EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-165 LHCb-PAPER-2014-037

Measurement of the $\kern 2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}$
meson lifetime in $D_{s}^{+}\pi^{-}$ decays

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

We present a measurement of the ratio of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson lifetime, in the flavor-
specific decay to $D_{s}^{+}\pi^{-}$, to that of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson. The $pp$ collision data
used correspond to an integrated luminosity of 1 fb-1, collected with the LHCb
detector, at a center-of-mass energy of 7 TeV. Combining our measured value of
$1.010\pm 0.010\pm 0.008$ for this ratio with the known $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime, we determine the flavor-
specific $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime to be
$\tau(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})=1.535\pm 0.015\pm
0.014{\rm\,ps}$, where the uncertainties are statistical and systematic,
respectively. This is the most precise measurement to date, and is consistent
with previous measurements and theoretical predictions.

Submitted to Physical Review Letters

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

Lifetimes of $b$-flavored hadrons show the effects of all processes governing
their weak decays. In the case of neutral mesons, the decay rates are not
purely exponential, but are modified by flavor mixing and charge parity
($C\\!P$) violation. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson’s decay width
$\Gamma_{s}$ differs for the heavy and light mass eigenstates, by an amount
$\Delta\Gamma_{s}$ that has been measured to be significantly different from
zero [1]. This gives rise to a rich phenomenology of mixing and $C\\!P$
violation. Precision measurement of the lifetime $\tau_{s}=\hbar/\Gamma_{s}$
is therefore an important benchmark. The ratio of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetimes is well predicted in the
heavy quark expansion (HQE) model [2], which is used to extract values of the
quark-mixing parameters $|V_{cb}|$ and $|V_{ub}|$, and thus lifetime
measurements provide a precision test of the theory.

In this Letter we measure the lifetime of the decay
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}_{s}^{0}\rightarrow
D_{s}^{\pm}\pi^{\mp}$ by summing over ${B}^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ states. Since $C\\!P$
violation in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing is
negligible [3], the final state receives equal contributions from light and
heavy mass eigenstates. Consequently, the decay rate is given by the sum of
two exponentials and can be fitted by a single exponential with the measured
flavor-specific lifetime $\tau_{\rm fs}$ related to the decay width. Expanding
in terms of $\Delta\Gamma_{s}/\Gamma_{s}$ [4],111We use natural units where
$\hbar=c=1$.

$\tau_{\rm
fs}\approx\frac{1}{\Gamma_{s}}\frac{1+\left(\frac{\Delta\Gamma_{s}}{2\Gamma_{s}}\right)^{2}}{1-\left(\frac{\Delta\Gamma_{s}}{2\Gamma_{s}}\right)^{2}}~{}.$
(1)

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ time-dependent
decay rate is measured with respect to the well-measured lifetimes of the
$B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons, which
are reconstructed in final states with similar topology and kinematic
properties.222Reference to a given decay mode implies the use of the charge-
conjugate mode as well.

The LHCb detector [5] 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 [6], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes [7] placed
downstream of the magnet. The tracking system provides a measurement of
momentum, $p$, with a relative uncertainty that varies from 0.4% at low
momentum to 0.6% at 100 GeV. 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 GeV. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors [8].
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
[9].

The trigger 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 signal candidates are hardware triggered if
there is at least one track having a large transverse energy deposit, then the
track is required in software to have a transverse momentum $\mbox{$p_{\rm
T}$}>1.7$ GeV and an impact parameter $\chi_{\rm IP}^{2}$ with respect to the
primary vertex (PV) greater than 16, where $\chi_{\rm IP}^{2}$ is defined as
the difference in $\chi^{2}$ of a given PV reconstructed with and without the
considered particle included. In addition a vertex detached from the PV must
be formed with either two, three or four tracks, with a scalar $p_{\rm T}$ sum
of the tracks that must exceed a threshold that varies with the track
multiplicity.

The advantage of measuring the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime using the ratio with
respect to well-measured lifetimes is that the decay time acceptances
introduced by the trigger and selection almost cancel, and only small
corrections are required to the ratio of the decay time acceptances, which are
taken from simulation. Thus, we reconstruct signals not only in the ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}\pi^{-}}$,
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ (denoted $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]) decay mode, but also in
the topologically similar channels ($i$) ${B^{-}}\rightarrow{D^{0}\pi^{-}}$,
$D^{0}\rightarrow K^{-}\pi^{+}$ ($B^{-}_{[K\pi]}$), ($ii$)
${B^{-}}\rightarrow{D^{0}\pi^{-}}$, $D^{0}\rightarrow
K^{-}\pi^{-}\pi^{+}\pi^{+}$ ($B^{-}_{[K\pi\pi\pi]}$), and ($iii$) ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}}\rightarrow{D^{+}\pi^{-}}$,
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$).

These decay modes are selected using some common criteria. All of the tracks
coming from candidate $D$ meson decays are required to have $\chi^{2}_{\rm
IP}>9$. Pions arising from $\overline{B}$ meson decays have a more selective
requirement $\chi^{2}_{\rm IP}>36$ and they are required to be inconsistent
with being identified as muons. The $D$ candidates are required to have masses
within 25 MeV of their known values [1], be reconstructed downstream of the
PV, and have $\chi^{2}_{\rm IP}>4$. The $D$ vertex separation from the
$\overline{B}$ vertex should satisfy $\chi^{2}_{\rm VS}>2$, where
$\chi^{2}_{\rm VS}$ is the increase in $\chi^{2}$ of the parent $\overline{B}$
vertex fit when the $D$ decay products are constrained to come from the
$\overline{B}$ vertex, relative to when they are allowed to come from a
separate vertex.

$B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates are
required to have $\chi^{2}_{\rm IP}<16$ with respect to the PV and masses in
the ranges $5100-5600$ MeV, while for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates the mass range is
changed to $5200-5700$ MeV. The cosine of the angle between the $\overline{B}$
momentum and its direction of flight is required to be greater than 0.9999.
All signal candidates are refitted taking both $D$ mass and vertex constraints
into account [10]. All charged particles are required to be identified as
either pions or kaons. Efficiencies are evaluated with a data-driven method
using large samples of $D^{0}\rightarrow K^{-}\pi^{+}$ events, where the
kinematic distributions of kaons and pions from the calibration sample are
reweighted to match those of the $\overline{B}$ decays under study.

We eliminate $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$
decay candidates that result from other similar decays, the ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}\pi^{-}}$,
$D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+}$ and ${\mathchar
28931\relax^{0}_{b}}\rightarrow{\mathchar 28931\relax^{+}_{c}\pi^{-}}$,
$\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ modes, if the
invariant mass of the particles forming the $D^{+}$ candidate, with
appropriately swapped mass assignments, is compatible within 30 MeV with
either of the known $D^{+}_{s}$ or $\mathchar 28931\relax^{+}_{c}$ masses.
Similar vetoes are applied for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ] candidates, where cross-
feed from ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}}\rightarrow{D^{+}\pi^{-}}$,
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$, and ${\mathchar
28931\relax^{0}_{b}}\rightarrow{\mathchar 28931\relax^{+}_{c}\pi^{-}}$,
$\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ can happen if
misidentification occurs. The combined efficiencies of the particle
identification requirements and the mass vetoes depend on the specific decay
mode considered, ranging from 80% to 90%, while more than 95% of cross-feed
backgrounds are rejected.

The $\overline{B}$ candidate mass distributions for the four decay modes
considered are shown in Fig. 1, along with the results of binned maximum
likelihood fits. Signal shapes are parameterized using modified Gaussian
functions (Cruijff) with independent tail shapes on both sides [11,
*delAmoSanchez:2010ae]. All signal parameters are allowed to vary in the fit.
A residual component of $\overline{B}\rightarrow DK^{-}$ misidentified events
is also included, with its yield constrained to that determined by an
independent analysis where the kaon is positively identified. Partially
reconstructed backgrounds, where a pion or a photon are missed in
reconstruction, are modeled using a sum of parametric empirical functions
convolved with resolution functions. The unique decay kinematics of each of
the modes, mostly determined by the polarization amplitudes, is taken into
account. The combinatorial background is parameterized by a linear term. The
fitted signal yields are $179\,623\pm 467$, $82\,880\pm 339$, $109\,670\pm
378$ and $21\,058\pm 245$ for $B^{-}_{[K\pi]}$, $B^{-}_{[K\pi\pi\pi]}$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ] decays, respectively.

