# Near-Deterministic Weak-Value Metrology via Collective non-Linearity

Muthumanimaran Vetrivelan<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology-Bombay, Powai, Mumbai 400076, India  Sai Vinjanampathy
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology-
Bombay, Powai, Mumbai 400076, India  Centre for Quantum Technologies, National
University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore

###### Abstract

Weak-value amplification employs postselection to enhance the measurement of
small parameters of interest. The amplification comes at the expense of
reduced success probability, hindering the utility of this technique as a tool
for practical metrology. Following other quantum technologies that display a
quantum advantage, we formalize a quantum advantage in the success probability
and present a scheme based on non-linear collective Hamiltonians that shows a
super-extensive growth in success probability while simultaneously displaying
an extensive growth in the weak value. We propose an experimental
implementation of our scheme.

## Introduction.—

Quantum metrology employs non-classical correlations and Hamiltonian
interactions to enhance parameter estimation Giovannetti  _et al._ (2011);
Degen  _et al._ (2017). An example of a metrology scheme with demonstrable
advantages to measure weak signals in the presence of technical noise sources
Jordan  _et al._ (2014) and also for saturated signals Xu  _et al._ (2020) is
weak-value metrology. Weak-value amplification (WVA) uses postselection to
enhance measurements of small parameters via amplification Aharonov  _et al._
(1988); Ritchie  _et al._ (1991); Kofman  _et al._ (2012); Dressel  _et al._
(2014); Kedem and Vaidman (2010). Such postselection involves rejecting
measurement outcomes and there exists a trade-off between the postselection
probability of the desired measurement and the amplification produced by the
apparatus. This trade-off has always limited the applicability of weak-value
metrology. Given its practical applications Dixon  _et al._ (2009); Salazar-
Serrano  _et al._ (2015); Pati  _et al._ (2020); Ghosh  _et al._ (2019), it is
desirable to have (at least) linear scaling in both quantities, since this
promises that a modest increase in the number of atoms participating in the
collective interaction will amplify the signal while simultaneously making the
desired measurements more probable. Such a solution was lacking until now. In
this manuscript, we show that non-linear collective Hamiltonians can enhance
existing WVA schemes substantially. We demonstrate a linear enhancement in the
weak-value $\mathcal{A}_{w}$ and a simultaneous quadratic enhancement in the
success probability $P_{s}$, both defined below. Furthermore, we discuss a
near-deterministic WVA scheme that shows a quadratic enhancement of
$\mathcal{A}_{w}$. Finally, we propose an experimental scheme that realizes
the initial state preparation, the final system state measurement and an
implementation of the non-linear WVA Hamiltonian to realize this advantage
using existing experimental resources.

## Success Probability and Weak Value.—

Consider the bipartite Hamiltonian $H_{I}=gA\otimes B\delta(t-t^{\prime})$,
where $A$($B$) is a Hermitian operator defined in the system(meter) Hilbert
space and $g$ is the extremely small parameter of interest. The initial state
$|{\Psi_{i}}\rangle=|{\psi_{i}}\rangle\otimes|{\phi_{i}}\rangle$ is allowed to
weakly interact for a small time and is then postselected in a final system
state $|{\psi_{f}}\rangle$. The resulting postselected or “kicked” meter state
is given by $|{\phi_{f}}\rangle=M|{\phi_{i}}\rangle/||M|{\phi_{i}}\rangle||$,
where $M=\exp(-ig\mathcal{A}_{w}B)$ is the non-unitary evolution defined in
terms of the weak value $\mathcal{A}_{w}$ Ritchie  _et al._ (1991), given by

$\mathcal{A}_{w}=\frac{\langle{\psi_{f}}|A|{\psi_{i}}\rangle}{\langle{\psi_{f}}|\psi_{i}\rangle}.$
(1)

We will refer to the choice of the initial state of the system, the final
state of the system and the initial state of the meter as the weak value
strategy. Note that the $\mathcal{A}_{w}$ is inversely proportional to the
square-root of the success probability
$P_{s}=|\langle{\psi_{f}}|\psi_{i}\rangle|^{2}$. A small success probability
hence implies a large amplification. WVA hence allows for the small value $g$
to be amplified due to the effect of the measurement of the system. In general
$\mathcal{A}_{w}$ can be a complex number and greater than the expectation
value of the observable $A$. For unbiased estimators, when the weak value is
chosen to be purely imaginary Pang  _et al._ (2014) the expectation value of a
meter observable $R$ is approximately given by
$\langle{R}\rangle_{|{\phi_{f}}\rangle}\approx
2g[Im(\mathcal{A}_{w})Re(\alpha)]$ (where
$\alpha=\langle{RB}\rangle_{|{\phi_{i}}\rangle}$, offering the potential
amplification of $g$ in the measured signal Pang  _et al._ (2014). For
example, if $A=\sigma_{x}$, $|{\psi_{i}}\rangle=|{\downarrow}\rangle$ and
$|{\psi_{f}}\rangle=|{\theta}\rangle=\sin(\theta)|{\downarrow}\rangle+\exp(i\frac{\pi}{2})\cos(\theta)|{\uparrow}\rangle$,
then $\mathcal{A}_{w}\approx i/\theta$ for small $\theta$. This implies that
the choice of the postselected state to be nearly orthogonal to the initial
state amplifies $g$ to a larger number which can then be read from the kicked
meter state. The central trade-off of the not deterministic WVA technique is
that enhancing weak value comes at a corresponding loss of the success
probability of the postselection event. This hinders the utility of the WVA
procedure in practice, since the successful detector clicks are always
competing with the amplification. One may seek to remedy this by having
multiple probes.

If measurements are performed on $2j$ independent copies, the success
probability of at least one positive click scales extensively in the number of
independent ancilla Brun  _et al._ (2008) since

$P_{s}^{(2j)}=1-(1-P_{s})^{2j}\approx 2jP_{s},$ (2)

where we denote the success probability associated with multiple probes as
$P_{s}^{(2j)}$. This presupposes that at the end of these $2j$ detector
clicks, we are satisfied with just one of the experimental runs returning an
amplification. If we require that all $2j$ independent measurements
simultaneously produce weak value amplification, then the corresponding
success probability decreases exponentially. A natural question that arises is
whether this extensive scaling of $P_{s}$ can be improved while enhancing the
weak value. Since entanglement provides an advantage in computing and allows
us to go beyond shot-noise variance in metrology, it is reasonable to expect
an entanglement advantage in WVA as well. Furthermore, since an advantage due
to collective interactions has been observed in disparate technologies such as
metrology Boixo  _et al._ (2008); Zwierz  _et al._ (2010) and energy storage
Binder  _et al._ (2015); Campaioli  _et al._ (2017); Rossini  _et al._ (2019);
Andolina  _et al._ (2019), it is reasonable to expect collective interactions
to lead to an advantage in weak-value metrology. This was explored by Pang and
Brun Pang  _et al._ (2014) whose proposal leads to sub-linear scaling in
$\mathcal{A}_{w}$ with a simultaneous linear enhancement in $P_{s}$, which we
briefly review.

Figure 1: Circuit QED analogue of our model with the energy levels of the atom
(transmon) and the first two levels of the resonator modes denoted. The
resonator mode at $\omega_{c}$ is two-photon detuned from $\omega_{01}$ as
shown. Interaction with another far-detuned readout mode is indicated at the
detuning $\delta_{r}$.

## Optimized Trade-off for Linear Hamiltonians.—

The postselection probability, the weak value and the variance of the
Hamiltonian in the initial state are related to each other Pang  _et al._
(2014). Hence an optimization of $P_{s}^{(2j)}$ or $\mathcal{A}_{w}$ can be
performed separately, each time at the expense of the other. To optimize
$P_{s}^{(2j)}$, Eq. (1) can be modified to read
$\langle{\psi_{f}}|(A-\mathcal{A}_{w})|{\psi_{i}}\rangle=0$. The postselected
state is hence orthogonal to the state
$(A-\mathcal{A}_{w})|{\psi_{i}}\rangle$. The optimal postselected state can be
found using Gram-Schmidt orthogonalization and the corresponding maximum
$P_{s}^{(2j)}$ is

$\max
P_{s}^{(2j)}\approx\frac{Var(A)_{|{\psi_{i}}\rangle}}{|\mathcal{A}_{w}|^{2}},$
(3)

where
$Var(A)_{|{\psi_{i}}\rangle}=\langle{\psi_{i}}|A^{2}|{\psi_{i}}\rangle-[\langle{\psi_{i}}|A|{\psi_{i}}\rangle]^{2}$.
This equation is valid for $|\mathcal{A}_{w}|\gg\max\langle A\rangle$ . For
the collective operator $A=J_{z}$, maximizing the variance over an arbitrary
initial state $|{\psi_{i}}\rangle$ yields Giovannetti  _et al._ (2004) $\max
Var(J_{z})_{|{\psi_{i}}\rangle}=j^{2}$, corresponding to the state

$|{\psi_{i}}\rangle\propto|{j,j}\rangle+|{j,-j}\rangle,$ (4)

where normalization constant has been omitted. This initial state is well
known for its advantage in various quantum metrology protocols Giovannetti
_et al._ (2011). For a fixed value of $\mathcal{A}_{w}$, the postselection
probability hence scales as $j^{2}$. This is $j$ times better than using
uncoupled ancillas which shows linear scaling in $j$. Using the
orthogonalisation argument presented above, the final postselection state for
linear collective Hamiltonians takes of the form

$|{\psi_{f}}\rangle\propto(j+\mathcal{A}_{w})|{j,j}\rangle+(j-\mathcal{A}_{w})|{j,-j}\rangle.$
(5)

Hence the entanglement of the initial/final state and the collective
Hamiltonian interactions are leveraged to produce a quantifiable advantage in
the success probability. To compare collective and individual strategies for
weak value amplification, we define the success probability advantage of
collective strategies as

$\sigma:=\frac{P_{s}^{(2j)}}{2jP_{s}}.$ (6)

While an uncorrelated weak-value strategy yields an advantage $\sigma\propto
1$, the aforementioned entanglement strategy offers $\sigma\propto j$ for a
fixed $\mathcal{A}_{w}$. Clearly the limitation of this strategy is the
extensive scaling of $\sigma$ comes at the cost of holding $\mathcal{A}_{w}$
fixed, though a large number of particles are participating in the dynamics.

