the fact that $0\leq T\leq
I_{H_{\tilde{L}}}$, we have

$\displaystyle\sum_{i,j}\left\langle
a_{i},K\left(s_{i},s_{j}\right)a_{j}\right\rangle_{H}$
$\displaystyle=\sum_{i,j}\left\langle
a_{i},V_{L}\left(s_{i}\right)^{*}TV_{L}\left(s_{j}\right)a_{j}\right\rangle_{H}$
$\displaystyle=\left\|T^{1/2}\sum_{i}V_{L}\left(s_{i}\right)a_{i}\right\|_{H_{\tilde{L}}}^{2}\leq\left\|\sum_{i}V_{L}\left(s_{i}\right)a_{i}\right\|_{H_{\tilde{L}}}^{2}$
$\displaystyle=\sum_{i,j}\left\langle
a_{i},V_{L}\left(s_{i}\right)^{*}V_{L}\left(s_{j}\right)a_{j}\right\rangle_{H}$
$\displaystyle=\sum_{i,j}\left\langle
a_{i},L\left(s_{i},s_{j}\right)a_{j}\right\rangle_{H}$

and so $K\leq L$. ∎

######  Corollary 3.5.

Suppose $K,L:S\times S\rightarrow\mathcal{L}\left(H\right)$ p.d., and $K\leq
L$. Let $T=dK/dL$ be the Radon-Nikodym derivative from Theorem 3.4.

Let $W_{L}\left(t\right)$ be the $H$-valued Gaussian process from (2.8), i.e.,

$W_{L}\left(t\right)=\sum_{i}\left(V_{L}\left(t\right)^{*}\varphi_{i}\right)Z_{i}$
(3.6)

Set

$W_{K}\left(t\right):=\sum_{i}\left(V_{L}\left(t\right)^{*}T^{1/2}\varphi_{i}\right)Z_{i}.$
(3.7)

Then $K$ admits the following decomposition

$K\left(s,t\right)=\int_{\Omega}\left|W_{K}\left(s\right)\left\rangle\right\langle
W_{K}\left(t\right)\right|d\mathbb{P}.$ (3.8)

###### Proof.

Recall that (3.8) is equivalent to

$\mathbb{E}\left[\left\langle
a,W_{K}\left(s\right)\right\rangle_{H}\left\langle
W_{K}\left(t\right),b\right\rangle_{H}\right]=\left\langle
a,K\left(s,t\right)b\right\rangle_{H},$ (3.9)

for all $a,b\in H$ and $s,t\in S$.

Given $L:S\times S\rightarrow\mathcal{L}\left(H\right)$ p.d., recall that

$L\left(s,t\right)=V_{L}\left(s\right)^{*}V_{L}\left(t\right)$

where $V_{L}\left(t\right):H\rightarrow H_{\tilde{L}}=$ the RKHS of
$\tilde{L}$.

Let $\left(\varphi_{i}\right)$ be an ONB in $H_{\tilde{L}}$, and apply the
identity

$I_{H_{\tilde{L}}}=\sum_{i}\left|\varphi_{i}\left\rangle\right\langle\varphi_{i}\right|$

we then get

$\displaystyle L\left(s,t\right)$
$\displaystyle=V_{L}\left(s\right)^{*}I_{H_{\tilde{L}}}V_{L}\left(t\right)$
$\displaystyle=\sum_{i}\left|V_{L}\left(s\right)^{*}\varphi_{i}\left\rangle\right\langle
V_{L}\left(t\right)^{*}\varphi_{i}\right|.$

Similarly, using $V_{K}\left(t\right)=T^{1/2}V_{L}\left(t\right)$ from (3.5),

$K\left(s,t\right)=\sum_{i}\left|V_{L}\left(s\right)^{*}T^{1/2}\varphi_{i}\left\rangle\right\langle
V_{L}\left(t\right)^{*}T^{1/2}\varphi_{i}\right|.$ (3.10)