Figure 1: Fits to the invariant mass spectra of candidates for the decays (a)
${B^{-}}\rightarrow{D^{0}[K\pi]\pi^{-}}$, (b)
${B^{-}}\rightarrow{D^{0}[K\pi\pi\pi]\pi^{-}}$, (c) ${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}}\rightarrow{D^{+}[K\pi\pi]\pi^{-}}$,
(d) ${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}[KK\pi]\pi^{-}}$.
The points are the data and the superimposed curves show the fit components.
The solid (blue) curve gives the total. The $DK^{-}$ component is not visible,
but is included.

The decay time, $t$, is derived from a flight-length measurement between
production and decay points of the $\overline{B}$ particle, given by

$t=m\cdot\frac{\vec{d}\cdot\vec{p}}{|p|^{2}}\,\mathrm{,}$ (2)

where $m$ is the reconstructed invariant mass, $\vec{p}$ the momentum and
$\vec{d}$ the distance vector of the particle from its production to decay
vertices. Prior to this determination, the PV position is refitted excluding
the tracks forming the signal candidate, and the $\overline{B}$ meson is
further constrained to come from the PV. The decay time distribution of the
signal $D_{\rm T}(t)$ can be described by an exponential function convolved
with a decay time resolution function, $G(t,\sigma)$, and multiplied by an
acceptance function $A(t)$:

$D_{\rm T}(t)=A(t)\times[e^{-t^{\prime}/\tau}\otimes G(t-t^{\prime},\sigma)].$
(3)

The ratio of the measured decay time distributions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $B^{-}$ (we denote the use of
either $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $B^{-}$ modes by
the symbol $B_{x}$) can be written as

$R(t)=\frac{A_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}(t)\times[e^{-t^{\prime}/\tau_{\kern
0.89996pt\overline{\kern-0.89996ptB}{}^{0}_{s}}}\otimes
G(t-t^{\prime},\sigma_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}})]}{A_{B_{x}}(t)\times[e^{-t^{\prime}/\tau_{B_{x}}}\otimes
G(t-t^{\prime},\sigma_{B_{x}})]}.$ (4)

Resolutions are evaluated using simulated events and they are found to be 38,
37, 39 and 36 $\rm\,fs$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ], $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$, $B^{-}_{[K\pi]}$ and
$B^{-}_{[K\pi\pi\pi]}$, respectively. Since the resolution is very similar in
all the modes, and much smaller than our 0.5 ps bin width, the resolution
effects cancel [13], and we are left with a ratio of two exponentials times
the ratio of acceptance functions,

$R(t)=\frac{A_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}(t)}{A_{B_{x}}(t)}e^{-t(1/\tau_{\kern
0.89996pt\overline{\kern-0.89996ptB}{}^{0}_{s}}-1/\tau_{B_{x}})}=\frac{A_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}(t)}{A_{B_{x}}(t)}e^{-t\Delta_{{B}^{0}_{s}B_{x}}},$
(5)

where $\Delta_{{B}^{0}_{s}B_{x}}\equiv 1/\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}-1/\tau_{B_{x}}$. Acceptance
functions are evaluated by simulation. The effective lifetime, $\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}$, can then be calculated from
$\Delta_{{B}^{0}_{s}B_{x}}$ using the well-known $B_{x}$ lifetimes. The
current world average values are $\tau_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}}=1.519\pm 0.007$ ps and
$\tau_{B^{-}}=1.641\pm 0.008$ ps [1].

The signal yields are determined in each decay time bin by fitting the mass
distribution in each bin with the same shapes as used in the full fits, with
the signal shape parameters fixed to those of the full fit as they are
independent of the decay time. The yields are shown in Fig. 2.

Figure 2: Decay time distributions for
${B^{-}}\rightarrow{D^{0}[K\pi]\pi^{-}}$ shown as triangles (blue),
${B^{-}}\rightarrow{D^{0}[K\pi\pi\pi]\pi^{-}}$ shown as inverted triangles
(cyan), ${B^{0}}\rightarrow{D^{+}[K\pi\pi]\pi^{-}}$ shown as squares (red),
${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}[KK\pi]\pi^{-}}$
shown as circles (magenta). For most entries the error bars are smaller than
the point markers.

The signal yields are then corrected by the relative decay time acceptance
ratio, obtained by simulation, and shown in Fig. 3. Then the efficiency-
corrected yield ratios are fitted with a single exponential function to
extract $\Delta_{{B}^{0}_{s}B_{x}}$. Fits are performed in the 1–8${\rm\,ps}$
region. The 0–1${\rm\,ps}$ region is excluded since the ratio of acceptances
varies significantly here, due to the differences between the lifetimes and
track multiplicities in the $D$ decays.

Figure 3: Ratio of the decay time acceptances between ${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}[KK\pi]\pi^{-}}$
and ${B^{-}}\rightarrow{D^{0}[K\pi]\pi^{-}}$ shown as triangles (blue),
${B^{0}}\rightarrow{D^{+}[K\pi\pi]\pi^{-}}$ shown as squares (red), and
${B^{-}}\rightarrow{D^{0}[K\pi\pi\pi]\pi^{-}}$ shown as inverted triangles
(cyan). The vertical axis is shown in an arbitrary scale, different for each
mode ratio to improve clarity.

The full analysis is also applied to the control decay modes and the $B^{-}$
lifetime is measured relative to that of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ meson. Given their well-known
lifetimes, this provides a robust check on the validity of the procedure. We
then measure the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/B_{y}$
lifetime ratio for each of the three samples. The exponential fits for the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/B_{y}$ lifetime ratios
are shown in Fig. 4, with the results given in Table 1.

Table 1: Measured lifetime ratios, compared with the known values, and the difference (fitted – known), as well as the resulting measured lifetime $\tau_{\rm meas}$. Errors are statistical only. $B_{x}$ and $B_{y}$ indicate the modes used. Value |  $B^{-}_{[K\pi]}$/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$ |  $B^{-}_{[K\pi\pi\pi]}$/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$ |  $B^{-}_{[K\pi\pi\pi]}$/$B^{-}_{[K\pi]}$
---|---|---|---
Measured $\Delta_{B_{x}B_{y}}$ (${\rm\,ps^{-1}}$) | $-0.0451\pm 0.0033$ | $-0.0452\pm 0.0039$ | $0.0011\pm 0.0034$
Known $\Delta_{B_{x}B_{y}}$ (${\rm\,ps^{-1}}$) [1] | $-0.0489\pm 0.0042$ | $-0.0489\pm 0.0042$ | 0
Difference (${\rm\,ps^{-1}}$) |  $0.0038\pm 0.0054$ |  $0.0037\pm 0.0057$ | $0.0011\pm 0.0034$
$\tau_{\rm meas}(B^{-})$ (ps) |  $1.631\pm 0.009$ |  $1.631\pm 0.010$ | $1.638\pm 0.009$
Value |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$B^{-}_{[K\pi]}$ |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$ |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$B^{-}_{[K\pi\pi\pi]}$
Fitted $\Delta_{{B}^{0}_{s}B_{y}}$ | $0.0402\pm 0.0062$ | $-0.0063\pm 0.0065$ | $0.0418\pm 0.0066$
$\tau_{B_{y}}$ (ps) [1] | $1.641\pm 0.008$ |  $1.519\pm 0.007$ | $1.641\pm 0.008$
$\tau_{\rm meas}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})$ (ps) | $1.540\pm 0.015$ |  $1.535\pm 0.015$ | $1.535\pm 0.016$

In each case good agreement with the known values of the light $\overline{B}$
meson lifetime ratio is found, and the three values of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime are consistent.

Figure 4: Efficiency-corrected yield ratios of ${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}}\rightarrow{D^{+}_{s}[KK\pi]\pi^{-}}$
relative to ${B^{-}}\rightarrow{D^{0}[K\pi]\pi^{-}}$ shown as triangles
(blue), ${\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}}\rightarrow{D^{+}[K\pi\pi]\pi^{-}}$
shown as squares (red), and ${B^{-}}\rightarrow{D^{0}[K\pi\pi\pi]\pi^{-}}$
shown as inverted triangles (cyan). The simulation uncertainties are not
included. The exponential fits are also shown. The vertical axis is shown in
an arbitrary scale, different for each case to improve clarity.