On the other hand, WVA can be optimized by holding $\sigma$ fixed. Consider a
postselection state $|{\psi_{f}}\rangle$ of the form

$|{\psi_{f}}\rangle=\sqrt{P_{s}^{(2j)}}|{\psi_{i}}\rangle+\sqrt{1-P_{s}^{(2j)}}|{\psi_{i}^{\perp}}\rangle,$
(7)

where $|{\psi_{i}^{\perp}}\rangle$ is a state that is orthogonal to the
initial state $|{\psi_{i}}\rangle$. The weak value in Eq. (1) becomes

$\mathcal{A}_{w}=\langle{\psi_{i}}|A|{\psi_{i}}\rangle+\sqrt{\frac{1-P_{s}^{(2j)}}{P_{s}^{(2j)}}}\langle{\psi_{i}^{\perp}}|A|{\psi_{i}}\rangle.$
(8)

For small $P_{s}$, it can be shown that

$\max|\mathcal{A}_{w}|\approx\sqrt{\frac{Var(A)_{|{\psi_{i}}\rangle}}{P_{s}^{(2j)}}}.$
(9)

As noted in Eq. (3), the maximum weak value depends only on the variance of
observable $A$ with respect to initial state. By fixing $\sigma\propto 1$, the
maximum attainable weak-value scaling is $\mathcal{A}_{w}\propto\sqrt{j}$.

To summarize the case of linear collective Hamiltonians, there is a definite
trade off between either maximizing post-measurement quantum advantage
$\sigma$ or gaining a collective advantage in the weak value
$\mathcal{A}_{w}$. Different strategies involving linear Hamiltonians can
either produce an extensive success probability advantage for fixed
$\mathcal{A}_{w}$ or no advantage for the case where
$\mathcal{A}_{w}\propto\sqrt{j}$. We now show that non-linear Hamiltonian
metrology offers a simultaneous collective advantage $\sigma$ and an
enhancement in $\mathcal{A}_{w}$ that scales favorably with $j$.

Figure 2: Deterministic Dicke state preparation involves the action of the
unitary operators $U_{j,m_{1}}$ and $U_{j,m_{2}}$ on the computational state
prepare Dicke states. The state after passing through the control-SWAP gate is
projected onto $|{\mu}\rangle\langle{\mu}|\otimes
I\otimes|{\zeta}\rangle\langle{\zeta}|$. The non-zero overlaps
$\langle{j,m_{1}|\zeta}\rangle$ and $\langle{j,m_{2}|\zeta}\rangle$ are used
to define the state $|{\mu}\rangle$ which produces the desired initial state
as shown.

## Non-Linearity and Joint Optimization.—

Consider the Hamiltonian

$H=g({J}^{2}-{J}_{z}^{2})\otimes{a}^{\dagger}{a},$ (10)

whose physical implementation is discussed below. The maximum and minimum
eigenvalues of the system Hamiltonian are $j(j+1)$ and $j$ respectively
Bärtschi and Eidenbenz (2019) correspond to the eigenstates $|{j,0}\rangle$
and $|{j,\pm j}\rangle$. Choosing $|{j,-j}\rangle$ as the lower eigenstate,
the initial state is taken as the equal superposition of these eigenstates,
namely

$|{\psi_{i}}\rangle\propto|{j,0}\rangle+|{j,-j}\rangle.$ (11)

Following the analysis outlined for linear Hamiltonians, the maximum variance
with $A=J^{2}-J_{z}^{2}$ is

$\max Var(A)_{|{\psi_{i}}\rangle}=\frac{j^{4}}{4}.$ (12)

Since the maximum value of $\mathcal{A}_{w}$ in Eq. (9) depends on the square
root of variance over the postselection probability, if we fix
$P_{s}^{(2j)}=\kappa j^{2}\ll 1$, the maximum weak value will scale as
$\mathcal{A}_{w}\propto j$. This means that simultaneously we have achieved
superextensive scaling in the success probability denoted by $\sigma\propto j$
and extensive weak value amplification due to collective non-linearity. This
is one of the main results of our manuscript. The corresponding postselected
state (see Appendix A) can be evaluated to be

$|{\psi_{f}}\rangle\propto(\sqrt{\kappa}j+1)|{j,0}\rangle+(\sqrt{\kappa}j-1)|{j,-j}\rangle.$
(13)

## Near-Deterministic Strategy.—

On the other hand, we can optimize the final state differently to obtain a
near-deterministic weak-value metrology protocol. Let us consider Eq. (8).
once again, but seek a near-deterministic protocol by setting
$P_{s}^{(2j)}=1-\epsilon$ . If we choose
$|{\psi_{f}}\rangle\propto(1+\sqrt{\epsilon})|{j,0}\rangle+(1-\sqrt{\epsilon})|{j,-j}\rangle$,
with the same initial state as before, we obtain a nearly deterministic
protocol with quadratic scaling in $\mathcal{A}_{w}$ simultaneously. This
quantum advantage is predicated on (a) an implementation of the non-linear
Hamiltonian in Eq. (10) and (b) preparation of the initial state and
measurement in the final system state. We now detail an experimental proposal
to implement this.

## Implementation of non-Linear Hamiltonian.—

Consider the two-photon Tavis-Cummings Hamiltonian namely

$H_{0}=g_{0}(J_{+}a^{2}e^{i\delta_{-}t}+J_{-}a^{{\dagger}^{2}}e^{-i\delta_{-}t}).$
(14)

Such a two-photon collective Hamiltonian can be implemented by generalizing
existing two-photon Jaynes-Cummings Garbe  _et al._ (2017, 2020) models to
include several atoms or modifying an existing Dicke model implementation to
induce a two-photon term Fink  _et al._ (2009); Mlynek  _et al._ (2014). We
propose the second method, wherein $2j$ transmon atoms Song  _et al._ (2019)
are coupled to a resonator mode whose fundamental frequency is at
$\omega_{c}$. Since the cavity frequency is close to half the transmon’s
frequency $\omega_{01}$, one needs to ensure that any higher modes of the
cavity are sufficiently detuned from $\omega_{01}$. This can be achieved by
either using lumped-element resonators Vijay  _et al._ (2011) which
effectively only have one mode, or by choosing a quarter-wave transmission
line resonator Pozar (1990) whose second harmonic is at three times the
fundamental frequency. This is depicted in Fig. (1) alongside the energy level
of a (different) cavity used to perform the qubit transformations highlighted
in the next section.

In order to derive the Hamiltonian from a two-photon Rabi model Felicetti  _et
al._ (2018), the frequency $\omega_{01}$ and the cavity frequency $\omega_{c}$
are related to the bare coupling via $g_{0}<|\delta_{-}|\ll\delta_{+}$. Here
$\delta_{\mp}=\omega_{0}\mp 2\omega_{c}$ is the two-photon detuning (sum
frequency) and we have already dropped the counter-rotating terms owing to the
magnitude of $\delta_{+}$ to arrive at Eq. (14). In the far-detuned regime
$g_{0}\ll\delta_{-}$, we can write the effective Hamiltonian James and Jerke
(2007); Shao  _et al._ (2017) as Eq.(10) with the identification
$g\equiv-4g_{0}^{2}/\delta_{-}$. This Hamiltonian can be understood to induce
a (collective) state dependent Bloch-Siegert Forn-Díaz  _et al._ (2010) like
shift in the harmonic spectrum. Such a Hamiltonian represents a non-linear
interaction between atoms mediated by a cavity and is crucially different from
the Tavis-Cummings model where twisting Hamiltonians Ma  _et al._ (2011);
Degen  _et al._ (2017) are produced by adiabatically eliminating the cavity.
We note that the coupling in Eq. (14) is significantly small and hence the
spectral collapse Garbe  _et al._ (2020) of the Hamiltonian in ultra-strong
coupling regime is avoided. A small magnetic field would displace the energy
levels represented by $\omega_{01}$ linearly, which can be inferred from a
corresponding weak-value amplified shift in the measured value of
$4g_{0}^{2}/\delta_{-}$.

Figure 3: The time-evolved system-meter state $|{\Psi(t)}\rangle$ enters the
circuit where $|{\mu}\rangle$ and $|{\zeta}\rangle$ are ancilla and reference
states defined in the text. The final state is projected onto the joint
quantum state $|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle$
which results in the postselection of the system in the state
$|{\psi_{f}}\rangle$.

## State Preparation & Measurement.—

Now we propose a scheme to prepare the initial state and perform the
measurement in the final state. The solution is presented as multi-qubit
circuit implementations, which may be implemented by using an additional
resonator mode. The initial state for all of the weak value strategies
considered is the superposition of two Dicke states, which we can write as
$\alpha|{j,m_{1}}\rangle+\beta|{j,m_{2}}\rangle$. Dicke states can be prepared
with an efficient deterministic quantum algorithm Bärtschi and Eidenbenz
(2019). We combine this with a probabilistic protocol for implementing
superposition of two pure states Oszmaniec  _et al._ (2016); Li  _et al._
(2017) if the overlap with a reference state $|{\zeta}\rangle$ is known. The
choice of the reference state $|{\zeta}\rangle$ is arbitrary and constrained
only to have non-zero overlap with the two states being superposed. The two
states $|{j,m_{1,2}}\rangle$ are input into the device shown in Fig. (2). An
ancilla is initialized in the state $\alpha|{0}\rangle+\beta|{1}\rangle$ and
the control-SWAP gate is implemented. Following this, the second output is
projected onto the state $|{\zeta}\rangle$ and the ancilla qubit is projected
onto a state whose coefficients are drawn from $|{\zeta}\rangle$ Oszmaniec
_et al._ (2016) (see Appendix B). Since a projector onto the state
$|{\zeta}\rangle$ has to be implemented, a judicious choice of the state
incorporates ease of implementation. To prepare the initial state in Eq. (4)
we choose $|{\zeta}\rangle=|{+}\rangle^{\otimes 2j}$. Although the success
probability of this state preparation is $2^{-4j}\binom{2j}{j}^{2}$, this
procedure can serve as an offline resource for the initial state. Once
prepared, the success probability of the metrology scheme follows the scaling
arguments we outline.