Thus,

$\displaystyle\text{l.h.s.}_{\left(\ref{eq:c8}\right)}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\left\langle
a,V_{L}\left(s\right)^{*}T^{1/2}\varphi_{i}\right\rangle_{H}\left\langle
V_{L}\left(t\right)^{*}T^{1/2}\varphi_{j},b\right\rangle_{H}\underset{=\delta_{i,j}}{\underbrace{\mathbb{E}\left[Z_{i}Z_{j}\right]}}$
$\displaystyle=$ $\displaystyle\sum_{i}\left\langle
a,V_{L}\left(s\right)^{*}T^{1/2}\varphi_{i}\right\rangle_{H}\left\langle
V_{L}\left(t\right)^{*}T^{1/2}\varphi_{i},b\right\rangle_{H}$ $\displaystyle=$
$\displaystyle\left\langle
a,\left(\sum_{i}\left|V_{L}\left(s\right)^{*}T^{1/2}\varphi_{i}\left\rangle\right\langle
V_{L}\left(t\right)^{*}T^{1/2}\varphi_{i}\right|\right)b\right\rangle_{H}$
$\displaystyle\underset{\left(\ref{eq:c9}\right)}{=}$
$\displaystyle\left\langle a,K\left(s,t\right)b\right\rangle_{H}.$

∎

##  4\. Completely positive maps

The role in physics of completely positive maps includes the following: It is
the kind of transformation resulting, for example, from passing a beam in a
certain mixed state through some device thereby producing another beam in a
different mixed state, hence allowing for dissipative effects. Such
transformations must map states into states and hence be positive. But
complete positivity is stronger. Under standard assumptions of unitary
dynamics as a whole, completely positive maps arise as restrictions of
representations realized in the bigger system, e.g., beam plus transformer. In
mathematical terms, this corresponds to the form given in Stinespring’s
theorem. Recall that Stinespring’s theorem (see (4.1)) states that each
completely positive map can be realized as a compression of a unitary
dynamics, hence is experimentally realizable.

Let $\mathfrak{A}$ and $\mathfrak{B}$ be unital $C^{*}$-algebras. A map
$\psi:\mathfrak{A}\rightarrow\mathfrak{B}$ is said to be completely positive
(CP) if $\psi\otimes I_{n}:\mathfrak{A}\otimes
M_{n}\rightarrow\mathfrak{B}\otimes M_{n}$ is positive for all
$n\in\mathbb{N}$.

Let $\mathfrak{A}$ be a unital $C^{*}$-algebra. Suppose
$\psi:\mathfrak{A}\rightarrow\mathcal{L}\left(H\right)$ is completely positive
and $\psi\left(I\right)=I_{H}$. Stinespring’s dilation theorem states that,
there exists a Hilbert space $\mathscr{K}$, a representation
$\pi:\mathfrak{A}\rightarrow\mathcal{L}\left(\mathscr{K}\right)$, and an
isometric embedding $V:H\rightarrow\mathscr{K}$ such that

$\psi\left(A\right)=V^{*}\pi\left(A\right)V.$ (4.1)

Further, $\left(\pi,V,\mathscr{K}\right)$ may be chosen to be minimal, i.e.,
$\mathscr{K}=\overline{\pi\left(\mathfrak{A}\right)VH}$. In that case, the
dilation is unique up to unitary equivalence.

We sketch a proof of (4.1) as an application of results in Sections 2–3.

######  Corollary 4.1.

Let $\psi:\mathfrak{A}\rightarrow\mathcal{L}\left(H\right)$ be as above, i.e.,
completely positive, and $\psi\left(I\right)=I$. Let
$K=K_{\varphi}:\mathfrak{A}\times\mathfrak{A}\rightarrow\mathcal{L}\left(H\right)$
be given by

$K\left(A,B\right):=\psi\left(A^{*}B\right),\quad A,B\in\mathfrak{A},$

as a $\mathcal{L}\left(H\right)$-valued p.d. kernel on
$\mathfrak{A}\times\mathfrak{A}$. Let $\mathscr{K}=H_{\tilde{K}}=$ the RKHS of
the associated scalar-valued kernel $\tilde{K}$. Define
$V=V\left(I\right):H\rightarrow\mathscr{K}$ by

$Vh=\tilde{K}_{\left(I,h\right)}:\mathfrak{A}\times H\rightarrow\mathbb{C};$

set $\pi:\mathfrak{A}\rightarrow\mathcal{L}\left(\mathscr{K}\right)$ by

$\pi\left(A\right)\tilde{K}_{\left(B,h\right)}=\tilde{K}_{\left(AB,h\right)}$

for all $\left(B,h\right)\in\mathfrak{A}\times H$. Then (4.1) holds, and
$\left(\pi,V,\mathscr{K}\right)$ is minimal.

###### Proof.

This is a direct “translation” of Theorem 2.1 to the setting of CP maps.