The sources of systematic uncertainties on $\Delta_{{B}^{0}_{s}B_{x}}$ are
summarized in Table 2. The statistical precision on the relative acceptance is
the largest source of systematic uncertainty. The uncertainties due to the
background description are estimated by comparing the nominal result to that
obtained when the linear background slope is allowed to float separately in
each decay time bin; in addition an exponential shape is used, and the largest
deviation is assigned as a systematic uncertainty. Using a different signal
shape to fit the data (double Crystal Ball function [14]) leads to small
changes. There is also an uncertainty due to the decay time range and binning
used. These uncertainties are ascertained by changing the fit range limits
down to 0.5${\rm\,ps}$ and changing the size of the bins from 0.3${\rm\,ps}$
to 1${\rm\,ps}$. The relative measurements with respect to the three control
samples agree within 0.005${\rm\,ps}$, and this is conservatively added to the
total systematic uncertainty.

Table 2: Systematic uncertainties for $\Delta_{{B}^{0}_{s}B_{x}}$.  Source |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$B^{-}_{[K\pi]}$ |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}$ |  $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$[KKπ]/$B^{-}_{[K\pi\pi\pi]}$
---|---|---|---
Lifetime acceptance | 0.003 | 0.004 | 0.005
Background model | 0.002 | 0.002 | 0.002
Signal shape | 0.0004 | 0.0005 | 0.0005
Binning schemes | 0.003 | 0.001 | 0.005
Total | 0.005 | 0.005 | 0.007

Using the known lifetimes of the $B^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons and the three different
normalization channels, the flavor-specific $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime is determined as

$\begin{array}[]{rcl}\tau_{\rm fs}&=&1.540\pm 0.015\pm 0.012\pm
0.008{\rm\,ps}~{}[B^{-}_{[K\pi]}]\\\ \tau_{\rm fs}&=&1.535\pm 0.015\pm
0.012\pm 0.007{\rm\,ps}~{}[\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{[K\pi\pi]}]\\\ \tau_{\rm
fs}&=&1.535\pm 0.016\pm 0.018\pm 0.008{\rm\,ps}~{}[B^{-}_{[K\pi\pi\pi]}],\\\
\end{array}$

where the first uncertainty is statistical, the second systematic and the
third the uncertainty due to the input decay lifetimes of the $B^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons, 0.008${\rm\,ps}$
for the $B^{-}$ meson and 0.007${\rm\,ps}$ for the $B^{0}$ [1]. As the results
are fully correlated, that with the smallest uncertainty is chosen

$\tau_{\rm fs}=1.535\pm 0.015\pm 0.012\pm 0.007{\rm\,ps}.$

This is the most precise measurement to date and it is consistent with
previously available flavor-specific measurements [15, 1], and measurements of
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetimes in $C\\!P$
eigenstate modes [16]. The lifetime ratio ${\tau_{\rm fs}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s})}/{\tau(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})}={1}/\left({1+\tau(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})\Delta_{{B}^{0}_{s}B^{0}}}\right)$
is determined as $1.010\pm 0.010\pm 0.008$, where we assign the uncertainty
due to the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime as
purely systematic. A rather precise prediction of $\Gamma_{d}/\Gamma_{s}$ is
given using the HQE model [2]. To compare with our measured lifetime ratio we
apply a 0.8% correction from Eq. (1) resulting in a corrected prediction for
our measured lifetime ratio of $1.009\pm 0.004$, in excellent agreement with
our measurement, lending credence to this model.

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); SFI (Ireland);
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

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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, 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, 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. Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46, H. Brown52,
A. Bursche40, G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f,
M. Calvi20,k, M. Calvo Gomez36,p, P. Campana18,38, D. Campora Perez38, A.
Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, L. Carson50,
K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38, M.
Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, M.
Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26,
K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38,
H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38,
A. Comerma-Montells11, A. Contu15, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C.
Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, 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, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori38, A. Di Canto38, 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,
RF 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, M. Fontana10, F. Fontanelli19,j, R. Forty38, O.
Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21,38, E.
Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22, S.
Gambetta19,j, M. Gandelman2, P. Gandini59, Y. Gao3, J. García Pardiñas37, J.
Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L.
Gavardi9, G. Gavrilov30, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38,
C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E. Graugés36,
G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P. Griffith45, L.
Grillo11, O. Grünberg62, 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. Head38, V. Heijne41, K. Hennessy52, P.
Henrard5, L. Henry8, J.A. Hernando Morata37, E. van Herwijnen38, M. Heß62, A.
Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55,
N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58,
F. Jing3, M. John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, M. Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,
S. Karodia51, M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C.
Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O. Kochebina7, M. Kolpin11,
I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A.
Kozlinskiy41, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk20,26,38,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, S. Leo23, O.
Leroy6, T. Lesiak26, B. Leverington11, Y. Li3, T. Likhomanenko63, M. Liles52,
R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, S. Lohn38, I. Longstaff51,
J.H. Lopes2, N. Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,r, H. Luo50,
A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31,
F. Maciuc29, O. Maev30, S. Malde55, A. Malinin63, G. Manca15,e, G.
Mancinelli6, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin Benito36, P.
Marino23,t, R. Märki39, J. Marks11, G. Martellotti25, A. Martens8, A. Martín
Sánchez7, M. Martinelli39, D. Martinez Santos42, F. Martinez Vidal64, D.
Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A.
Mazurov16,f, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
McSkelly52, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes8,
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. Oggero41, S. Ogilvy51,
O. Okhrimenko44, R. Oldeman15,e, G. Onderwater65, M. Orlandea29, J.M. Otalora
Goicochea2, P. Owen53, A. Oyanguren64, 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. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P.
Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen66, 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, E.
Polycarpo2, A. Popov35, D. Popov10, B. Popovici29, C. Potterat2, E. Price46,
J. Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig
Navarro39, G. Punzi23,s, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, S.
Richards46, M. Rihl38, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P.
Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls64, J.J. Saborido Silva37, N. Sagidova30,
P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B.
Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37, E.
Santovetti24,l, A. Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, 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. Shevchenko63,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, 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, A. Sparkes50, P. Spradlin51, S. Sridharan38, F.
Stagni38, M. Stahl11, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
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D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, A. Zvyagin38.

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 to2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated
to31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to36
65KVI - University of Groningen, Groningen, The Netherlands, associated to41
66Celal Bayar University, Manisa, Turkey, associated to38

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


# Bell nonlocality is not sufficient for the security of standard device-
independent quantum key distribution protocols

Máté Farkas<EMAIL_ADDRESS>ICFO – Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain
Maria Balanzó-Juandó  ICFO – Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels, Spain  Karol
Łukanowski  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warszawa, Poland
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
Jan Kołodyński  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warszawa, Poland
Antonio Acín  ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute
of Science and Technology, 08860 Castelldefels, Spain  ICREA-Institució
Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona,
Spain

###### Abstract

Device-independent quantum key distribution is a secure quantum cryptographic
paradigm that allows two honest users to establish a secret key, while putting
minimal trust in their devices. Most of the existing protocols have the
following structure: first, a bipartite nonlocal quantum state is distributed
between the honest users, who perform local measurements to establish nonlocal
correlations. Then, they announce the implemented measurements and extract a
secure key by post-processing their measurement outcomes. We show that no
protocol of this form allows for establishing a secret key when implemented on
any correlation obtained by measuring local projective measurements on certain
entangled nonlocal states, namely on a range of entangled two-qubit Werner
states. To prove this result, we introduce a technique for upper-bounding the
asymptotic key rate of device-independent quantum key distribution protocols,
based on a simple eavesdropping attack. Our results imply that either
different reconciliation techniques are needed for device-independent quantum
key distribution in the large-noise regime, or Bell nonlocality is not
sufficient for this task.

_Introduction.—_ Device-independent quantum key distribution (DIQKD) is the
strongest form of quantum cryptographic protocols Acín  _et al._ (2007);
Pironio  _et al._ (2009). DIQKD security proofs are based on the assumption
that quantum theory is correct, and on physically observable quantities. Most
importantly, the honest users do not need to make any assumptions about the
inner workings of their devices and, hence, they do not need to trust the
preparation by the manufacturer. These facts make DIQKD a promising paradigm
for providing complete security, in that DIQKD protocols are not vulnerable to
implementation flaws that may be exploited in hacking attacks Zhao  _et al._
(2008); Lydersen  _et al._ (2010); Gerhardt  _et al._ (2011).

In DIQKD protocols, two distant honest parties aim at sharing a cryptographic
key, while assuring it to be unknown to any eavesdropper limited by quantum
theory. To achieve this, a quantum state is distributed between them in each
round of the protocol, and in each of these rounds they measure the part of
the state that is available to them. The resulting set of measurement outcomes
is the raw data from which they extract the secure key. This data is
characterised by the set of measurement outcome probabilities, called the
correlation. If the observed correlation violates a Bell inequality, the
information of any quantum eavesdropper about the outcomes is limited, which
opens up the possibility of extracting a secure key.