Finally, we address the task of measurement in the states $|{\zeta}\rangle$
and $|{\psi_{f}}\rangle$. The circuit presented in Fig. (3) implements the
projector $|{\psi_{f}}\rangle\langle{\psi_{f}}|$ on the time-evolved state
(see Appendix C for a detailed derivation). This circuit is the standard
modification of the superposition circuit that exchanges sources with
detectors. Our modified scheme hence starts with the initial state
$|{R_{1}}\rangle=|{\mu}\rangle\otimes|{\Psi(t)}\rangle\otimes|{\zeta}\rangle$,
where $|{\Psi(t)}\rangle$ is the time evolved system-meter state. We then pass
it through the circuit shown in Fig. (3). The evolved state $|{R_{2}}\rangle$
is projected onto the system basis
$|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle$. The
successful projection onto $|{\psi_{f}}\rangle$ happens with probability
$\tilde{P}_{s}\propto[\alpha^{*}\langle{\Psi(t)|j,m_{1}}\rangle+\beta^{*}\langle{\Psi(t)|j,m_{2}}\rangle][\alpha\langle{j,m_{1}|\Psi(t)}\rangle+\beta\langle{j,m_{2}|\Psi(t)}\rangle]$.
For the state in Eq. (13), we choose $m_{1}=0$ and $m_{2}=-j$ which yields the
same scaling $\tilde{P}_{s}\propto j^{2}$ as $P^{(2j)}_{s}$.

## Fisher Analysis.—

postselecting on a rare outcome always involves discarding many measurement
results. The question that arises is to establish under what conditions the
postselected states have a majority of the information available from the full
measurement of the joint pre-measurement state. The information content is
quantified by the quantum Fisher information $I(g)$ (QFI) associated with the
joint state is Alves  _et al._ (2015); Braunstein and Caves (1994) given by

$I(g)=4\frac{d\langle{\Psi_{g}}|}{dg}\frac{d|{\Psi_{g}}\rangle}{dg}-4\bigg{|}{\frac{d\langle{\Psi_{g}}|}{dg}|{\Psi_{g}}\rangle}\bigg{|}^{2},$
(15)

where $|{\Psi_{g}}\rangle=\exp(-igH)|{\Psi_{i}}\rangle$. For an unbiased
estimator, it is well known Braunstein and Caves (1994) that the variance of
the unknown parameter scales inversely with the QFI namely $Var(g)\propto
I^{-1}(g)$. For the bipartite Hamiltonian $gA\otimes B$ the QFI is evaluated
as
$I(g)=4[\langle{A^{2}}\rangle\langle{B^{2}}\rangle-(\langle{A}\rangle\langle{B}\rangle)^{2}]$.
For the initial state of the system given by Eq. (11) and meter being in the
coherent state $|{\phi_{i}}\rangle=|{\eta}\rangle$, this is evaluated to be

$\displaystyle I(g)$ $\displaystyle=$ $\displaystyle
4\bigg{[}\frac{1}{2}(j^{4}+2j^{3}+2j^{2})(|\eta|^{4}+|\eta|^{2})$ (16)
$\displaystyle-\frac{1}{4}(j^{4}+4j^{3}+4j^{2})|\eta|^{4}\bigg{]},$
$\displaystyle\approx$ $\displaystyle 2j^{4}|\eta|^{2},$

for $|\eta|\ll 1$.

Furthermore, we also investigate if the QFI present in the measurement of the
state in Eq. (13) saturates the total QFI available to the initial state. Let
$I^{{}^{\prime}}(g)$ denote the quantum Fisher information of the kicked meter
state after postselection of the system state. The ratio of the post-
measurement Fisher information to the total QFI in this case is evaluated to
be

$\frac{I^{{}^{\prime}}(g)}{I(g)}\approx\frac{1}{2}(1-|\eta
g\mathcal{A}_{w}|^{2}),$ (17)

where $|\eta g\mathcal{A}_{w}|\ll 1$. Hence though only half of all of the
information is available from the rarely postselected states from the full
measurement, this is a constant reduction in the total QFI independent of the
number of particles. A Fisher analysis of the near-deterministic scheme
provides a similar scaling.

## Conclusions.—

One of the drawbacks of WVA is that detector clicks are competing with the
desired amplification. In this work, we describe a metrology scheme that
simultaneously enhances the success probability and weak value. Furthermore,
we demonstrate a collective advantage from non-linear Hamiltonians and prove
that the success probability advantage scales linearly $\sigma\propto j$. This
extends the well-known advantage of non-linear Hamiltonians Boixo  _et al._
(2008) to WVA metrology. This super-extensive enhancement of the success
probability implies that with a modest enhancement in the number of spins
$2j$, there can be a dramatic enhancement of the success probability and the
amplified weak value. We propose experimental implementations of the crucial
steps of the protocol, namely the preparation of the initial states, the
measurement in the final state and the implementation of the proposed
interaction Hamiltonian. The initial and reference states are prepared as an
offline resource which does not impact the success probability of the
metrology scheme. Our scheme makes it possible to perform weak-value metrology
with linear enhancement in weak-value and quadratic enhancement in success
probability. Such an enhancement can be further generalized to other higher
order non-linear Hamiltonians using our theoretical approach and the
corresponding experimental proposal. An information theoretic analysis of both
our schemes show that approximately half of the available Fisher information
is collected by our scheme, a ratio that does not scale with the number of
atoms. Finally, we formalize the quantum advantage due to collective
interactions and non-linear Hamiltonians via the success probability advantage
$\sigma$.

This simultaneous enhancement can be used for practical metrology protocols
which still retain the known advantages of WVA metrology, in the amplifying
exceedingly small parameters, in the presence of technical noise sources and
in the presence of detector saturation while boosting the underlying
probabilities. Such enhancement will find applications in sensing various
physical parameters of interest.

## Acknowledgements.—

S.V. acknowledges support from the DST-SERB Early Career Research Award
(ECR/2018/000957) and DST-QUEST grant number DST/ICPS/QuST/Theme-4/2019. S.V.
thanks Yaron Kedem, Prasanna B. Venkatesh and R.Vijay for insightful
discussions.

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##  Appendix A A: Proof of Simultaneous Optimization

The final state can be written as

$|{\psi_{f}}\rangle=\sqrt{P_{s}^{(2j)}}|{\psi_{i}}\rangle+\sqrt{1-P_{s}^{(2j)}}|{\psi_{i}^{\perp}}\rangle,$
(18)

where $P_{s}^{(2j)}$ is small. In Eq. (18) $|{\psi_{i}^{\perp}}\rangle$ is
orthogonal to the state $|{\psi_{i}}\rangle$. The orthogonal state can be
written in terms of initial state $|{\psi_{i}}\rangle$ as

$|{\psi_{i}^{\perp}}\rangle=\frac{({A}-\langle{A}\rangle_{|{\psi_{i}}\rangle})|{\psi_{i}}\rangle}{\sqrt{Var(A)_{|{\psi_{i}}\rangle}}}.$
(19)

If we fix $P_{s}^{(2j)}=\kappa j^{2}$, the equation above can be rewritten as

$|{\psi_{f}}\rangle=\sqrt{\kappa
j^{2}}\frac{1}{\sqrt{2}}(|{j,0}\rangle+|{j,-j}\rangle)+\frac{\sqrt{1-\kappa
j^{2}}}{\sqrt{Var(A)_{|{\psi_{i}}\rangle}}}\frac{1}{\sqrt{2}}\bigg{[}\frac{j^{2}}{2}|{j,0}\rangle-\frac{j^{2}}{2}|{j,-j}\rangle\bigg{]}.$
(20)

The variance of $A$ goes as $j^{4}/4$, so Eq. (20) becomes

$|{\psi_{f}}\rangle\propto\sqrt{\kappa}j(|{j,0}\rangle+|{j,-j}\rangle)+\sqrt{1-\kappa
j^{2}}(|{j,0}\rangle-|{j,-j}\rangle).$ (21)

Since $P_{s}^{(2j)}$ is small, $\sqrt{1-\kappa j^{2}}$ is approximated to
unity. Finally we get the postselection state as

$|{\psi_{f}}\rangle\propto(\sqrt{\kappa}j+1)|{j,0}\rangle+(\sqrt{\kappa}j-1)|{j,-j}\rangle.$
(22)

This is Eq. (13) in the main text.

##  Appendix B B: Preparation of Initial State

We propose a non-deterministic scheme for preparing the initial state of the
system. The initial state can be constructed from the circuit diagram given in
Fig. (2). The Dicke states $|{j,m_{1}}\rangle$ and $|{j,m_{2}}\rangle$ are
created by the action of unitaries $U_{j,m_{1}}$ and $U_{j,m_{2}}$ on the
computational states $|{0}\rangle^{\otimes j-m_{1}}\otimes|{1}\rangle^{\otimes
j+m_{1}}$ and $|{0}\rangle^{\otimes j-m_{2}}\otimes|{1}\rangle^{\otimes
j+m_{2}}$. Thus the input state of the circuit is

$|{R_{1}}\rangle=|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle.$
(23)

The ancilla state $|{\nu}\rangle$ is taken to be
$\alpha|{0}\rangle+\beta|{1}\rangle$. Setting $\alpha=\beta=1/\sqrt{2}$. The
control-SWAP gate gets activated when state $|{1}\rangle$ passes through the
first terminal. The state after the control-SWAP gate is

$|{R_{2}}\rangle=\frac{1}{\sqrt{2}}(|{0}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle+|{1}\rangle\otimes|{j,m_{2}}\rangle\otimes|{j,m_{1}}\rangle).$
(24)

Finally the state is projected onto $K=|{\mu}\rangle\langle{\mu}|\otimes
I\otimes|{\zeta}\rangle\langle{\zeta}|$. For convenience, $|{\mu}\rangle$ is
set to
$|\langle{j,m_{1}|\zeta}\rangle||{0}\rangle+|\langle{j,m_{2}|\zeta}\rangle||{1}\rangle$.
The only condition required is the overlaps $\langle{j,m_{1}|\zeta}\rangle$
and $\langle{j,m_{2}|\zeta}\rangle$ are non-zero. After measurement the state
$|{R_{2}}\rangle$ becomes

$|{R_{3}}\rangle=|{\mu}\rangle\otimes{\bigg{[}\frac{1}{\sqrt{2}}\frac{\langle{\zeta|j,m_{2}}\rangle}{|\langle{\zeta|j,m_{2}}\rangle|}|{j,m_{1}}\rangle+\frac{1}{\sqrt{2}}\frac{\langle{\zeta|j,m_{1}}\rangle}{|\langle{\zeta|j,m_{1}}\rangle|}|{j,m_{2}}\rangle\bigg{]}}\otimes|{\zeta}\rangle.$
(25)

For our problem, we choose $m_{1}=0$ and $m_{2}=-j$ and the reference state
$|{\zeta}\rangle$ as $|{+}\rangle^{\otimes 2j}$. The overlaps in Eq. (25) are
$\langle{\zeta|j,0}\rangle=|\langle{\zeta|j,0}\rangle|=(1/\sqrt{2})^{2j}\binom{2j}{j}$
and
$\langle{\zeta|j,-j}\rangle=|\langle{\zeta|j,-j}\rangle|=(1/\sqrt{2})^{2j}$.
Finally we get the input state from the output of second terminal as

$|{\psi_{i}}\rangle=\frac{1}{\sqrt{2}}(|{j,0}\rangle+|{j,-j}\rangle).$ (26)

The success probability of this state preparation is evaluated to be

$P=\langle{R_{2}}|K^{\dagger}K|{R_{2}}\rangle=|\langle{j,m_{1}|\zeta}\rangle|^{2}|\langle{j,m_{2}|\zeta}\rangle|^{2}.$
(27)

For the initial state defined in Eq. (4), the success probability of state
preparation is $P=(1/2)^{4j}\binom{2j}{j}^{2}$.