Specifically, when $K\left(A,B\right)=\psi\left(A^{*}B\right)$, it factors
into

$K\left(A,B\right)=V^{*}\left(A\right)V\left(B\right)$

as in (2.2). That is,

$K\left(A,B\right)h=V^{*}\left(A\right)V\left(B\right)h=V^{*}\left(A\right)\tilde{K}_{\left(B,h\right)}=\psi\left(A^{*}B\right)h.$

By setting $A=I$ and $V:=V\left(I\right)$, this reduces to

$\displaystyle K\left(I,B\right)h$
$\displaystyle=V^{*}V\left(B\right)h=V^{*}\tilde{K}_{\left(B,h\right)}$
$\displaystyle=V^{*}\pi\left(B\right)\tilde{K}_{\left(I,h\right)}$
$\displaystyle=V^{*}\pi\left(B\right)Vh=\psi\left(B\right)h,$

which is (4.1).

Moreover, since

$\displaystyle\mathscr{K}=H_{\tilde{K}}$
$\displaystyle=\overline{span}\left\\{\tilde{K}_{\left(A,h\right)}:A\in\mathfrak{A},h\in
H\right\\}$
$\displaystyle=\overline{span}\left\\{\pi\left(A\right)Vh:A\in\mathfrak{A},h\in
H\right\\}=\overline{\pi\left(\mathfrak{A}\right)VH},$

the dilation is minimal. The assertion of uniqueness (up to unitary
equivalence) is immediate. ∎

In the above setting, there is also a Radon–Nikodym type theorem that
characterizes all CP maps $\varphi,\psi$ for which $\varphi\leq\psi$, in the
sense that $\psi-\varphi$ is CP. In term of the operator-valued kernels, say
$K_{\varphi}$ and $K_{\psi}$, we have $\varphi\leq\psi$ $\Longleftrightarrow$
$K_{\varphi}\leq K_{\psi}$, and the latter is in the sense of Definition 3.3.

We sketch below that this result can be derived as an application of Theorem
3.4.

######  Corollary 4.2.

Let $\varphi,\psi:\mathfrak{A}\rightarrow\mathcal{L}\left(H\right)$ be CP
maps. Let $\left(\pi_{\psi},V_{\psi},\mathscr{K}_{\psi}\right)$ be the minimal
Stinespring dilation from Corollary 4.1, i.e.,
$\mathscr{K}_{\psi}=H_{\tilde{K}_{\psi}}=$ the RKHS of $\tilde{K}_{\psi}$, and

$\tilde{K}\left(\left(A,a\right),\left(B,b\right)\right)=\left\langle
a,\psi\left(A^{*}B\right)b\right\rangle_{H}$

for all $A,B\in\mathfrak{A}$, and $a,b\in H$.

Then $\varphi\leq\psi$ if and only if there exists a unique positive
selfadjoint operator $T$ in the commutant
$\pi_{\psi}\left(\mathfrak{A}\right)^{\prime}$, $0\leq T\leq
I_{\mathscr{K}_{\psi}}$, such that

$\varphi\left(A\right)=V_{\psi}^{*}T^{1/2}\pi_{\psi}\left(A\right)T^{1/2}V_{\psi},\quad
A\in\mathfrak{A}.$ (4.2)

###### Proof.

Suppose $\varphi\leq\psi$ (i.e., $K_{\varphi}\leq K_{\psi}$).

Set $K_{\psi}\left(A,B\right)=\psi\left(A^{*}B\right)$ as before, so it
factors into

$K_{\psi}\left(A,B\right)=V_{\psi}\left(A\right)^{*}V_{\psi}\left(B\right).$

By Theorem 3.4, there exists a unique $T$, $0\leq T\leq
I_{H_{\tilde{K}_{\psi}}}$, such that (see (3.4))

$K_{\varphi}\left(A,B\right)=V_{\psi}\left(A\right)^{*}TV_{\psi}\left(B\right).$
(4.3)

Set $A=I$ in (4.3), then

$\varphi\left(B\right)=K_{\varphi}\left(I,B\right)=V_{\psi}\left(I\right)^{*}TV_{\psi}\left(B\right)=V_{\psi}T\pi_{\psi}\left(B\right)V_{\psi},$
(4.4)

where $V_{\psi}:=V_{\psi}\left(I\right)$, and
$\pi_{\psi}:\mathfrak{A}\rightarrow\mathcal{L}\left(\mathscr{K}_{\psi}\right)$
is as in Corollary 4.1. In particular,