The first quantitative security proofs were achieved based on the violation of
the Clauser–Horne–Shimony–Holt (CHSH) Bell inequality Clauser  _et al._
(1969), using one-way public communication for key extraction Acín  _et al._
(2007); Pironio  _et al._ (2009). The original proofs, valid under collective
attacks, were later extended to the most powerful coherent attacks Vazirani
and Vidick (2014); Arnon-Friedman (2020); Arnon-Friedman  _et al._ (2018).
Furthermore, moderate improvements on key rates were recently achieved using
the more general biased CHSH inequalities Acín  _et al._ (2012), and applying
noisy pre-processing Woodhead  _et al._ (2020); Ho  _et al._ (2020); Sekatski
_et al._ (2020). Similarly, advantage distillation—a protocol for key
distillation using two-way public communication—has been shown to be useful
under the assumption of collective attacks Tan  _et al._ (2020).

While nonlocality of the observed correlations is necessary for DIQKD, it
remains an open question whether it is also sufficient. Recently, methods have
been proposed for upper-bounding secure key rates in DIQKD, even if two-way
communication is allowed Kaur  _et al._ (2020); Christandl  _et al._ (2020);
Arnon-Friedman and Leditzky (2020). However, all the bounds constructed thus
far remain strictly positive for nonlocal correlations, suggesting that
nonlocality may be sufficient for DIQKD.

In this work, we show this not to be the case for DIQKD protocols consisting
of the following two steps: (i) nonlocal correlations are established by
applying local measurements on an entangled quantum state; (ii) the
implemented measurements are announced and the key is constructed by
classically post-processing the outcomes. Most of the existing DIQKD protocols
have this form and, hence, in what follows we refer to such protocols as
_standard_ protocols. To prove the result, we provide a generic tool for
upper-bounding key rates in DIQKD. We then apply our tool to standard
protocols implemented on a two-qubit Werner state Werner (1989) using an
arbitrary number of projective measurements. We show that for a range of
visibilities for which the Werner state is known to be nonlocal, the upper
bound on the key rate is zero, and therefore no standard DIQKD protocol can be
secure. This means that there exist nonlocal correlations that cannot be used
for standard DIQKD, and, furthermore, that there exist nonlocal quantum states
that cannot be used for standard DIQKD with projective measurements. We also
show how the provable region of insecurity can be enlarged when fixing the
number of measurements in the protocol. In particular, we compute visibilities
for which the commonly used protocols based on the (biased) CHSH inequality
Acín  _et al._ (2007); Pironio  _et al._ (2009); Arnon-Friedman  _et al._
(2018); Woodhead  _et al._ (2020); Ho  _et al._ (2020); Sekatski  _et al._
(2020); Tan  _et al._ (2020) all become insecure despite the correlations
still being nonlocal.

_Methods.—_ Formally, in a DIQKD protocol two parties, Alice and Bob, have
access to a bipartite quantum state, $\rho_{AB}$, represented by a positive
semidefinite operator with unit trace on the tensor product Hilbert space,
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. The protocol consists of several
rounds, in each of which Alice and Bob choose a particular quantum measurement
to measure their part of a fresh copy of $\rho_{AB}$. In particular, Alice
chooses a measurement labelled by $x\in\\{0,1,\ldots,n_{A}-1\\}\equiv[n_{A}]$,
and Bob chooses a measurement labelled by $y\in[n_{B}]$. Without loss of
generality, we assume that each of Alice’s (Bob’s) measurements has $k_{A}$
($k_{B}$) possible outcomes. According to quantum theory,
$k_{A}$-($k_{B}$-)outcome measurements correspond to a set of $k_{A}$
($k_{B}$) positive semidefinite operators on $\mathcal{H}_{A}$
($\mathcal{H}_{B}$), adding up to the identity operator $\mathbb{I}_{A}$
($\mathbb{I}_{B}$). We denote these measurement operators by $A^{x}_{a}$ and
$B^{y}_{b}$, where $x\in[n_{A}]$, $a\in[k_{A}]$, $y\in[n_{B}]$ and
$b\in[k_{B}]$. Then, the correlation shared by Alice and Bob reads

$p_{AB}(a,b|x,y)=\operatorname{tr}[\rho_{AB}(A^{x}_{a}\otimes B^{y}_{b})],$
(1)

specifying the probabilities of observing the outcomes $a$ and $b$, given that
the measurements $x$ and $y$ were selected.

The raw data in the protocol corresponds to the pair of strings held by Alice
and Bob containing the measurement outcomes and the implemented measurements,
collected over all protocol rounds. Since individually Alice and Bob only have
access to their marginal statistics, they publicly reveal the measurement
settings and outcomes for a fraction of this data to estimate the joint
statistics and detect its nonlocality. This part of the dataset is discarded.
The secret key is distilled by classically post-processing the remaining
dataset with the help of public communication, so that they finally hold
identical strings that must appear perfectly random to any third party.

As stated earlier, in this work we consider what we call _standard_ protocols,
in which the measurements implemented by Alice and Bob are announced in the
key distillation part. Apart from this constraint, the rest of the protocol is
arbitrary. This family is quite broad and covers most DIQKD protocols
introduced so far Acín  _et al._ (2007); Pironio  _et al._ (2009); Arnon-
Friedman  _et al._ (2018); Woodhead  _et al._ (2020); Ho  _et al._ (2020);
Sekatski  _et al._ (2020); Tan  _et al._ (2020), with only a few exceptions
proposed to date (see e.g. Ref. Acín  _et al._ (2006)).

In order to upper-bound the key rate for a given DIQKD protocol, it suffices
to consider a particular model of the eavesdropper, Eve. Here, we restrict the
analysis to individual attacks that do not require any quantum memory Pironio
_et al._ (2013). In device-independent protocols, Alice and Bob have no
knowledge of the form of the state $\rho_{AB}$ and the measurements
$\\{A^{x}_{a}\\}$, $\\{B^{y}_{b}\\}$, and it is precisely this lack of
knowledge that Eve makes use of in her attack. In particular, we assume that
she knows the precise form of the measurement operators and that she is the
one distributing the quantum state (therefore effectively distributing quantum
correlations) to Alice and Bob in each round.

In our convex combination (CC) attack—originally considered for eavesdroppers
limited only by the no-signalling principle Acín  _et al._ (2006, 2006)—Eve
distributes local deterministic correlations with certain probabilities that
give rise to a local correlation $p^{\mathcal{L}}_{AB}(a,b|x,y)$ with overall
probability $q_{\mathcal{L}}$, and she distributes a nonlocal quantum
correlation $p^{\mathcal{N\\!L}}_{AB}(a,b|x,y)$ with probability
$1-q_{\mathcal{L}}$. While presented in this form for the sake of simplicity,
Eve can equally implement the attack by fixing the measurements of Alice and
Bob and preparing a unique quantum state $\rho_{ABE}$. Eventually, the
observed correlation of Alice and Bob takes the form

$p_{AB}(a,b|x,y)=q_{\mathcal{L}}\cdot
p^{\mathcal{L}}_{AB}(a,b|x,y)+(1-q_{\mathcal{L}})\cdot
p^{\mathcal{N\\!L}}_{AB}(a,b|x,y),$ (2)

and we call $q_{\mathcal{L}}\in[0,1]$ the local weight. Since nonlocality is
necessary for secure DIQKD, in the CC attack Eve maximises $q_{\mathcal{L}}$
for the given observed correlation $p_{AB}(a,b|x,y)$ and a judiciously chosen
nonlocal quantum correlation $p^{\mathcal{N\\!L}}_{AB}(a,b|x,y)$.

We apply the CC attack to the standard DIQKD protocols introduced above. Since
Alice and Bob announce their inputs for every round, Eve knows their outcomes
in all rounds in which she distributes a local correlation. We represent this
knowledge by the classical variable $e$, and we write $e=(a,b)$ for the local
rounds. On the other hand, we assume in what follows that Eve is not
correlated to the nonlocal part of the correlation of Alice and Bob, denoted
by $e=?$. Therefore, for any combination of inputs $x$ and $y$, Alice, Bob and
Eve share correlated random variables distributed as

$\begin{split}p_{ABE}(a,b,e|x,y)&\left.=q_{\mathcal{L}}\cdot
p^{\mathcal{L}}_{AB}(a,b|x,y)\cdot\delta_{e,(a,b)}\right.\\\
&\left.+(1-q_{\mathcal{L}})\cdot
p^{\mathcal{N\\!L}}_{AB}(a,b|x,y)\cdot\delta_{e,?},\right.\end{split}$ (3)

where $\delta$ is the Kronecker delta.