##  Appendix C C: Implementation of Measurement

The post-measurement can be implemented using the circuit in Fig. (3). The
input state of the circuit is

$|{R_{1}}\rangle=|{\mu}\rangle\otimes|{\Psi(t)}\rangle\otimes|{\zeta}\rangle.$

The ancilla state is chosen to be
$|{\mu}\rangle=\alpha|{0}\rangle+\beta|{1}\rangle$. The control-SWAP gate gets
activated when $|{1}\rangle$ passes through the first terminal. The state
after control-SWAP gate is

$|{R_{2}}\rangle=\alpha|{0}\rangle\otimes|{\Psi(t)}\rangle\otimes|{\zeta}\rangle+\beta|{1}\rangle\otimes|{\zeta}\rangle\otimes|{\Psi(t)}\rangle.$

We finally measure the states with a joint quantum state
$|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle$ with the
measurement operator defined as

$K=(|{\nu}\rangle\langle{\nu}|\otimes|{j,m_{1}}\rangle\langle{j,m_{1}}|\otimes|{j,m_{2}}\rangle\langle{j,m_{2}}|).$
(28)

The state after measurement $K$ is

$|{R_{3}}\rangle=\frac{(|{\nu}\rangle\langle{\nu}|\otimes|{j,m_{1}}\rangle\langle{j,m_{1}}|\otimes|{j,m_{2}}\rangle\langle{j,m_{2}}|)|{R_{2}}\rangle}{\sqrt{\langle{R_{2}}|K^{\dagger}K|{R_{2}}\rangle}},$

For convenience, we assume
$\langle{\nu|0}\rangle=\lambda\langle{j,m_{1}|\zeta}\rangle$ and
$\langle{\nu|1}\rangle=\lambda\langle{j,m_{2}|\zeta}\rangle$, we get

$|{R_{3}}\rangle=\frac{\lambda\langle{j,m_{1}|\zeta}\rangle\langle{j,m_{2}|\zeta}\rangle[\alpha\langle{j,m_{1}}|+\beta\langle{j,m_{2}}|]|{\Psi(t)}\rangle|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle}{\sqrt{\langle{R_{2}}|K^{\dagger}K|{R_{2}}\rangle}}.$
(29)

$\lambda$ can be obtained by normalizing $|{\nu}\rangle$. On normalizing
$|{\nu}\rangle$, we get

$\lambda=\frac{1}{\sqrt{|\langle{\zeta|j,m_{1}}\rangle|^{2}+|\langle{\zeta|j,m_{2}}\rangle|^{2}}}.$
(30)

The probability of the measurement Eq. (28) for a particular outcome can be
obtained from evaluating
$\tilde{P}_{s}=\langle{R_{2}}|K^{\dagger}K|{R_{2}}\rangle$ as follows

$\displaystyle\tilde{P}_{s}$ $\displaystyle=$
$\displaystyle\langle{R_{2}}|K^{\dagger}K|{R_{2}}\rangle,$ $\displaystyle=$
$\displaystyle\bigg{(}\langle{j,m_{2}}|\otimes\langle{j,m_{1}}|\otimes\langle{\nu}|\bigg{[}\langle{\Psi(t)}|[\alpha^{*}|{j,m_{1}}\rangle+\beta^{*}|{j,m_{2}}\rangle]\frac{\langle{\zeta|j,m_{1}}\rangle\langle{\zeta|j,m_{2}}\rangle}{\sqrt{|\langle{\zeta|j,m_{1}}\rangle|^{2}+|\langle{\zeta|j,m_{2}}\rangle|^{2}}}\bigg{]}\bigg{)}$
$\displaystyle\times\bigg{(}\bigg{[}\frac{\langle{j,m_{1}|\zeta}\rangle\langle{j,m_{2}|\zeta}\rangle}{\sqrt{|\langle{\zeta|j,m_{1}}\rangle|^{2}+|\langle{\zeta|j,m_{2}}\rangle|^{2}}}[\alpha\langle{j,m_{1}}|+\beta\langle{j,m_{2}}|]|{\Psi(t)}\rangle\bigg{]}|{\nu}\rangle\otimes|{j,m_{1}}\rangle\otimes|{j,m_{2}}\rangle\bigg{)},$
$\displaystyle\tilde{P}_{s}$ $\displaystyle=$
$\displaystyle\bigg{[}\frac{|\langle{j,m_{1}|\zeta}\rangle|^{2}|\langle{j,m_{2}|\zeta}\rangle|^{2}}{|\langle{\zeta|j,m_{1}}\rangle|^{2}+|\langle{\zeta|j,m_{2}}\rangle|^{2}}\bigg{]}[\alpha^{*}\langle{\Psi(t)|j,m_{1}}\rangle+\beta^{*}\langle{\Psi(t)|j,m_{2}}\rangle][\alpha\langle{j,m_{1}|\Psi(t)}\rangle+\beta\langle{j,m_{2}|\Psi(t)}\rangle].$
(31)

For postselecting onto the state given in Eq. (13), we set
$\alpha=(\sqrt{\kappa}j+1)/\sqrt{2(1+\kappa j^{2})}$ and
$\beta=(\sqrt{\kappa}j-1)/\sqrt{2(1+\kappa j^{2})}$, $m_{1}=0$ and $m_{2}=-j$.
Choosing reference state as the equal superposition of Dicke states (i.e,
$|{\zeta}\rangle=1/\sqrt{2}[|{j,0}\rangle+|{j,-j}\rangle]$), which can be
prepared with the circuit in the Fig. (2) with low probability by choosing the
reference state as $|{+}\rangle^{\otimes 2j}$ and using it as an offline
resource. Therefore the overlaps are
$|\langle{j,m_{1}|\zeta}\rangle|=|\langle{j,m_{2}|\zeta}\rangle|=1/\sqrt{2}$.
The input state of the circuit is $|{\Psi(t)}\rangle=\exp(-igA\otimes
B)|{\psi_{i}}\rangle\otimes|{\phi_{i}}\rangle\approx(1-igA\otimes
B)|{\psi_{i}}\rangle\otimes|{\phi_{i}}\rangle$ for small values of $g$. Before
evaluating the probability, we first calculate the following overlaps

$\displaystyle\alpha^{*}\langle{\Psi(t)|j,m_{1}}\rangle+\beta^{*}\langle{\Psi(t)|j,m_{2}}\rangle$
$\displaystyle\approx$
$\displaystyle\langle{\phi_{i}}|\bigg{(}\frac{1}{\sqrt{1+\kappa
j^{2}}}(\sqrt{\kappa}j+i\frac{gB}{2}[j(j+1)(\sqrt{\kappa}j+1)+j(\sqrt{\kappa}j-1)])\bigg{)},$
(32) $\displaystyle\approx$
$\displaystyle\langle{\phi_{i}}|\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}(1+i\frac{gB}{2}[j^{2}+2j+j/\sqrt{\kappa}])\bigg{)},$ $\displaystyle=$
$\displaystyle\langle{\phi_{i}}|\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}\exp\bigg{(}i\frac{g}{2}\bigg{[}j^{2}+2j+\frac{j}{\sqrt{\kappa}}\bigg{]}B\bigg{)}\bigg{)}.$

Since the assumption $\kappa j^{2}$ is very small, $j/\sqrt{\kappa}$ is the
dominant term in the exponential. So the overlap can be written as

$\alpha^{*}\langle{\Psi(t)|j,m_{1}}\rangle+\beta^{*}\langle{\Psi(t)|j,m_{2}}\rangle\approx\langle{\phi_{i}}|\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}\exp(ig\mathcal{A}_{w}B)\bigg{)}.$ (33)

Similarly the other overlap can also be evaluated in the same fashion

$\displaystyle\alpha\langle{j,m_{1}|\Psi(t)}\rangle+\beta\langle{j,m_{2}|\Psi(t)}\rangle$
$\displaystyle\approx$ $\displaystyle\bigg{(}\frac{1}{\sqrt{1+\kappa
j^{2}}}(\sqrt{\kappa}j-i\frac{gB}{2}[j(j+1)(\sqrt{\kappa}j+1)+j(\sqrt{\kappa}j-1)])\bigg{)}|{\phi_{i}}\rangle,$
$\displaystyle\approx$
$\displaystyle\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}(1-i\frac{gB}{2}[j^{2}+2j+j/\sqrt{\kappa}])\bigg{)}|{\phi_{i}}\rangle,$
$\displaystyle=$ $\displaystyle\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}\exp\bigg{(}-i\frac{g}{2}\bigg{[}j^{2}+2j+\frac{j}{\sqrt{\kappa}}\bigg{]}B\bigg{)}\bigg{)}|{\phi_{i}}\rangle,$
$\displaystyle\alpha\langle{j,m_{1}|\Psi(t)}\rangle+\beta\langle{j,m_{2}|\Psi(t)}\rangle$
$\displaystyle\approx$
$\displaystyle\bigg{(}\frac{\sqrt{\kappa}j}{\sqrt{1+\kappa
j^{2}}}\exp(-ig\mathcal{A}_{w}B)\bigg{)}|{\phi_{i}}\rangle.$ (34)

Substituting Eq. (33) and Eq. (34) in Eq. (31), we get

$\tilde{P}_{s}\propto\frac{\kappa j^{2}}{4(1+\kappa j^{2})}.$ (35)

Approximating $1+\kappa j^{2}$ to unity, the probability of measurement is
$\tilde{P}_{s}\propto\kappa j^{2}$, which has the same quadratic scaling in
$j$ as $P_{s}^{(2j)}$.


# Realization of a Lie algebra and classifying space of crossed modules

Yves Félix  Institut de Mathématiques et Physique
Université Catholique de Louvain-la-Neuve
Louvain-la-Neuve
Belgique<EMAIL_ADDRESS>and  Daniel Tanré  Département de
Mathématiques, UMR-CNRS 8524
Université de Lille, F-59000 Lille<EMAIL_ADDRESS>
###### Abstract.