$V_{\psi}\left(A\right)=\pi_{\psi}\left(A\right)V_{\psi},\quad
A\in\mathfrak{A}.$ (4.5)

It remains to show that $T\in\pi_{\psi}\left(\mathfrak{A}\right)^{\prime}$.
For this, one checks that

$\displaystyle\varphi\left(B\right)$ $\displaystyle=$
$\displaystyle\varphi\left(\left(B^{*}\right)^{*}I\right)$ (4.6)
$\displaystyle=$ $\displaystyle K_{\varphi}\left(B^{*},I\right)$
$\displaystyle\underset{\left(\ref{eq:d2}\right)}{=}$ $\displaystyle
V_{\psi}\left(B^{*}\right)^{*}TV_{\psi}\left(I\right)$
$\displaystyle\underset{\left(\ref{eq:d5}\right)}{=}$
$\displaystyle\left(\pi_{\psi}\left(B^{*}\right)V_{\psi}\right)^{*}TV_{\psi}$
$\displaystyle=$ $\displaystyle
V_{\psi}^{*}\pi_{\psi}\left(B\right)TV_{\psi}.$

Combining (4.4) and (4.6), we conclude that
$\pi_{\psi}\left(B\right)T=T\pi_{\psi}\left(B\right)$, for all
$B\in\mathfrak{A}$. Thus, $T\in\pi_{\psi}\left(\mathfrak{A}\right)^{\prime}$
and (4.2) follows.

Conversely, given (4.2), it is clear that $\varphi$ is CP. See e.g., the proof
of “(2)$\Rightarrow$(1)” in Theorem 3.4. ∎

In view of the correspondence between operator-valued p.d. kernels and Hilbert
space-valued Gaussian processes, we have:

######  Corollary 4.3.

Every CP map $\psi:\mathfrak{A}\rightarrow\mathcal{L}\left(H\right)$ admits a
direct integral decomposition

$\psi\left(A^{*}B\right)=\int_{\Omega}\left|W_{\psi}\left(A\right)\left\rangle\right\langle
W_{\psi}\left(B\right)\right|d\mathbb{P}$ (4.7)

where $\left\\{W_{\psi}\left(A\right)\right\\}_{A\in\mathfrak{A}}$ is a mean-
zero, $H$-valued Gaussian process, realized in some probability space
$\left(\Omega,\mathbb{P}\right)$.

###### Proof.

Let $\left(V_{\psi},\pi_{\psi},\mathscr{K}_{\psi}\right)$ be the minimal
Stinespring dilation in Corollary 4.1, i.e.,
$\mathscr{K}_{\psi}=H_{\tilde{K}_{\psi}}=$ the RKHS of $\tilde{K}_{\psi}$.
Then,

$\displaystyle\psi\left(A^{*}B\right)$
$\displaystyle=V_{\psi}^{*}\pi_{\psi}\left(A^{*}B\right)V_{\psi}$
$\displaystyle=\left(\pi_{\psi}\left(A\right)V_{\psi}\right)^{*}\left(\pi_{\psi}\left(B\right)V_{\psi}\right)$
$\displaystyle=\sum_{i}\left|\left(\pi_{\psi}\left(A\right)V_{\psi}\right)^{*}\varphi_{i}\left\rangle\right\langle\left(\pi_{\psi}\left(B\right)V_{\psi}\right)^{*}\varphi_{i}\right|$

where $\left(\varphi_{i}\right)$ is an ONB in $\mathscr{K}_{\psi}$. Setting

$W_{\psi}\left(A\right):=\sum_{i}\left(\left(\pi_{\psi}\left(A\right)V_{\psi}\right)^{*}\varphi_{i}\right)Z_{i}$

with $\left\\{Z_{i}\right\\}$ i.i.d. $N\left(0,1\right)$ as above, we get

$\left\langle
a,\psi\left(A^{*}B\right)b\right\rangle_{H}=\mathbb{E}\left[\left\langle
a,W_{\psi}\left(A\right)\right\rangle_{H}\left\langle
W_{\psi}\left(B\right),b\right\rangle_{H}\right]$