Well-established results in classical cryptography prove that the asymptotic
key rate $r$ extractable from a dataset of strings distributed according to
$p_{ABE}(a,b,e)$ is upper-bounded by the intrinsic information Maurer and Wolf
(1997); Christandl  _et al._ (2007),

$I(A,B\downarrow E)=\min_{E\rightarrow F}I(A:B|F),$ (4)

where $I(A:B|F)=\sum_{f}p_{F}(f)\cdot I(A:B|F=f)$ is the conditional mutual
information of $p_{ABF}(a,b,f)$, and the minimisation is taken over all
stochastic maps $E\rightarrow F$ that map the variable $E$ (with values $e$)
to a new variable $F$ (with values $f$), such that the alphabet size of $F$ is
at most that of $E$ Christandl  _et al._ (2003). While this minimisation may
be hard, any candidate stochastic map provides a valid upper bound.

When applying this bound to the CC attack, the key rate is upper-bounded by

$r\leq\sum_{x,y}p_{xy}\cdot I_{xy}(A:B\downarrow E),$ (5)

where the sum runs over all those settings $(x,y)$ from which the key is
distilled, $p_{xy}$ is the probability of Alice and Bob choosing the settings
$x$ and $y$, respectively, and $I_{xy}(A:B\downarrow E)$ is the intrinsic
information of the distribution in Eq. (3). Note that the bound in Eq. (5) is
based only on the observed correlation, without any assumption on the state or
the measurements.

_Nonlocality is not sufficient for DIQKD.—_ In what follows, we prove that
there exist nonlocal correlations that cannot be used for secure key
extraction with standard DIQKD. We do this by applying the CC attack on any
correlation obtained by performing arbitrary projective measurements on the
two-qubit Werner state Werner (1989) with _visibility_ $v\in[0,1]$,

$\rho^{v}_{AB}=v|\hskip 0.7pt\psi_{-}\rangle\langle\psi_{-}\hskip
0.7pt|+(1-v)\frac{\mathbb{I}}{4},$ (6)

where $|\hskip 0.7pt\psi_{-}\rangle=(|\hskip 0.7pt01\rangle-|\hskip
0.7pt10\rangle)/\sqrt{2}$. It is known that for arbitrary (even infinitely
many) projective measurements $\\{A^{x}_{a}\\}$ and $\\{B^{y}_{b}\\}$, the
correlation
$p^{v}_{AB}(a,b|x,y):=\operatorname{tr}[\rho^{v}_{AB}(A^{x}_{a}\otimes
B^{y}_{b})]$ is local whenever the visibility is at most $v\leq
v^{w}_{\mathcal{L}}:=999\cdot 689\cdot 10^{-6}\cdot\cos^{4}(\pi/50)\approx
0.6829$, see Ref. Hirsch  _et al._ (2017). On the other hand, it is also known
that there exist projective measurements that give rise to nonlocal
correlations for $v\geq v^{w}_{\mathcal{N\\!L}}:\approx 0.6964$, see Ref.
Diviánszky  _et al._ (2017).

Let us consider all DIQKD protocols that use correlations obtained by
implementing arbitrarily many projective measurements on the Werner state. The
measurements can be written as
$A^{x}_{a}=\frac{1}{2}[\mathbb{I}+(-1)^{a}\vec{\alpha}_{x}\cdot\vec{\sigma}]$
and
$B^{y}_{b}=\frac{1}{2}[\mathbb{I}+(-1)^{b}\vec{\beta}_{y}\cdot\vec{\sigma}]$,
where $a,b\in\\{0,1\\}$, $\vec{\alpha}_{x}$ and $\vec{\beta}_{y}$ are unit
vectors in $\mathbb{R}^{3}$, and $\vec{\sigma}=(X,Y,Z)$ is a vector containing
the Pauli matrices. It is easy to verify that

$\begin{bmatrix}p^{v}_{AB}(0,0|x,y)&p^{v}_{AB}(0,1|x,y)\\\
p^{v}_{AB}(1,0|x,y)&p^{v}_{AB}(1,1|x,y)\end{bmatrix}=\frac{1}{2}\begin{bmatrix}s^{v}_{xy}&1-s^{v}_{xy}\\\
1-s^{v}_{xy}&s^{v}_{xy}\end{bmatrix},$ (7)

where
$s^{v}_{xy}=\frac{1}{2}(1-v\vec{\alpha}_{x}\cdot\vec{\beta}_{y})\in[0,1]$.

The CC attack we consider is rather intuitive: it uses the nonlocal
correlation $p_{AB}^{\mathcal{N\\!L}}=p_{AB}^{v=1}$, and local deterministic
correlations that sum up to the correlation for the provable local Werner
state, $p_{AB}^{\mathcal{L}}=p_{AB}^{v=v_{\mathcal{L}}^{w}}$. It is easy to
verify that in this case we have that
$q_{\mathcal{L}}=q^{w}_{\mathcal{L}}:=(1-v)/(1-v_{\mathcal{L}}^{w})$. For now,
let us assume that $s^{1}_{xy}\equiv s^{v=1}_{xy}\geq\frac{1}{2}$. This
implies that in the ideal ($v=1$) case, the outcomes of Alice and Bob are
correlated, i.e., they observe $a=b$ more often than $a\neq b$ [see Eq. (7)].
For this reason, in her stochastic relabelling $E\to F$, Eve will attempt to
become as correlated to the $a=b$ events as possible, that is, she picks $f=a$
whenever the correlation is local and $a=b$. In order to reduce the
conditional mutual information of all the other events, she sets $f=?$ for all
the remaining cases. The resulting distribution reads

$\begin{split}p_{ABF}(a,b,f|x,y)&\left.=q^{w}_{\mathcal{L}}\cdot
p^{v=v_{\mathcal{L}}^{w}}_{AB}(a,b|x,y)\right.\\\
&\left.\times\big{[}\delta_{a,b}\cdot\delta_{f,a}+(1-\delta_{a,b})\cdot\delta_{f,?}\big{]}\right.\\\
&\left.+(1-q^{w}_{\mathcal{L}})\cdot
p^{v=1}_{AB}(a,b|x,y)\cdot\delta_{f,?}.\right.\end{split}$ (8)

Note that a similar distribution can be introduced for the case of
$s^{1}_{xy}<\frac{1}{2}$, in which case Eve becomes correlated with the $a\neq
b$ events. For the distribution in Eq. (8) we have that $I_{xy}(A:B|F=a)=0$
for all $a$, so the final bound on the key rate is given by

$r\leq\sum_{x,y}p_{xy}\cdot p_{F}(?|x,y)\cdot I_{xy}(A:B|F=?),$ (9)

where $I_{xy}(A:B|F=f)$ is the mutual information of the distribution
$p_{AB|F}(a,b|f,x,y)=p_{ABF}(a,b,f|x,y)/p_{F}(f|x,y)$.

To compute the upper bound, we need to calculate the terms $I_{xy}(A:B|F=?)$
in Eq. (9), that is, the mutual information of the distribution

$p_{AB|F}(a,b|?,x,y)=\frac{1}{2(1-q^{w}_{\mathcal{L}}\cdot
s_{xy}^{w})}\begin{bmatrix}(1-q^{w}_{\mathcal{L}})s^{1}_{xy}&(1-q^{w}_{\mathcal{L}})(1-s^{1}_{xy})+q^{w}_{\mathcal{L}}(1-s_{xy}^{w})\\\
(1-q^{w}_{\mathcal{L}})(1-s^{1}_{xy})+q^{w}_{\mathcal{L}}(1-s_{xy}^{w})&(1-q^{w}_{\mathcal{L}})s^{1}_{xy}\end{bmatrix},$
(10)

where $s^{w}_{xy}=s^{v=v^{w}_{\mathcal{L}}}_{xy}$ and
$s^{1}_{xy}=s^{v=1}_{xy}$. The mutual information is clearly zero whenever
$(1-q^{w}_{\mathcal{L}})s^{1}_{xy}=(1-q^{w}_{\mathcal{L}})(1-s^{1}_{xy})+q^{w}_{\mathcal{L}}(1-s_{xy}^{w})$,
which is equivalent to

$v=\frac{v^{w}_{\mathcal{L}}(2s^{1}_{xy}-1)+1}{v^{w}_{\mathcal{L}}(1-2s^{1}_{xy})+4s^{1}_{xy}-1}=:v_{xy}.$
(11)

Note that while $I_{xy}(A:B|F=?)$ is in general positive for $v<v_{xy}$, a
slight modification of the CC attack leads to $I_{xy}(A:B|F=?)=0$ for any
$v\leq v_{xy}$. This is achieved by a stochastic relabelling $E\to F$ in which
Eve only maps some fraction $\lambda_{xy}$ of her variables $e=(a,b)$ with
$a\neq b$ to $f=?$, and leaves the remaining fraction $1-\lambda_{xy}$
invariant. It is straightforward to verify that with a properly chosen
$\lambda_{xy}$, this relabelling leads to $I_{xy}(A:B|F)=0$ for any $v\leq
v_{xy}$.