Complete differential graded Lie algebras appear to be a wonderful tool for
giving nice algebraic models for the rational homotopy type of non-simply
connected spaces. In particular, there is a realization functor of any Lie
algebra as a simplicial set. In a previous work, we considered the particular
case of a complete graded Lie algebra, $L$, concentrated in degree 0 and
proved that its geometric realization is isomorphic to the usual bar
construction on the Malcev group associated to $L$.

Here we consider the case of a complete differential graded Lie algebra, $L$,
concentrated in degrees 0 and 1. We first establish that $L$ has the structure
of a crossed module and prove that the geometric realization of $L$ is
isomorphic to the simplicial classifying space of this crossed module.

###### Key words and phrases:

Rational homotopy theory. Realization of Lie algebras. Lie models of
simplicial sets. Model category.

###### 2010 Mathematics Subject Classification:

Primary: 55P62, 17B55; Secondary: 55U10

The authors are partially supported by the MINECO-FEDER grant MTM2016-78647-P.
The second author is partially supported by the ANR-11-LABX-0007-01 “CEMPI”

## Introduction

In this text, we pursue the study of the rational homotopy type of spaces with
models in complete differential graded Lie algebras (henceforth cdgl), as
developed in the text, _Lie algebras in Topology_ , [3]. We emphasize that in
this approach, there is no requirements concerning simply connectivity or
nilpotency. Those models are very useful. For instance to any finite
simplicial complex is associated a cdgl $M_{X}$ whose homology in degree $0$
is the Malcev completion of $\pi_{1}(X)$ ([3, Theorem 10.5]).

One of the main tools in the theory is given by a cosimplicial cdgl
$(\mathfrak{L}_{n})$, where $\mathfrak{L}_{0}$ is the free Lie algebra on a
Maurer Cartan element in degree -1, and $\mathfrak{L}_{1}$ is the Lawrence-
Sullivan interval (see below for more details). This cosimplicial cdgl plays a
role similar to the simplicial algebra of PL-forms on the $\Delta^{n}$. It
permits to construct a realization functor from the category of complete
differential graded Lie algebras to the category of simplicial sets,
$\langle-\rangle\colon{\mathbf{cdgl}}\to{\mathbf{Sset}}$, defined by

$\langle
L\rangle_{\bullet}:=\mbox{Hom}_{{{\mathbf{cdgl}}}}(\mathfrak{L}_{\bullet},L).$

If the Lie algebra is concentrated in degree 0, $L=L_{0}$, we prove in [5]
that the realization functor $<L>$ is _isomorphic_ to the usual bar
construction on the group $\mathrm{exp}\,L$, constructed on the set $L$ with
the Baker-Campbell-Hausdorff product.

Here we consider the next step: _what happens for a connected cdgl with non-
trivial homology only in degrees 0 and 1?_ Geometrically, this corresponds to
the notion of homotopy 2-types and, by analogy, a connected cdgl $L$ such that
$H_{*}L=H_{0}L\oplus H_{1}L$ is called a _2-type cdgl._ From the work of
Whitehead ([11]), we know that homotopy 2-types are algebraically classified
by crossed modules. We will follow this way.

First of all, if $L=L_{\geq 0}$ and $H_{\geq 2}(L)=0$, then the Lie subalgebra
$I=L_{\geq 2}\oplus dL_{2}$ is an ideal because if $a\in L_{0}$ and $b\in
L_{2}$, then $da=0$ and $[a,d(h)]=d[a,h]$. Moreover $I$ is acyclic and the
quotient map is a quasi-isomorphism

$\varphi\colon(L,d)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(L/I,\overline{d}).$

Therefore, since the realization functor $<\;\;>$ preserves the quasi-
isomorphisms of connected cdgl ([3, Corollary 8.2]), we get a quasi-
isomorphism

$<\varphi>\colon<L,d>\to<L/I,\overline{d}>.$

We are thus reduced to consider only cdgl’s $L$ of the form $L=L_{0}\oplus
L_{1}$. We show in Proposition 2.3 that $d\colon L_{1}\to L_{0}$ inherits a
natural structure of crossed module ${\mathcal{C}}(L)$. This is done by
defining in Proposition 2.2 a product in $L_{1}$.

Any crossed module ${\mathcal{C}}$ has a classifying space $B{\mathcal{C}}$.
We describe its construction in Section 3. Therefore, our main result, which
extends [5], states as follows.

######  Theorem .

Let $L$ be a complete differential graded Lie algebra such that $L=L_{0}\oplus
L_{1}$. Then its geometric realization $\langle L\rangle$ is isomorphic to the
classifying simplicial set $B{\mathcal{C}}(L)$.

More generally we can consider a connected cdgl $L$ such that $H_{\geq
n+1}L=0$. Using the ideal $J=L_{\geq n+1}\oplus dL_{n+1}$, the same argument
gives a quasi-isomorphism

$<\varphi>\colon<L,d>\to<L/J,\overline{d}>.$

We conjecture that the differential $d$ gives a structure of $n$-cat-group
${\mathcal{C}}(L)$ (in the sense of Loday in [8]) and that the geometrical
realization $<L/J,\overline{d}>$ is isomorphic to the realization of this
$n$-cat-group.

Our program for the proof of the Theorem is carried out in Sections 1-5 below,
whose headings are self-explanatory.

###### Contents

  1. 1 Background on Lie models
  2. 2 Crossed modules and cdgl’s
  3. 3 The classifying space of a crossed module
  4. 4 The classifying space functor $\overline{W}$ and twisting functions
  5. 5 Proof of the Theorem

## Conventions and notations

In a graded Lie algebra $L$, the group of elements of degree $i$ is denoted
$L_{i}$. A differential of a Lie algebra decreases the degree; i.e.,
$dL_{i}\subset L_{i-1}$. If $x\in L$, we denote by $\mathrm{ad}_{x}$ the Lie
derivation of $L$ defined by $\mathrm{ad}_{x}(y)=[x,y]$.

If there is no ambiguity, the product of two elements $m$, $m^{\prime}$ of a
group $M$ is denoted $mm^{\prime}$. Sometimes, if several laws are involved,
we can use some specific notations, as $m\perp m^{\prime}$ or $m*m^{\prime}$,
to avoid confusion. An action of a group $N$ on a group $M$ is always a left
action and is denoted $(n,m)\mapsto{}^{n}m$. We denote then by $M\rtimes N$
the semi-direct product whose multiplication law is defined by

$(m,n)(m^{\prime},n^{\prime})=(m\,^{n}m^{\prime},nn^{\prime}).$

##  1\. Background on Lie models

A _complete differential graded Lie algebra_ (henceforth cdgl) is a
differential graded Lie algebra $L$ equipped with a decreasing filtration of
differential Lie ideals, such that $F^{1}=L$, $[F^{p}L,F^{q}L]\subset
F^{p+q}L$ and

$L=\varprojlim_{n}L/F^{n}L.$

If no filtration is specified, it is understood that we consider the
decreasing central series.

Let $V=\oplus_{i\in\mathbb{Z}}V_{i}$ be a rational graded vector space. We
denote by $\mathbb{L}(V)$ the free graded Lie algebra on $V$, and by
$\mathbb{L}^{\geq n}(V)$ the ideal of $\mathbb{L}(V)$ generated by the
brackets of length greater than or equal to $n$. The _completion of
$\mathbb{L}(V)$,_ is the inverse limit,

$\widehat{\mathbb{L}}(V)=\varprojlim_{n}\mathbb{L}(V)/\mathbb{L}^{\geq n}(V).$

This is a cdgl for the filtration given by the ideals
$G^{n}=\mbox{ker}\,(\widehat{\mathbb{L}}(V)\to\mathbb{L}(V)/\mathbb{L}^{>n}(V))$.
We call $\widehat{\mathbb{L}}(V)$ the _free complete graded Lie algebra on
$V$._

In any cgl $L$, the sub Lie algebra $L_{0}$ admits a group structure whose
multiplication law is given by the Baker-Campbell-Hausdorff product.

Now we recall the first properties of the cosimplicial cdgl
$(\mathfrak{L}_{n})$ ([3, Chapter 6]). Denote as usual by
$\underline{\Delta}^{n}$ the simplicial set in which
$\underline{\Delta}^{n}_{p}$ is the set of $p+1$-uples of integers
$(j_{0},\dots,j_{p})$ such that $0\leq j_{0}\leq\dots\leq j_{p}\leq n$. We
also denote by $\Delta^{n}$ the simplicial complex formed by the non-empty
subsets of $\\{0,\dots,n\\}$. The subcomplex ${\dot{\Delta}}^{n}$ of
$\Delta^{n}$ is the simplicial complex containing the non-empty subsets of
$\\{0,\dots,n\\}$ except $(0,\dots,n)$.

Finally $s^{-1}C_{*}\underline{\Delta}^{n}$ denotes the desuspension of the
simplicial chain complex on $\underline{\Delta}^{n}$ and
$s^{-1}C_{*}{\Delta}^{n}$ the complex of simplicial chains on $\Delta^{n}$.
This last one is isomorphic to the complex of non-degenerate chains on
$\underline{\Delta}^{n}$, $s^{-1}C_{*}{\Delta}^{n}\cong
s^{-1}N_{*}\underline{\Delta}^{n}$. Then, as a graded Lie algebra (without
differential), we set

$\mathfrak{L}_{n}=\widehat{\mathbb{L}}(s^{-1}C_{*}{\Delta}^{n}).$

In other words, $\mathfrak{L}_{n}$ is the free complete graded Lie algebra on
the elements $a_{i_{0}\dots i_{k}}$ of degree $|a_{i_{0}\dots i_{k}}|=k-1$,
with $0\leq i_{0}<\dots<i_{k}\leq n$.