for all $a,b\in H$, and $A,B\in\mathfrak{A}$, which is (4.7). ∎

## References

  * [AA22] Mukhamed Aleroev and Temirkhan Aleroev, _On positive definite kernels of integral operators corresponding to the boundary value problems for fractional differential equations_ , Entropy 24 (2022), no. 4, Paper No. 515, 7. MR 4414884
  * [AB97] D. Alpay and V. Bolotnikov, _On tangential interpolation in reproducing kernel Hilbert modules and applications_ , Topics in interpolation theory (Leipzig, 1994), Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 37–68. MR 1473250
  * [AD93] Daniel Alpay and Harry Dym, _On a new class of structured reproducing kernel spaces_ , J. Funct. Anal. 111 (1993), no. 1, 1–28. MR 1200633
  * [AD06] D. Alpay and C. Dubi, _Some remarks on the smoothing problem in a reproducing kernel Hilbert space_ , J. Anal. Appl. 4 (2006), no. 2, 119–132. MR 2223568
  * [AFMP94] Gregory T. Adams, John Froelich, Paul J. McGuire, and Vern I. Paulsen, _Analytic reproducing kernels and factorization_ , Indiana Univ. Math. J. 43 (1994), no. 3, 839–856. MR 1305949
  * [Ahm13] Nasiruddin Ahmed, _A note on Radon-Nikodym theorem for operator valued measures and its applications_ , Commun. Korean Math. Soc. 28 (2013), no. 2, 285–295. MR 3054037
  * [AJ15] Daniel Alpay and Palle Jorgensen, _Spectral theory for Gaussian processes: reproducing kernels, boundaries, and $L^{2}$-wavelet generators with fractional scales_, Numer. Funct. Anal. Optim. 36 (2015), no. 10, 1239–1285. MR 3402823
  * [AJ21] Daniel Alpay and Palle E. T. Jorgensen, _New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry_ , Opuscula Math. 41 (2021), no. 3, 283–300. MR 4302453
  * [AJ22] Daniel Alpay and Palle Jorgensen, _mu-Brownian motion, dualities, diffusions, transforms, and reproducing kernel Hilbert spaces_ , J. Theoret. Probab. 35 (2022), no. 4, 2757–2783. MR 4509085
  * [AL95] Daniel Alpay and Juliette Leblond, _Traces of Hardy functions and reproducing kernel Hilbert spaces_ , Arch. Math. (Basel) 64 (1995), no. 6, 490–499. MR 1329822
  * [Alp91] Daniel Alpay, _Some reproducing kernel spaces of continuous functions_ , J. Math. Anal. Appl. 160 (1991), no. 2, 424–433. MR 1126127
  * [Aro48] N. Aronszajn, _Reproducing and pseudo-reproducing kernels and their application to the partial differential equations of physics_ , Harvard University, Graduate School of Engineering, Cambridge, MA, 1948, Studies in partial differential equations. Technical report 5, preliminary note. MR 31663
  * [Aro50] by same author, _Theory of reproducing kernels_ , Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437
  * [Arv10] William Arveson, _Dilation theory yesterday and today_ , A glimpse at Hilbert space operators, Oper. Theory Adv. Appl., vol. 207, Birkhäuser Verlag, Basel, 2010, pp. 99–123. MR 2743416
  * [AS56] N. Aronszajn and K. T. Smith, _Functional spaces and functional completion_ , Ann. Inst. Fourier (Grenoble) 6 (1955/56), 125–185. MR 80878
  * [CY17] Rupak Chatterjee and Ting Yu, _Generalized coherent states, reproducing kernels, and quantum support vector machines_ , Quantum Inf. Comput. 17 (2017), no. 15-16, 1292–1306. MR 3700848
  * [Dav21] Oleg Davydov, _Approximation with conditionally positive definite kernels on deficient sets_ , Approximation theory XVI, Springer Proc. Math. Stat., vol. 336, Springer, Cham, [2021] ©2021, pp. 27–38. MR 4201338
  * [FS20] Martin Forde and Benjamin Smith, _The conditional law of the Bacry-Muzy and Riemann-Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces_ , Statist. Probab. Lett. 161 (2020), 108732, 11. MR 4073554
  * [GLL24] Erik J. Gustafson, Henry Lamm, and Felicity Lovelace, _Primitive quantum gates for an $SU(2)$ discrete subgroup: Binary octahedral_, Phys. Rev. D 109 (2024), no. 5, Paper No. 054503. MR 4733151