Also note that $v_{xy}$ is monotonically decreasing in $s^{1}_{xy}$, and
hence, it reaches its lowest possible value at $s^{1}_{xy}=1$. This gives rise
to the critical visibility of the Werner state,

$v^{w}_{\text{crit}}=\frac{v^{w}_{\mathcal{L}}+1}{3-v^{w}_{\mathcal{L}}}\approx
0.7263>v^{w}_{\mathcal{N\\!L}}\approx 0.6964.$ (12)

An analogous derivation yields the same critical visibility for
$s^{1}_{xy}<\frac{1}{2}$. From the above arguments, it follows that whenever
the visibility is $v^{w}_{\mathcal{N\\!L}}\leq v\leq v^{w}_{\text{crit}}$,
Alice and Bob cannot extract a secure key from correlations obtained from the
Werner state with any (even infinite) number of projective measurements, even
though the state is nonlocal (i.e., there exist projective measurements that,
measured on the state, give rise to nonlocal correlations) and distillable
Horodecki  _et al._ (1997). This means that the Bell nonlocality of the
observed correlation is in general not sufficient for DIQKD whenever Alice and
Bob announce their measurement settings, and, moreover, that there exist
nonlocal states that cannot be used for standard DIQKD with projective
measurements.

_CHSH-based protocols.—_ The most commonly used DIQKD protocols Acín  _et al._
(2007); Pironio  _et al._ (2009); Arnon-Friedman  _et al._ (2018); Woodhead
_et al._ (2020); Ho  _et al._ (2020); Sekatski  _et al._ (2020); Tan  _et al._
(2020) are based on the maximal violation of the biased CHSH inequality Acín
_et al._ (2012). In these protocols the shared state is $\rho_{AB}=|\hskip
0.7pt\psi_{-}\rangle\langle\psi_{-}\hskip 0.7pt|$, and Alice’s measurements
are described by $\vec{\alpha}_{0}=(0,0,-1)$ and $\vec{\alpha}_{1}=(-1,0,0)$,
while Bob’s measurements are described by
$\vec{\beta}^{\theta}_{0}=(\sin\theta,0,\cos\theta)$,
$\vec{\beta}^{\theta}_{1}=(-\cos\theta,0,\sin\theta)$ and
$\vec{\beta}^{\theta}_{2}=(0,0,1)$, where $0<\theta<\frac{\pi}{2}$. The
protocol based on the standard CHSH inequality is reproduced by setting
$\theta=\frac{\pi}{4}$ Acín  _et al._ (2007); Pironio  _et al._ (2009). The
noisy versions of these protocols with visibility $v$ can be described by
sharing a Werner state, and our results from the previous section readily
apply. In particular, $v^{w}_{\text{crit}}$ in Eq. (12) is a lower bound on
the critical visibility for CHSH-based protocols.

However, for a fixed protocol, the bound on the critical visibility can be
improved. This is because in the setting of the CHSH-based protocols, the
polytope of local correlations is completely characterised Collins and Gisin
(2004). One can verify that in the CHSH-based protocols, if Alice and Bob
observe a correlation that corresponds to the Werner state with visibility
$v$, then this correlation is local if and only if $v\leq
v^{\theta}_{\mathcal{L}}:=1/(\cos\theta+\sin\theta)$ 111See the Supplemental
Material.. Therefore, an improved bound on the critical visibility for the
CHSH-based protocols is given by

$v^{\theta}_{\text{crit}}=\frac{v^{\theta}_{\mathcal{L}}+1}{3-v^{\theta}_{\mathcal{L}}}>v^{\theta}_{\mathcal{L}}\quad\forall\theta\in(0,\pi/2).$
(13)

That is, for a range of visibilities for which the observed correlation is
nonlocal, Alice and Bob cannot extract a secure key. Note that the same
critical visibility holds for the recently introduced modification of the
standard CHSH-based protocol in Ref. Schwonnek  _et al._ (2020), where the
authors add a fourth setting for Bob. Indeed, since the local polytope is
completely characterised in this case as well Collins and Gisin (2004), one
can verify that the correlation becomes local at the same visibility
$v^{\theta}_{\mathcal{L}}$ Note (1).

Last, we note that in the CHSH-based protocols, Alice and Bob usually extract
their key from the setting pair $x=0$ and $y=2$, by setting $p_{02}$ in Eq.
(5) arbitrarily close to 1. In this case, it is possible to compute the upper
bound in Eq. (9) for any visibility $v\geq v^{\theta}_{\text{crit}}$, and we
get

$\begin{split}r^{\theta}(v)&\left.\leq
2(1-s^{\theta}q^{\theta}_{\mathcal{L}})+(1-q^{\theta}_{\mathcal{L}})\log_{2}\\!\left[\frac{1-q^{\theta}_{\mathcal{L}}}{2(1-s^{\theta}q^{\theta}_{\mathcal{L}})}\right]\right.\\\
&\left.+q^{\theta}_{\mathcal{L}}(1-s^{\theta})\log_{2}\\!\left[\frac{q^{\theta}_{\mathcal{L}}(1-s^{\theta})}{2(1-s^{\theta}q^{\theta}_{\mathcal{L}})}\right]\right.\end{split}$
(14)

where $s^{\theta}=\frac{1}{2}(1+v^{\theta}_{\mathcal{L}})$ and
$q^{\theta}_{\mathcal{L}}=(1-v)/(1-v^{\theta}_{\mathcal{L}})$. In Fig. 1 we
plot the bound for the standard CHSH protocol ($\theta=\frac{\pi}{4}$), and
show that it outperforms the recently derived upper bounds Kaur  _et al._
(2020); Arnon-Friedman and Leditzky (2020) near the critical visibility. In
the Supplemental Material we also describe a two-dimensional region in the set
of quantum correlations corresponding to correlations from the biased CHSH
protocol that are nonlocal but cannot be used to extract a key using standard
DIQKD Note (1).

Fig. 1: Upper bounds on the two-way key rate for the standard CHSH protocol in
terms of the visibility. The dotted line is the upper bound from Kaur  _et
al._ (2020), the dashed line is the upper bound from Arnon-Friedman and
Leditzky (2020), and the solid line is the bound in Eq. (14). Note that the
visibility can be converted into the CHSH violation $S$ via $S=2\sqrt{2}v$.
The shaded area represents the lower bound from Acín  _et al._ (2007).

_Discussion.—_ We introduced a generic tool for upper-bounding DIQKD key rates
using a simple eavesdropping attack. Using our tool, we showed that Bell
nonlocality is not sufficient for secure DIQKD when the honest parties
announce their measurement settings. Our results also imply that all the
commonly used DIQKD protocols become insecure in the noisy case already in the
nonlocal regime, even when assisted by arbitrary two-way communication. Our
analysis does not prove that the considered nonlocal correlations are useless
for secure key distribution, but it shows that the standard
reconciliation—where the settings are announced by both parties—does not work
for all nonlocal correlations.

Given the above, one possibility to lower the stringent requirements on noise
parameters is to employ protocols such as that of Ref. Acín  _et al._ (2006),
in which only one party announces their settings. Indeed, for this protocol we
were not able to find an upper bound that vanishes in the nonlocal regime.
Whether a secure key can be distilled from all nonlocal correlations using
these protocols is an open question that deserves further investigation.
Another possibility for improving the key rates extractable from a given
quantum state is to employ measurements that are not projective. However, we
note that no state is known thus far that is local for all arrangements of
projective measurements, while exhibiting nonlocality for some arrangement of
non-projective measurements. Hence, the critical visibilities of the Werner
state derived in this work also hold for all the hitherto studied arrangements
of non-projective measurements. Another question worth investigating is
whether tighter upper bounds can be derived using collective or coherent
attacks. Nonetheless, let us note that our (individual) CC attack can be
applied to a broad class of DIQKD protocols, and gives rise to bounds on the
critical visibility in experimentally relevant scenarios. We elaborate on
these findings in 222In preparation.