The family $\underline{\Delta}=(\underline{\Delta}^{n},n\geq 0)$ is a
cosimplicial object in the category of simplicial sets. It follows that the
family $(s^{-1}C_{*}\underline{\Delta}^{n})$ is a cosimplicial object in the
category of chain complexes. The identification $s^{-1}C_{*}{\Delta}^{n}\cong
s^{-1}N_{*}\underline{\Delta}^{n}$ makes $s^{-1}C_{*}{\Delta}^{n}$ a
cosimplicial object in the category of chain complexes. The extension of the
cofaces and codegeneracies as morphisms of Lie algebras gives morphisms of
complete graded Lie algebras
$\delta^{i}\colon\mathfrak{L}_{n}\to\mathfrak{L}_{n+1}$ and
$\sigma^{i}\colon\mathfrak{L}_{n}\to\mathfrak{L}_{n-1}$. More precisely, we
have

$\delta^{i}(a_{j_{0}\dots j_{p}})=a_{r_{0}\dots r_{p}}\vspace{5mm}\mbox{ ,
with }r_{k}=\left\\{\begin{array}[]{ll}j_{k}&\mbox{if }j_{k}<i,\\\
j_{k}+1&\mbox{if }j_{k}\geq i,\end{array}\right.$

and

$\sigma^{i}(a_{j_{0}\dots j_{p}})=a_{r_{0}\dots r_{p}}\vspace{5mm}\mbox{ ,
with }r_{k}=\left\\{\begin{array}[]{ll}j_{k}&\mbox{if }j_{k}\leq i,\\\
j_{k}-1&\mbox{if }j_{k}>i,\end{array}\right.$

if $r_{0}<\dots<r_{p}$. Otherwise, $\sigma^{i}(a_{j_{0}\dots j_{p}})=0$.

######  Proposition 1.1.

[3, Theorem 6.1] Each $\mathfrak{L}_{n}$ can be endowed with a differential
$d$ satisfying the following properties.

  1. (i)

The linear part $d_{1}$ of $d$ is given by

$d_{1}a_{i_{0}\dots i_{p}}=\sum_{j=0}^{p}(-1)^{j}a_{i_{0}\dots\hat{i}_{j}\dots
p}.$

  2. (ii)

The generators $a_{i}$ are Maurer-Cartan elements; i.e.,
$da_{i}=-1/2[a_{i},a_{i}]$.

  3. (iii)

The cofaces $\delta^{i}$ and the codegeneracies $\sigma^{i}$ are cdgl
morphisms.

  4. (iv)

For $n\geq 2$,

$da_{0\dots n}=[{a_{0}},a_{0\dots n}]+\Phi,$

with $\Phi\in\widehat{\mathbb{L}}(s^{-1}C_{*}{\dot{\Delta}}^{n})$.

Thus, in particular, the family $\mathfrak{L}_{\bullet}$ is a cosimplicial
cdgl.

Let us specify the cdgl $\mathfrak{L}_{n}$ in low dimensions.

  1. $\bullet$

$\mathfrak{L}_{0}=(\mathbb{L}(a_{0}),d)$ is the free Lie algebra on a Maurer-
Cartan element $a_{0}$.

  2. $\bullet$

$\mathfrak{L}_{1}=(\widehat{\mathbb{L}}(a_{0},a_{1},a_{01}),d)$ is the
Lawrence-Sullivan interval (see [7]) with

$da_{01}=[a_{01},a_{1}]+\frac{\mathrm{ad}_{a_{01}}}{e^{\mathrm{ad}_{a_{01}}}-1}(a_{1}-a_{0}).$

  3. $\bullet$

$\mathfrak{L}_{2}=(\widehat{\mathbb{L}}(a_{0},a_{1},a_{2},a_{01},a_{02},a_{12},a_{012}),d)$
is a model of the triangle (see [3, Proposition 5.14]) with the differential

$d(a_{012})=a_{01}*a_{12}*a_{02}^{-1}-[a_{0},a_{012}].$

##  2\. Crossed modules and cdgl’s

For general developments on crossed modules, we send the reader to the
historical papers of Whitehead ([11], [9]) or to more modern presentations, as
[1], [2] or [8] for instance. We only recall the basics we need.

######  Definition 2.1.

A crossed module ${\mathcal{C}}=(\mu\colon M\to N)$ is a morphism of groups
$\mu$ together with an action of $N$ on $M$, satisfying two conditions:

  1. (1)

for all $m\in M$ and $n\in N$, $\mu({}^{n}m)=n\mu(m)n^{-1}$,

  2. (2)

for all $m\in M$, $m^{\prime}\in M$,
${}^{\mu(m)}m^{\prime}=mm^{\prime}m^{-1}$.

If the group $N$ acts on itself by conjugation, the first property means that
$\mu$ is compatible with the $N$-actions. It also implies that the group
$\mu(M)$ is a normal subgroup of $N$ and that ker$\,\mu$ is a sub $N$-module
of $M$.

On the other hand, we remark that if $\mu(m)=1$, the second property implies
$mm^{\prime}=m^{\prime}m$ which means that $\mathrm{ker}\,\mu$ is included in
the center of $M$. The same property shows that Im$\,\mu$ acts trivially on
ker$\,\mu$ and induces thus an action of coker$\,\mu$ on ker$\mu$.

Now let $L=L_{0}\oplus L_{1}$ be a cdgl. We will prove that $d\colon L_{1}\to
L_{0}$ is a crossed module. The first step consists to define a group
structure on $L_{1}$. This construction was originally done in [3, Definition
6.14].

######  Proposition 2.2.

For any cdgl $(L,d)$, with $L=L_{0}\oplus L_{1}$, $L_{1}$ admits a natural
group structure preserved by the differential ${d}$. The corresponding law,
denoted $\perp$, coincides with the abelian structure on $L_{1}$ for any pair
$(a,b)$ of cycles.

###### Proof.

The different possibilities for a definition of this law are described in [3,
Section 6.5]. We recall here the construction for the convenience of the
reader. Let begin with the “universal” example, the cdgl
$L^{\prime}=\widehat{\mathbb{L}}(u_{1},u_{2},du_{1},du_{2})$, with $u_{i}$ in
degree $1$. Since $HL^{\prime}=0$ there is an element $\omega$ in
$L^{\prime}_{1}$ such that

$d\omega=du_{1}*du_{2}.$ (2.1)

Of course such an element is not unique. If $\omega^{\prime}$ is another
element satisfying (2.1), the difference $\omega-\omega^{\prime}$ is a
boundary since $H_{\geq 1}L^{\prime}=0$. This shows that the class of $\omega$
is well defined in the cdgl quotient $(L^{\prime}/(L^{\prime}_{\geq 2}\oplus
dL_{2}^{\prime}),\overline{d})$. We denote this class by $u_{1}\perp u_{2}$.
By construction, it verifies $\overline{d}(u_{1}\perp u_{2})=du_{1}*du_{2}$.

Among all the different possible choices for $\omega$, one consists to start
with the Baker-Campbell-Hausdorff series for $du_{1}*du_{2}$. Replacing in
each term one and only one $du_{i}$ by $u_{i}$ we get an element $\omega$ with
$d\omega=d\omega_{1}*d\omega_{2}$. This gives,

$\displaystyle\omega=\omega_{1}+\omega_{2}+\frac{1}{2}[\omega_{1},d\omega_{2}]+\dots$
(2.2)

Now, let $L$ be a cdgl with $L=L_{0}\oplus L_{1}$, $e_{1},e_{2}\in L_{1}$, and
$f\colon L^{\prime}\to L$ the unique cdgl map sending $u_{i}$ to $e_{i}$.
Therefore the element $e_{1}\perp e_{2}:=f(u_{1}\perp u_{2})$ is a well
defined element in $L_{1}$. By construction, if $e_{1}$ and $e_{2}$ are
cycles, using the image of the formula (2.2) in $L$, we have $e_{1}\perp
e_{2}=e_{1}+e_{2}$.

For the associativity of $\perp$, we consider
$L^{\prime\prime}=\widehat{\mathbb{L}}(u_{1},u_{2},u_{3},du_{1},du_{2},du_{3})$
and notice that in $L^{\prime\prime}_{1}/dL^{\prime\prime}_{2}$ we have
$(u_{1}\perp u_{2})\perp u_{3}=u_{1}\perp(u_{2}\perp u_{3})$ because both have
the same boundary. The same is thus true in $L_{1}$.

∎

With this group structure on $L_{1}$ we can now prove that $L=L_{0}\oplus
L_{1}$ is a crossed module.

######  Proposition 2.3.

Let $(L,d)$ be a connected complete differential graded Lie algebra such that
$L=L_{0}\oplus L_{1}$. Then

$d\colon(L_{1},\perp)\to(L_{0},*),$

is a crossed module. Here $*$ is the Baker-Campbell-Hausdorff product and
$\perp$ the product defined in Proposition 2.2.

###### Proof.

The group $L_{0}$ acts on $L_{1}$ by

${}^{x}z=e^{\mathrm{ad}_{x}}(z)\,,\quad\text{for all }x\in L_{0},z\in L.$

From [3, Corollary 4.12]), we deduce that this is a group action. Indeed, for
any $x\in L_{0}$, $y\in L_{0}$, $z\in L_{1}$, we have

${}^{(x*y)}z=e^{\mathrm{ad}_{x*y}}(z)=e^{\mathrm{ad}_{x}}(e^{\mathrm{ad}_{y}}z)={}^{x}({}^{y}z).$

Moreover, as $x$ is a cycle, by [3, Proposition 4.10] we have

$d({}^{x}z)=e^{\mathrm{ad}_{x}}(dz)=x*dz*x^{-1},$

and Property (1) is satisfied.

To prove that the application $y\mapsto{}^{x}y$ is a group homomorphism, as in
Proposition 2.2, we consider a universal example. Let
$E=\widehat{\mathbb{L}}(x,z,t,dz,dt)$ with $x$ in degree $0$, $z$ and $t$ in
degree $1$, and $dx=0$. Since the injection $\mathbb{L}(x)\to E$ is a quasi-
iso, we have $H_{\geq 1}(E)=0$. Now we notice that in $E/(E_{\geq 2}\oplus
dE_{2})$ we have

$\begin{array}[]{lcl}d(^{x}(z\perp t)&=&e^{\mathrm{ad}(x)}(d(z\perp
t))=e^{\mathrm{ad}(x)}(dz*dt)\\\ &=&x*dz*dt*x^{-1}=x*dz*x^{-1}*x*dt*x^{-1}\\\
&=&e^{\mathrm{ad}(x)}(dz)*e^{\mathrm{ad}(x)}(dt)\\\
&=&d(e^{\mathrm{ad}(x)}z)*d(e^{\mathrm{ad}(x)}t)=d({}^{x}z\perp{}^{x}t).\end{array}$

Thus in $E_{1}/dE_{2}$, we get

${}^{x}(z\perp t)={}^{x}z\perp{}^{x}t.$

The same is therefore also true in $L_{1}$.