  * [Gue22a] J. C. Guella, _On Gaussian kernels on Hilbert spaces and kernels on hyperbolic spaces_ , J. Approx. Theory 279 (2022), Paper No. 105765, 36\. MR 4414825
  * [Gue22b] by same author, _Operator-valued positive definite kernels and differentiable universality_ , Anal. Appl. (Singap.) 20 (2022), no. 4, 681–735. MR 4439542
  * [JT23] Palle E. T. Jorgensen and James Tian, _Operator-valued Gaussian processes and their covariance kernels_ , Infinite Dimensional Analysis, Quantum Probability and Related Topics 0 (2023), no. 0, 2350020.
  * [JT24a] Palle E. T. Jorgensen and James Tian, _Canonical data-reconstructions via kernels, hilbert space-valued gaussian processes, and quantum states_ , arXiv:2405.02796 (2024).
  * [JT24b] by same author, _Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure_ , arXiv: 2404.14685 (2024).
  * [Kat19] Victor Katsnelson, _On the completeness of Gaussians in a Hilbert functional space_ , Complex Anal. Oper. Theory 13 (2019), no. 3, 637–658. MR 3940383
  * [KM20] Vít Kubelka and Bohdan Maslowski, _Filtering of Gaussian processes in Hilbert spaces_ , Stoch. Dyn. 20 (2020), no. 3, 2050020, 22. MR 4101087
  * [Kod23] Kazunori Kodaka, _Morita equivalences for completely positive linear maps and GNS-C*-correspondences_ , Bull. Iranian Math. Soc. 49 (2023), no. 1, Paper No. 1, 17. MR 4537434
  * [Kre19] Johannes T. N. Krebs, _The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions_ , J. Multivariate Anal. 173 (2019), 620–639. MR 3959187
  * [LCH24] B. Lari, W. S. Chung, and H. Hassanabadi, _Quantum gates based on two strongly coupled harmonic oscillators in thermal non-equilibrium conditions_ , Phys. A 637 (2024), Paper No. 129561, 13. MR 4703397
  * [LM17] Tijana Levajković and Hermann Mena, _Equations involving Malliavin calculus operators_ , SpringerBriefs in Mathematics, Springer, Cham, 2017, Applications and numerical approximation. MR 3699787
  * [Lop23] Oleh Lopushansky, _Bernstein-Jackson inequalities on Gaussian Hilbert spaces_ , J. Fourier Anal. Appl. 29 (2023), no. 5, Paper No. 58, 21. MR 4641110
  * [LYA23] Binglin Li, Changwon Yoon, and Jeongyoun Ahn, _Reproducing kernels and new approaches in compositional data analysis_ , J. Mach. Learn. Res. 24 (2023), Paper No. [327], 34. MR 4690276
  * [MPR20] Darian McLaren, Sarah Plosker, and Christopher Ramsey, _On operator valued measures_ , Houston J. Math. 46 (2020), no. 1, 201–226. MR 4137283
  * [PR16] Vern I. Paulsen and Mrinal Raghupathi, _An introduction to the theory of reproducing kernel Hilbert spaces_ , Cambridge Studies in Advanced Mathematics, vol. 152, Cambridge University Press, Cambridge, 2016. MR 3526117
  * [PY24] Jens Palsberg and Nengkun Yu, _Optimal implementation of quantum gates with two controls_ , Linear Algebra Appl. 694 (2024), 206–261. MR 4737207
  * [Sak65] Shôichirô Sakai, _A Radon-Nikodým theorem in $W^{\ast}$-algebras_, Bull. Amer. Math. Soc. 71 (1965), 149–151. MR 174992
  * [SCC21] Katarzyna Siudzińska, Sagnik Chakraborty, and Dariusz Chruściński, _Interpolating between positive and completely positive maps: a new hierarchy of entangled states_ , Entropy 23 (2021), no. 5, Paper No. 625, 10. MR 4287855
  * [Sch52] Laurent Schwartz, _Théorie des noyaux_ , Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, RI, 1952, pp. 220–230. MR 45307
  * [Tho24] Ian Thompson, _An approximate unique extension property for completely positive maps_ , J. Funct. Anal. 286 (2024), no. 1, Paper No. 110193, 32\. MR 4654017
  * [ZD22] Haiyan Zhang and Yanni Dou, _Fixed points of completely positive maps and their dual maps_ , J. Inequal. Appl. (2022), Paper No. 163, 9. MR 4526376
  * [ZHL24] Shihao Zhang, Kai Huang, and Lvzhou Li, _Depth-optimized quantum circuit synthesis for diagonal unitary operators with asymptotically optimal gate count_ , Phys. Rev. A 109 (2024), no. 4, Paper No. 042601. MR 4740581