_Acknowledgements.—_ We thank Erik Woodhead, Stefano Pironio, Jędrzej
Kaniewski, Karol Horodecki, Filip Rozpędek and Goh Koon Tong for fruitful
discussions, and Marco Túlio Quintino for pointing out Ref. Diviánszky  _et
al._ (2017). We acknowledge support from the Government of Spain
(FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundació Cellex, Fundació
Mir-Puig, Generalitat de Catalunya (CERCA, AGAUR SGR 1381 and QuantumCAT), the
ERC AdG CERQUTE, the AXA Chair in Quantum Information Science, the EU Quantum
Flagship project QRANGE, and the Foundation for Polish Science within the
“Quantum Optical Technologies” project carried out within the International
Research Agendas programme cofinanced by the European Union under the European
Regional Development Fund. MBJ acknowledges funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 847517.

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Supplemental Material

##  Appendix A Local visibility for CHSH-based protocols

Consider the correlation
$p^{\theta,v}_{AB}(a,b|x,y)=\operatorname{tr}\\{\rho^{v}_{AB}[A^{x}_{a}\otimes
B^{y}_{b}(\theta)]\\}$, obtained by measuring the Werner state

$\rho^{v}_{AB}=v|\hskip 0.7pt\psi_{-}\rangle\langle\psi_{-}\hskip
0.7pt|+(1-v)\frac{\mathbb{I}}{4}$ (15)

with the projective measurements
$A^{x}_{a}=\frac{1}{2}[\mathbb{I}+(-1)^{a}\vec{\alpha}_{x}\cdot\vec{\sigma}]$
and
$B^{y}_{b}(\theta)=\frac{1}{2}[\mathbb{I}+(-1)^{b}\vec{\beta}^{\theta}_{y}\cdot\vec{\sigma}]$
with $a,b\in\\{0,1\\}$, given by

$\begin{split}\vec{\alpha}_{0}=(0,0,-1),\quad\vec{\alpha}_{1}=(-1,&\left.0,0)\right.\\\
\vec{\beta}_{0}^{\theta}=(\sin\theta,0,\cos\theta),\quad\vec{\beta}_{1}^{\theta}=(-\cos\theta,0,&\left.\sin\theta),\quad\vec{\beta}_{2}^{\theta}=(0,0,1).\right.\end{split}$
(16)

Note that with the above notation, we have that
$p^{\theta,v}_{AB}(a,b|x,y)=\frac{1}{4}[1-v(-1)^{a+b}\vec{\alpha}_{x}\cdot\vec{\beta}_{y}^{\theta}]$.

In this section, we show that $p^{\theta,v}_{AB}(a,b|x,y)$ is local if and
only if

$v\leq v^{\theta}_{\mathcal{L}}=\frac{1}{\cos\theta+\sin\theta}.$ (17)

In order to prove this, we note that for the case of two binary measurements
on Alice’s side and three binary measurements on Bob’s side, the polytope of
local correlations is completely characterised [28]. In particular, all the
facets that correspond to non-trivial constraints (i.e., do not correspond to
the positivity and normalisation of the probabilities) are of the CHSH-type:

$-2\leq\langle A_{x_{0}}B_{y_{0}}\rangle+\langle
A_{x_{0}}B_{y_{1}}\rangle+\langle A_{x_{1}}B_{y_{0}}\rangle-\langle
A_{x_{1}}B_{y_{1}}\rangle\leq 2\quad
x_{0},x_{1}\in\\{0,1\\},\,y_{0},y_{1}\in\\{0,1,2\\},x_{0}\neq x_{1},y_{0}\neq
y_{1},$ (18)

where

$\langle
A_{x}B_{y}\rangle=p_{AB}(0,0|x,y)+p_{AB}(1,1|x,y)-p_{AB}(0,1|x,y)-p_{AB}(1,0|x,y)$
(19)

are the correlators. In other words, a correlation $p_{AB}(a,b|x,y)$ in this
setting is local if and only if it satisfies all the inequalities in Eq. (18).

Let us denote the correlators of $p^{\theta,v}_{AB}(a,b|x,y)$ by $\langle
A_{x}B_{y}\rangle^{\theta,v}$. Note that $\langle
A_{x}B_{y}\rangle^{\theta,v}=-v\vec{\alpha}_{x}\cdot\vec{\beta}_{y}^{\theta}$,
and hence $\langle A_{x}B_{y}\rangle^{\theta,v}=v\cdot\langle
A_{x}B_{y}\rangle^{\theta,1}$. Therefore, the statement that
$p^{\theta,v}_{AB}(a,b|x,y)$ is local is equivalent to

$-2\leq v\cdot\mathcal{S}^{\theta}_{x_{0},x_{1},y_{0},y_{1}}\leq 2\quad\forall
x_{0},x_{1}\in\\{0,1\\},\,y_{0},y_{1}\in\\{0,1,2\\},x_{0}\neq x_{1},y_{0}\neq
y_{1},$ (20)

where we have defined

$\mathcal{S}^{\theta}_{x_{0},x_{1},y_{0},y_{1}}=\langle
A_{x_{0}}B_{y_{0}}\rangle^{\theta,1}+\langle
A_{x_{0}}B_{y_{1}}\rangle^{\theta,1}+\langle
A_{x_{1}}B_{y_{0}}\rangle^{\theta,1}-\langle
A_{x_{1}}B_{y_{1}}\rangle^{\theta,1}.$ (21)

A straightforward computation yields

$\begin{split}\mathcal{S}^{\theta}_{0,1,0,1}&=2(\cos\theta+\sin\theta)\\\
\mathcal{S}^{\theta}_{0,1,0,2}&=1+\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{0,1,1,0}&=-\cos\theta-\sin\theta\\\
\mathcal{S}^{\theta}_{0,1,1,2}&=1-\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{0,1,2,0}&=1+\cos\theta-\sin\theta\\\
\mathcal{S}^{\theta}_{0,1,2,1}&=1+\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{1,0,0,1}&=0\\\
\mathcal{S}^{\theta}_{1,0,0,2}&=-1+\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{1,0,1,0}&=2(-\cos\theta+\sin\theta)\\\
\mathcal{S}^{\theta}_{1,0,1,2}&=-1-\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{1,0,2,0}&=1-\cos\theta+\sin\theta\\\
\mathcal{S}^{\theta}_{1,0,2,1}&=1-\cos\theta-\sin\theta.\end{split}$ (22)

By comparing Eqs. (20) and (22), it is straightforward to verify that
$p^{\theta,v}_{AB}(a,b|x,y)$ is local if and only if $v\leq
v^{\theta}_{\mathcal{L}}=1/(\cos\theta+\sin\theta)$.

This argument naturally extends to the CHSH-based protocol recently introduced
in Ref. [30]. This protocol is based on the state in Eq. (15) and the
measurements in Eq. (16) (with $\theta=\frac{\pi}{4}$, but our argument works
for arbitrary $\theta$), with an added setting for Bob, described by
$\vec{\beta}^{\theta}_{3}=(1,0,0)$. These new correlations trivially satisfy
Eq. (22), and additionally, it is easy to verify that

$\mathcal{S}^{\theta}_{x_{0},x_{1},y,3}=\mathcal{S}^{\theta}_{x_{1},x_{0},y,2}\quad\text{and}\quad\mathcal{S}^{\theta}_{x_{0},x_{1},3,y}=\mathcal{S}^{\theta}_{x_{0},x_{1},2,y}.$
(23)

The local polytope is completely characterised in this scenario as well [28],
and all the non-trivial facets are of the CHSH-type:

$-2\leq\langle A_{x_{0}}B_{y_{0}}\rangle+\langle
A_{x_{0}}B_{y_{1}}\rangle+\langle A_{x_{1}}B_{y_{0}}\rangle-\langle
A_{x_{1}}B_{y_{1}}\rangle\leq 2\quad
x_{0},x_{1}\in\\{0,1\\},\,y_{0},y_{1}\in\\{0,1,2,3\\},x_{0}\neq
x_{1},y_{0}\neq y_{1}.$ (24)

Hence, it is clear that these new correlations are also local in the noisy
case if and only if $v\leq v^{\theta}_{\mathcal{L}}$.