For Property (2), we use once again the universal example
$L^{\prime}=\mathbb{L}(u_{1},u_{2},du_{1},du_{2})$ already considered in the
proof of Proposition 2.2. Since in $L^{\prime}_{1}/dL^{\prime}_{2}$ we have

$d({}^{du_{1}}u_{2})=du_{1}*du_{2}*du_{1}^{-1}=d(u_{1}\perp u_{2}\perp
u_{1}^{-1}),$

we deduce

${}^{du_{1}}u_{2}=u_{1}\perp u_{2}\perp u_{1}^{-1},$

and the same is true in $L_{1}$. ∎

######  Remark 2.4.

From the basic properties of crossed modules, we deduce that on
$H_{1}L=\mathrm{ker}\,d$, the group structures $\perp$ and $+$ coincide.

On the other hand, we note that the action of $H_{0}L=$ coker$\,d$ on $H_{1}L$
induced by the exponential of the adjoint derivation coincides with the
geometrical realization as shown in [3, Theorem 12.41].

##  3\. The classifying space of a crossed module

Let ${\mathcal{C}}=(d\colon(M,\perp)\to(N,*))$ be a crossed module. Its
classifying space is by definition the classifying space of the nerve of the
categorical group associated to ${\mathcal{C}}$.

For recall, a categorical group is a group object in the category of groups
(see [8, Section 1.1]),

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.93124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.29062pt\raise 8.81248pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{s}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.93124pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.66736pt\raise-5.15277pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15279pt\hbox{$\scriptstyle{t}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.93124pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{N}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $N$ is a subgroup of $G$, $s$ and $t$ are homomorphisms such that
$s|_{N}=b|_{N}=\mbox{id}_{N}$ and $[\ker s,\ker t]=1$.

To a crossed module ${\mathcal{C}}$ we associate a categorical group as
follows:

  * •

$G=M\rtimes N$ is the product $M\times N$ with the semi-direct product given
by the action of $N$ on $M$. Thus, the product of $(m^{\prime},n^{\prime})$
and $(m,n)$ in $G$ is

$(m^{\prime},n^{\prime})\bullet(m,n)=(m^{\prime}\perp{}^{n^{\prime}}m,n^{\prime}*n).$
(3.1)

  * •

An element $(m,n)$ of $G$ has for source and target, respectively,

$s(m,n)=dm\ast n\quad\text{and}\quad t(m,n)=n.$ (3.2)

Thus, the group $N$ is interpreted as the group of objects viewed in $G$ as
$\\{1\\}\times N$. The group $G=M\rtimes N$ is the group of arrows, the
morphisms $s$ and $t$ giving the source and the target. Two elements
$(m^{\prime},n^{\prime})$ and $(m,n)$ are composable if
$n^{\prime}=t(m^{\prime},n^{\prime})=s(m,n)=dm*n$. In this case the
composition is defined by

$(m^{\prime},n^{\prime})\circ(m,n)=(m^{\prime}\perp m,n).$ (3.3)

We deduce easily from Property (1) that $s$ and $t$ are group homomorphisms.
We also verify that the source of a composite is the source of the first one
and the target is the target of the second one:

$\begin{array}[]{lcl}s(m^{\prime}\perp m,n)&=&d(m^{\prime}\perp
m)*n=dm^{\prime}*dm*n=dm^{\prime}*n^{\prime}\\\
&=&s(m^{\prime},n^{\prime}),\\\ t(m^{\prime}\perp m,n)&=&n=t(m,n).\end{array}$

Finally, the composition is a group homomorphism, see [8, Lemma 2.2].

The usual nerve of a category is a simplicial set. When the category is a
categorical group, we obtain naturally a simplicial group
${\mathrm{Ner}}_{*}$.

In low dimensions, for the categorical group
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{t}$$\textstyle{N}$
associated to a crossed module, we have

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
33.6777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{Ner}}_{1}=M\rtimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.856pt\raise 10.38054pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{d_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.6777pt\raise
4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.856pt\raise-6.07498pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{d_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.6777pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
57.6777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathrm{Ner}}_{0}=N}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

with $d_{0}(m,n)=t(m,n)=n$, $d_{1}(m,n)=s(m,n)=dm*n$ and for degeneracy
$s_{0}\colon{\mathrm{Ner}}_{0}\to{\mathrm{Ner}}_{1}$ the canonical injection
$N\to M\rtimes N$.

An element of ${\mathrm{Ner}}_{k}$ is a sequence $(m_{i},n_{i})_{1\leq i\leq
k}$ such that $n_{i}=t(m_{i},n_{i})=s(m_{i-1},n_{i-1})=dm_{i-1}*n_{i-1}$. As
the $n_{i}$, $i\geq 2$, are defined by $n_{1}$ and the family $(m_{i})_{1\leq
i\leq k}$, the sequence $(m_{i},n_{i})_{i\leq k}$ can be identified with the
sequence

$(m_{k},m_{k-1},\dots,m_{1},n_{1})\in M^{k}\times N.$

In particular, we can identify

${\mathrm{Ner}}_{k}=M^{k}\times N.$ (3.4)

Each ${\mathrm{Ner}}_{k}$ is a group, the multiplication being given component
wise. With the identification (3.4), this product is given by

$((m_{i})_{1\leq i\leq k},n)\bullet((m^{\prime}_{i})_{1\leq i\leq
k},n^{\prime})=((m_{i}\perp^{d(\perp_{j=1}^{i-1}m_{j})*n}m^{\prime}_{i})_{1\leq
i\leq k},n*n^{\prime}).$ (3.5)

The boundary and degeneracy maps of ${\mathrm{Ner}}_{*}$ are morphisms of
groups defined as usual by:

$\left\\{\begin{array}[]{ll}&d_{0}(m_{k},\dots,m_{1},n)=(m_{k},\dots
m_{2},d(m_{1})*n),\\\
0<i<k,&d_{i}(m_{k},\dots,m_{1},n)=(m_{k},\dots,m_{i+1}\perp
m_{i},\dots,m_{1},n),\\\
&d_{k}(m_{k},\dots,m_{1},n)=(m_{k-1},\dots,m_{1},n),\\\ 0\leq i\leq
k,&s_{i}(m_{k},\dots,m_{1},n)=(m_{k},\dots,m_{i},1,m_{i-1},\dots,m_{1},n).\end{array}\right.$

The identity $e_{k}\in{\mathrm{Ner}}_{k}$ is the element $(1,\dots 1,1)$.

Finally, the classifying space $B{\mathcal{C}}$ of the crossed module
${\mathcal{C}}$ is the space obtained by applying the classifying functor
$\overline{W}$ to ${\mathrm{Ner}}_{*}$ (see [4, Definition 3.20] or [6, Page
255]) :

$B{\mathcal{C}}=\overline{W}({\mathrm{Ner}}_{*}).$

By definition of $\overline{W}$, we have

$(B{\mathcal{C}})_{k}=\\{(h_{k-1},\dots,h_{0})\mid
h_{i}\in{\mathrm{Ner}}_{i}\\}.$

The boundaries and degeneracies are given by

$\left\\{\begin{array}[]{lrcl}&d_{0}(h_{k-1},\dots,h_{0})&=&(h_{k-2},\dots,h_{0}),\\\
0<i<k-1,&d_{i}(h_{k-1},\dots,h_{0})&=&(d_{i-1}h_{k-1},\dots,d_{0}h_{k-i}\bullet
h_{k-i-1},h_{k-i-2},\dots,h_{0}),\\\
&d_{k}(h_{k-1},\dots,h_{0})&=&(d_{k-1}h_{k-1},\dots,d_{1}h_{1})\\\
&s_{0}(h_{k-1},\dots,h_{0})&=&(1,h_{k-1},\dots,h_{0})\\\ 0<i\leq
k,&s_{i}(h_{k-1},\dots,h_{0})&=&(s_{i-1}h_{k-1},\dots,s_{0}h_{k-i},1,h_{k-i-1},\dots,h_{0}).\end{array}\right.$

In particular, in low dimensions, we have $B{\mathcal{C}}_{0}=1$,
$B{\mathcal{C}}_{1}=N$, $B{\mathcal{C}}_{2}=(M\rtimes N)\times N$,
$B{\mathcal{C}}_{3}=(M^{2}\rtimes N)\times(M\rtimes N)\times N$.

##  4\. The classifying space functor $\overline{W}$ and twisting functions

Let $A_{*}$ be a simplicial set. By [10, Corollary 27.2], there is a bijective
correspondence between morphisms of simplicial sets $\varphi\colon
A_{*}\to\overline{W}({\mathrm{Ner}}_{*})$ and twisting functions

$\tau=\\{\tau_{k}\colon A_{k}\to{\mathrm{Ner}}_{k-1}\\}_{k\geq 1}.$

For recall ([10, Definition 18.3]) a twisting function $\tau$ is a family of
linear maps $\tau_{k}\colon A_{k}\to{\mathrm{Ner}}_{k-1}$ satisfying the
following properties for $x\in A_{k}$:

$\left\\{\begin{array}[]{lrcl}&d_{0}\tau x&=&\tau d_{1}x\bullet(\tau
d_{0}x)^{-1},\\\ i>0,&d_{i}\tau x&=&\tau d_{i+1}x,\\\ i\geq 0,&s_{i}\tau
x&=&\tau s_{i+1}x,\\\ &\tau
s_{0}x&=&e_{k}\in{\mathrm{Ner}}_{k}.\end{array}\right.$

The simplicial map $\varphi_{k}\colon A_{k}\to(B{\mathcal{C}})_{k}$ associated
to the twisting function $\tau$ is given by

$\displaystyle\varphi_{k}x=(\tau x,\tau d_{0}x,\dots,\tau d_{0}^{k-1}x).$
(4.1)

Conversely ([10, Page 88]) the twisting function $\tau$ associated to a
simplicial morphism $\varphi\colon A_{*}\to\overline{W}({\mathrm{Ner}}_{*})$
is defined by

$\tau=\tau({\mathrm{Ner}}_{*})\circ\varphi,$

where $\tau({\mathrm{Ner}}_{*})$ is the twisting function associated to the
identity on $\overline{W}({\mathrm{Ner}}_{*})$,

$\tau({\mathrm{Ner}}_{*})\colon\overline{W}({\mathrm{Ner}}_{*})_{k}\to{\mathrm{Ner}}_{k-1}\,,$

defined by

$\tau({\mathrm{Ner}}_{*})(g_{n-1},\dots,g_{0})=g_{n-1}.$

##  5\. Proof of the Theorem

First of all we describe the simplicial set $\langle
L\rangle_{\bullet}=\hom_{{\mathbf{cdgl}}}({\mathfrak{L}}_{\bullet},L)$ in the
case $L=L_{0}\oplus L_{1}$.