##  Appendix B A two-dimensional region of quantum correlations with zero key

In this section, we describe a two-dimensional region in the set of quantum
correlations that is nonlocal but cannot be used to extract a secure key using
standard DIQKD. This region corresponds to correlations based on the biased
CHSH inequality (see the main text), and is depicted in Fig. 2. The points in
the figure are based on correlations with two inputs and two outputs,
$p_{AB}(a,b|x,y)=\operatorname{tr}[\rho_{AB}(A^{x}_{a}\otimes B^{y}_{b})]$,
such that $a,b,x,y\in\\{0,1\\}$. The full set of these correlations can be
embedded in an 8-dimensional real vector space [28]. To see this, let us
define the observables

$A_{x}\equiv A^{x}_{0}-A^{x}_{1}\quad\text{and}\quad B_{y}\equiv
B^{y}_{0}-B^{y}_{1}.$ (25)

The original measurement operators can be recovered from the observables via

$A^{x}_{a}=\frac{1}{2}[\mathbb{I}+(-1)^{a}A_{x}]\quad\text{and}\quad
B^{y}_{b}=\frac{1}{2}[\mathbb{I}+(-1)^{b}B_{y}].$ (26)

In terms of these observables, we define the marginals

$\langle
A_{x}\rangle=\operatorname{tr}[\rho_{AB}(A_{x}\otimes\mathbb{I})]\quad\text{and}\quad\langle
B_{y}\rangle=\operatorname{tr}[\rho_{AB}(\mathbb{I}\otimes B_{y})]$ (27)

and the correlators

$\langle A_{x}B_{y}\rangle=\operatorname{tr}[\rho_{AB}(A_{x}\otimes B_{y})]$
(28)

[notice that these correlators are the same as those in Eq. (19)]. Indeed, any
correlation can be written in terms of the marginals and the correlators as

$p_{AB}(a,b|x,y)=\operatorname{tr}[\rho_{AB}(A^{x}_{a}\otimes
B^{y}_{b})]=\frac{1}{4}\big{[}1+(-1)^{a}\langle A_{x}\rangle+(-1)^{b}\langle
B_{y}\rangle+(-1)^{a+b}\langle A_{x}B_{y}\rangle\big{]}.$ (29)

In order to describe a 2-dimensional slice of the set of correlations in this
scenario, we need to describe all correlators and marginals in terms of affine
functions of two real variables, which we will denote by $s$ and $t$. The
slice we are interested in is given by

$\begin{split}&\left.\langle A_{x}\rangle=\langle B_{y}\rangle=0\quad\forall
x,y\right.\\\ &\left.\langle A_{0}B_{0}\rangle=s\right.\\\ &\left.\langle
A_{0}B_{1}\rangle=t\right.\\\ &\left.\langle A_{1}B_{0}\rangle=t\right.\\\
&\left.\langle A_{1}B_{1}\rangle=-s.\right.\end{split}$ (30)

The set of quantum correlations in this case is bounded by the relation
$s^{2}+t^{2}\leq 1$, which corresponds to the maximal possible quantum
violation of the biased CHSH inequalities [10]. The local set is bounded by
four constraints, $\pm s\pm t\leq 1$, which are various relabellings of the
CHSH inequality, and the inequality $s+t\leq 1$ corresponds to the standard
CHSH inequality

$\langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle
A_{1}B_{0}\rangle-\langle A_{1}B_{1}\rangle\leq 2.$ (31)

Using this parametrisation, the local points on the CHSH facet correspond to
$s+t=1$, with the two endpoints being $(s,t)=(1,0)$ and $(s,t)=(0,1)$. The
boundary of the quantum set (with all points being extremal) corresponds to
the curve $s^{2}+t^{2}=1$. We will be interested in the region $s+t>1$, i.e.,
correlations violating the standard CHSH inequality, while still being
quantum, i.e., $s^{2}+t^{2}\leq 1$. It is convenient to parametrise this
region with the polar coordinates $(s,t)=(v\cos\theta,v\sin\theta)$:

$\begin{split}&\left.\langle A_{x}\rangle=\langle B_{y}\rangle=0\quad\forall
x,y\right.\\\ &\left.\langle A_{0}B_{0}\rangle=v\cos\theta\right.\\\
&\left.\langle A_{0}B_{1}\rangle=v\sin\theta\right.\\\ &\left.\langle
A_{1}B_{0}\rangle=v\sin\theta\right.\\\ &\left.\langle
A_{1}B_{1}\rangle=-v\cos\theta,\right.\end{split}$ (32)

where $0<\theta<\frac{\pi}{2}$ and $v^{\theta}_{\mathcal{L}}<v\leq 1$ [let us
recall that $v^{\theta}_{\mathcal{L}}=1/(\cos\theta+\sin\theta)$, see the
previous section].

The region depicted in Fig. 2 corresponds to DIQKD protocols with two binary
measurements on Alice’s side and three binary measurements on Bob’s side. The
correlation observed in the protocol is given by Eq. (32) with an added binary
measurement, $y=2$, on Bob’s side, that gives rise to the final correlation

$\begin{split}&\left.\langle A_{x}\rangle=\langle B_{y}\rangle=0\quad\forall
x,y\right.\\\ &\left.\langle A_{0}B_{0}\rangle=v\cos\theta\right.\\\
&\left.\langle A_{0}B_{1}\rangle=v\sin\theta\right.\\\ &\left.\langle
A_{0}B_{2}\rangle=v\right.\\\ &\left.\langle
A_{1}B_{0}\rangle=v\sin\theta\right.\\\ &\left.\langle
A_{1}B_{1}\rangle=-v\cos\theta,\right.\\\ &\left.\langle
A_{1}B_{2}\rangle=0.\right.\end{split}$ (33)

It is clear that this correlation can be obtained by measuring the state in
Eq. (15) with the projective measurements described in Eq. (16).

The region depicted in Fig. 2 corresponds to the set of correlations of the
form (33) with $0<\theta<\frac{\pi}{2}$ and $v^{\theta}_{\mathcal{L}}<v\leq 1$
[this region is depicted using the coordinates
$(s,t)=(v\cos\theta,v\sin\theta)$]. The black dashed line corresponds to
$v=v^{\theta}_{\mathcal{L}}$, and the blue curved boundary on the edge
corresponds to $v=1$. The green line is given by $\theta=\frac{\pi}{4}$. The
red region corresponds to $v^{\theta}_{\mathcal{L}}<v\leq
v^{\theta}_{\text{crit}}=(v^{\theta}_{\mathcal{L}}+1)/(3-v^{\theta}_{\mathcal{L}})$.
As explained in the main text, Alice and Bob cannot extract a secure key if
they observe a correlation in the red region and publicly announce their
settings for each round, even though these correlations are nonlocal.

Fig. 2: We depict a 2-dimensional slice of the set of correlations in the
scenario with two binary measurements on Alice’s side and three binary
measurements on Bob’s side. We can see that there are three different regions:
in red, we show the region of nonlocal quantum correlations that cannot be
used for key extraction if the honest parties announce their settings. In
yellow, we depict (part of) the region of local correlations, and the dashed
black line represents the CHSH facet of the local polytope. In light blue, we
show the region of nonlocal quantum correlations for which our upper bound on
the key rate is non-zero. The dark blue curve corresponds to the maximal
possible quantum violation of the biased CHSH inequalities, that is, the ideal
correlations for $0<\theta<\pi/2$. Finally, the green line corresponds to the
correlations obtained from the noisy standard CHSH protocol.


is Theorem 6.19. Hence $\mathbb{P}_{\mu}(\mathbf{O})=0$ since $\mathbf{O}$ is
the countable union of sets of measures $0$.

We consider an example where the limit is equal to 1.

######  Remark 6.21.

Notice if we consider $\mu_{t}$ from Example 2.4 and set
$\varphi(x,y)\coloneqq R(x,y)$, the edge relation, we see that

$\displaystyle\lim_{n\to\infty}\frac{1}{n!}\sum_{\sigma\in\operatorname{Sym}(n)}\mathbb{P}_{\mu_{t}}(O_{n}^{\varphi}(x_{\sigma(1)},...,x_{\sigma(n)}))=\lim_{n\to\infty}\frac{1}{n!}\sum_{\sigma\in\operatorname{Sym}(n)}1=1,$

because $\mu$ is $M$-excellent (i.e. self-commutes) and
$\mathbb{P}_{\mu_{t}}(O_{n}^{R}(x_{1},...,x_{n}))=1$.

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