Since any morphism $\mathfrak{L}_{k}\to L$ vanishes on the elements $a_{i}$,
$0\leq i\leq k$ we denote by $({\mathfrak{L}}^{red}_{k})$ the cosimplicial
cdgl, quotient of $({\mathfrak{L}}_{k})$ by the ideal generated by the
elements of degree $-1$.

We then denote by $(\overline{\mathfrak{L}}_{k})$ the cosimplicial cdgl
quotient of $({\mathfrak{L}}^{red}_{k})$ by the elements of degrees $\geq 2$
and their boundaries. Therefore, from the differentials of the triangle, we
define

$\overline{\mathfrak{L}}_{k}=\widehat{\mathbb{L}}(a_{ij},a_{rst})/I_{k},\;\text{
with }\;0\leq i<j\leq k\;\text{ and }\;0\leq r<s<t\leq k.$

Here the generators $a_{ij}$ are of degree 0 and the $a_{rst}$ of degree 1.
Their differentials are given by $da_{ij}=0$ and
$da_{rst}=a_{rs}*a_{st}*a_{rt}^{-1}$. The ideal $I_{k}$ is the ideal generated
by the elements

$a_{rst}\perp a_{rtv}\perp a_{rsv}^{-1}\perp{}^{a_{rs}}(a_{stv}^{-1}),$

one such relation for each sequence $0\leq r<s<t<v\leq k$. Therefore the
following relations are satisfied in the quotient:

$a_{rst}\perp a_{rtv}\perp a_{rsv}^{-1}={}^{a_{rs}}a_{stv}.$

In particular,

$\overline{\mathfrak{L}_{k}}=(\widehat{\mathbb{L}}(a_{ij},a_{0rs}),d)\hskip
5.69054pt\mbox{with }0\leq i<j\leq k\hskip 5.69054pt\mbox{and }0<r<s\leq k.$

######  Proposition 5.1.

Let $L=L_{0}\oplus L_{1}$, then the morphism

$\Phi\colon\hom_{{\mathbf{cdgl}}}(\mathfrak{L}_{k},L)\to L_{0}^{k}\times
L_{1}^{\frac{k(k-1)}{2}}$

given by $\Phi(f)=((f(a_{0i})_{0<i\leq k}),(f(a_{0jr})_{0<j<r\leq k})$ is an
isomorphism.

###### Proof.

First of all the map $\Phi$ is injective since, for $0<i<j\leq k$, we have

$f(a_{ij})=f(a_{0i})^{-1}*df(a_{0ij})*f(a_{0j}),$

and for $0<i<j<r\leq k$,

$f(a_{ijr})={}^{a_{0i}^{-1}}\left(f(a_{0ij})\perp f(a_{0jr})\perp
f(a_{0ir})^{-1}\,\right).$

In particular all the $f(a_{ij})$ and $f(a_{ijk})$ are completely determined
by the $f(a_{0r})$ and the $f(a_{0rs})$.

Conversely, the only conditions for having an element of
$\hom_{{\mathbf{cdgl}}}(\mathfrak{L}_{k},L)$ are that for $0\leq i<j<r\leq k$
we have $df(a_{ijr})=f(a_{ij})*f(a_{jr})*f(a_{rj})^{-1}$. Thus, each sequence
$(\alpha_{0i},\beta_{0ij})\in L_{0}^{k}\times L_{1}^{\frac{k(k-1)}{2}}$
determines a unique morphism $f$ with $f(a_{0i})=\alpha_{0i}$ and
$f(a_{0ij})=\beta_{0ij}$. ∎

######  Proposition 5.2.

Let $L=L_{0}\oplus L_{1}$, then the morphism

$\Psi\colon\hom_{{\mathbf{cdgl}}}(\mathfrak{L}_{k},L)(\mathfrak{L}_{k},L)\to
L_{0}^{k}\times L_{1}^{\frac{k(k-1)}{2}}$

given by $\Psi(f)=((f(a_{r\,r+1})_{0\leq r<k}),(f(a_{r,r+1,s})_{r+1<s\leq
k}))$ is an isomorphism.

###### Proof.

For sake of simplicity write for $i<j$, $a_{ji}=a_{ij}^{-1}$, and for $0\leq
i<j<r\leq k$,

$\begin{array}[]{ll}a_{irj}=a_{ijr}^{-1},&\\\
a_{rij}={}^{a_{ri}}a_{ijr}={}^{a_{ir}^{-1}}a_{ijr}\mbox{},&a_{jir}={}^{a_{ji}}a_{irj}={}^{a_{ij}^{-1}}a_{ijr}^{-1},\\\
a_{jri}={}^{a_{ji}}a_{ijr}={}^{a_{ij}^{-1}}a_{ijr},\hskip
14.22636pt\mbox{}&a_{rji}={}^{a_{ri}}a_{irj}={}^{a_{ir}^{-1}}a_{ijr}^{-1}.\end{array}$

With those notations, when the integers $i,j,r$ are all different from each
other and between $0$ and $k$, we have

$da_{ijr}=a_{ij}*a_{jr}*a_{ri}.$

Suppose that the elements $f(a_{r,r+1})$ and $f(a_{r,r+1,t})$, with $r+1<t$,
are defined. Then the other elements, $f(a_{r,r+s})$ and $f(a_{r,r+s,t})$ with
$r+s<t$, can be derived by induction on $s$ from the following formulas

$f(a_{r,r+s+1})=df(a_{r,r+1,r+s+1})^{-1}*f(a_{r,r+1})*f(a_{r+1,r+s+1}),$

and

$f(a_{r,r+s+1,t})={}^{f(a_{r,r+1})}\left(f(a_{r+1,r,r+s+1})\perp
f(a_{r+1,r+s+1,t})\perp f(a_{r+1,t,r})\,\right).$

This shows that $\Psi$ is injective. The same construction process shows that
$\Psi$ is also surjective. ∎

The isomorphism of our main theorem is based on a family $\tau$ of linear maps
$\tau_{k}\colon\hom_{{\mathbf{cdgl}}}(\mathfrak{L}_{k},L)\to{\mathrm{Ner}}_{k-1},k\geq
1$, defined by

$\tau_{k}f=(m_{k-1},\dots,m_{1},n)\in M^{k-1}\times N,$

with $n=f(a_{01})$, $m_{1}=(f(a_{012}))^{-1}$ and
$m_{i}=(f(a_{01(i+1)}))^{-1}\perp f(a_{01i})$, for $i\geq 2$.

In low dimensions, this gives:

$\left\\{\begin{array}[]{l}\tau_{1}f=f(a_{01})\in N,\\\
\tau_{2}f=(f(a_{012})^{-1},f(a_{01}))\in M\times N,\\\
\tau_{3}f=(f(a_{013})^{-1}\perp f(a_{012}),f(a_{012})^{-1},f(a_{01}))\in
M^{2}\times N.\end{array}\right.$

######  Proposition 5.3.

The family $\tau$ is a twisting function.

###### Proof.

Let us notice $m_{i+1}\perp m_{i}=f(a_{01(i+2)})\perp f(a_{01i})$. Thus, the
indice $i+1$ disappears in the expression of $d_{i}\tau_{k}f$ and we get
$d_{i}\tau_{k}f=\tau_{k-1}d_{i+1}f$ for $0<i<k-1$. A similar argument gives
also the result for $d_{k-1}$. We are reduced to the more delicate equality
with $d_{0}$. We use an induction, supposing the result is true for
$\tau_{j}$, $j<k$, and consider $\tau_{k}$. Grants to the inductive step, we
can concentrate the computations on the left hand factor. From the
definitions, we have

$\begin{array}[]{lcl}\tau_{k-1}d_{1}f&=&(f(a_{02k}))^{-1}\perp
f(a_{02(k-1)},\dots,f(a_{02})),\\\ \tau_{k-1}d_{0}f&=&((f(a_{12k})^{-1}\perp
f(a_{12(k-1)}),\dots,f(a_{12})),\\\ d_{0}\tau_{k}f&=&((f(a_{01k})^{-1}\perp
f(a_{01(k-1)}),\dots,(df(a_{012}))^{-1}*f(a_{01})).\end{array}$

We determine the product of the two last terms,

$d_{0}\tau_{k}f\bullet\tau_{k-1}d_{0}f=(f(a_{01k})^{-1}\perp
f(a_{01(k-1)})\perp\;^{\gamma}(f(a_{12k})^{-1}\perp f(a_{12(k-1)})),\dots),$

where $\gamma=dm_{k-2}\ast dm_{k-1}\ast\dots\ast dm_{1}\ast
n=f(a_{0(k-1)})*(f(a_{1(k-1)}))^{-1}$. For obtaining the equality with
$\tau_{k-1}d_{1}f$, we calculate in $\overline{\mathfrak{L}_{k}}$:

$\displaystyle d(a_{01k}^{-1}\perp
a_{01(k-1)}\perp\;^{a_{0(k-1)}*a_{1(k-1)}^{-1}}(a_{12k}^{-1}\perp
a_{12(k-1}))$ $\displaystyle=$ $\displaystyle
a_{0k}*a_{2k}^{-1}*a_{2(k-1)}*a_{0(k-1)}^{-1}$ $\displaystyle=$ $\displaystyle
d(a_{02k}^{-1}\perp a_{02(k-1)}).$

Similar computations give the equalities for degeneracy maps. ∎

Denote by $\varphi$ the morphism of simplicial sets induced by the previous
twisting function $\tau$,

$\varphi\colon\hom_{{\mathbf{cdgl}}}({\mathfrak{L}},L)\to(B{\mathcal{C}}).$

######  Proposition 5.4.

The morphism $\varphi$ is an isomorphism of simplicial sets.

###### Proof.

Recall from (4.1) that

$\varphi_{k}f=(\tau f,\tau d_{0}f,\dots,\tau d_{0}^{k-1}f).$

On the other hand, using $d_{0}f=f\delta^{0}$, we get $\tau
d_{0}f=(m^{\prime}_{k-2},\dots,m^{\prime}_{1},n^{\prime})$, with
$n^{\prime}=f(a_{12})$, $m^{\prime}_{1}=f(a_{123})^{-1}$ and for $i>1$,
$m^{\prime}_{i}=f(a_{12(i+2)})^{-1}\perp f(a_{12(i+1)})$. By iteration from
$(d_{0})^{\ell}f=f(\delta^{0})^{\ell}$, we deduce that the image of
$\varphi_{k}$ is the linear subspace generated by the elements $f(a_{r,r+1})$,
for $0\leq r<k$, and $f(a_{r,r+1,s})$, for $r+1<s\leq k$. The result follows
thus from Proposition 5.2. ∎

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